Relative entropy in quantum information theory
Benjamin Schumacher, Michael D. Westmoreland
我们回顾了量子相对熵函数的性质，并讨论了它在经典和量子信息传输问题以及量子数据压缩中的应用。然后，我们概述了进一步使用相对熵量化量子纠缠和分析其操纵。
We review the properties of the quantum relative entropy function and discuss its application to problems of classical and quantum information transfer and to quantum data compression. We then outline further uses of relative entropy to quantify quantum entanglement and analyze its manipulation.
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Mixing Property of Quantum Relative Entropy
推导出量子相对熵的量子熵混合性质的模拟，并将其应用于理想测量的最终状态和第二密度算符的谱形式。讨论了三种状态在相对熵有向直线上的情形。
Characterization of several kinds of quantum analogues of relative entropy
量子相对熵$D(ρ\|σ)\defeq\Tr ρ(\log ρ- \log σ)$在量子信息及相关领域中起着重要的作用。然而，有许多量子类似物的相对熵。本文从信息几何的角度刻画了这些类似物。我们还考虑了量子相对熵在这些类似物中的自然性。
Monogamous property of generalized W states in three-qubit systems in terms of relative entropy of entanglement
由于纠缠相对熵的解析表达式难以得到，研究三比特纯态的纠缠相对熵的单偶关系成为一个难题。然而，通过计算三量子比特系统中广义W态约化态的相对熵，我们发现所有的广义W态都具有纠缠相对熵的一夫一妻性质.
On divergence, relative entropy and the substate property
本文研究了量子态的三种可分性度量--发散度、相对熵与子态性质之间的关系。
All Inequalities for the Relative Entropy
两个n方量子态的相对熵是一个重要的量，例如，表现出两个态不同的程度。通过将两个n方减少到更小数目的$m$方而形成的状态的相对熵总是小于或等于两个原始n方状态的相对熵。这就是相对熵的单调性。利用凸几何的技巧，我们证明了限制下的单调性是相对熵满足的唯一一般不等式。在这样做的时候，我们与一般的访问结构的秘密共享计划。一个令人惊讶的结果是，多方状态子集的允许相对熵值的结构比允许熵值的结构简单得多。而允许的相对熵值的结构（不同于熵的结构）对于经典概率分布和量子态是相同的。
On a parametrically extended entanglement-measure due to Tsallis relative entropy
在\cite{FYK}中，我们主要研究了Tsallis相对熵关于密度算子的数学性质。作为它的一个应用，我们采用了一种基于Tsallis相对熵的参数化扩展纠缠测度来度量纠缠度。然后我们研究了它的性质与参数$q$出现在Tsallis熵。此外，还研究了它与纠缠的相对熵之间的关系。
Operator monotones, the reduction criterion and the relative entropy
引入算子单调函数理论，导出了一个新的量子相对熵与量子条件熵的关系不等式。我们提出了这个新的不等式的应用，特别是我们证明了一个新的下界的相对熵的纠缠和其他性质的纠缠措施。
Modified relative entropy of entanglement for multi-party systems
我们给出了多方系统的修正的相对熵，它是由相对密度矩阵的基和每一方的约化密度矩阵的直积的线性组合所构成的相对密度矩阵.对于纯态的三量子比特系统，我们导出了相对密度矩阵的显式封闭表达式，它是三个量子比特的态矢量分量的函数。通过对一些重要和有趣的例子的修正相对熵的计算，展示了修正相对熵的主要行为和基本性质，并与广义形成纠缠进行了比较。此外，我们还提出了一个辅助的上界作为修正的相对熵。
Imaginarity measures induced by relative entropy
该研究为量子虚性资源理论提出两种分别由α-z-Rényi相对熵和Tsallis相对算子熵诱导的度量，并探讨了不同度量间的关系与性质。
Gaussian relative entropy of entanglement
对于给定关联矩阵的两个高斯态，为了使它们之间的相对熵实际上是可计算的，本文描述了在指数密度算符中如何将关联矩阵转化为矩阵。高斯态相对于可分高斯态集的最小相对熵称为纠缠的高斯相对熵。证明了当可分高斯态处于可分高斯态集和不可分高斯态集的边界时，纠缠的高斯相对熵达到。对于双模高斯态，纠缠态的高斯相对熵的计算由寻找一个10个待定参数的矩阵简化为3个变量。将双模高斯态分为四种类型，数值结果表明，每种类型的纠缠态的高斯相对熵都是由同一类型的可分态实现的，对于对称高斯态，严格证明了它是由对称高斯态实现的.
