The Quantum Formalization Drive

Given two Banach spaces $X$ and $Y$, we introduce and study a concept of norm-attainment in the space of nuclear operators $\mathcal{N}(X,Y)$ and in the projective tensor product space $X \widehat{\otimes}_\pi Y$. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which…

We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so-called short quantum…

We present an algorithm for the explicit numerical calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients, based on the Gelfand-Tsetlin pattern calculus. Our algorithm is well-suited for numerical implementation; we include a computer code in an appendix. Our exposition presumes only familiarity with the representation theory of SU(2).

We describe a new method for approximating an arbitrary $n$ qubit unitary with precision $\varepsilon$ using a Clifford and T circuit with $O(4^{n}n(\log(1/\varepsilon)+n))$ gates. The method is based on rounding off a unitary to a unitary over the ring $\mathbb{Z}[i,1/\sqrt{2}]$ and employing exact synthesis. We also show that any $n$ qubit unitary over the ring …

A state on a tripartite quantum system $A \otimes B \otimes C$ forms a Markov chain if it can be reconstructed from its marginal on $A \otimes B$ by a quantum operation from $B$ to $B \otimes C$. We show that the quantum conditional mutual information $I(A: C | B)$ of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus…

We find a regularized formula for the entanglement-assisted (EA) capacity region for quantum multiple access channels (QMAC). We illustrate the capacity region calculation with the example of the collective phase-flip channel which admits a single-letter characterization. On the way, we provide a first-principles proof of the EA coding theorem based on a…

This paper is essentially a lecture from the author's course on quantum information theory, which is devoted to the result of C. H. Bennett, P. W. Shor, J. A. Smolin and A. V. Thapliyal (quant-ph/0106052) concerning entanglement-assisted classical capacity of a quantum channel. A modified proof of this result is given and relation between…

This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.

Collins and Popescu realized a powerful analogy between several resources in classical and quantum information theory. The Collins-Popescu analogy states that public classical communication, private classical communication, and secret key interact with one another somewhat similarly to the way that classical communication, quantum communication, and…

We define classical-quantum multiway channels for transmission of classical information, after recent work by Allahverdyan and Saakian. Bounds on the capacity region are derived in a uniform way, which are analogous to the classically known ones, simply replacing Shannon entropy with von Neumann entropy. For the single receiver case (multiple access…

The entanglement-assisted classical capacity of a noisy quantum channel is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that…

We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a bipartite state \rho is equal to the regularized entanglement of formation of \rho.

A new protocol for quantum broadcast channels based on the fully quantum Slepian-Wolf protocol is presented. The protocol yields an achievable rate region for entanglement-assisted transmission of quantum information through a quantum broadcast channel that can be considered the quantum analogue of Marton's region for classical broadcast channels. The…

The problem of converting noisy quantum correlations between two parties into noiseless classical ones using a limited amount of one-way classical communication is addressed. A single-letter formula for the optimal trade-off between the extracted common randomness and classical communication rate is obtained for the special case of classical-quantum…

Can entangled noise erase entanglement more efficiently than separable noise? Standard robustness restricts the added noise to separable states; generalized robustness allows any state. We prove that the two costs coincide for every two-qubit state and give an explicit full-rank qubit--qutrit state with a strict gap. Positivity under partial transpose (PPT)…

An uncertainty relation bounds how sharply two incompatible measurements can both be predicted on one quantum state. The entropic form of Maassen and Uffink (1988), $H(X) + H(Z) \ge -\log_2 c_{\max}$, measures unpredictability by the Shannon entropies of the outcomes of two measurement bases $\mathbb X$ and $\mathbb Z$, and incompatibility by their maximal…

Many proposed quantum optimization and machine-learning methods (variational quantum eigensolvers, QAOA, quantum classifiers) tune the real parameters of a circuit by gradient descent on an objective $f:\mathbb R^d\to\mathbb R$ that is only accessible as a black box. Classically, a finite-difference estimate of $\nabla f$ costs on the order of $d$ evaluations of $f$…

Decoded quantum interferometry (DQI) is a quantum algorithm, introduced by Jordan, Shutty, Wootters, Zalcman, Schmidhuber, King, Isakov, Khattar and Babbush (arXiv:2408.08292), that turns an approximate optimization problem over a finite field into a decoding problem for a linear code. It applies to max-LINSAT: given linear constraints over $\mathbb F_p$,…

Two-player zero-sum matrix games are the simplest model of strict competition, and computing their value is equivalent to linear programming. Many algorithms in optimization and learning (boosting, online learning with experts, approximate LP and SDP solvers) are, at their core, procedures that approximately solve such a game. For large games the cost of…

The Quantum Approximate Optimization Algorithm (QAOA) of Farhi, Goldstone and Gutmann (arXiv:1411.4028, 2014) is a family of shallow quantum circuits for combinatorial optimization, indexed by a depth $p$ and $2p$ real angles. It has become one of the standard candidate algorithms for near-term quantum hardware, and its MaxCut instance is the usual benchmark on…

The quantum approximate optimization algorithm (QAOA) of Farhi, Goldstone and Gutmann (arXiv:1411.4028, 2014) is a family of shallow quantum circuits, indexed by an integer depth $p \ge 1$, that prepare states whose computational-basis measurements are biased toward good solutions of a combinatorial optimization problem. It became the standard candidate for…

This mission develops Chapter 2 of An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation by Daniel Gottesman. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 7 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 5 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 4 of A Survey of Quantum Learning Theory by Srinivasan Arunachalam and Ronald de Wolf. The chapter's selected results describe the mathematical guarantees below.