Quantum Entropy Inequalities Research Articles

On the foundations and extremal structure of the holographic entropy cone

The holographic entropy cone (HEC) is a polyhedral cone first introduced in the study of a class of quantum entropy inequalities. It admits a graph-theoretic description in terms of minimum cuts in weighted graphs, a characterization which naturally generalizes the cut function for complete graphs. Unfortunately, no complete facet or extreme-ray representation of the HEC is known. In this work, starting from a purely graph-theoretic perspective, we develop a theoretical and computational foundation for the HEC. The paper is self-contained, giving new proofs of known results and proving several new results as well. These are also used to develop two systematic approaches for finding the facets and extreme rays of the HEC, which we illustrate by recomputing the HEC on $5$ terminals and improving its graph description. We also report on some partial results for $6$ terminals. Some interesting open problems are stated throughout.

Hypergraph min-cuts from quantum entropies

The von Neumann entropy of pure quantum states and the min-cut function of weighted hypergraphs are both symmetric submodular functions. In this article, we explain this coincidence by proving that the min-cut function of any weighted hypergraph can be approximated (up to an overall rescaling) by the entropies of quantum states known as stabilizer states. We do so by constructing a novel ensemble of random quantum states, built from tensor networks, whose entanglement structure is determined by a given hypergraph. This implies that the min-cuts of hypergraphs are constrained by quantum entropy inequalities, and it follows that the recently defined hypergraph cones are contained in the quantum stabilizer entropy cones, which confirms a conjecture made in the recent literature.

Computing Quantum Channel Capacities

The capacity of noisy quantum channels characterizes the highest rate at which information can be reliably transmitted and it is therefore of practical as well as fundamental importance. Capacities of classical channels are computed using alternating optimization schemes, called Blahut-Arimoto algorithms. In this work, we generalize classical Blahut-Arimoto algorithms to the quantum setting. In particular, we give efficient iterative schemes to compute the capacity of channels with classical input and quantum output, the quantum capacity of less noisy channels, the thermodynamic capacity of quantum channels, as well as the entanglement-assisted capacity of quantum channels. We give rigorous a priori and a posteriori bounds on the estimation error by employing quantum entropy inequalities and demonstrate fast convergence of our algorithms in numerical experiments.

A remainder term for Hölder’s inequality for matrices and quantum entropy inequalities

We prove a sharp remainder term for Holder’s inequality for traces as a consequence of the uniform convexity properties of the Schatten trace norms. We then show how this implies a novel family of Pinsker type bounds for the quantum Renyi entropy. Finally, we show how the sharp form of the usual quantum Pinsker inequality for relative entropy may be obtained as a fairly direct consequence of uniform convexity.

Quantum Information and the Representation Theory of the Symmetric Group

A number of important results in quantum information theory can be connected quite elegantly to the representation theory of the symmetric group through a quantum analogue of the classical information-theoretic method of types that arises naturally from the Schur-Weyl duality. We will give a brief introduction to this connection and brie y discuss some of the results that follow from it, such as quantum source compression rates, entanglement concentration rates, quantum entropy inequalities, and the admissisble spectra of partial density matrices from pure, multipartite entangled states.

A Strengthened Monotonicity Inequality of Quantum Relative Entropy: A Unifying Approach Via Rényi Relative Entropy

We derive a strengthened monotonicity inequality for quantum relative entropy by employing properties of $\alpha$-R\'{e}nyi relative entropy. We develop a unifying treatment towards the improvement of some quantum entropy inequalities. In particular, an emphasis is put on a lower bound of quantum conditional mutual information (QCMI) as it gives a Pinsker-like lower bound for the QCMI. We also give some improved entropy inequalities based on R\'{e}nyi relative entropy. The inequalities obtained, thus, extends some well-known ones. We also obtain a condition under which a tripartite operator becomes a Markov state. As a by-product we provide some trace inequalities of operators, which are of independent interest.

Remainder terms for some quantum entropy inequalities

We consider three von Neumann entropy inequalities: subadditivity; Pinsker's inequality for relative entropy; and the monotonicity of relative entropy. For these we state conditions for equality, and we prove some new error bounds away from equality, including an improved version of Pinsker's inequality.

A note on quantum entropy inequalities and channel capacities

Quantum entropy inequalities are studied. Some quantum entropy inequalities are obtained by several methods. For entanglement breaking channel, we show that the entanglement-assisted classical capacity is upper bounded by $\log d$. A relationship between entanglement-assisted and one-shot unassisted capacities is obtained. This relationship shows the entanglement-assisted channel capacity is upper bounded by the sum of $\log d$ and the one-shot unassisted classical capacity.