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 equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein's elegant proof of the relevant concavity theorem of Lieb.
A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality
我们介绍了一个推广的相对熵来自Wigner-Yanase-Dyson熵，并给出了一个简单的，自足的证明，它是凸的。此外，特殊情况产生了相对熵的联合凸性，并且对于映射（A，B）--> Tr K^* A^p K B^{1-p}，0 < p < 1时是Lieb的联合凸性，1 < p < 2时是Ando的联合凸性。这种方法使我们能够获得这些情况下的平等条件，以及由此产生的一些不平等的平等条件。其中包括部分迹下的单调性，以及由Lieb和Carlen证明的关于混合（p，q）范数的Minkowski型矩阵不等式。在所有情况下，等式条件与p无关;对于三个空间的扩张，它们与相对熵的强次可加性中的等式条件相同。
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.
We derive an inequality for the linear entropy, that gives sharp bounds for all finite dimensional systems. The derivation is based on generalised Bloch decompositions and provides a strict improvement for the possible distribution of purities for all finite dimensional quantum states. It thus extends the widely used concept of entropy inequalities from the asymptotic to the finite regime, and should also find applications in entanglement detection and efficient experimental characterisations of quantum states.
We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entr…
We explore the role of sandwiched Renyi relative entropy in AdS/CFT and in finite-dimensional models of holographic quantum error correction. In particular, in the context of operator algebra quantum error correction, we discuss a suitable generalization of sandwiched Renyi relative entropy over finite-dimensional von Neumann algebras. It is then shown that the equality of bulk and boundary sandwiched relative Renyi entropies is equivalent to algebraic encoding of bulk and boundary states, the Ryu-Takayanagi formula, the equality of bulk and boundary relative entropy, and subregion duality. Th…
In this paper we propose a time-independent \textit{equality} and time-dependent \textit{inequality}, suitable for an experimental test of the hypothesis of realism. The derivation of these relations is based on the concept of conditional probability and on Bayes' theorem in the framework of Kolmogorov's axiomatics of probability theory. The equality obtained is intrinsically different from the well known GHZ-equality and its variants, because violation of the new equality might be tested in experiments with only two microsystems in a maximally entangled Bell state $\big\vert\, Ψ^-\,\big\rangl…
We numerically investigate entropic Bell inequalities for a pair of entangled qutrits using information-theoretic distances. We show that for this class of inequalities Tsallis entropy is more suitable than Shannon as it reveals non-classicality for a larger set of quantum states. Finally, we find that like probability based inequalities, entropic ones are maximally violated by the non-maximally entangled qutrit state.
We investigate monogamy of correlations and entropy inequalities in the Bloch representation. Here, both can be understood as direct relations between different correlation tensor elements and thus appear intimately related. To that end we introduce the split Bloch basis, that is particularly useful for representing quantum states with low dimensional support and thus amenable to purification arguments. Furthermore, we find dimension dependent entropy inequalities for the Tsallis 2-entropy. In particular, we present an analogue of the strong subadditivity and a quadratic entropy inequality. Th…
We extend our studies of holographic entropy inequalities to gapped phases of matter. For any number of regions, we determine the linear entropy inequalities satisfied by systems in which the entanglement entropy satisfies an exact area law. In particular, we find that all holographic entropy inequalities are valid in such systems. In gapped systems with topological order, the "cyclic inequalities" derived recently for the holographic entanglement entropy generalize the Kitaev-Preskill formula for the topological entanglement entropy. Finally, we propose a candidate linear inequality for gener…
Significant work has gone into determining the minimal set of entropy inequalities that determine the holographic entropy cone. Holographic systems with three or more parties have been shown to obey additional inequalities that generic quantum systems do not. We consider a two dimensional conformal field theory that is a single boundary of a holographic system and find four additional non-linear inequalities which are derived from strong subadditivity and the formula for the entanglement entropy of a region on the conformal field theory. We also present an equality obtained by application of a…
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.
A graphical framework for proving holographic entanglement entropy inequalities in multipartite systems
Center for Fundamental Physics, School of Physical Science and Technology, ShanghaiTech University（上海科技大学物理科学与技术学院基础物理中心）；Department of Physics, Ateneo de Manila University（马尼拉雅典大学物理系）
