On divergence, relative entropy and the substate property
Rahul Jain, Jaikumar Radhakrishnan, Pranab Sen
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。
In this article we study relationship between three measures of distinguishability of quantum states called as divergence, relative entropy and the substate property.
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The Quantum Substate Theorem due to Jain, Radhakrishnan, and Sen (2002) gives us a powerful operational interpretation of relative entropy, in fact, of the observational divergence of two quantum states, a quantity that is related to their relative entropy. Informally, the theorem states that if the observational divergence between two quantum states rho, sigma is small, then there is a quantum state rho' close to rho in trace distance, such that rho' when scaled down by a small factor becomes a substate of sigma. We present new proofs of this theorem. The resulting statement is optimal up to …
A theorem about relative entropy of quantum states with an application to privacy in quantum communication
我们证明了关于量子态的相对熵的下列定理。and the相对熵S（rho| sigma）= Tr rho（log rho - log sigma）= c.则对于所有的φ> 0，存在状态ρ '，使得迹距离||罗罗||_t = Tr sqrt{（rho ′- rho）^2} <= σ，rho ′/2^{O（c/ε ^2）} <= σ。“它指出，如果rho和sigma的相对熵很小，则存在接近rho的状态rho'，即具有小的迹线距离||罗罗||_t，当按因子2^{O（c）}缩小时，它“位于”sigma的“内部”，或者成为sigma的“子状态”。这一结果在量子通信复杂性和密码学中有几个应用。利用子态定理，我们导出了两方量子通信模型中集员问题的隐私折衷。这里Alice被给予[n]的子集A，Bob在[n]中输入i，他们需要确定i是否在A中。集合成员的隐私权衡：在任何两方量子通信协议中，如果鲍勃只透露关于他的输入的k比特信息，那么爱丽丝必须透露关于她的输入的至少n/2^{O（k）}比特信息。“我们还讨论了在子状态定理的背景下自然产生的各种信息理论量之间的关系。
A Separation between Divergence and Holevo Information for Ensembles
概率分布集合的发散信息的概念是由Jain、Radhakrishnan和Sen在“子状态定理”的背景下引入的。从那时起，发散度被认为是量子和经典通信中几种情况下更自然的信息度量。我们构建的概率分布，发散信息可能会显着小于更标准的Holevo信息的合奏。其结果是，我们建立了以前显示的Holevo信息的下限弱于类似的发散信息。
One-shot information theory entertains a plethora of entropic quantities, such as the smooth max-divergence, hypothesis testing divergence and information spectrum divergence, that characterize various operational tasks and are used to prove the asymptotic behavior of various tasks in quantum information theory. Tight inequalities between these quantities are thus of immediate interest. In this note we use a minimax approach (appearing previously for example in the proofs of the quantum substate theorem), to simplify the quantum problem to a commutative one, which allows us to derive such ineq…
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-s…
The regularized channel Rényi divergence is continuous
本文证明正则化信道Rényi散度在α=1处连续，并由此推导出信道鉴别的指数强逆、子信道平滑渐近等分性质、相对熵累积定理及广义量子Stein引理的新证明，同时表明正则化信道相对熵可计算。
Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states
我们讨论了相对熵作为混合量子态之间距离的度量的替代方案。所提出的量是概率分布之间的詹森-香农发散（JSD）的量子理论领域的扩展。JSD有几个有趣的属性。它出现在信息论中，与Kullback-Leibler分歧不同，它是对称的，总是定义良好且有界的。我们表明，量子JSD（QJSD）的份额与相对熵的大部分物理相关的属性，特别是那些需要一个“好”的量子可测量性。我们把它与其他已知的量子距离，我们建议可能的应用领域的量子信息理论。
Quantum Algorithms for Computing Maximal Quantum $f$-divergence and Kubo-Ando means
针对量子信息论量的计算需求，本文提出可计算最大量子$f$-散度与Kubo-Ando均值的量子算法，该框架具备通用性，可覆盖Renyi熵、矩阵均值等特例。
The Holevo quantity provides an upper bound for the mutual information between the sender of a classical message encoded in quantum carriers and the receiver. Applying the strong sub-additivity of entropy we prove that the Holevo quantity associated with an initial state and a given quantum operation represented in its Kraus form is not larger than the exchange entropy. This implies upper bounds for the coherent information and for the quantum Jensen--Shannon divergence. Restricting our attention to classical information we bound the transmission distance between any two probability distributi…
Logarithmic divergence of the block entanglement entropy for the ferromagnetic Heisenberg model
最近的研究表明，在量子模型中，纠缠熵的对数发散度作为子系统大小的函数是临界性的标志。我们证明了对于自旋为1/2的铁磁海森堡链，在每个位置具有固定磁化强度$y$的扇区中，$ n$位置的基态纠缠熵增长为${1/2}\log_{2} \frac{n(L-n)}{L}C(y)$，其中$C(y)=2πe({1/4}-y^{2})$
Benign Projective Landscapes for Measured Quantum Divergences
本文研究测量量子散度的非凸优化，证明在光滑生成元下投影局部最优即全局最优，并给出分块判据，应用于二进制可访问信息与Rényi散度，解决相关猜想。
