The quantum relative entropy is frequently used as a distance, or distinguishability measure between two quantum states. In this paper we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More precisely, we derive lower and upper bounds on the relative entropy in terms of various distance measures for the difference of the states based on unitarily invariant norms. The upper bounds can be considered as statements of continuity of the relative entropy distance in the sense of Fannes. We employ methods from optimisation theory to obtain bounds that are as sharp as possible.
We study the continuity of the maximum-entropy inference map for two observables in finite dimensions. We prove that the continuity is equivalent to the strong continuity of the set-valued inverse numerical range map. This gives a continuity condition in terms of analytic eigenvalue functions which implies that discontinuities are very rare. It shows also that the continuity of the MaxEnt inference method is independent of the prior state.
We show that a positive linear map preserves local continuity (convergence) of the entropy if and only if it preserves finiteness of the entropy, i.e. transforms operators with finite entropy to operators with finite entropy. The last property is equivalent to the boundedness of the output entropy of a map on the set of pure states.
Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative $q$-entropy is generally unbounded for $q>1$. Upper bounds on the quantum relative $q$-entropy in terms of norm distances between its arguments are obtained in finite-dimensional context. These bounds characterize a continuity property in the sense of Fannes.
Continuity properties of the output entropy of positive linear maps between Banach spaces of trace class operators are investigated with the special attention to the classes of quantum channels and operations. It is shown that finiteness of the output entropy of a positive map on the whole input state space implies its continuity. Sufficient conditions for this property expressed in terms of the Kraus representation of quantum channels and operations are presented. The characterization of a positive map "preserving continuity of the entropy" (in the sense that continuity of the entropy on a se…
A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex to…
Given an arbitrary quantum state ($σ$), we obtain an explicit construction of a state $ρ^*_\varepsilon(σ)$ (resp. $ρ_{*,\varepsilon}(σ)$) which has the maximum (resp. minimum) entropy among all states which lie in a specified neighbourhood ($\varepsilon$-ball) of $σ$. Computing the entropy of these states leads to a local strengthening of the continuity bound of the von Neumann entropy, i.e., the Audenaert-Fannes inequality. Our bound is local in the sense that it depends on the spectrum of $σ$. The states $ρ^*_\varepsilon(σ)$ and $ρ_{*,\varepsilon}(σ)$ depend only on the geometry of the $\var…
A well-known result due to Fannes is a certain upper bound on the modulus of continuity of the von Neumann entropy with respect to the trace distance between density matrices; this distance is the maximum probability of distinguishing between the corresponding quantum states. Much more recently, Audenaert obtained an exact expression of this modulus of continuity. In the present note, Audenaert's result is extended to a broad class of entropy functions indexed by arbitrary continuous convex functions $f$ in place of the Shannon--von Neumann function $x\mapsto x\log_2x$. The proof is based on t…
We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entangleme…
In this short note, I show how a recent result of Alhejji and Smith [arXiv:1909.00787] regarding an optimal uniform continuity bound for classical conditional entropy leads to an optimal uniform continuity bound for quantum conditional entropy of classical--quantum states. The bound is optimal in the sense that there always exists a pair of classical--quantum states saturating the bound, and so no further improvements are possible. An immediate application is a uniform continuity bound for entanglement of formation that improves upon the one previously given by Winter in [arXiv:1507.07775]. Tw…
The Renyi entropy plays an essential role in quantum information theory. We study the continuity estimation of the Renyi entropy. An inequality relating the Renyi entropy difference of two quantum states to their trace norm distance is derived. This inequality is shown to be tight in the sense that equality can be attained for every prescribed value of the trace norm distance. It includes the sharp Fannes inequality for von Neumann entropy as a special case.
We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One re…
The quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory. We have derived many such inequalities in our previous work (K.M.R. Audenaert and J. Eisert, J. Math. Phys. 46, 102104 (2005)). The present paper is a follow-up on this, and provides sharp upper bounds on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned. The results obtained here are more general than …
We show that an entanglement measure called relative entropy of entanglement satisfies a strong continuity condition. If two states are close to each other then so are their entanglements per particle pair in this measure. It follows in particular, that the measure is appropriate for the description of entanglement manipulations in the limit of an infinite number of pairs of particles.
We prove a simple uniform continuity bound for the sandwiched Rényi conditional entropy for $α\in [1/2, 1) \cup (1, \infty]$, which is independent of the dimension of the conditioning system.
