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Quantum Information·Captain: Goku

No advantage from classical assistance for energy-constrained bosonic dephasing channelsResearch Paper

Motivation

We prove that four capacities of single-mode bosonic dephasing channels coincide under an average photon-number constraint: the unassisted quantum capacity, the unassisted private capacity, the two-way classically assisted quantum capacity, and the two-way public-classically assisted private capacity. The assistance may be interactive: the parties can exchange classical messages between channel uses and choose subsequent inputs based on earlier outcomes. For every finite value of the constraint, their common value is the coherent information optimized over Fock-diagonal input states satisfying the constraint. The proof uses a relative entropy comparison with the completely dephasing channel to derive chain rules for the Rains relative entropy and the relative entropy of entanglement. These chain rules give an upper bound on the rates achievable by adaptive protocols whose mean input photon number is constrained on average over all channel uses. We also show that both chain rules are attained by canonical purifications of finite-energy Fock-diagonal states. △ Less

This mission is the statement-level formalization of No advantage from classical assistance for energy-constrained bosonic dephasing channels (arXiv:2610.13854v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

1 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Fock dephasing absorbs bosonic dephasing on either side (Lemma 2.2, Eq. (23), PDF p. 6). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

13 results of the paper are not stated yet, because the library lacks an interface they need:

  • Abstract / Eq. (1): All four operational finite-energy capacities and the countable Fock-diagonal coherent-information supremum. (hard)
  • Lemma 2.1: Degradable CPTP channels on arbitrary separable output/environment spaces, complementary dilation, output-entropy supremum, and operational Q, P, Cp. (hard)
  • Lemma 2.3: Base-2 extended von Neumann entropy of positive trace-one operators on l²(N), complementary environment mixture, and extended mean photon number. (hard)
  • Corollary 2.1: Both the non-diagonal-input coherent-information comparison and the operational capacity formula over all countable photon distributions. (hard)
  • Lemma 3.1: Infinite-dimensional Umegaki relative entropy between normalized states and identity-tensored dephasing on a general separable reference, including pure-input equality. (hard)
  • Lemma 3.2: Countable strongly complete orthogonal pinching on general separable Hilbert spaces and signed extended relative entropy with arbitrary positive trace-class second argument. (hard)
  • Theorem 3.1: Identity-tensored channels on general trace-class operators and signed extended relative entropy for unrestricted positive sigma. (hard)
  • Theorem 3.2: Rains infimum with Hilbert–Schmidt partial transpose and extended trace norm, before and after the moving Alice/Bob cut. (hard)
  • Theorem 3.3: Two-way adaptive operational quantum capacity with countably decomposable LOCC and unconditional average energy. (hard)
  • Theorem 3.4: Relative entropy of entanglement over the trace-norm-closed product-state convex hull, on arbitrary separable local memories and the moving cut. (hard)
  • Theorem 3.5: Operational assisted private capacity with arbitrary retained shields, twisting, squared fidelity, and public environments/transcripts, together with operational Q and P. (hard)
  • Corollary 3.1: Constructed normalized countable canonical purification, its dephasing-channel output, and both infinite-dimensional entanglement infima. (hard)
  • Lemma A.1: Countable maximally correlated states on paired orthonormal families, arbitrary extending dephasing bases, finite marginal entropy, and the infinite-dimensional Rains infimum. (hard)

Selected references

3 thms2 active usersReviewed
Completed
Quantum Information·Captain: Goku

Coherent Control of Channel Dilations Activate Temporal Bell NonclassicalityResearch Paper

Motivation

The temporal Clauser-Horne-Shimony-Holt (CHSH) inequality witnesses the nonclassicality of temporal correlations, but its violation is generally degraded by environmental noise. Here, we show that violation of the temporal CHSH inequality can be revived through coherent control of noisy quantum evolutions. We compare two physically distinct implementations: coherent control of noisy evolutions induced by interaction of the system with independent environments, and coherent control of two physically distinct, unitarily equivalent Stinespring dilations of the same noisy channel. Although these constructions generate identical deterministic system dynamics, they induce distinctly different post-selected evolutions. With a focus on the amplitude damping channel (ADC), we show that coherent control of equivalent dilations extend the range of temporal CHSH inequality violation well beyond both the incoherently controlled, or deterministic, scenario and what is achievable with independent environments. Under setting-independent post-selection of the coherent control implementation, the resulting violation further certifies that the channel is not strongly CHSH nonlocality-breaking. Our results identify the choice of Stinespring dilation as an operationally relevant resource in coherently controlled tests of temporal quantum correlations. △ Less

This mission is the statement-level formalization of Coherent Control of Channel Dilations Activate Temporal Bell Nonclassicality (arXiv:2607.05643v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

14 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Dilation choice strictly enlarges equatorial temporal-CHSH activation (Abstract / §§IV–V: dilation choice strictly enlarges equatorial temporal-CHSH activation, PDF p. 1). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

1 results of the paper are not stated yet, because the library lacks an interface they need:

  • Abstract / §V / Appendix A certification clause: An explicitly defined original-channel strong CHSH nonlocality-breaking predicate and a valid bridge from this dilation-access post-selection to its allowed system-only SLOCC/filter operations. (source-ambiguity)

Selected references

18 thms2 active usersReviewed
Quantum Information·Captain: Goku

Maximal coherence of quantum measurement and the resource theory of sharpnessResearch Paper

Motivation

A resource theory of quantum measurement can be addressed in terms of quantum coherence and measurement sharpness, respectively. The former analyzes the off-diagonal structure of POVM elements in a predetermined basis while the latter analyzes the deviation from trivial, state-independent, measurements. We establish a direct connection between the two resource theories by identifying measurement sharpness as the maximal coherence that is achievable under all possible unitary changes of the reference basis. For a broad class of POVMs whose elements share a common eigenbasis, we show that the maximal distance-based coherence of measurement coincides exactly with the corresponding distance-based sharpness monotone. We further extend this equivalence, with element-additive distances, to POVMs whose elements admit a common mutually unbiased basis structure. These results provide a measurement-theoretic analogue of the maximal-coherence & purity correspondence for quantum states. We also show that the maximal coherence of measurement is faithful with respect to trivial measurements and is monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant preprocessing channels. Finally, we illustrate the operational meaning and limitations of the equivalence through qubit POVMs, single-photon phase sensing, and noisy photon-number resolving detection. In particular, the maximal Fisher information in a Mach-Zehnder interferometer is shown to be determined by the squared maximal coherence of the measurement, while in an imperfect photon-number resolving detector the maximal coherence behaves as a proper sharpness monotone, unlike conventional PVM-based unsharpness measures. △ Less

This mission is the statement-level formalization of Maximal coherence of quantum measurement and the resource theory of sharpness (arXiv:2607.05847v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

6 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Common-MUB equivalence for spectral distance (§III B, Eq. (32), PDF p. 6). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

11 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 2, Eq. (17): A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • Theorem 1, Eq. (16): A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • Eqs. (18)–(21): A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • Corollary 3, Eq. (26): A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • Corollary 4, Eq. (33): A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • §III C, trivial-noise mixing: A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • Theorem 5, Eq. (34): A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • Appendix A, Lemma 2: A source-pinned extremal-value convention or sufficient regularity/attainment guarantee for the general distance class. (source-ambiguity)
  • Corollary 6: A condition ensuring that the covariant channel preserves incoherent POVMs, or another valid monotonicity mechanism. (source-ambiguity)
  • §IV A and Figure 1: Specified tetrahedral Bloch vectors and their outcome correspondence with the joint X–Y measurement, plus a consistent sharpness curve. (source-ambiguity)
  • §IV C, Eq. (47), Figure 3: Finite photon-number/energy restriction or a defined infinite-dimensional extension of all distances and normalization rules. (source-ambiguity)

Selected references

9 thms2 active usersReviewed
Completed
Quantum Information·Captain: Goku

Anomaly fluctuation theorem for quantum coherence dynamicsResearch Paper

Motivation

Quantum coherence is a central resource in quantum information science, yet general frameworks and constraints governing its dynamics remain limited. Although quantum coherence lacks a general monotonicity law, we establish an exact integral fluctuation theorem (FT) for coherence dynamics, formulated in terms of Kirkwood-Dirac (KD) quasiprobability trajectories and valid for arbitrary initial states and dynamics. This integral FT deviates from the standard unit-valued form, and the complex deviation, termed the anomaly, quantifies a weighted overlap between the residual final-state coherence and the coherence generated from the dephased input. The real part of the anomaly yields bounds on coherence change, while its imaginary part constrains the second-order moment of the stochastic coherence change weighted by the imaginary parts of the KD quasiprobabilities. Our results establish general statistical constraints on coherence dynamics, demonstrating the utility and broad applicability of FTs for studying quantum-resource dynamics. △ Less

This mission is the statement-level formalization of Anomaly fluctuation theorem for quantum coherence dynamics (arXiv:2609.06632v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

8 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Anomaly bounded by output coherence and cohering power (Eq.(5), PDF p. 2). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

