Strong Subadditivity Research Articles

On mutual information in multipartite quantum states and equality in strong subadditivity of entropy

The challenge of equality in the strong subadditivity inequality of entropy is approached via a general additivity of correlation information in terms of nonoverlapping clusters of subsystems in multipartite states (density operators). A family of tripartite states satisfying equality is derived.

Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so–called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information.

Strong subadditivity inequality for quantum entropies and four-particle entanglement

The strong subadditivity inequality for a three-particle composite system is an important inequality in quantum information theory which can be studied via a four-particle entangled state. We use two three-level atoms in $\ensuremath{\Lambda}$ configuration interacting with a two-mode cavity and the Raman adiabatic passage technique for the production of the four-particle entangled state. Using this four-particle entanglement, we study various aspects of the strong subadditivity inequality.

The inequalities of quantum information theory

Let /spl rho/ denote the density matrix of a quantum state having n parts 1, ..., n. For I/spl sube/N={1, ..., n}, let /spl rho//sub I/=Tr/sub N/spl bsol/I/(/spl rho/) denote the density matrix of the state comprising those parts i such that i/spl isin/I, and let S(/spl rho//sub I/) denote the von Neumann (1927) entropy of the state /spl rho//sub I/. The collection of /spl nu/=2/sup n/ numbers {S(/spl rho//sub I/)}/sub I/spl sube/N/ may be regarded as a point, called the allocation of entropy for /spl rho/, in the vector space R/sup /spl nu//. Let A/sub n/ denote the set of points in R/sup /spl nu// that are allocations of entropy for n-part quantum states. We show that A~/sub n/~ (the topological closure of A/sub n/) is a closed convex cone in R/sup /spl nu//. This implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies. Lieb and Ruskai (1973) have established a number of inequalities for multipartite quantum states (strong subadditivity and weak monotonicity). We give a finite set of instances of these inequalities that is complete (in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations) and independent (in the sense that none of them can be deduced from the others by taking positive linear combinations). Let B/sub n/ denote the polyhedral cone in R/sup /spl nu// determined by these inequalities. We show that A~/sub n/~=B/sub n/ for n/spl les/3. The status of this equality is open for n/spl ges/4. We also consider a symmetric version of this situation, in which S(/spl rho//sub I/) depends on I only through the number i=/spl ne/I of indexes in I and can thus be denoted S(/spl rho//sub i/). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(/spl rho//sub i/)}/sub 0/spl les/i/spl les/n/ in R/sup n+1/.

Partial traces and entropy inequalities

The partial trace operation and the strong subadditivity property of entropy in quantum mechanics are explained in linear algebra terms.

MONOTONICITY OF QUANTUM RELATIVE ENTROPY REVISITED

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden–Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly indicated.

Inequalities for quantum entropy: A review with conditions for equality

This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), 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 Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. 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 article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.

Monotonicity with volume of entropy and of mean entropy for translationally invariant systems as consequences of strong subadditivity

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 limiting mean entropy and related questions. In particular, we prove that as the sizes of boxes in Z^n or R^n increase in the sense of set inclusion, (1) their mean entropy decreases monotonically and (2) their entropy increases monotonically. Our proof of (2) uses the notion of m-point entropies which we introduce and which generalize the notion of index of correlation (see e.g. R. Horodecki, Phys. Lett. A 187 p145 1994). We mention a number of further results and questions concerning monotonicity of mean entropy for more general shapes than boxes and for more general translationally invariant (/homogeneous) lattices and spaces than Z^n or R^n.

Limitation on the amount of accessible information in a quantum channel.

We prove a new result limiting the amount of accessible information in a quantum channel. This generalizes Kholevo's theorem and implies it as a simple corollary. Our proof uses the strong subadditivity of the von Neumann entropy functional $S(\ensuremath{\rho})$ and a specific physical analysis of the measurement process. The result presented here has application in information obtained from ``weak'' measurements, such as those sometimes considered in quantum cryptography.

BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

On certain properties of the relative entropy of states of operator algebras

The aim of the present paper is to establish certain properties of the relative entropy functional in the general context of C*-algebras. Most of these properties have been observed under more restrictive conditions in earlier papers. (The celebrated strong subadditivity of the entropy gives such an example). The general approach is applied to prove the existence of the mean relative entropy for a noncommutative stationary Markov chain with conditional expectations

Irreversibility, reduction, and entropy increase in quantum measurements

Using the strong subadditivity property of entropy, it is shown that the interaction of the measuring device with the environment brings about the reduction process characteristic of a quantum measurement. This entails an entropy increase which has an inviolable lower limit. This limit is found and explicitly calculated for illustrative examples.

A counterexample to the strong subadditivity of extremal plurisubharmonic functions

A simple example is given which shows that one way have $$h_E (z^0 ) + h_F (z^0 ) > h_{E \cup F} (z^0 ) + h_{E \cap F} (z^0 )$$

General properties of entropy

It is rather paradoxical that, although entropy is one of the most important quantities in physics, its main properties are rarely listed in the usual textbooks on statistical mechanics. In this paper we try to fill this gap by discussing these properties, as, for instance, invariance, additivity, concavity, subadditivity, strong subadditivity, continuity, etc., in detail, with reference to their implications in statistical mechanics. In addition, we consider related concepts such as relative entropy, skew entropy, dynamical entropy, etc. Taking into account that statistical mechanics deals with large, essentially infinite systems, we finally will get a glimpse of systems with infinitely many degrees of freedom.

Completely positive maps and entropy inequalities

It is proved that the relative entropy for a quantum system is non-increasing under a trace-preserving completely positive map. The proof is based on the strong sub-additivity property of the quantum-mechanical entropy.

Expectations and entropy inequalities for finite quantum systems

We prove that the relative entropy is decreasing under a trace-preserving expectation inB(K1), and we show the connection between this theorem and the strong subadditivity of the entropy. It is also proved that a linear, positive, trace-preserving map Φ ofB(K) into itself such that ‖Φ‖≦1 decreases the value of any convex trace function.

Proof of the strong subadditivity of quantum-mechanical entropy

We prove several theorems about quantum-mechanical entropy, in particular, that it is strongly subadditive.

Convex trace functions and the Wigner-Yanase-Dyson conjecture

Several convex mappings of linear operators on a Hilbert space into the real numbers are derived, an example being A ⊸ − Tr exp(L + In A) . Some of these have applications to physics, specifically to the Wigner-Yanase-Dyson conjecture which is proved here and to the strong subadditivity of quantum mechanical entropy which will be proved elsewhere.

A Fundamental Property of Quantum-Mechanical Entropy

We have proved the strong subadditivity of quantum-mechanical entropy and the Wigner-Yanase-Dyson conjecture.

Capacities of Borelian Sets and the Continuity of Potentials

One of the most important problems in the potential theory is the one of capacitability, that is, whether the inner capacity of an arbitrary borelian subset B is equal to the outer capacity of B. As for the capacities induced by the Newtonian potentials and other classical potentials, Choquet [5] has shown that every borelian and, more generally, every analytic set are capacitable. He goes on as follows : first he shows that, for the Newtonian capacity f, the inequality of strong subadditivity holds, that is,and then, using this inequality, he shows that the outer capacity f* has the analogous property to one of the outer measure, more precisely, if an increasing sequence {An} of arbitrary subsets converges to A, then f*(A) = limf*(An)- This property plays an important role in his proof.