Rupert L. Frank, Elliott H. Lieb
We show that a recent definition of relative Rényi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of Müller-Lennert et al.
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The R{é}nyi entropy is one of the important information measures that generalizes Shannon's entropy. The quantum R{é}nyi entropy has a fundamental role in quantum information theory, therefore, bounding this quantity is of vital importance. Another important quantity is R{é}nyi relative entropy on which R{é}nyi generalization of the conditional entropy, and mutual information are defined based. Thus, finding lower bound for R{é}nyi relative entropy is our goal in this paper. We use matrix inequalities to prove new bounds on the entropy of type $β$, R{é}nyi entropy.
It has been shown that the $α-z$ R{é}nyi relative entropy satisfies the Data Processing Inequality (DPI) for a certain range of $α$'s and $z$'s. Moreover, the range is completely characterized by Zhang in `20. We prove necessary and algebraically sufficient conditions to saturate the DPI for the $α-z$ R{é}nyi relative entropy whenever $1<α\leq 2$ and $\fracα{2}\leq z\leqα$. Moreover, these conditions coincide whenever $α=z$.
New monotonicity property of the quantum relative entropy
本文证明量子相对熵的局部不连续跳变在量子信道与操作作用下具有不增加的新单调性。
The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, arXiv:1410.0664] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our pap…
We derive a strengthened monotonicity inequality for quantum relative entropy by employing properties of $α$-Rényi 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 Rényi relative entropy. The inequalities obtained, thus, extends some well-known ones. We also obtain a condition under which a tripartite operator becomes a Marko…
On Uhlmann's proof of the Monotonicity of the Relative Entropy
本文自洽地介绍了 A. Uhlmann 关于相对熵在完全正且保迹映射下单调性的著名定理，以更一般的形式呈现并提供有意义的示例。
We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the affirmative a natural question that has been open for a long time, as monotonicity had previously only been shown to hold under additional assumptions, such as complete positivity or Schwarz-positivity of the adjoint map. The first step in our proof is to show monotonicity of the sandwiched Renyi divergences under positive trace-preserving maps, extending a proof of the data processing inequality by Be…
We consider some questions concerning the monotonicity properties of entropy and mean entropy of states on translationally invariant systems (classical lattice, quantum lattice and quantum continuous). By taking the property of strong subadditivity, which for quantum systems was proven rather late in the historical development, as one of four primary axioms (the other three being simply positivity, subadditivity and translational invariance) we are able to obtain results, some new, some proved in a new way, which appear to complement in an interesting way results proved around thirty years ago…
On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches
本文在统一的有限维算子理论框架下重新审视了相对熵在量子信道作用下的单调性，比较了Petz和Uhlmann的证明方法，指出了Petz原证明的缺陷及修正方式，并发展了适用于不可逆密度算子的Uhlmann方法，阐明了其数学结构。
The quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a "recovery map" that exactly reverses the effects of the quantum channel on both states. In this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination …
Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction
本文以量子相对熵的正定性和单调性为核心，系统介绍其与热力学第二定律相关的不等式，并推广到量子涨落定理、Hatano-Sasa不等式、数据处理不等式及带反馈控制的广义第二定律。
Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras
本文证明Araki-Uhlmann相对熵的Uhlmann单调性定理并推导相关不等式，同时利用KMS态无界扰动理论将相对自由能的双侧Bogoliubov不等式推广至任意von Neumann代数。
In [Berta et al., J. Math. Phys. 56, 022205 (2015)], we recently proposed Renyi 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 Renyi quantity is monotone with respect to local operations on system $A$). Here we define a Renyi squashed entanglement and a Renyi quantum discord based on a R…
Monotonicity of the logarithmic energy for random matrices
本文研究随机矩阵对数能量随矩阵维数的单调性，验证于高斯酉系综、复吉尼布雷系综和方拉盖尔酉系综，并给出显式公式，数值模拟暗示其普遍性。
We present a genuine coherence measure based on a quasi-relative entropy as a difference between quasi-entropies of the dephased and the original states. The measure satisfies non-negativity and monotonicity under genuine incoherent operations (GIO). It is strongly monotone under GIO in two- and three-dimensions, or for pure states in any dimension, making it a genuine coherence monotone. We provide a bound on the error term in the monotonicity relation in terms of the trace distance between the original and the dephased states. Moreover, the lower bound on the coherence measure can also be ca…
A new operator extension of strong subadditivity of quantum entropy
本研究针对量子熵强次可加性相关的弱单调性不等式，提出了一种新的算子不等式，其取态的期望值可退化为弱单调性不等式，并给出了含两个独立密度矩阵的推广及Rényi推广形式。
This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equalit…
In this paper a general definition of quantum conditional entropy for infinite-dimensional systems is given based on recent work of Holevo and Shirokov arXiv:1004.2495 devoted to quantum mutual and coherent informations in the infinite-dimensional case. The properties of the conditional entropy such as monotonicity, concavity and subadditivity are also generalized to the infinite-dimensional case.
Several ways have been proposed in the literature to define a coherence measure based on Tsallis relative entropy. One of them is defined as a distance between a state and a set of incoherent states with Tsallis relative entropy taken as a distance measure. Unfortunately, this measure does not satisfy the required strong monotonicity, but a modification of this coherence has been proposed that does. We introduce three new Tsallis coherence measures coming from a more general definition that also satisfy the strong monotonicity, and compare all five definitions between each other. Using three c…
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 new form of zero-discord state via Petz's monotonicity condition on relative entropy with equality has been derived systematically. A generalization of symmetric zero-discord states is presented and the related physical implications are discussed.
The Donsker-Varadhan rate function for occupation-time fluctuations has been seen numerically to exhibit monotone return to stationary nonequilibrium [Phys. Rev. Lett. 107, 010601 (2011)]. That rate function is related to dynamical activity and, except under detailed balance, it does not derive from the relative entropy for which the monotonicity in time is well understood. We give a rigorous argument that the Donsker-Varadhan function is indeed monotone under the Markov evolution at large enough times with respect to the relaxation time, provided that a "normal linear-response" condition is s…
We show how recent results of Lieb and Seiringer [math-ph/0412009; Phys. Rev. A 71, 062329 (2005)] can be obtained from repeated use of the monotonicity of relative entropy under partial traces, and explain how to use their approach to obtain tighter bounds in many situations.
We consider a quantum quasi-relative entropy $S_f^K$ for an operator $K$ and an operator convex function $f$. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the $f$-divergences (i.e. $K=I$). We also provide an error term for a class of operator inequalities, that generalize operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner-Yanase-Dyson skew in…
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