8 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 1 / Eq.(4): Arbitrary-input exponential moment on the active KD trajectory support (source-ambiguity)
  • KD mean after Eq.(3): Finite stochastic coherence variable on nonzero KD weights with zero output eigenvalues (source-ambiguity)
  • Corollary 2: Anomaly-free exponential FT for maximally incoherent channels (source-ambiguity)
  • Sharpness following Theorem 3: Quantifiers and family witnessing sharpness of d-2 (source-ambiguity)
  • Theorem 4 / Eq.(7): Active-support negativity mass and bounded stochastic coherence change (source-ambiguity)
  • Nonnegative-real-KD specialization of Eq.(7): Domain of the nonnegative-real-KD specialization (source-ambiguity)
  • Imaginary zeroth and first moments, pp.3–4: First imaginary logarithmic moment on nonzero complex KD weights (source-ambiguity)
  • Theorem 5 / Eq.(9): Imaginary logarithmic second moment and bounded remainder domain (source-ambiguity)

Selected references

12 thms2 active usersReviewed
Quantum Information·Captain: Goku

Hierarchy of Rényi Coherent Information in Stabilizer CodesResearch Paper

Motivation

Rényi coherent information, a computable proxy for the von Neumann coherent information, is widely used to study mixed-state phases of matter and decodability transitions in noisy quantum error-correcting codes. However, being a difference of two Rényi entropies, it need not be monotonic in the Rényi index, and lacks the operational meaning of its von Neumann counterpart. Here we address both issues for stabilizer codes. First, for Pauli noise generated by independent Bernoulli events, we prove that the Rényi-nn coherent information is nondecreasing in n∈Z+n \in \mathbb{Z}^+. This follows from a general theorem: if independent random bits are mapped linearly to a fine label TT and a coarse label CC, then the Rényi entropy difference Hn(C)−Hn(T)H_n(C)-H_n(T) is nondecreasing in nn. For stabilizer codes, TT is the joint syndrome--logical class and CC is the syndrome, and the difference is the Rényi-nn coherent information up to a constant. The same theorem covers classical linear codes and independent detector error models. Second, for arbitrary stochastic Pauli noise, we give the Rényi-nn coherent information an operational meaning via postselection on matching syndromes between one data block and n−1n-1 auxiliary blocks. We determine when this defines a quantum channel and show that saturation of the Rényi-nn coherent information is equivalent to asymptotically perfect recovery of the postselected channel. Moreover, the Rényi-nn coherent information also upper-bounds the ordinary coherent information achievable after any syndrome-conditioned recovery. △ Less

This mission is the statement-level formalization of Hierarchy of Rényi Coherent Information in Stabilizer Codes (arXiv:2609.11930v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

12 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Corollary 1.1 / abstract: stabilizer-code Rényi hierarchy (Corollary 1.1 / abstract: stabilizer-code Rényi hierarchy, PDF p. 2). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

18 results of the paper are not stated yet, because the library lacks an interface they need:

  • Unnumbered Pauli matching channel and Choi state, Eqs.(36)–(37): code-derived syndrome instrument and retained flagged matching map (hard)
  • Proposition 1 (Matched-syndrome recovery under Pauli noise): attained optimal entanglement fidelity over arbitrary syndrome-conditioned CPTP recoveries (hard)
  • Unnumbered absolute saturation iff full-block recovery: sequence of actual matched channels and their optimal full-block recovery fidelities (hard)
  • Unnumbered nonpositive coherent-information certificate and fidelity ceiling: ordinary coherent information after arbitrary flagged recovery and its optimal fidelity (hard)
  • Appendix B.1: general-noise replica ratio and optimal state recovery: normalized branch Choi states, syndrome-resolved state, moments and attained recovery optimum (hard)
  • Appendix B.2: balance, channel, and affine normalization iff: input-by-input normalization of a retained flagged quantum operation and its affinity on density matrices (hard)
  • Appendix B.2: branchwise trace scaling suffices for every-order balance: branch adjoint effects coupled to the actual syndrome-instrument maps (hard)
  • Appendix B.2: balance without branchwise trace scaling: explicit basis-measurement and sector-preparation instrument (hard)
  • Appendix B.2: finite Vandermonde criterion for every-order balance: positive probability fibers and their matrix-valued effects for the constructed syndrome instrument (hard)
  • Appendix B.2: amplitude-damping failure and endpoint balance: repetition-code amplitude-damping syndrome instrument (hard)
  • Appendix B.3: recovery controls uniform relative acceptance variation: uniform acceptance variation, operator norm, reference-basis transpose and optimal recovery on the same branches (hard)
  • Proposition S1 (Finite-size recovery bounds): general-noise recovery optimum and source-coupled A_n,U_n,B_n,Delta_n (hard)
  • Appendix C.1: dimension-independent linear recovery-error bound: actual optimal general-noise fidelity and syndrome-resolved entropy deficit (hard)
  • Appendix C.2: dimension-uniform converse with mixed output marginals: maximally mixed output-marginal predicate on normalized branch states and optimal recovery (hard)
  • Appendix C.2: exactly balanced identity/replacer example: explicit identity/replacer flagged instrument parametrized by d,n,q and its optimized moments (hard)
  • Appendix C.2: recovery does not uniformly force saturation: dimension-varying exactly balanced identity/replacer instruments (hard)
  • Appendix C.2: sharp dimension scaling of the converse class: uniform converse class over genuine instruments and its identity/replacer witness family (hard)
  • Appendix C.2 final remark: rate-normalized saturation is insufficient: full-block optimal CPTP recovery of identity tensor complete dephasing (hard)

Selected references

15 thms2 active usersReviewed
Completed
Quantum Information·Captain: Goku

Coherence-generating power deviation: Fluctuations and input selectivity beyond average coherence generationResearch Paper

Motivation

Coherence-generating power (CGP) tells how much coherence a quantum process produces on average from incoherent inputs, but an average can hide where that coherence comes from. We introduce the coherence-generating power deviation (CGPD), the standard deviation of the generated Hilbert--Schmidt coherence over the uniform simplex of incoherent states. CGPD turns coherence generation into a landscape: a small value signals robust production across input populations, whereas a large value reveals selective conversion of particular population imbalances. By representing unitary coherence generation as a quadratic response on population space, we obtain exact first and second moments in arbitrary finite dimension. We prove that every nontrivial unitary generator must fluctuate, and we separate this unavoidable isotropic fluctuation from excess anisotropy. The resulting distinction between average strength and input selectivity has direct consequences for benchmarking, reliability guarantees, input-ensemble susceptibility, and four-copy measurement protocols; it also extends naturally to unital quantum channels. Matched quantum-walk and quasiperiodic-transport examples show that topology and localization can leave CGP unchanged while substantially changing CGPD, revealing structure invisible to the mean alone. △ Less

This mission is the statement-level formalization of Coherence-generating power deviation: Fluctuations and input selectivity beyond average coherence generation (arXiv:2609.19046v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

34 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Theorem 3, Exact coherence moments (original paper seed) (Theorem 3, Exact coherence moments (original paper seed), PDF p. 4). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

1 results of the paper are not stated yet, because the library lacks an interface they need:

  • §VIII.C, Eqs.(96)–(100); Appendix E, Table II: Exact selected-root definitions or brackets, and numerical error tolerances for the matched transport values and approximate variance ratio. (source-ambiguity)

Selected references

39 thms2 active usersReviewed
Quantum Information·Captain: Goku

A complete characterization of eventually entanglement breaking unital quantum channelsResearch Paper

Motivation

We obtain a complete characterization of eventually entanglement breaking (EEB) unital channels in terms of some simple commutation properties of their eigenvectors. Specifically, the following properties are equivalent for a unital channel: (i) it is EEB; (ii) it is eventually positive partial transpose (PPT); (iii) it is eventually mixed twisted dephasing; and (iv) its eigenvectors corresponding to non-zero eigenvalues commute with all its peripheral eigenvectors. Similar results hold for one-parameter semigroups of unital quantum channels, where continuity forces such semigroups to become primitive. We employ a matrix integral technique inspired by a result of Watrous. We also characterize the extreme points of the convex set of unital EB channels and, using this characterization and the theory of equiangular tight frames (ETFs), identify several extreme points beyond the twisted dephasing channels. △ Less

This mission is the statement-level formalization of A complete characterization of eventually entanglement breaking unital quantum channels (arXiv:2609.24168v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

49 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Theorem 6.11: seven equivalent eventual channel properties (Theorem 6.11, PDF p. 24). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

1 results of the paper are not stated yet, because the library lacks an interface they need:

  • Corollary 5.9: A consistent basis-covariant interpretation of the restricted-unitary affine representation in Corollary 5.9 (source-ambiguity)

Selected references

53 thms2 active usersReviewed
Completed
Quantum Information·Captain: Goku

The Structure of Higher-Order Quantum Resources with an Application to CoherenceResearch Paper

Motivation

We develop a general framework for higher-order quantum resources that treats the resourcefulness of quantum states, channels, and supermaps (also called processes) in a unified manner. By generalizing well-established concepts from resource theories of states and channels, we introduce and study supermap resource monotones and supermap distillation and dilution. As free supermaps, we consider free networks (supermaps that can be implemented with free channels), compatible supermaps (sets of supermaps closed under arbitrary composition), and maximally free superchannels. Specializing to approximate channel-to-channel conversions, we show that under mild assumptions, the conversion error with respect to the diamond distance and the worst-case infidelity coincide. We then apply the general framework to resource theories of dynamical coherence, where we establish the existence of largest sets of compatible supermaps. By characterizing these sets with semidefinite constraints, we show that they are strictly larger than the corresponding free networks but do not contain all maximally free superchannels. We finish by investigating the operational differences of these sets in one-shot channel distillation and dilution, providing solutions for both tasks in the parallel setting and for dilution in the adaptive setting. △ Less