We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $α$-Rényi entropy, $S_α$, for $α\in (0,1)$. We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood ($\varepsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\varepsilon$-ball. This minimal state …
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 prove a tight uniform continuity bound for Arimoto's version of the conditional $α$-Rényi entropy, for the range $α\in [0, 1)$. This definition of the conditional Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with sid…
We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows us to expand the scope of possible applications of the method. We prove two general dominated convergence theorems and the theorem about preserving local continuity under convex mixtures. By using these theorems we obtain several convergence conditions for the quantum relative entropy and for the mutual informati…
We study the inverse problem of inferring the state of a finite-level quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximum-entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support, while this map restricts to a smooth parametrization of a Gibbsian family of fully supported states. Here we prove for arbitrary ranking functions that the inference is continuous up to boundary points. This foll…
We describe the class (semigroup) of quantum channels mapping states with finite entropy into states with finite entropy. We show, in particular, that this class is naturally decomposed into three convex subclasses, two of them are closed under concatenations and tensor products. We obtain asymptotically tight universal continuity bounds for the output entropy of two types of quantum channels: channels with finite output entropy and energy-constrained channels preserving finiteness of the entropy.
Phase-space features of the Wigner flow are examined so to provide a set of continuity equations that describe the flux of quantum information in the phase-space. The reported results suggest that the non-classicality profile of anharmonic (periodic) quantum systems can be consistently obtained in terms of the fluxes of probability, purity and von Neumann entropy. Extensions of the such phase-space quantifiers are also investigated in the context of the so-called Rényi entropy.
Continuity of Regularized Channel Rényi Divergences
Zyphra；QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（量子信息与计算倡议，计算与数据科学学院，香港大学）
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…
Fidelity and relative entropy are two significant quantities in quantum information theory. We study the quantum fidelity and relative entropy under unitary orbits. The maximal and minimal quantum fidelity and relative entropy between two unitary orbits are explicitly derived. The potential applications in quantum computation and information processing are discussed.
We prove a tight uniform continuity bound for the conditional Shannon entropy of discrete finitely supported random variables in terms of total variation distance.
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…
A useful kind of continuity of quantum states functions in asymptotic regime is so-called asymptotic continuity. In this paper we provide general tools for checking if a function possesses this property. First we prove equivalence of asymptotic continuity with so-called it robustness under admixture. This allows us to show that relative entropy distance from a convex set including maximally mixed state is asymptotically continuous. Subsequently, we consider it arrowing - a way of building a new function out of a given one. The procedure originates from constructions of intrinsic information an…
We introduce a notion of majorization flow, and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for entropic functions relevant in information theory. In particular, for the case of the alpha-Rényi entropy, whose connections to thermodynamics are discussed in this article, majorization flow yields a Lipschitz continuity bound for the case alpha > 1, thus resolving an open problem and providing a substantial improvement over previously known bounds.
In this paper we consider the $χ$-function (the Holevo capacity of constrained channel) and the convex closure of the output entropy for arbitrary infinite dimensional channel. It is shown that the $χ$-function of an arbitrary channel is a concave lower semicontinuous function on the whole state space, having continuous restriction to any set of continuity of the output entropy. The explicit representation for the convex closure of the output entropy is obtained and its properties are explored. It is shown that the convex closure of the output entropy coincides with the convex hull of the outp…
In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between majorization and the trace distance between quantum states. Namely, the majorization pre-order attains a minimum over $ε$-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance $ε$, we obtain the path of "majorization flow". This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which ex…
The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$σ$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results…
We axiomatically characterize the Tsallis entropy of a finite quantum system. In addition, we derive a continuity property of Tsallis entropy. This gives a generalization of the Fannes inequality.
After recalling definition, monotonicity, concavity, and continuity of a channel's entropy with respect to a state (finite dimensional cases only), I introduce the roof property, a convex analytic tool, and show its use in treating an example. Full proofs and more examples will appear elsewhere. The relation (a la Benatti) to accessible information is mentioned.
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.
An analogue of the mixing property of quantum entropy is derived for quantum relative entropy.It is applied to the final state of ideal measurement and to the spectral form of the second density operator. Three cases of states on a directed straight line of relative entropy are discussed.
Quantum relative entropy $D(ρ\|σ)\defeq\Tr ρ(\log ρ- \log σ)$ plays an important role in quantum information and related fields. However, there are many quantum analogues of relative entropy. In this paper, we characterize these analogues from information geometrical viewpoint. We also consider the naturalness of quantum relative entropy among these analogues.
Because of the difficulty in getting the analytic formula of relative entropy of entanglement, it becomes troublesome to study the monogamy relations of relative entropy of entanglement for three-qubit pure states. However, we find that all generalized W states have the monogamous property for relative entropy of entanglement by calculating the relative entropy of entanglement for the reduced states of the generalized W states in three-qubit systems.
In this article we study relationship between three measures of distinguishability of quantum states called as divergence, relative entropy and the substate property.
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.
The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party to a smaller number $m$ of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy. Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with …
In the previous paper \cite{FYK}, we mainly studied the mathematical properties of Tsallis relative entropy with respect to the density operators. As an application of it, we adopt a parametrically extended entanglement-measure due to Tsallis relative entropy in order to measure the degree of entanglement. Then we study its properies with respect to the parameter $q$ appearing in Tsallis entropies. In addition, the relation between it and the relative entropy of entanglement is studied.
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…