Relative entropy of entanglement of rotationally invariant states
本文计算了自旋为1/2的粒子与任意自旋为$j$的粒子以及自旋为1的粒子与自旋为$j$的粒子与整数$j$的旋转不变态纠缠的相对熵。给出了自旋1粒子和自旋$j$为半整数的自旋$j$粒子的旋转不变态的相对熵的一个下界和纠缠度的一个上界.
Information, Relative Entropy of Entanglement and Irreversibility
先前提出的纠缠度，如纠缠的形成和援助，被证明是特殊情况下的相对熵的纠缠。这些措施的混合状态的合奏之间的差异示出依赖于特定成员的合奏的经典信息的可用性。在此基础上，导出了纠缠的相对熵与互信息的关系。
Calculating the relative entropy of entanglement
我们将Vedral和Plenio的定理（Phys.Rev.A57，1619中的定理3）推广到更一般的情况，得到了一类混合态纠缠的相对熵，这个结果也可以从Rains的定理9（Phys.Rev.A60，179）得到.
Tripartite entanglement and quantum relative entropy
建立了三体纯态纠缠与两体纠缠相对熵的可加性之间的关系。我们的结果涉及到大量的副本状态的本地操作的渐近极限。我们发现，相对熵的可加性意味着至少有两种不等价的渐近三方纠缠。所使用的方法包括应用一些有用的引理，使我们能够解析计算的相对熵的一些类的二分态。
We provide an upper bound on the quasi-relative entropy in terms of the trace distance. The bound is derived for two cases: 1) any operator monotone decreasing function and full rank mixed qubit or classical states; 2) a large class of operator monotone decreasing function and any mixed qubit or classical states. Moreover, we derive an upper bound for the Umegaki and Tsallis relative entropies in the case of any finite-dimensional states. The bound for the relative entropy improves the known bounds for some states in any dimensions larger than four. The bound for the Tsallis entropy improves t…
We employ the relative entropy as a measure to quantify the difference of eigenmodes between Hermitian and non-Hermitian systems in elliptic optical microcavities. We have found that the average value of the relative entropy in the range of the collective Lamb shift is large, while that in the range of self-energy is small. Furthermore, the weak and strong interactions in the non-Hermitian system exhibit rather different behaviors in term of the relative entropy, and thus it displays an obvious exchange of eigenmodes in the elliptic microcavity.
Modification of relative entropy of Entanglement
为了提高纠缠相对熵的可计算性，避免形成纠缠似乎大于蒸馏纠缠的问题，提出了修正的纠缠相对熵（MRE）.对于两量子比特系统，我们导出了一个明确的和“弱”的封闭表达式的MRE依赖于纯态分解的情况下的混合态。对于更多量子比特的系统，我们原则上得到了计算MRE的算法。MRE显著提高了纠缠相对熵的可计算性，降低了对纯态分解的依赖性和敏感性。而且它还能继承纠缠态相对熵的大部分重要物理特性。此外，还对作为Werner态的推广的一类态进行了建设性的讨论。
Relative Entropy of Entanglement for Two-Qubit State with $z$-directional Bloch Vectors
任意两比特态纠缠的相对熵至今还没有一个封闭的计算公式。本文提出了一种方法，它保证了大多数具有$z$ -方向Bloch矢量的态的相对纠缠熵的推导。结果表明，这些状态的最近可分状态也有$z$ -方向布洛赫矢量，虽然有少数例外。
The study of quantum correlations in High-dimensional bipartite systems is crucial for the development of quantum computing. We propose relative entropy as a distance measure of correlations may be measured by means of the distance from the quantum state to the closest classical-classical state. In particular, We establish relations between relative entropy and quantum discord quantifiers obtained by means of orthogonal projective measurements. We show that for symmetrical X-states density matrices the quantum discord is equal to relative entropy. At the end of paper, various examples of X-sta…
Monotonicity of quantum relative entropy revisited
粗粒化下的单调性是量子相对熵的一个重要性质。本文研究了单调性定理的等价条件及其推论，如冯诺伊曼熵的强次可加性、Golden-Thompson迹不等式和Holevo量的单调性，并简要地指出了它们与量子马尔可夫态的关系。
Based on the resource theory for quantifying the coherence of quantum channels, we introduce a new coherence quantifier for quantum channels via maximum relative entropy. We prove that the maximum relative entropy for coherence of quantum channels is directly related to the maximally coherent channels under a particular class of superoperations, which results in an operational interpretation of the maximum relative entropy for coherence of quantum channels. We also introduce the conception of sub-superchannels and sub-superchannel discrimination. For any quantum channels, we show that the adva…
It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched Rényi relative $α$-entropy with $α>1$ is also equivalent to this property. For the proof, we use an interpolating family of $L_p$-norms with respect to a state.