This mission is the statement-level formalization of The Structure of Higher-Order Quantum Resources with an Application to Coherence (arXiv:2609.40186v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

9 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Lemma 18: four explicit DIO channels used as compatibility testers (Lemma 18, PDF p. 25). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

20 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 10: Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre (hard)
  • Proposition 4: A real-valued convention for infimum over an empty fixed-signature compatible fibre (source-ambiguity)
  • Theorem 5: General closed convex free-channel fibres, maximally preserving deterministic superchannels, and the complete extended-valued standard/hypothesis-testing/min/affine resource optimization interfaces (hard)
  • Unnumbered extension (37): General closed convex free-channel fibres, maximally preserving deterministic superchannels, and the complete extended-valued standard/hypothesis-testing/min/affine resource optimization interfaces (hard)
  • Theorem 6: General closed convex free-channel fibres, maximally preserving deterministic superchannels, and the complete extended-valued standard/hypothesis-testing/min/affine resource optimization interfaces (hard)
  • Corollary 7: General closed convex free-channel fibres, maximally preserving deterministic superchannels, and the complete extended-valued standard/hypothesis-testing/min/affine resource optimization interfaces (hard)
  • Corollary 8: General closed convex free-channel fibres, maximally preserving deterministic superchannels, and the complete extended-valued standard/hypothesis-testing/min/affine resource optimization interfaces (hard)
  • Unnumbered result (C40): Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre (hard)
  • Proposition 11: Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre (hard)
  • Theorem 12: Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre; both adaptive N-slot evaluation and parallel N-fold tensor input, including optimizer versus infimum distinctions (hard)
  • Unnumbered results (58)–(59): Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre; one-shot Fourier target dimension cost for both diamond and fidelity metrics (hard)
  • Theorem 13: The explicitly constrained compatible/maximal Fourier-distillation functionals (60), (63), and (C225), linked to independently defined compatible/maximally free transformations (hard)
  • Unnumbered result (62): The explicitly constrained compatible/maximal Fourier-distillation functionals (60), (63), and (C225), linked to independently defined compatible/maximally free transformations (hard)
  • Proposition 14: The explicitly constrained compatible/maximal Fourier-distillation functionals (60), (63), and (C225), linked to independently defined compatible/maximally free transformations (hard)
  • Proposition 15: The explicitly constrained compatible/maximal Fourier-distillation functionals (60), (63), and (C225), linked to independently defined compatible/maximally free transformations (hard)
  • Proposition 16: Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre; two-use compatible slot substitution and the parallel tensor-power channel (hard)
  • Lemma 17: Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre (hard)
  • Unnumbered Appendix A result (A26): Signature-indexed deterministic combs of every order, causally lawful partial link products, and compatible families with exactly the designated channel fibre (hard)
  • Corollary 25: The explicitly constrained compatible/maximal Fourier-distillation functionals (60), (63), and (C225), linked to independently defined compatible/maximally free transformations (hard)
  • Proposition 26: The explicitly constrained compatible/maximal Fourier-distillation functionals (60), (63), and (C225), linked to independently defined compatible/maximally free transformations (hard)

Selected references

14 thms2 active usersReviewed
Quantum Information·Captain: Goku

Feasibility Ordering of Entanglement-Source Placement for Qubit ChannelsResearch Paper

Motivation

This work studies the placement of an entanglement source along a communication line formed by two noisy qubit channels. Recent work argued, on analytical and numerical grounds, that midpoint placement should be at least as favorable as endpoint placement. Here it is shown that, for arbitrary qubit channels, if a sequential composition can preserve entanglement, then the corresponding parallel action cannot annihilate all entanglement. The proof uses the transpose-factorization criterion introduced in that recent work. Quantum Sinkhorn scaling converts every strictly positive qubit channel into a unital representative, and the special normal form of unital qubit channels then yields the required factorization of the transposed map through the original channel. Depolarizing regularization and the closedness of the set of entanglement-breaking channels extend the result to arbitrary channels. Consequently, Λ1⊗Λ2∈EAΛ_1 \otimes Λ_2 \in \mathrm{EA} implies Λ2∘Λ1∈EBΛ_2 \circ Λ_1 \in \mathrm{EB}, and, by exchanging the two channels, the same holds for the opposite composition order. This proves the recent conjecture that midpoint placement is optimal for all qubit channels, in the feasibility sense in which that optimality was originally defined. △ Less

This mission is the statement-level formalization of Feasibility Ordering of Entanglement-Source Placement for Qubit Channels (arXiv:2609.18803v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

8 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Parallel EA implies sequential EB (Theorem 1, PDF p. 4). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Selected references

10 thms2 active usersReviewed
Completed
Quantum Information·Captain: Goku

Static classical-quantum-entanglement trade-offs: an entropic converse and single-letter characterizationsResearch Paper

Motivation

The direct static capacity region of a bipartite quantum state describes its asymptotic conversion into classical communication, quantum communication, and entanglement, with all communication directed from Alice to Bob. This paper gives a unified entropic converse for this region. The proof treats generated and consumed resources simultaneously and derives all three information inequalities for one local instrument. Its main ingredients are no-signalling, the chain rule, strong subadditivity, and continuity of conditional entropy. A supporting-hyperplane characterization leads to the static capacity formula, which is helpful in determining single-letterization of the capacity region. For a pure state of entanglement entropy hh subjected to erasure with probability pp, the complete region is the convex hull of 00 and (0,−ph,(1−p)h)(0,-ph,(1-p)h), plus the unit-resource cone. This statement holds for arbitrary collective instruments by using subset-entropy inequalities related to earlier methods used for the dynamic capacity region of the erasure channel. Complete characterizations also hold for symmetrically extendible states and locally flagged pure-state mixtures. For maximally correlated states, the optimization over local instruments admits an exact matrix formulation and an additive outer bound. An explicit qubit example shows that a nonorthogonal discarded quantum memory can outperform every efficient instrument and every instrument with conditionally commuting discarded states. For states obtained by qubit dephasing of Schmidt-aligned pure states, an exact threshold characterizes when the static capacity formula vanishes, at every blocklength and after regularization. Full single-letter characterizations for general Hadamard states and for the remaining trade-offs for dephased states remain open, with sufficient additivity conditions identified. △ Less

This mission is the statement-level formalization of Static classical-quantum-entanglement trade-offs: an entropic converse and single-letter characterizations (arXiv:2609.23298v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

11 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is §8.2.4: matrix score as a weighted coherence difference (§8.2.4: matrix score as a weighted coherence difference, PDF p. 36). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

36 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 4.1: finite-block simultaneous entropic converse (seed paper result): finite-block code and encoder-induced instrument (hard)
  • Theorem 3.1: direct static capacity theorem: operational catalytic and conventional capacity regions (hard)
  • §3.1 and Appendix A: generating rate point is achievable: operational primitive-point achievability (hard)
  • §5.1: exact finite-support directions and fixed-instrument duality: extended-real support of an actual instrument polyhedron (hard)
  • §5.2: superadditivity, boundedness, and regularization limit: unrestricted-instrument support and its tensor-power sequence (hard)
  • Theorem 5.1: static capacity half-space characterization: operational capacity and regularized unrestricted support (hard)
  • §5.3: self-additivity implies a one-copy region: operational region and one-copy instrument union (hard)
  • Theorem 6.1: symmetrically extendible states: operational region of a symmetrically extendible resource (hard)
  • Theorem 6.2: erased-state support at every blocklength: arbitrary-dimension erased-state unrestricted support (hard)
  • Theorem 6.3: erased-state complete static region: operational erased-state region (hard)
  • §6.2.3: five-facet form and limiting regions: operational erased-state region with endpoint regimes (hard)
  • §6.2.4: exact two erased-state capacity slices: operational entanglement slices and maximum attainment (hard)
  • Theorem 6.5: locally flagged pure-state mixtures: flagged-state operational region and unrestricted support (hard)
  • Proposition 7.1: Hadamard realization iff separable complementary marginal: Hadamard realization with support isometries and pure ancillas (hard)
  • Theorem 7.2: additive coherent-information face of Hadamard states: Hadamard-generated resource and unrestricted support (hard)
  • Lemma 7.3: Hadamard score decomposition: Hadamard instrument state and penalty (hard)
  • §7.2: optimized penalty and exact additivity criterion: full Hadamard penalty infimum and unrestricted support (hard)
  • §7.2: exact score change when retaining discarded output: retention operation on the actual dilated instrument (hard)
  • Theorem 8.1: canonical instrument normal form: full instrument information union and canonical subunion (hard)
  • Theorem 8.3: exact attained matrix support formula: unrestricted support versus attained mean-diagonal matrix optimization (hard)
  • §8.2.2: complete one-copy matrix information region: one-copy instrument union and constrained primitive matrix ensemble set (hard)
  • §8.2.3: exact block formula and outcome bounds: block unrestricted support and attained matrix ensemble reduction (hard)
  • §8.3.2 and Appendix D.5: commuting memories cannot improve efficient support: conditional discarded states of arbitrary rectangular instruments (hard)
  • Proposition 8.5: maximally correlated coherent-information face: unrestricted collective support for maximally correlated states (hard)
  • Theorem 8.6: strict discarded quantum-memory advantage: full efficient/commuting instrument comparison and achievable primitive region (hard)
  • §8.5.2: exact restricted bottleneck identity and outer relaxation: flagged-pure memory class and canonical-instrument correspondence (hard)
  • Corollary 8.9: additive outer bound for operational region: operational capacity and full block static support (hard)
  • Conjecture 8.10: memory reduction (deferred open statement): source-proved memory-reduction equality (hard)
  • Theorem 9.1: exact one-copy qubit-dephasing support: unrestricted one-copy support and attained constrained qubit ensemble (hard)
  • §9.1: symmetric support simplification and complete one-copy region: one-copy unrestricted information region and joint primitive ensemble (hard)
  • Theorem 9.3: exact vanishing threshold, every block and regularization: unrestricted block support and regularized support of aligned state (hard)
  • §9.3: exact threshold supporting planes and endpoint regions: operational supporting faces and endpoint regions (hard)
  • Corollary 9.4: exact entanglement slope at zero quantum supply: operational zero-classical-rate entanglement-slice supremum (hard)
  • §9.4: collective outer bounds on two dephasing slices: operational slice membership for the fixed symmetric dephasing resource (hard)
  • §9.5: arbitrary pure inputs, exact coherent face and primitive point: arbitrary-input Hadamard realization and primitive achievability (hard)
  • Appendix F: feasible finite-family capacity-slice construction: operational feasibility and attained finite-sample primal optimization (hard)