To this day, von Neumann definition of entropy remains the most popular measure of quantum entanglement. Much of the literature on entanglement entropy, particularly in the context of field theory, has focused on isolating the UV divergences. Zero-mode divergences of the entanglement entropy are less studied in this context, and apart from being easier to isolate, they offer an interesting insight into the physics of the system. To gain a better understanding of the system in this limit, we develop the free particle approximation of Harmonic oscillator, with which we investigate the properties…
Black hole entropy has been shown by 't Hooft to diverge at the horizon. The region near the horizon is in a thermal state, so entropy is linear to energy which consequently also diverges. We find a similar divergence for the energy of the reduced density matrix of relativistic and non-relativistic field theories, extending previous results in quantum mechanics. This divergence is due to an infinitely sharp division between the observable and unobservable regions of space, and it stems from the position/momentum uncertainty relation in the same way that the momentum fluctuations of a precisely…
We investigate the cause of the divergence of the entanglement entropy for the free scalar fields in $(1+1)$ and $(D + 1)$ dimensional space-times. In a canonically equivalent set of variables, we show explicitly that the divergence in the entanglement entropy of the continuum field in $(1 + 1)-$ dimensions is due to the accumulation of large number of near-zero frequency modes as opposed to the commonly held view of divergence having UV origin. The feature revealing the divergence in zero modes is related to the observation that the entropy is invariant under a hidden scaling transformation e…
We examine the correlations between divergences in ground state entanglement entropy and emergent zero-modes of the underlying Hamiltonian in the context of one-dimensional Bosonic and Fermionic chains. Starting with a pair of coupled Bosonic degrees of freedom, we show that zero modes are necessary, but not sufficient for entanglement entropy divergences. We then list sufficient conditions that identify divergences. Next, we extend our analysis to Bosonic chains, where we demonstrate that zero modes of the entanglement Hamiltonian provide a signature for divergences independent of the entangl…
Black hole entropy has been shown by 't Hooft to diverge at the horizon, whereas entanglement entropy in general does not. We show that because the region near the horizon is a thermal state, entropy is linear to energy, and energy at a barrier is inversely proportional to barrier slope, and diverges at an infinitely sharp barrier as a result of position/momentum uncertainty. We show that 't Hooft's divergence at the black hole is also an example of momentum/position uncertainty, as seen by the fact that the "brick wall" which corrects it in fact smooths the sharp boundary into a more gradual …
Recently it has been demonstrated that the Shannon entropy or the von Neuman entropy are the only entropy functions that generate a local Bregman divergences as long as the state space has rank 3 or higher. In this paper we will study the properties of Bregman divergences for convex bodies of rank 2. The two most important convex bodies of rank 2 can be identified with the bit and the qubit. We demonstrate that if a convex body of rank 2 has a Bregman divergence that satisfies sufficiency then the convex body is spectral and if the Bregman divergence is monotone then the convex body has the sh…
Perturbative calculation of field space entanglement entropy
本文提出了一种计算场空间纠缠熵的方法，解决了微扰计算中出现的发散问题，并通过简单例子验证了该方法的有效性。
The subject is the overview of the use of quasi-entropy in finite dimensional spaces. Matrix monotone functions and relative modular operators are used. The origin is the relative entropy and the f-divergence, monotone metrics, covariance and the chi-square-divergence are the most important particular cases. The extension of the monotone metric to two variables is a new concept.