In [Gallego and Aolita, Physical Review X 5, 041008 (2015)], the authors proposed a definition for the relative entropy of steering and showed that the resulting quantity is a convex steering monotone. Here we advocate for a different definition for relative entropy of steering, based on well grounded concerns coming from quantum Shannon theory. We prove that this modified relative entropy of steering is a convex steering monotone. Furthermore, we establish that it is uniformly continuous and faithful, in both cases giving quantitative bounds that should be useful in applications. We also cons…
We present a family of coherence quantifiers based on the generalized $α-z-$relative R$\acute{e}$nyi entropy. These quantifiers satisfy all the standard criteria for well-defined measures of coherence, and include some existing coherence measures as special cases.
A linearized variant of relative entropy is used to quantify in a unified scheme the different kinds of correlations in a bipartite quantum system. As illustration, we consider a two-qubit state with parity and exchange symmetries for which we determine the total, classical and quantum correlations. We also give the explicit expressions of its closest product state, closest classical state and the corresponding closest product state. A closed additive relation, involving the various correlations quantified by linear relative entropy, is derived.
Asymptotics of Quantum Relative Entropy From Representation Theoretical Viewpoint
本文证明了量子相对熵$D(σ\| ρ)$可以由给定的POVM序列的概率的Kullback-Leibler发散渐近地得到。POVM的顺序取决于$ρ$，但与$σ$的选择无关。
Bounds for the reduced relative entropies
本文利用变分表达式给出了约化相对熵的下界，定义了约化 Tsallis 相对熵并证明其凸性，最后给出了其上界。
Relative Entropy and Single Qubit Holevo-Schumacher-Westmoreland Channel Capacity
将经典信道容量的相对熵描述应用于单比特信道.推导了量子比特密度矩阵在Bloch球表象中的相对熵的一个简单公式。将该公式与King-Ruskai-Szarek-Werner量子比特通道椭球图相结合，详细分析了几种单位和非单位量子比特通道。给出了另一种证明，即单量子比特单位信道的最佳HSW信号状态是具有最小信道输出熵的那些状态。推导是基于量子位相对熵公式和King-Ruskai-Szarek-Werner量子位通道椭球图像的对称性。证明了（量子比特或非量子比特）量子信道的任何一组最佳HSW信号态的平均输出密度矩阵是唯一的。
In this paper we formulate a kind of new geometric measure of quantum correlations. This new measure is in terms of the quantum Tsallis relative entropy and can be viewed as a one-parameter extension quantum discordlike measure that satisfies all requirements of a good measure of quantum correlations. It is of an elegant analytic expression and contains several existing good quantum correlation measures as special cases.
The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the Rényi $α$-relative entropy which present in [Phys. Rev. A 94, 052336, 2016]. We show that the Rényi $α$-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement which presents in [Phys. Rev. A 93, 0321…
Relative entropy is an exact measure of non-Gaussianity
我们证明了，通过相对熵，任意$N$模场态最接近的高斯态是由给定态的协方差矩阵和平均位移建立的。因此，一个$N$模式状态与其关联的高斯状态的相对熵是非高斯性的精确距离型度量。为了说明这一发现，我们讨论了$N$模式Fock对角态的一般性质，并计算了它们的非高斯熵的精确值。
Max- Relative Entropy of Entanglement, alias Log Robustness
研究了纠缠的最大相对熵的性质，讨论了在特定量子操作下，纠缠的最大相对熵作为理想纠缠稀释的一次速率上界的意义.结果表明，它实际上等于另一个已知的纠缠单调，即对数鲁棒性。已知后者不是渐近连续的，并且不知道它是否是弱可加的。然而，通过适当地修改纠缠的最大相对熵，我们得到了一个被认为满足这两个性质的量。事实上，修正后的量被证明是等于正规化的相对熵的纠缠。
Continuity of Relative Entropy of Entanglement
我们证明了称为纠缠相对熵的纠缠度量满足强连续性条件。如果两个状态彼此接近，那么在这种测量中，它们的每个粒子对的纠缠度也是如此。特别是，它遵循的措施是适当的纠缠操纵的描述在无限数量的粒子对的限制。
Relative entropy of entanglement of a kind of two qubit entangled states
本文严格证明了某个具有六个非零元素的块对角化两量子比特纠缠态通过具有相同矩阵结构的可分态达到其量子相对熵纠缠。纠缠态包括作为特殊情况的局部滤波结果态。