Selected references

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Beyond Maximal Entanglement : Exact Resources for Multiparty Encrypted Quantum CloningResearch Paper

Motivation

Encrypted quantum cloning distributes an unknown kk-qubit state among mm encrypted clones so that no individual clone reveals the input, yet the state can be recovered from any one clone together with a common quantum key. We ask the inverse question: for a fixed encoding architecture, which multipartite pure states can serve as exact resources for this task? For m≥2m\ge2, we completely characterize the pure resources compatible with a sector-wise two-Pauli encoder, with necessity holding for arbitrary completely positive trace-preserving (CPTP) recovery maps. For even mm, exact recovery requires maximal entanglement across the signal--noise cut. For odd mm, less entanglement can suffice, provided that the surviving signal correlations have the structure selected by the encoder. We further show, without fixing the encoder, that exact recovery from every authorized subsystem already implies perfect concealment of each individual signal and requires at least (m−1)k(m-1)k ebits of signal--noise entanglement. For graph states, the resource classification reduces to an exact condition on the kernel of the signal--noise cut matrix, leading to binary certification and constructive Clifford recovery. We identify rank-deficient graph resources that attain the architecture-independent entanglement bound and prove that the entire Dicke family, including WW states, is excluded for every sector-wise two-Pauli encoder. Our results show that encrypted recovery depends not only on how much entanglement a resource contains, but also on how its correlations are organized relative to the encoder. △ Less

This mission is the statement-level formalization of Beyond Maximal Entanglement : Exact Resources for Multiparty Encrypted Quantum Cloning (arXiv:2609.28241v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

40 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Theorem 1 (seed): Exact fixed-encoder pure-resource criterion (Theorem 1 (seed): Exact fixed-encoder pure-resource criterion, PDF p. 4). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

2 results of the paper are not stated yet, because the library lacks an interface they need:

  • S4.3(b), Eq.(S62), grouped finite certificates: An unambiguous chronological/operator order for the printed P24 CNOT word, consistent with MP=B^(-T). (source-ambiguity)
  • Table II / S4.3(e–f): The two complete fixed 24-vertex sampled graph realizations and deterministic generator specification. (source-ambiguity)

Selected references

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A sharp norm inequality for entanglement-breaking channelsResearch Paper

Motivation

For a channel ΦΦ on MdM_d written in the generalised Bloch parameterisation r↦Ar+cr \mapsto Ar+c, we prove that entanglement breaking implies ∥A∥∗2+d(d−1)2∣c∣2≤(d−1)2\|A\|_*^2 + \frac{d(d-1)}{2}|c|^2 \le (d-1)^2, where ∥⋅∥∗\|\cdot\|_* denotes the nuclear norm. The bound is attained in every dimension by the completely dephasing channel, and at d=2d=2 by an explicit family with c≠0c\neq0. The qubit case reads ∥A∥∗2+∣c∣2≤1\|A\|_*^2+|c|^2\le1 and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound ∥A∥∗≤1\|A\|_*\le1 by retaining the POVM completeness relation ∑kfk=0\sum_k f_k=0, which her argument discards; this recentres AA from a second moment into a cross-covariance, and is the entire content of the improvement. △ Less

This mission is the statement-level formalization of A sharp norm inequality for entanglement-breaking channels (arXiv:2609.27906v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

7 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Theorem 1 (§3): sharp EB nuclear-norm inequality, including its qubit specialization (Theorem 1 (§3): sharp EB nuclear-norm inequality, including its qubit specialization, PDF p. 2). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

2 results of the paper are not stated yet, because the library lacks an interface they need:

  • §4 unnumbered: claimed necessary equality conditions (i)–(iv): Correct necessary equality characterization allowing zero output variance. (source-ambiguity)
  • §6 unnumbered: claimed strict improvement off the unital set: A quantified, valid scope for strict improvement of integer EB-index crossing times away from unitality. (source-ambiguity)

Selected references

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Complete Classification of Reversible Entanglement Transformations of Two QutritsResearch Paper

Motivation

Suppose Alice and Bob can transform one shared mixed state into another by local operations and classical communication, and can also recover the first state from the second. What can change in such a conversion? We give a complete classification of reversible conversions between two-qutrit states. We consider exact deterministic LOCC transformations of a single copy, allowing finite protocols and the countable protocols specified here. Local flags are known to allow a separable part to be replaced by local preparation while an entangled block is retained. We prove that such a decomposition is necessary for two mutually convertible entangled states that are not locally unitarily equivalent. The two parts are distinguishable by local measurements. The entangled blocks have the same weight and are locally unitarily equivalent after normalization. Each entangled state has a unique smallest retained block. Two entangled states are reversibly convertible if and only if these unnormalized blocks are locally unitarily equivalent. For a state with negative partial transpose, the smallest block can be found using the negative eigenspace of its partial transpose and at most three matrix zero tests. △ Less

This mission is the statement-level formalization of Complete Classification of Reversible Entanglement Transformations of Two Qutrits (arXiv:2610.05234v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

17 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Lemma E1: joint-kernel obstruction (Lemma E1: joint-kernel obstruction, PDF p. 5). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

16 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 1: complete four-case classification (seed on p.1): Source-specific operational LOCC protocol and execution interfaces (hard)
  • Theorem 2: canonical unnormalized-block criterion: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Proposition 1: core rigidity and its channel form: Source-specific operational LOCC protocol and execution interfaces (hard)
  • S6, Eq. (S42): Linked LOCC comparison semantics for the complete NPT decision procedure (hard)
  • Unnumbered: PPT entangled rigidity and PPT/NPT separation: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Unnumbered: reversible conversions have finite flag witnesses: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Unnumbered: reversible conversions force finite-prefix monotone equality: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Unnumbered: Jordan parts and NPT auxiliary states are transported separately: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Unnumbered: exact algebraic-data decision procedure and 81-term separability test: Effective exact algebraic data and a computable real-algebraic feasibility decision interface, including the complete 81 pure product projector formulation. (hard)
  • Example (12), S7.1: reversible non-LU pair with different global ranks: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Example S7.2: retained weights and Schmidt coefficients both matter: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Example S7.3: same negative part and weight do not determine a class: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Example S7.4: exterior coherence test cannot be omitted: Source-specific operational LOCC protocol and execution interfaces (hard)
  • Unnumbered: canonical flag identity, Eq. (S54): Source-specific operational LOCC protocol and execution interfaces (hard)
  • Unnumbered: maximum exact canonical-state preparation probability, Eq. (S55): Source-specific operational LOCC protocol and execution interfaces (hard)
  • S6, rank-one canonical-block comparison: Source-specific operational LOCC protocol and execution interfaces (hard)

Selected references

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On PPT entanglement distillationResearch Paper

Motivation

We study entanglement distillation under operations that remain completely positive under partial-transpose conjugation, a.k.a. PPT channels, in the standard quantum Shannon theory regime of asymptotically vanishing but non-zero error. Our main results are: (a) a regularised formula for the PPT distillable entanglement in terms of a measured-relative-entropy-like quantity; and (b) two different single-letter converses, one based on operator quadratic forms and the other on Hirschman's strengthening of the Hadamard three-line theorem. As an immediate application of (b), we show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, thereby resolving an open problem in the theory of entanglement manipulation [Regula et al., NJP 21:103017, 2019]. A gap appears already for 3×33\times 3 Werner states: at antisymmetric weight 25/2625/26, a certified upper bound of 0.621070.62107 ebits lies below the (regularised) Rains bound, which equals 2526log⁡25−log⁡23≈0.64766\frac{25}{26} \log_2 5 - \log_2 3 \approx 0.64766 ebits. On a different note, (a) implies a faithful lower bound on the PPT distillable entanglement in terms of the entanglement negativity, which provides a quantitative counterpart to the qualitatively known fact that any NPT state is PPT distillable [Eggeling et al., PRL 87:257902, 2001]. △ Less

This mission is the statement-level formalization of On PPT entanglement distillation (arXiv:2610.12454v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

15 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Regularized formula for PPT distillable entanglement (Theorem1, PDF p. 5). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

