The Quantum Formalization Drive

Quantum error correction protects quantum information from noise by storing it in a code and undoing the damage with a recovery operation. The textbook theory, the Knill–Laflamme conditions, characterizes codes that can be corrected perfectly. Real noise rarely admits perfect correction, and codes that are only approximately correctable can perform better…

Given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the "partial information" one system needs conditioned on it's prior information. It turns out to be given by an extremely simple formula, the conditional entropy. In the…

It is proven that recently introduced states with perfectly secure bits of cryptographic key (private states representing secure bit) [K. Horodecki et al., Phys. Rev. Lett. 94, 160502 (2005)] as well as its multipartite and higher dimension generalizations always represent distillable entanglement. The corresponding lower bounds on distillable entanglement…

Consider many instances of an arbitrary quadripartite pure state of four quantum systems ABCD. Alice holds the AC part of each state, Bob holds B, while D represents all other parties correlated with ABC. Alice is required to redistribute the C systems to Bob while asymptotically preserving the overall purity. We prove that this is possible using Q qubits…

In this paper, we present a new entanglement monotone for bipartite quantum states. Its definition is inspired by the so-called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions. We derive certain properties of the new measure which we call "squashed…

In finite dimensional quantum information, a state of a full matrix algebra can be restricted to a smaller algebra of observables. The resulting entropy depends on which observables are retained. For a maximal commutative subalgebra, the retained observables are equivalent to measurements in a fixed orthonormal basis. The restriction has an ordinary…

Permutation operators encode the symmetries of many identical quantum systems. When one tensor factor is transposed, the resulting operators occur in questions about the commutant of mixed tensor representations and about partially transposed quantum states. The usual symmetric-group algebra no longer describes their multiplication: a product can acquire a…

Quantum devices rarely prepare the state they are asked for. A procedure meant to output a pure state $|\psi\rangle$ on $d$ levels often outputs a mixed state $\rho$, the result of noise during preparation. If the procedure can be run $n$ times, the copies $\rho^{\otimes n}$ can be processed jointly into a single output closer to $|\psi\rangle$. This task is quantum…

Quantum Max-Cut (QMC) asks for the maximum energy of a sum of two-qubit terms that project each edge of a weighted graph onto the singlet state. Its maximum-energy state is the ground state of the antiferromagnetic Heisenberg model, a standard model of magnetism, and computing that energy is QMA-hard (Piddock–Montanaro 2017, Cubitt–Montanaro 2016). QMC has…

The cyclotomic Birman–Wenzl–Murakami algebra is an algebra of braids and contractions equipped with an additional cyclotomic relation. It connects diagram algebras, representation theory, and knot theoretic constructions. For calculations over a coefficient ring, one needs to know whether the diagrammatic spanning elements remain independent after…

Photosynthetic light-harvesting complexes move an electronic excitation from the pigment that absorbed a photon to a reaction centre, where its energy is stored chemically. In the Fenna–Matthews–Olson (FMO) complex this transfer is close to lossless, and two-dimensional spectroscopy showed long-lived quantum coherence during it (Engel et al. 2007). This…

Prior entanglement between sender and receiver, which exactly doubles the classical capacity of a noiseless quantum channel, can increase the classical capacity of some noisy quantum channels by an arbitrarily large constant factor depending on the channel, relative to the best known classical capacity achievable without entanglement. The enhancement factor…

How correlated are two quantum systems from the perspective of a third? We answer this by providing an optimal quantum state redistribution protocol for multipartite product sources. Specifically, given an arbitrary quantum state of three systems, where Alice holds two and Bob holds one, we identify the cost, in terms of quantum communication and…

The information carrying capacity of the d-dimensional depolarizing channel is computed. It is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space, and using measurements which are products of projections onto this same orthonormal basis. In other words, neither…

The quantum analog of the classical erasure channel provides a simple example of a channel whose asymptotic capacity for faithful transmission of intact quantum states, with and without the assistance of a two-way classical side channel, can be computed exactly. We derive the quantum and classical capacities for the quantum erasure channel and related…

Based on the monogamy of entanglement, we develop the technique of quantum conditioning to build an {\it additive} entanglement measure: the conditional entanglement of mutual information. Its {\it operational} meaning is elaborated to be the minimal net "flow of qubits" in the process of partial state merging. The result and conclusion can also be…

In quantum mechanics, a system that interacts with an environment is described by a density matrix: a positive semidefinite complex matrix of trace one. When nothing is known about the joint state of the system and its environment, a natural model is to take that joint state to be a uniformly random pure state and to trace out the environment. The resulting…

Topological quantum computing represents a gate operation by moving anyons around one another. A compiler must turn a requested single-qubit unitary into a finite braid or gate word, with enough accuracy and a length that grows slowly as the error tolerance shrinks. For Fibonacci anyons, the matrices generated by the standard elementary braids have…

A Hilbert space has an orthonormal basis, so every Hilbert space is isometric to some $\ell^2(I)$. Hilbert C$^*$-modules, the analogue of Hilbert spaces in which the inner product takes values in an operator algebra, have no such basis in general. Over a von Neumann algebra $\mathcal M$, Paschke showed that the self-dual Hilbert C$^*$-modules are exactly the…

The Temperley–Lieb algebra is a family of algebras built from generators that satisfy local relations. It appears in diagrammatic representation theory and provides a tractable setting in which to study cellular bases and Gram determinants. A Gram determinant tests whether the bilinear form on a module is degenerate after specializing its parameter. Thus an…

In a system driven away from equilibrium, a force changes a reference state and a dynamics, while a corresponding flux records the transported quantity. Statistical mechanics asks how the mean flux changes under a small force. A second question is whether the matrix of these changes is symmetric when the underlying dynamics admit time reversal. Jakšić,…

A quantum channel is a completely positive, trace-preserving linear map between matrix algebras; it models the most general physical transformation of a quantum system. When a group $G$ acts on the input and output systems, a channel is $G$-covariant if it commutes with the group action. Covariant channels appear whenever a physical process respects a symmetry:…

Operators on $n$ copies of $\mathbb C^d$ that commute with every $U^{\otimes n}$, $U$ unitary, are exactly the linear combinations of permutation operators $V(\sigma)$, $\sigma\in S(n)$ (Schur–Weyl duality). Many protocols in quantum information carry a different symmetry: the first $n-1$ systems transform by $U$ and the last by the complex conjugate $\bar U$. It is the…

Operators made from a complex matrix and its conjugate occur in the non-chiral sector of gauge theories. Their correlation functions involve many possible pairings of matrix entries, and those pairings become difficult to organize as the number of entries grows. Kimura and Ramgoolam use the walled Brauer algebra to label these operators and to describe…

Positive linear maps on complex matrix algebras provide tests for quantum entanglement: applying such a map to one part of a bipartite state can expose a state that cannot be written as a mixture of product states. The distinction between ordinary positivity, positivity with an auxiliary matrix system, and complete positivity is therefore substantive.…