We consider the information flow on a system's observable $X$ corresponding to a positive-operator valued measure under a quantum measurement process $Y$ described by a completely positive instrument from the viewpoint of the relative entropy. We establish a sufficient condition for the relative-entropy conservation law which states that the averaged decrease in the relative entropy of the system's observable $X$ equals the relative entropy of the measurement outcome of $Y$, i.e. the information gain due to measurement. This sufficient condition is interpreted as an assumption of classicality …
Evaluation of relative entropy of entanglement and derivation of optimal Lewenstein-Sanpera decomposition of Bell decomposable states via convex optimization
利用半定规划方法给出了Bell可分解态的最优Lewenstein-Sanpera分解的解析表达式。利用Karush-Kuhn-Tucker最优化方法，计算了Bell可分解态的最小纠缠相对熵，结果表明：位于可分Bell可分解态凸集边界的同一可分Bell可分解态，既优化了Lewenstein-Sanpera分解，又优化了纠缠的相对熵. { \bf关键词：纠缠的最小相对熵，半定规划，凸优化，Lewenstein-Sanpera分解，Bell可分解态。{ \bf PAC索引：03.65.Ud}
On the quantum f-relative entropy and generalized data processing inequalities
研究了量子f-相对熵的基本性质，其中f（.）是算子凸函数。本文给出了单调性和联合凸性下的等式条件，这些条件比，更一般，因为它们对一类算子凸函数成立，而对f（t）= -ln（t），与以前已知的条件不同。用量子f-相对熵定义了量子f-熵，并研究了它的性质，给出了某些情况下的等式条件。证明了Holevo信息、纠缠辅助容量和相干信息的f-推广也满足数据处理不等式，并给出了f-相干信息的等价条件.
Bounds on relative entropy of entanglement for multi-party systems
根据两体纠缠的形成和升华以及各子系统的熵，给出了多方系统纠缠的相对熵的上、下界。我们指出了这些结果对多方纯态局域可逆可转换性的意义，并从信息删除的角度讨论了它们的物理基础。
We employ the conditional version of sandwiched Tsallis relative entropy to determine 1:N-1 separability range in the noisy one-parameter families of pseudopure and Werner-like N-qubit W, GHZ states. The range of the noisy parameter, for which the conditional sandwiched Tsallis relative entropy is positive, reveals perfect agreement with the necessary and sufficient criteria for the separability in the 1:N-1 partition of these one parameter noisy states.
Connections between relative entropy of entanglement and geometric measure of entanglement
由于两个最重要的纠缠度量--形成纠缠和蒸馏纠缠--到目前为止仅限于两体系统，因此研究多体系统的其他纠缠度量显得很有必要。在这里，其他两个纠缠措施之间的联系-纠缠的相对熵和纠缠的几何措施-进行了调查。结果发现，对于任意纯态，后者会导致前者的下界。对于某些纯态，有些是二体态，有些是多体态，这个下界是饱和的，因此它们的纠缠相对熵可以根据它们已知的纠缠几何度量解析地找到。对于特定的混合态，还建立了纠缠的相对熵的上界。数值证据强烈表明，这些上限是紧的，即，它们实际上是纠缠的相对熵。
The relative entropy of entanglement of Schmidt correlated states and distillation
我们证明了混合态$ρ=\sum_{mn}a_{mn}|m> < n|$可以由纯态$\{p_{k}, |ϕ_{k} > \}$的系综实现，其中$|ϕ_{k}>=\sum_{m}\sqrt{a_{mm}}e^{iθ_{m}^{k}}|m>$。利用这种形式，我们讨论了Schmidt关联态纠缠的相对熵.此外，我们还计算了一类混合态的纠缠度。
We show that the von Neumann--Lueders collapse rules in quantum mechanics always select the unique state that maximises the quantum relative entropy with respect to the premeasurement state, subject to the constraint that the postmeasurement state has to be compatible with the knowledge gained in the measurement. This way we provide an information theoretic characterisation of quantum collapse rules by means of the maximum relative entropy principle.
We consider the gravity dual of the modular Hamiltonian associated to a general subregion of a boundary theory. We use it to argue that the relative entropy of nearby states is given by the relative entropy in the bulk, to leading order in the bulk gravitational coupling. We also argue that the boundary modular flow is dual to the bulk modular flow in the entanglement wedge, with implications for entanglement wedge reconstruction.