18 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem2: HS operator calculus and the all-copy conditional infimum (11)–(13) (hard)
  • Theorem3: Support-aware divergences, admissible analytic reference families, and J of (19) (hard)
  • Theorem4: All three optimized converses E_Q, U, E_A and regularized Rains entropy (hard)
  • Lemma10: V_alpha on finite direct sums and Hilbert–Schmidt adjoints/order (hard)
  • Lemma11: Singular-input V_alpha for correlated principal/ancillary states (hard)
  • Proposition12: R_ext, R, R_infty, and E_Q with the correct unnormalized reference (hard)
  • Corollary13: HS functional calculus for noncommuting left/right multiplication and Gamma (hard)
  • Proposition15: U and V_2 on all tensor powers with an arbitrary-operator form bound (hard)
  • Corollary18: Fully defined E_Q, E_A and regularized Rains bound (hard)
  • loccCombinedConverse: Local-instrument LOCC protocols and squashed entanglement in addition to E_Q/E_A (hard)
  • eq74-75: Jointly convex V_alpha and its finite-direct-sum functional calculus (hard)
  • eq102-104: U, its regularized infimum and exponential strong-converse rate predicate (hard)
  • eq117-118: Matrix analytic-family testing functional and support-aware sandwiched Renyi entropy (hard)
  • AppendixA: Singular-input HS support logarithm and regularized weighted geometric means (hard)
  • AppendixB-B1: Spectral truncation on HS tensor powers at singular ancillary marginals (hard)
  • AppendixB-B5: Extended-parameter geometric mean K #_(1+s) L_omega (hard)
  • AppendixB-B9: All-copy admissibility, HS fractional powers and the PPT decay exponent (hard)
  • AppendixE: Admissible Mobius operator family, J residuals, and operational strong converse (hard)

Selected references

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Separability criteria and lower bounds of entanglement via realignmentResearch Paper

Motivation

The detection and quantification of quantum entanglement are central problems in quantum information science. Based on the realignment of density matrices and the vectorization of reduced density matrices, we construct a family of enhanced separability criteria for bipartite and multipartite quantum systems. Our criteria recover the previously proposed realignment-based criteria as special cases and detect more entanglement than the earlier ones. Moreover, we derive tighter lower bounds for the concurrence and the convex-roof extended negativity, criteria for detecting the genuine tripartite entanglement, and lower bounds on genuine-multipartite-entanglement concurrence. The advantages of our results are demonstrated with detailed examples. △ Less

This mission is the statement-level formalization of Separability criteria and lower bounds of entanglement via realignment (arXiv:2610.15664v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

12 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Arbitrary-rank bipartite separability criterion (Theorem 1, PDF p. 3). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

3 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 5, Eq.(16) and its p.10 continuation: A well-defined decomposition-independent multipartite matrix QR in Eq.(16) for arbitrary rank r. (source-ambiguity)
  • Example 3; Figures 1–2 p.8: An exact quantified comparative claim for the concurrence and CREN curves in Figures 1–2. (source-ambiguity)
  • Example 5; Figure 3 and Table IV: A valid literal inclusive endpoint for the displayed GHZ detection witness. (source-ambiguity)

Selected references

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Universal Improvement of Channel Fidelity via Entanglement AssistanceResearch Paper

Motivation

Communication using quantum channels generally requires encoding and decoding on many identical uses of a quantum channel. In practical settings, decoherence may severely limit our ability to do so, potentially rendering each channel use nonidentical. Given nn nonidentical channels, we present an entanglement-assisted encoding and decoding strategy that yields a channel with higher worst-case fidelity than all of the nn given channels. The strategy is universal, in the sense that the improvement holds regardless of what channels are given, as long as they satisfy mild assumptions on their initial entanglement fidelities. Our protocol is optimal when the given channels are depolarizing channels, and is essentially the unique one that is universal. This idea can also be extended to classical channels, where shared randomness between the sender and the receiver allows universal enhancement of the probability of correct transmission. We then consider general quantum resource theories, providing a necessary condition and a sufficient condition for the possibility of such universal improvement of quantum resources. △ Less

This mission is the statement-level formalization of Universal Improvement of Channel Fidelity via Entanglement Assistance (arXiv:2610.05669v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

7 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Unnumbered: shared-randomness protocol and identity-biased improvement (Unnumbered: shared-randomness protocol and identity-biased improvement, PDF p. 6). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

25 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 1: universal entanglement-assisted improvement: Explicit n-register entanglement-assisted Bell agreement instrument, its local separated-party realization, and reference-assisted worst-case squared fidelity. (hard)
  • Unnumbered: untwirled entanglement-assisted protocol, Eqs.15–22: Concrete raw Bell-agreement instrument with N-tensor-id weights and transpose corrections. (hard)
  • Proposition 2: optimal fidelity upper bound for generalized Pauli inputs: Heralded NS-assisted superinstrument predicate retaining both no-signalling marginals and action on arbitrary input maps. (hard)
  • Unnumbered corollary: optimal depolarizing fidelity and single-use obstruction: Heralded NS model and the concrete EA construction, parameterized by entanglement fidelity rather than depolarizing noise rate. (hard)
  • Proposition 3: optimal success probability at optimal two-channel fidelity: Heralded NS model, exact optimal-output fidelity, and input-independent success scalar on strict interior depolarizing inputs. (hard)
  • Proposition 4: uniqueness of the universal quantum protocol: NS instrument plus supermap input/output Haar twirls and explicit twirled EA branch. (hard)
  • Proposition 5: classical NS fidelity upper bound: NS instrument on classical embeddings, classical minimum-diagonal fidelity and the extended column ratio. (hard)
  • Unnumbered corollary: optimal d-ary symmetric fidelity: NS-assisted upper bound and achievable fully specified shared-randomness instrument on symmetric inputs. (hard)
  • Proposition 6: uniqueness of the universal classical protocol: NS instrument and equality of fully classically twirled supermaps with the SR successful branch. (hard)
  • Unnumbered example: two entanglement-breaking channels yield a non-breaking output: Physical EA Bell construction joined to the accepted finite-dimensional entanglement-breaking predicate. (hard)
  • Theorem 7: sufficient conditions for universal resource distillation: Theorem 7: sufficient conditions for universal resource distillation; source domain/provenance (source-ambiguity)
  • Proposition 8: multiplicativity from product-generated free sets: A paper-local generalized channel twirl and compatible one-system and two-system free-channel sets, including convex-hull and affine-hull-within-CPTP product generation. (hard)
  • Theorem 9: generic uniqueness up to twirls and rescaling: Theorem 9: generic uniqueness up to twirls and rescaling; source domain/provenance (source-ambiguity)
  • Theorem 10: obstructions to universal resource distillation: Theorem 10: obstructions to universal resource distillation; source domain/provenance (source-ambiguity)
  • Theorem 11 [22, Thm.1]: Choi representation of completely CPP supermaps: Theorem 11 [22, Thm.1]: Choi representation of completely CPP supermaps; source domain/provenance (hard)
  • Unnumbered supplementary result: output twirling makes success state-independent: Twirling of a whole physical superinstrument and its conditional reference-assisted fidelity functional. (hard)
  • Lemma 1: robustness equality and the free isotropic interval: Lemma 1: robustness equality and the free isotropic interval; source domain/provenance (source-ambiguity)
  • Background: capacity and one-shot coding results: Background: capacity and one-shot coding results; source domain/provenance (hard)
  • Background: process-tomography query lower bound: Background: process-tomography query lower bound; source domain/provenance (hard)
  • Background: universal purification no-go results: Background: universal purification no-go results; source domain/provenance (hard)
  • Background: realization, NS/RNG equivalence and generalized twirl realization: Background: realization, NS/RNG equivalence and generalized twirl realization; source domain/provenance (hard)
  • Background: Haar twirling and entanglement-breaking depolarizing criterion: Background: Haar twirling and entanglement-breaking depolarizing criterion; source domain/provenance (hard)
  • Background: static projective robustness conversion criteria: Background: static projective robustness conversion criteria; source domain/provenance (hard)
  • Background: resource-theory examples with free twirls or equal robustness: Background: resource-theory examples with free twirls or equal robustness; source domain/provenance (hard)
  • Background: open-to-closed threshold universality step: Background: open-to-closed threshold universality step; source domain/provenance (hard)

Selected references

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Conditions for strictly purity-decreasing quantum Markovian dynamicsResearch Paper

Motivation

The purity, Tr(rho^2), measures how pure or mixed a quantum state rho is. It is well known that quantum dynamical semigroups that preserve the identity operator (which we refer to as unital) are strictly purity-decreasing transformations. Here we provide an almost complete characterization of the class of strictly purity-decreasing quantum dynamical semigroups. We show that in the case of finite-dimensional Hilbert spaces a dynamical semigroup is strictly purity-decreasing if and only if it is unital, while in the infinite dimensional case, unitality is only sufficient.