We characterize the functions for which the corresponding Bregman divergence is jointly convex on matrices. As an application of this characterization, we derive a sharp inequality for the quantum Tsallis entropy of a tripartite state, which can be considered as a generalization of the strong subadditivity of the von Neumann entropy. (In general, the strong subadditivity of the Tsallis entropy fails for quantum states, but it holds for classical states.) Furthermore, we show that the joint convexity of the Bregman divergence does not imply the monotonicity under stochastic maps, but every mono…
Audenaert and Datta recently introduced a two-parameter family of relative Rényi entropies, known as the $α$-$z$-relative Rényi entropies. The definition of the $α$-$z$-relative Rényi entropy unifies all previously proposed definitions of the quantum Rényi divergence of order $α$ under a common framework. Here we will prove that the $α$-$z$-relative Rényi entropies are a proper generalization of the quantum relative entropy by computing the limit of the $α$-$z$ divergence as $α$ approaches one and $z$ is an arbitrary function of $α$. We also show that certain operationally relevant families of…
We consider a family of quantum conditional entropies based on the concept of quantum $f$-divergences. First, we explicitly formulate conditions under which the notion of quantum conditional entropy is well defined in this way. In particular, we demand that the value of conditional entropy be independent of any extension of the principal Hilbert space. Using fundamental properties of the quantum $f$-divergences, several interesting relations for such conditional entropies are formulated. We separately examine an especially important case of quantum conditional entropies related to the Tsallis …
Induced Quantum Divergence: A New Lens on Communication and Source Coding
本文提出由母量子相对熵导出的诱导量子散度，用以替代基于位置解码中的假设检验散度，从而简化量子通信与状态重分配分析并获得更紧界。
Thermodynamics describes large-scale, slowly evolving systems. Two modern approaches generalize thermodynamics: fluctuation theorems, which concern finite-time nonequilibrium processes, and one-shot statistical mechanics, which concerns small scales and finite numbers of trials. Combining these approaches, we calculate a one-shot analog of the average dissipated work defined in fluctuation contexts: the cost of performing a protocol in finite time instead of quasistatically. The average dissipated work has been shown to be proportional to a relative entropy between phase-space densities, to a …
We study many-party correlations quantified in terms of the Umegaki relative entropy (divergence) from a Gibbs family known as a hierarchical model. We derive these quantities from the maximum-entropy principle which was used earlier to define the closely related irreducible correlation. We point out differences between quantum states and probability vectors which exist in hierarchical models, in the divergence from a hierarchical model and in local maximizers of this divergence. The differences are, respectively, missing factorization, discontinuity and reduction of uncertainty. We discuss gl…
Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective
本文提出一种扩展Umegaki相对熵的量子相对-α熵，展现非线性凸性，补充了Petz-Renyi熵在α>1时的凸性结果，且在张量积下可加，具有几何性质。
We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order wh…
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz…
Integral formula for quantum relative entropy implies data processing inequality
本文建立了量子相对熵及von Neumann熵高阶导数的积分表示，简化证明了Holevo界和量子相对熵单调性等数据处理不等式，并展示了其在量子散度及信息论概率模型中的应用。
It has long been conjectured that the entropy of quantum fields across boundaries scales as the boundary area. This conjecture has not been easy to test in spacetime dimensions greater than four because of divergences in the von Neumann entropy. Here we show that the Renyi entropy provides a convergent alternative, yielding a quantitative measure of entanglement between quantum field theoretic degrees of freedom inside and outside hypersurfaces. For the first time, we show that the entanglement entropy in higher dimensions is proportional to the higher dimensional area. We also show that the R…
Motivated by the geometrical structures of quantum mechanics, we introduce an almost-complex structure $J$ on the product $M\times M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a metric-like tensor on $M\times M$ from a divergence function. These tensors may be pulled back to $M$, and we compute them in the case of an N-dimensional symplex with respect to the Kullback-Leibler relative entropy, and in the case of (a suitable unfolding space of) the manifold of faithful density operators with respect to the von Neumann-Umegaki relative e…
