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arXiv:1403.6102 · Journal of Mathematical Physics

Rényi generalizations of the conditional quantum mutual information

Mario Berta, Kaushik Parasuram Seshadreesan, Mark M. Wilde

Published Feb 1, 2015• 86 citations
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Abstract

The conditional quantum mutual information I(A; B|C) of a tripartite state ρABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B|C) = I(A; B|D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Rényi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different α-Rényi generalizations Iα(A; B|C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit α → 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Rényi conditional mutual informations defined here with respect to the Rényi parameter α. We prove that this conjecture is true in some special cases and when α is in a neighborhood of one.

DOI
10.1063/1.4908102

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The conditional quantum mutual information I(A; B|C) of a tripartite state ρABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B|C) = I(A; B|D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Rényi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different α-Rényi generalizations Iα(A; B|C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit α → 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Rényi conditional mutual informations defined here with respect to the Rényi parameter α. We prove that this conjecture is true in some special cases and when α is in a neighborhood of one.
The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205), we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333), we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set . Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.
Quantum conditional mutual information (QCMI) is a fundamental measure of conditional independence, with central applications throughout quantum information theory and many-body physics. Despite its importance, finding a fully quantum Rényi generalization that satisfies all desirable structural properties has remained an open question. In this work, we introduce a quantum Rényi conditional mutual information possessing these desirable properties: it is monotone under local quantum channels, additive under tensor products, monotone in the Rényi parameter, and converges to the QCMI in the limit α→1α\to 1α→1. We establish bounds on this quantity in terms of relative entropies to the Petz-recovered state, showing that it provides a distinct measure of approximate conditional independence. As a consequence, we derive recovery-based upper and lower bounds on the QCMI in terms of quantum generalizations of the Kullback-Leibler divergence, improving upon the best known results in certain regimes. Finally, we establish bounds in terms of relative entropies to the sets of Markov and separable states and derive Rényi chain rules, which appear to be novel even in the classical setting.
The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205 ), we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A ). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B , we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333 ), we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set . Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.
Quantum information measures such as the entropy and the mutual information find applications in physics, e.g., as correlation measures. Generalizing such measures based on the R\'enyi entropies is expected to enhance their scope in applications. We prescribe R\'enyi generalizations for any quantum information measure which consists of a linear combination of von Neumann entropies with coefficients chosen from the set $\left\{\ensuremath{-}1,0,1\right\}$. As examples, we describe R\'enyi generalizations of the conditional quantum mutual information, some quantum multipartite information measures, and the topological entanglement entropy. Among these, we discuss the various properties of the R\'enyi conditional quantum mutual information and sketch some potential applications. We conjecture that the proposed R\'enyi conditional quantum mutual informations are monotone increasing in the R\'enyi parameter, and we have proof of this conjecture for some special cases.
A short quantum Markov chain is a tripartite state such that system A can be recovered perfectly by acting on system C of the reduced state Such states have conditional mutual information equal to zero and are the only states with this property. A quantum channel is sufficient for two states ρ and σ if there exists a recovery channel using which one can perfectly recover ρ from and σ from The relative entropy difference is equal to zero if and only if is sufficient for ρ and σ . In this paper, we show that these properties extend to Rényi generalizations of these information measures which were proposed in (Berta et al 2015 J. Math. Phys. 56 022205 ; Seshadreesan et al 2015 J. Phys. A: Math. Theor. 48 395303 ), thus providing an alternate characterization of short quantum Markov chains and sufficient quantum channels. These results give further support to these quantities as being legitimate Rényi generalizations of the conditional mutual information and the relative entropy difference. Along the way, we solve some open questions of Ruskai and Zhang, regarding the trace of particular matrices that arise in the study of monotonicity of relative entropy under quantum operations and strong subadditivity of the von Neumann entropy.
There are no universally accepted definitions of Rényi conditional entropy and Rényi mutual information, although motivated by different applications, several definitions have been proposed in the literature. In this paper, we consider a family of two-parameter Rényi conditional entropy and a family of two-parameter Rényi mutual information. By performing a change of variables for the parameters, the two-parameter Rényi conditional entropy we study coincides precisely with the definition introduced by Hayashi and Tan [IEEE Trans. Inf. Theory, 2016], and it also emerges naturally as the classical specialization of the three-parameter quantum Rényi conditional entropy recently put forward by Rubboli, Goodarzi, and Tomamichel [arXiv:2410.21976 (2024)]. We establish several fundamental properties of the two-parameter Rényi conditional entropy, including monotonicity with respect to the parameters and variational expression. The associated two-parameter Rényi mutual information considered in this paper is new and it unifies three commonly used variants of Rényi mutual information. For this quantity, we prove several important properties, including the non-negativity, additivity, data processing inequality, monotonicity with respect to the parameters, variational expression, as well as convexity and concavity. Finally, we demonstrate that these two-parameter Rényi information quantities can be used to characterize the strong converse exponents in privacy amplification and soft covering problems under Rényi divergence of order α∈(0,∞)α\in (0, \infty)α∈(0,∞).