This mission is the statement-level formalization of Conditions for strictly purity-decreasing quantum Markovian dynamics (arXiv:quant-ph/0411119v2). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

4 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Finite-dimensional purity decrease is equivalent to unitality (Theorem 2, PDF p. 2). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

2 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 1; III.A(a),(b); footnotes [14],[15] on p.4: An actual unbounded GKSL generator on trace-class operators, with common dense operator/product domains, convergent commutator form, trace-norm graph core, minimal strongly continuous CP solution, and right differentiation on Dom(G). (hard)
  • Section IV, F=a†aa† example: A concrete minimal full-Fock evolution for F=a†aa† and its natural full generator domain, with trace loss and purity nonincrease, distinguished from the paper’s trace-annihilating graph-core assumption. (source-ambiguity)

Selected references

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On the coincidence of discrete and continuous definitions of Entanglement of FormationResearch Paper

Motivation

It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the lifting a Borel ensemble of pure bipartite states to a Borel probability measure on the unit sphere in the space of Hilbert--Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction. △ Less

This mission is the statement-level formalization of On the coincidence of discrete and continuous definitions of Entanglement of Formation (arXiv:2610.18816v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

1 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Domination by a state gives an actual square-root-range preimage of norm at most one (§3.6, range remark, equation (42), PDF p. 13). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

8 results of the paper are not stated yet, because the library lacks an interface they need:

  • Abstract / §2, Corollary 2: coincidence of the two entanglement-of-formation roofs: Trace-norm state space, natural-log extended marginal entropy, and the two actual convex roofs (hard)
  • §2, Theorem 1: quantitative exact discretization: Borel pure-state ensembles with trace-class Bochner barycenters and finite or countable exact decompositions (hard)
  • §3.3, Lemma 3: convex spectral functional and all four bounds: Hilbert–Schmidt operator space with infinite singular-value sequence and the concrete spectral J and E (hard)
  • §3.4.2, Lemma 4: countable Borel cells controlling J-differences: Concrete finite-entropy unit-amplitude Borel space U and its spectral J-difference functional (hard)
  • §3.5, Lemma 5: exact extraction with a positive ensemble remainder: Trace-class rank-one Bochner integral on U, homogeneous entropy, and finite Borel residual ensembles (hard)
  • §3.6, Corollary 6: lower semicontinuity, zero on separable states, nonattainment: Trace-norm separability and countable nondecomposability together with the actual discrete entropy roof (hard)
  • §3.1, unnumbered: vector–amplitude identification and marginal identity: Completed tensor product, concrete Hilbert–Schmidt coefficient operator, and trace-class partial trace (hard)
  • §3.4.1, unnumbered: explicit Borel phase section and ensemble lift: Trace-norm pure-state Borel space, the first-coordinate section and its Hilbert–Schmidt pushforward with computed cost (hard)

Selected references

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Degradability of Bosonic Gaussian channelsResearch Paper

Motivation

The notion of weak-degradability of quantum channels is introduced by generalizing the degradability definition given by Devetak and Shor. Exploiting the unitary equivalence with beam-splitter/amplifier channels we then prove that a large class of one-mode Bosonic Gaussian channels are either weakly degradable or anti-degradable. In the latter case this implies that their quantum capacity Q is null. In the former case instead, this allows us to establish the additivity of the coherent information for those maps which admit unitary representation with single-mode pure environment.

This mission is the statement-level formalization of Degradability of Bosonic Gaussian channels (arXiv:quant-ph/0603257v4). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

13 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Single-environment Gaussian-channel threshold classification (§III, Table I; q=0,1 on p.4, PDF p. 3). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

1 results of the paper are not stated yet, because the library lacks an interface they need:

  • Abstract; §II p.2; Table I p.3; §V pp.4-5: A source-pinned optimized coherent-information functional, its admissible bosonic input states, and its tensor-power/single-letter capacity convention. (source-ambiguity)

Selected references

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Classical capacity of the lossy bosonic channel: the exact solutionResearch Paper

Motivation

The classical capacity of the lossy bosonic channel is calculated exactly. It is shown that its Holevo information is not superadditive, and that a coherent-state encoding achieves capacity. The capacity of far-field, free-space optical communications is given as an example.

This mission is the statement-level formalization of Classical capacity of the lossy bosonic channel: the exact solution (arXiv:quant-ph/0308012v2). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

4 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Vacuum loss contracts coherent states and preserves purity (Discussion, coherent-state purity, PDF p. 2). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

9 results of the paper are not stated yet, because the library lacks an interface they need:

  • Eq.(4), multimode classical capacity: Eq.(4), multimode classical capacity with its admissibility/approximation convention (source-ambiguity)
  • Discussion, no superadditivity: Discussion, no superadditivity with its admissibility/approximation convention (source-ambiguity)
  • Discussion, coherent-state capacity attainment: Discussion, coherent-state capacity attainment with its admissibility/approximation convention (source-ambiguity)
  • Eq.(11), conventional receiver rates: Eq.(11), conventional receiver rates with its admissibility/approximation convention (source-ambiguity)
  • Discussion, operational high-occupation limit: Discussion, operational high-occupation limit with its admissibility/approximation convention (source-ambiguity)
  • Eq.(15), uniform-loss broadband capacity: Eq.(15), uniform-loss broadband capacity with its admissibility/approximation convention (source-ambiguity)
  • Eqs.(16)-(17), far-field capacity: Eqs.(16)-(17), far-field capacity with its admissibility/approximation convention (source-ambiguity)
  • Eq.(18) and receiver comparisons: Eq.(18) and receiver comparisons with its admissibility/approximation convention (source-ambiguity)
  • Narrowband channel, Eq.(14): Executed unassisted energy-constrained classical codes on full Fock trace-class states, with actual memoryless channel outputs and collective measurements (hard)

Selected references

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Generalized compactness in linear spaces and its applicationsResearch Paper

Motivation

The class of subsets of locally convex spaces called μ\mu-compact sets is considered. This class contains all compact sets as well as several noncompact sets widely used in applications. It is shown that many results well known for compact sets can be generalized to μ\mu-compact sets. Several examples are considered. The main result of the paper is a generalization to μ\mu-compact convex sets of the Vesterstrom-O'Brien theorem showing equivalence of the particular properties of a compact convex set (s.t. openness of the mixture map, openness of the barycenter map and of its restriction to maximal measures, continuity of a convex hull of any continuous function, continuity of a convex hull of any concave continuous function). It is shown that the Vesterstrom-O'Brien theorem does not hold for pointwise μ\mu-compact convex sets defined by the slight relaxing of the μ\mu-compactness condition. Applications of the obtained results to quantum information theory are considered.

This mission is the statement-level formalization of Generalized compactness in linear spaces and its applications (arXiv:1002.3610v3). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

33 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Theorem 1, pp.14–16: generalized Vesterstrom–O’Brien theorem (original paper seed) (Theorem 1, pp.14–16: generalized Vesterstrom–O’Brien theorem (original paper seed), PDF p. 14). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

14 results of the paper are not stated yet, because the library lacks an interface they need:

  • Remark 1: affine coercive families in the examples: Affine lower semicontinuous coercive families on all noncompact examples of §2.2, including trace-class operator and positive-map domains. (hard)
  • Unnumbered local compactness necessity after Proposition 8: Whether the convergent range includes its limit, or compactness is meant as relative compactness. (source-ambiguity)
  • Proposition 11: Schatten positive unit parts: Schatten p operator ideals on arbitrary separable complex Hilbert spaces, with actual p norms and positive closed unit parts. (hard)
  • Proposition 12: positive maps on trace class: Positive continuous complex-linear maps between trace-class Banach spaces, operator norm, and the strong topology of pointwise trace-norm convergence. (hard)
  • Proposition 16: Hilbert-cube dichotomy: An extension of pointwise mu-compactness to unbounded domains and laws without Pettis barycenters. (source-ambiguity)
  • Unnumbered consequences of Propositions 11–12: Trace-norm state domain; completely positive trace-nonincreasing and trace-preserving maps in strong topology; positive l1 maps and sub-Markov/Markov domains. (hard)
  • Theorem 2: continuous entanglement monotone: Exact arbitrary-LOCC protocol class in infinite-dimensional bipartite systems. (source-ambiguity)
  • Unnumbered attained roof identity in the proof of Theorem 2: Completed bipartite trace-class domain, trace-norm continuous partial trace, countable pure-state atomic roof, and distinct general-measure pure-state roof with attained optimizer. (hard)
  • Example 2: the trace-power family of monotones: The same infinite-dimensional arbitrary-LOCC model used by Theorem 2, plus its trace-power operator interface. (source-ambiguity)
  • Lemma 4: background finite-dimensional convex equivalences: A translated-cone convention, polar convention and nonempty-set hypothesis making all four equivalences correct. (source-ambiguity)
  • Proposition 19: finite-dimensional generalized µ-compactness characterization: A corrected version of all four finite-dimensional characterization clauses inherited from Lemma 4. (source-ambiguity)
  • Unnumbered weak-topology positive-cone examples: The p range, measure-space assumptions, and intended weak topology for the positive Lp cones. (source-ambiguity)
  • Unnumbered unbounded positive-cone examples: Generalized mu-compactness of the full positive trace-class cone, alongside the l1 and finite-measure cones. (hard)
  • Lemma 5: strong compactness of positive maps: Trace-class Banach spaces and positive continuous complex-linear maps with strong topology, uniform operator-norm bound, and faithful kernel-zero density operators. (hard)

Selected references

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The characterization of Gaussian operations and Distillation of Gaussian StatesResearch Paper

Motivation

We characterize the class of all physical operations that transform Gaussian states to Gaussian states. We show that this class coincides with that of all operations which can be performed on Gaussian states using linear optical elements and homodyne measurements. For bipartite systems we characterize the processes which can be implemented by local operations and classical communication, as well as those that can be implemented using positive partial transpose preserving maps. As an application, we show that Gaussian states cannot be distilled by local Gaussian operations and classical communication. We also define and characterize positive (but not completely positive) Gaussian maps.