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz…
No off-diagonal quantum focusing for Rényi divergences
研究量子聚焦猜想中关于Rényi散度的非对角部分，针对满足特定条件的Rényi型散度，通过相关证明对完整的Rényi聚焦表述是否成立这一问题给出否定答案。
In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard $f$-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal $f$-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dila…
In these decades, it has been revealed that there is rich information-theoretic structure in thermodynamics of out-of-equilibrium systems in both the classical and quantum regimes. This has led to the fruitful interplay among statistical physics, quantum information theory, and mathematical theories including matrix analysis and asymptotic probability theory. The main purpose of this book is to clarify how information theory works behind thermodynamics and to shed modern light on it. We focus on both of purely information-theoretic concepts and their physical implications: We present self-cont…
Quantum Noncommutativity Uniquely Determines Relative Entropy
本文证明，在二进制猜测游戏中，任何加性可区分性度量若尊重最优测量揭示的赔率，则必为Umegaki相对熵，这源于量子非对易性。
Renormalized Thermal Entropy in Field Theory
量子场论中的标准熵计算，当应用到一个子系统的确定体积，表现出面积依赖的紫外发散，使热力学解释麻烦。本文定义了一个与Newton-Wigner位置算符有关的重整化熵。因此，每当我们在空间区域上追踪时，我们追踪的自由度是根据牛顿-维格纳局部化而不是通常意义上的局部化的。我们考虑一个自由的标量场在d+1时空维度准备在热态，我们表明，我们的熵是免费的发散，并有一个完美的声音热力学行为。在高温/大体积极限下，我们的结果与标准QFT计算一致，只要从后者中减去发散贡献。在低温/小体积的极限下，熵趋于零，但对温度的依赖性不同。
Quantum Decoherence in a Four-Dimensional Black Hole Background
我们在四维中证明了存在爱因斯坦-杨-米尔斯黑洞的标量场的密度矩阵的对数发散。这种发散与先前发现的标量场熵的对数发散有关，这不能被吸收到黑洞的霍金-贝肯斯坦熵的重整化中。当后者衰减时，对数发散在量子刘维尔方程中引入一个非对易项$\nd{δH}ρ$，用于标量场的密度矩阵$ρ$，导致量子退相干。$\nd{δH}$的数量级是$μ^2/M_P$，其中$μ$是标量粒子的质量。
We describe the structure of the bijective transformations on the set of density operators which preserve the Bregman $f$-divergence for an arbitrary differentiable strictly convex function $f.$ Furthermore, we determine the preservers of the Jensen $f$-divergence in the case when the generating function $f$ belongs to a recently introduced function class called Matrix Entropy Class.
The optimized quantum $f$-divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched Rényi relative quasi-entropy as special cases. In this paper, we establish physically meaningful refinements of the data-processing inequality for the optimized $f$-divergence. In particular, the refinements state that the absolute difference between the optimized $f$-divergence and its channel-processed version is an upper bound on how well one can recover a quantum state acted upon by a quantum channel, whenever the recovery channel is taken to be a…
Resolution of Infrared Entanglement Divergences via the Extended Uncertainty Principle
本研究利用扩展不确定原理的几何修正，解决了量子系统中零模导致的红外纠缠发散问题，使纠缠熵饱和为有限值。
Jointly convex quantum Jensen divergences
本研究从联合凸性角度探究量子Jensen散度，证明生成正定矩阵上联合凸量子Jensen散度的函数集等价于Chen和Tropp近期提出的矩阵熵类。
Continuity of Regularized Channel Rényi Divergences
本文证明有限维量子信道的正则化夹心Rényi散度在阶数趋于1时收敛于正则化相对熵，并利用信道曲棍球棒散度导出指数强逆、零一检验定律和子信道渐近等分性质。
In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (in another name, sandwiched Rényi relative entropy) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.
We introduce an axiomatic approach for channel divergences and channel relative entropies that is based on three information-theoretic axioms of monotonicity under superchannels (i.e. generalized data processing inequality), additivity under tensor products, and normalization, similar to the approach given recently for the state domain. We show that these axioms are sufficient to give enough structure also in the channel domain, leading to numerous properties that are applicable to all channel divergences. These include faithfulness, continuity, a type of triangle inequality, and boundedness b…
We define a new divergence of von Neumann algebras using a variational expression that is similar in nature to Kosaki's formula for the relative entropy. Our divergence satisfies the usual desirable properties, upper bounds the sandwiched Renyi entropy and reduces to the fidelity in a limit. As an illustration, we use the formula in quantum field theory to compute our divergence between the vacuum in a bipartite system and an "orbifolded" -- in the sense of conditional expectation -- system in terms of the Jones index. We take the opportunity to point out entropic certainty relation for arbitr…
Tight relations and equivalences between smooth relative entropies
本文研究了光滑相对熵之间的关系与等价性，通过改进Datta和Renner的引理，建立了更精确的界和对偶关系，从而完善了光滑max相对熵与假设检验相对熵之间的紧密联系。
Regularized barycentric Rényi divergences
本文证明了对于任何可加且单调的量子相对熵，正则化重心Rényi散度在严格正输入上与最小重心Rényi散度一致，从而解决了可加性问题，并表明唯一可加的重心Rényi散度是最小散度。
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