arXiv subjects

Felix Leditzky

Publications and source records attributed to Felix Leditzky.

41 records · Page 3Linked to original sources

arXiv2016

Relative entropies and their use in quantum information theory

This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the $α$-sandwiched Rényi divergence ($α$-SRD). We first investigate the limit $α\to 0$ to determine whether this recovers the well-known $0$-Rényi relative divergence. We then prove various new results for entropic quantities derived from the $α$-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the $α$-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the $α$-SRD.

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arXiv2016

Data processing for the sandwiched Rényi divergence: a condition for equality

The $α$-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for $α\geq 1/2$. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the $α$-sandwiched Rényi divergence for all $α\geq 1/2$. For the range $α\in [1/2,1)$, our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace, and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb [J. Math. Phys. 54.12 (2013), p. 122201] using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki-Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb [Lett. Math. Phys. 101.1 (2012), pp. 1-11] about equality in the original Araki-Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation, and observe that it is attained by states saturating the Rényi version of the Araki-Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.

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arXiv2016

Strong converse theorems using Rényi entropies

We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [arXiv:1404.5940], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the $(e,q)$-plane, where $e$ and $q$ denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.

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arXiv2016

Second order asymptotics of visible mixed quantum source coding via universal codes

The simplest example of a quantum information source with memory is a mixed source which emits signals entirely from one of two memoryless quantum sources with given a priori probabilities. Considering a mixed source consisting of a general one-parameter family of memoryless sources, we derive the second order asymptotic rate for fixed-length visible source coding. Furthermore, we specialize our main result to a mixed source consisting of two memoryless sources. Our results provide the first example of second order asymptotics for a quantum information-processing task employing a resource with memory. For the case of a classical mixed source (using a finite alphabet), our results reduce to those obtained by Nomura and Han [IEEE Trans. on Inf. Th. 59.1 (2013), pp. 1-16]. To prove the achievability part of our main result, we introduce universal quantum source codes achieving second order asymptotic rates. These are obtained by an extension of Hayashi's construction [IEEE Trans. on Inf. Th. 54.10 (2008), pp. 4619-4637] of their classical counterparts.

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arXiv2013

A limit of the quantum Renyi divergence

Recently, an interesting quantity called the quantum Renyi divergence (or "sandwiched" Renyi relative entropy) was defined for pairs of positive semi-definite operators $ρ$ and $σ$. It depends on a parameter $α$ and acts as a parent quantity for other relative entropies which have important operational significances in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Renyi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Renyi entropy is obtainable from the quantum Renyi divergence only if $ρ$ and $σ$ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived - the quantum Renyi divergence with parameter $α\ge 1/2$, and the $α$-relative Rényi entropy with $α\in [0,1)$.

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