Thermodynamics is usually developed starting from entropy and the maximum entropy principle. We investigate here to what extent one can replace entropy with relative entropy which has several advantages, for example in the context of local quantum field theory. We find that the principle of maximum entropy can be replaced by a principle of minimum expected relative entropy. Various ensembles and their thermodynamic potentials can be defined through relative entropy. We also show that thermal fluctuations are in fact governed by a relative entropy. Furthermore we reformulate the third law of th…
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…
The operational characterization of quantum coherence is the corner stone in the development of resource theory of coherence. We introduce a new coherence quantifier based on max-relative entropy. We prove that max-relative entropy of coherence is directly related to the maximum overlap with maximally coherent states under a particular class of operations, which provides an operational interpretation of max-relative entropy of coherence. Moreover, we show that, for any coherent state, there are examples of subchannel discrimination problems such that this coherent state allows for a higher pro…
Dissipation, interaction and relative entropy
许多热力学关系涉及不等式，如果一个过程不涉及耗散，则是相等的。在这篇文章中，我们提供了等式，其中耗散贡献被证明涉及相对熵（a.k.a. Kullback-Leibler分歧）。所考虑的过程是经典力学和量子力学中的一般时间演化，初始状态有时是热的，有时是部分热的。作为应用，相对熵与输运系数有关。
Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. A…
Quantum fidelity and relative entropy between unitary orbits
保真度和相对熵是量子信息理论中的两个重要量。研究了幺正轨道下的量子保真度和相对熵。明确地导出了两个幺正轨道之间的最大和最小量子保真度以及相对熵。最后讨论了它在量子计算和信息处理中的潜在应用。
Symmetry-Resolved Relative Entropy of Random States
该研究利用大N图表技术计算Wishart系综对称随机态的相对熵，确定U(1)对称随机态的相对熵并推导对称分解Page曲线，揭示其普适统计行为，加深对随机态性质的理解。
It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for t…
Continuity of the relative entropy of resource
该研究建立资源相对熵局部连续性判据，推导其基本推论，并证明多体纠缠相对熵的局部连续性可由量子互信息的局部连续性得出。
We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these axioms induce sufficient structure to establish continuity in the interior of the probability simplex and meaningful upper and lower bounds, e.g., we find that every relative entropy must lie between the Rényi divergences of order $0$ and $\infty$. We further show simple conditions for positive definiteness of such relative entropies and a characte…
The relative entropy is a measure of the distinguishability of two quantum states. A great deal of progress has been made in the study of the relative entropy between an excited state and the vacuum state of a conformal field theory (CFT) reduced to a spherical region. For example, when the excited state is a small perturbation of the vacuum state, the relative entropy is known to have a universal expression for \textit{all} CFT's \cite{Faulk-GR-entanglement}. Specifically, the perturbative relative entropy can be written as the symplectic flux of a certain scalar field in an \textit{auxiliary…
We generalize our results in paper I in this series to quantum channels between general v. Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two v. Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.
The chain rule for the classical relative entropy ensures that the relative entropy between probability distributions on multipartite systems can be decomposed into a sum of relative entropies of suitably chosen conditional distributions on the individual systems. Here, we prove a similar chain rule inequality for the quantum relative entropy in terms of channel relative entropies. The new chain rule allows us to solve an open problem in the context of asymptotic quantum channel discrimination: surprisingly, adaptive protocols cannot improve the error rate for asymmetric channel discrimination…
Upper continuity bounds on the relative $q$-entropy for $q>1$
广义熵和相对熵是当前研究的热点。类似于标准相对熵，相对$q$ -熵对于$q>1$通常是无界的。在有限维情形下，得到了量子相对$q$ -熵的上界.这些界限的特点是在Fannes意义上的连续性。
Measurement-induced nonlocality based on the relative entropy
我们量化了测量引起的非定域性[Luo和Fu，Phys. Rev. Lett. 106，120401（2011）]从相对熵的角度。这种量化导致的measurementinduced nonlocality的操作解释，即，它是最大的熵增加后，局部不变的测量。非定域性的相对熵的上界是被测子系统的熵。在Hilbert- Schmidt范数的基础上，建立了非局域性的相对熵与几何非局域性之间的关系，并证明了它等于最大可压缩纠缠。得到了三重纯态的几种折衷关系。我们还给出了Bell对角态非定域性的相对熵的显式表达式。
Reduced relative quantum entropy
本文提出约化相对量子熵并证明其凸性，借此简化了 Lieb 与 Seiringer 相关定理的证明。
Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications
本文全面研究基于保真度的平滑最小相对熵，证明其基本性质，建立与其他信息量的联系，推导二阶渐近性，并应用于资源理论中的一次性界及随机性蒸馏，最后给出半定规划和双线性规划。
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