This mission is the statement-level formalization of The characterization of Gaussian operations and Distillation of Gaussian States (arXiv:quant-ph/0204085v4). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

5 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is The Gaussian entanglement scale is positive and attained (Appendix A, Eq.(A1): well-defined attained V for finite CMs, PDF p. 5). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

23 results of the paper are not stated yet, because the library lacks an interface they need:

  • Abstract; Sec. IVB: no Gaussian LOGCC distillation, including assistance: Actual Gaussian LOGCC instruments with adaptive classical communication and conditional normalized outputs (hard)
  • Sec. II, Eqs.(4)–(5): the finite EPR approximating family: Actual normalized squeezed states on bosonic Hilbert space and their characteristic functions (hard)
  • Sec. IIIA: full Gaussian Choi characterization and optical implementation: Admissible generalized Choi domain, convergence and exact-versus-limit optical implementation (source-ambiguity)
  • Sec. IIIA, Eqs.(9), (10a), (13): normalized conditional covariance: Finite-resource Gaussian Choi contraction on actual trace-class states (hard)
  • Sec. IIIA, Eq.(10b): printed displacement formula: Consistent displacement parameterization in Eqs.(8),(10b),(11),(12) (source-ambiguity)
  • Sec. IIIB: pure Gaussian displacement POVMs from homodyne: Weakly countably additive normalized displaced pure-seed POVM and homodyne implementation (hard)
  • Sec. IIIB, Eq.(15a): conditional covariance of a pure Gaussian projection: Actual normalized Gaussian conditional state of a partial pure-seed projection (hard)
  • Sec. IIIB, Eqs.(14), (15b): printed measurement means: Consistent centering, factor and measurement-label convention for Eqs.(14),(15b) (source-ambiguity)
  • Sec. IIIB: infinitely squeezed projection and pseudoinverse: Specified quadrature projection, limiting finite block, outcome scaling and convergence topology (source-ambiguity)
  • Sec. IIIB, pp.3–4: noise on a Gaussian operation is output displacement noise: Degenerate Gaussian displacement laws acting on actual normalized conditional outputs (hard)
  • Sec. IIIC: deterministic realization for known covariance: Trace-preserving Bell spectral instrument with covariance-dependent displacement feedback (hard)
  • Sec. IVA: Gaussian LOCC/PPT classification by the Choi state: Generalized Gaussian Choi object supporting separable/PPT/NPT classification (source-ambiguity)
  • Sec. IVA/IVB: separable Gaussian resource maps are products plus correlated noise: Finite-resource separability as actual state separability and its product-operation/noise implementation (hard)
  • Sec. IVB, Eqs.(16)–(17): Gaussian local monotonicity of V: Actual normalized branch action of all separable Gaussian completely positive operations (hard)
  • Sec. V: boundedness, self-adjointness, positivity and trace of Gaussian operators: Convergence domain, scalar-phase quotient/sign and corrected CCR trace normalization (source-ambiguity)
  • Sec. V: general Gaussian linear kernels and Weyl action: Analytic domain and admissible generalized Gaussian operator kernels (source-ambiguity)
  • Sec. V: adjoint preservation of Gaussian maps: Scalar phase quotient in adjoint-preservation criterion (source-ambiguity)
  • Sec. V, Eqs.(20)–(21): Gaussian positivity criterion: Generalized Schur range conditions and scalar positivity for admitted singular maps (source-ambiguity)
  • Sec. V: claimed numerical efficiency of the positivity test: Numerical input representation, tolerance, promise and complexity bound (source-ambiguity)
  • Appendix A(4): V detects PPT Gaussian entanglement: Actual Gaussian separability, partial transpose and trace-class negativity (hard)
  • Appendix A(5): V as a Gaussian entanglement quantity: Actual local Gaussian operations and independently defined Gaussian entanglement (hard)
  • Appendix A(6): asserted computability of V: Effective representation and certified approximation algorithm for V (source-ambiguity)
  • Reference note [23]: proposed regularized Choi/Bell limit: Normalized regularizer, compensating scaling and a valid Bell POVM limit (source-ambiguity)

Selected references

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Quantum Information·Captain: Goku

Gaussian Quantum InformationResearch Paper

Motivation

The science of quantum information has arisen over the last two decades centered on the manipulation of individual quanta of information, known as quantum bits or qubits. Quantum computers, quantum cryptography and quantum teleportation are among the most celebrated ideas that have emerged from this new field. It was realized later on that using continuous-variable quantum information carriers, instead of qubits, constitutes an extremely powerful alternative approach to quantum information processing. This review focuses on continuous-variable quantum information processes that rely on any combination of Gaussian states, Gaussian operations, and Gaussian measurements. Interestingly, such a restriction to the Gaussian realm comes with various benefits, since on the theoretical side, simple analytical tools are available and, on the experimental side, optical components effecting Gaussian processes are readily available in the laboratory. Yet, Gaussian quantum information processing opens the way to a wide variety of tasks and applications, including quantum communication, quantum cryptography, quantum computation, quantum teleportation, and quantum state and channel discrimination. This review reports on the state of the art in this field, ranging from the basic theoretical tools and landmark experimental realizations to the most recent successful developments.

This mission is the statement-level formalization of Gaussian Quantum Information (arXiv:1110.3234v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

5 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Coherent-state moments, completeness, and exact overlap (§II.B.2 Eqs.(28)–(30): coherent state moments and overlap, PDF p. 7). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

24 results of the paper are not stated yet, because the library lacks an interface they need:

  • §II.A.1, Eqs.(19)–(20): Gaussian states are completely characterized by first and second moments: Concrete multimode Weyl operators, mixed-state moment domain, and Gaussian density/Wigner representation (hard)
  • §II.A.1 Eqs.(17)–(18): uncertainty and strict covariance positivity: Actual multimode mixed-state quadrature first and symmetrized second moments with their finite form domain (hard)
  • §II.A.2 Eqs.(23)–(25): affine symplectic action: Concrete canonical Weyl representation and its affine symplectic unitary implementers (hard)
  • §II.B.1 Eqs.(26)–(27): fixed-energy entropy maximizer: Unique entropy-maximizing thermal density, its Gaussian characteristic function, and mixed-state moments (hard)
  • §II.B.5 Eq.(35): all one-mode Gaussian states: Actual displaced rotated squeezed thermal densities and Gaussian moment identification (hard)
  • §II.B.7: EPR states purify thermal states: Actual two-mode Gaussian pure density, partial trace and EPR observable moments in the Eq.(42) convention (hard)
  • §II.C.1 Eqs.(46)–(47): spectral entropy of Gaussian density operators: Identification of actual Gaussian density operators with their Williamson spectrum (hard)
  • §II.C.1 Eqs.(48)–(49): operator thermal decomposition: Gaussian density thermal tensor product and concrete Weyl/metaplectic conjugation (hard)
  • §II.C.1 Eq.(50): actual Gaussian purification: Actual Gaussian 2N-mode pure state with N-mode partial trace (hard)
  • §II.C.2 Eqs.(51)–(52): Euler/Bloch–Messiah decomposition: Metaplectic operator representation and tensor squeezing/passive-unitary decomposition (hard)
  • §II.E.3 Eqs.(64)–(65): Gaussian conditional covariance: Continuous homodyne and heterodyne instruments and almost-everywhere conditional normal states (hard)
  • §III.A.2.a Eqs.(72)–(78): Gaussian power overlap: Actual Gaussian density powers and trace pairing with their physical moment/Williamson data (hard)
  • §III.A.3 Eq.(80): one-mode Gaussian fidelity: Squared Uhlmann fidelity of actual Gaussian normal densities and their observable moments (hard)
  • §IV.A Eqs.(100)–(101): teleportation with Gaussian resources: Unity-gain bosonic Bell instrument, conditional feedforward, and outcome-averaged fidelity (hard)
  • §IV.B Eqs.(104)–(106): Gaussian cloner and anticlone: Actual three-mode symplectic cloner channel and its single-mode reduced densities (hard)
  • §V.A Eqs.(108)–(110): Gaussian channel action and CP: Full trace-class CPTP Gaussian channel, Weyl dual law, and mixed-state moment action (hard)
  • §V.A: Gaussian dilation with at most 2N environmental modes: Actual full-input Gaussian unitary Stinespring dilation with at most 2N pure environmental modes (hard)
  • §II.C.1: erroneous Hermiticity/singular-value recipe: Consistent source statement with a specified normalization and approximation/domain where applicable (source-ambiguity)
  • §II.B.7 Eqs.(39)–(42): incompatible EPR parameter/sign formulas: Consistent source statement with a specified normalization and approximation/domain where applicable (source-ambiguity)
  • §III.B.3 Eq.(97): inconsistent homodyne scale: Consistent source statement with a specified normalization and approximation/domain where applicable (source-ambiguity)
  • §V.B Eqs.(111)–(118), Table I: one-mode Gaussian canonical classification: Input/output Gaussian-unitary equivalence of actual one-mode channels and canonical dilations (hard)
  • §VI.B.5.a Eqs.(141)–(142): covariance/noise mismatch: Consistent source statement with a specified normalization and approximation/domain where applicable (source-ambiguity)
  • §VII.D Eqs.(174)–(176): Gaussian distortion sign/domain: Consistent source statement with a specified normalization and approximation/domain where applicable (source-ambiguity)
  • §VII.F.1 Eqs.(179)–(181): unquantified approximate cubic state: Consistent source statement with a specified normalization and approximation/domain where applicable (source-ambiguity)

Selected references

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Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its useResearch Paper

Motivation

We describe an universal method for quantitative continuity analysis of entropic characteristics of energy-constrained quantum systems and channels. It gives asymptotically tight continuity bounds for basic characteristics of quantum systems of wide class (including multi-mode quantum oscillators) and channels between such systems under the energy constraint. The main application of the proposed method is the advanced version of the uniform finite-dimensional approximation theorem for basic capacities of energy-constrained quantum channels.

This mission is the statement-level formalization of Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its use (arXiv:1907.02458v3). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

2 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Minimal entropy majorant and spectral asymptotics (Proposition 1(A,B), PDF p. 7). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

18 results of the paper are not stated yet, because the library lacks an interface they need:

  • Lemma 1: completed bipartite trace-class marginals and spectral marginal rank (hard)
  • Theorem 1: general multipartite finite-energy AFW function domains and optimized tightness modulus (hard)
  • Lemma 2: locally finite-rank multipartite state domain and trace-norm uniformly continuous extensions (hard)
  • Theorem 2: canonical extension through all finite-A-rank states, including nondense Hamiltonians (hard)
  • Example 1: canonical real extension of a difference of possibly infinite mutual informations (hard)
  • Corollary 1: Fock-space oscillator Hamiltonian and general AFW-function tightness (hard)
  • Example 2: full entropy continuity statement with oscillator specialization and optimal-t tightness (hard)
  • Example 3: extended conditional entropy on arbitrary completed bipartite spaces (hard)
  • Example 4: extended QCMI finite-projection supremum for all eight energy subsystems (hard)
  • Proposition 2: infinite-dimensional channel tensor action and extended output QCMI (hard)
  • Lemma 3: purification-defined mutual and coherent information of arbitrary trace-class channels (hard)
  • Proposition 3: channel information, complementary channels and degradability in the same infinite-state model (hard)
  • Proposition 4: generalized ensemble Holevo integral and three genuine ensemble metrics (hard)
  • Proposition 5: generalized-ensemble privacy with complementary CPTP channel and subclass predicates (hard)
  • Lemma 4: correlated n-copy CPTP truncation and actual output mutual-information variation (hard)
  • Theorem 3: boundary-inclusive least-dimension certificate at zero energy excess (source-ambiguity)
  • Theorem 3, positive-excess clause: six infinite-dimensional energy-constrained capacities and their input-subspace restrictions (hard)
  • Theorem 3, ground-energy existence clause: six genuine capacities restricted to the full finite ground eigenspace (hard)

Selected references

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Monotonicity of the Holevo quantity: a necessary condition for equality in terms of a channel and its applicationsResearch Paper

Motivation

A condition for reversibility (sufficiency) of a channel with respect to a given countable family of states with bounded rank is obtained. This condition shows that a quantum channel preserving the Holevo quantity of at least one (discrete or continuous) ensemble of states with rank ≤r\leq r has the r-partially entanglement-breaking complementary channel. Several applications of this result are considered. In particular, it is shown that coincidence of the constrained Holevo capacity and the quantum mutual information of a quantum channel at least at one full rank state implies that this channel is entanglement-breaking.

This mission is the statement-level formalization of Monotonicity of the Holevo quantity: a necessary condition for equality in terms of a channel and its applications (arXiv:1106.3297v6). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

3 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Lemma 1: isometric equivalence preserves reversibility (Lemma 1, PDF p. 9). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

27 results of the paper are not stated yet, because the library lacks an interface they need:

  • Theorem 2: equality for continuous ensembles: Construct extended spectral entropy/relative entropy on the accepted trace-class state model and the actual Bochner barycenter and measurable Holevo integral; then supply the node-specific rank/complement/support objects. (hard)
  • Theorem 1: reversibility of a total countable bounded-rank family: Construct common Stinespring complements, mixed closed Schmidt classes, r-PEB, spectral-domain inverse frame transformations and pseudo-diagonal forms, retaining all source clauses. (hard)
  • Corollary 1: equality for discrete ensembles: Construct extended spectral entropy/relative entropy on the accepted trace-class state model and the actual Bochner barycenter and measurable Holevo integral; then supply the node-specific rank/complement/support objects. (hard)
  • Proposition 1: four equivalent continuous equality criteria: Construct the normal predual channel characterized by the bounded sandwich identity (29) on faithful sigma/output support, together with extended relative entropy; allow nonfaithful rho. (hard)
  • Proposition 2: maximal finite-dimensional Holevo capacity: Connect the accepted finite entropy/channel objects to full pseudo-diagonal equivalence, common-dilation degradability, all-ensemble capacity or attained minimum output entropy as required by the node. (hard)
  • Corollary 2: positive minimum output entropy under covariance: Connect the accepted finite entropy/channel objects to full pseudo-diagonal equivalence, common-dilation degradability, all-ensemble capacity or attained minimum output entropy as required by the node. (hard)
  • Proposition 3: strict Holevo decrease under partial trace, A and B: Construct extended spectral entropy/relative entropy on the accepted trace-class state model and the actual Bochner barycenter and measurable Holevo integral; then supply the node-specific rank/complement/support objects. (hard)
  • Corollary 3: strict conditional entropy concavity: Construct extended spectral entropy/relative entropy on the accepted trace-class state model and the actual Bochner barycenter and measurable Holevo integral; then supply the node-specific rank/complement/support objects. (hard)
  • Proposition 4: constrained Holevo capacity equals mutual information: Construct general Holevo/barycenter and constrained-capacity functionals, canonical purification mutual information and common complements; retain source finiteness and attainment hypotheses. (hard)
  • Corollary 4: equality of energy-constrained capacities: Construct positive self-adjoint energy forms and expectation, feasible constraints, the single-use Holevo and assumed mutual-information capacity suprema and support-restricted channel. (hard)
  • Theorem 3: Petz theorem in infinite dimensions: Construct the normal predual channel characterized by the bounded sandwich identity (29) on faithful sigma/output support, together with extended relative entropy; allow nonfaithful rho. (hard)
  • Lemma 4: complementary Holevo difference: Construct general Holevo/barycenter and constrained-capacity functionals, canonical purification mutual information and common complements; retain source finiteness and attainment hypotheses. (hard)
  • Remark 1: convexity of entropy gain: Construct extended spectral entropy/relative entropy on the accepted trace-class state model and the actual Bochner barycenter and measurable Holevo integral; then supply the node-specific rank/complement/support objects. (hard)
  • Remark 2: strictness witnesses and support restriction: Construct common Stinespring complements, mixed closed Schmidt classes, r-PEB, spectral-domain inverse frame transformations and pseudo-diagonal forms, retaining all source clauses. (hard)
  • Remark 3: counterexample to sufficiency: Connect the accepted finite entropy/channel objects to full pseudo-diagonal equivalence, common-dilation degradability, all-ensemble capacity or attained minimum output entropy as required by the node. (hard)
  • Remark 4: reversibility without finite chi: Construct common Stinespring complements, mixed closed Schmidt classes, r-PEB, spectral-domain inverse frame transformations and pseudo-diagonal forms, retaining all source clauses. (hard)
  • Unnumbered consequence: unital qubit channels: Connect the accepted finite entropy/channel objects to full pseudo-diagonal equivalence, common-dilation degradability, all-ensemble capacity or attained minimum output entropy as required by the node. (hard)
  • Unnumbered continuous version of Corollary 3: Construct extended spectral entropy/relative entropy on the accepted trace-class state model and the actual Bochner barycenter and measurable Holevo integral; then supply the node-specific rank/complement/support objects. (hard)
  • Equation (19): constrained capacity <= mutual information: Construct general Holevo/barycenter and constrained-capacity functionals, canonical purification mutual information and common complements; retain source finiteness and attainment hypotheses. (hard)
  • Equation (23): complementary constrained-capacity identity: Construct general Holevo/barycenter and constrained-capacity functionals, canonical purification mutual information and common complements; retain source finiteness and attainment hypotheses. (hard)
  • Remark 6: finite-dimensional global-capacity equality: Connect the accepted finite entropy/channel objects to full pseudo-diagonal equivalence, common-dilation degradability, all-ensemble capacity or attained minimum output entropy as required by the node. (hard)
  • Unnumbered consequence of Lemma 4: optimal pure measures: Construct general Holevo/barycenter and constrained-capacity functionals, canonical purification mutual information and common complements; retain source finiteness and attainment hypotheses. (hard)
  • Unnumbered claim: output-entropy continuity supplies attainment: Supply a source-pinned compactness and entropy-control hypothesis for the feasible state set, or revise the source claim explicitly; continuity alone is insufficient. (source-ambiguity)
  • Example: conditional Gaussian-channel application: Pin a CCR convention, admissible oscillator Hamiltonian and optimizer support/faithfulness assumptions; derive only the justified support-restricted conclusion until then. (source-ambiguity)
  • Unnumbered appendix observation: generalized Holevo lsc: Construct trace-norm subnormalized samples, generalized relative entropy Tr(A log A-A log B+B-A), entropy S, computed barycenters, probability pushforwards and strong-topology quantum operations. (hard)
  • Equation (37): output Holevo entropy formula for operations: Construct trace-norm subnormalized samples, generalized relative entropy Tr(A log A-A log B+B-A), entropy S, computed barycenters, probability pushforwards and strong-topology quantum operations. (hard)
  • Unnumbered appendix observation: operation Holevo lsc: Construct trace-norm subnormalized samples, generalized relative entropy Tr(A log A-A log B+B-A), entropy S, computed barycenters, probability pushforwards and strong-topology quantum operations. (hard)

Selected references

  • Monotonicity of the Holevo quantity: a necessary condition for equality in terms of a channel and its applications, arXiv:1106.3297v6, https://arxiv.org/abs/1106.3297v6
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