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arXiv:1201.1093

On a conjectured property of the von Neumann entropy valid in the commutative case

Максим Евгеньевич Широков

Published Jan 5, 2012• 0 citations
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Abstract

It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian $H$ of the system satisfies the condition $\Tr\exp(-cH)0$. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is continuous, consists of states with bounded energy with respect to a particular Hamiltonian $H$ satisfying the above condition. It is shown that the classical analog of this conjecture is valid (i.e. it is valid for the Shannon entropy). It is also shown that this conjecture holds for some types of subsets consisting of non-commuting states, but its validity for all subsets of quantum states remains an open question.

DOI
10.48550/arxiv.1201.1093

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It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian HHH of the system satisfies the condition \Trexp⁡(−cH)0\Tr\exp(-cH)0\Trexp(−cH)0. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is continuous, consists of states with bounded energy with respect to a particular Hamiltonian HHH satisfying the above condition. It is shown that the classical analog of this conjecture is valid (i.e. it is valid for the Shannon entropy). It is also shown that this conjecture holds for some types of subsets consisting of non-commuting states, but its validity for all subsets of quantum states remains an open question.
Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].
The universal method of proving continuity of the von Neumann entropy on subsets of positive trace-class operators are considered. It makes possible to re-derive the known conditions of continuity of the entropy in the more general forms and to obtain the several new conditions. This method is based on the special approximation of the von Neumann entropy by the increasing sequence of concave continuous unitary invariant functions. Existence of this approximation is a corollary of the general property of the set of quantum states as a convex topological space called the strong stability property and considered in the first part of the paper.
We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha −Reˊnyiand-Rényi and−Reˊnyiand\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.
The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new methods might be needed. Here, a variety of different and complementary approaches are taken. First, I look at global properties. It is known that the von Neumann entropy region -- just like its classical counterpart -- forms a convex cone. I describe the symmetries of this cone and highlight geometric similarities and differences to the classical entropy cone. In a different approach, I utilize the local geometric properties of extremal rays of a cone. I show that quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have a very simple structure. As the set of all quantum states is very complicated, I look at a simple subset called stabilizer states. I improve on previously known results by showing that under a technical condition on the local dimension, entropies of stabilizer states respect an additional class of information inequalities that is valid for random variables from linear codes. In a last approach I find a representation-theoretic formulation of the classical marginal problem simplifying the comparison with its quantum mechanical counterpart. This novel correspondence yields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf. Theory, 48(7):1992-1995, 2002) in purely combinatorial terms.
In this note we wish to examine three ideas. First we consider the case of W(x,t)=Sum over n an exp(-iEn t) Wn(x) ((1)) (for n=0,1 as an example) and note that the von Neuman entropy is 0 for such a state. This is pointed out in (1) together with the fact that there exists a von Neumann density operator exp(-H/T) (H=Hamiltonian operator) which yields the same result as Shannon’s entropy S using P(ei) if one uses dQ/T=Sum over n en dP(en)/T = dS. It seems then that there is a contradiction if the usual von Neumann entropy is 0 yet using a density operator = exp(-H/T) yields nonzero results. A resolution is suggested in (2) which deals with information theory and we see if it applies to ((1)). Secondly, we consider the Shannon’s form P(e)ln(P(e)) in more detail. In Part I and in previous notes we argued that ln ensures that S=S1+S2 if one has two systems. We also argued that dE= Sum over n en dP(en) = TdS if dWork =0. In this note, we argue that a stronger idea may be at work, namely the idea that Eave= Sum over n en P(en/T) and TS are essentially the same function except for a function of T. This approach may be used to show that ln is the function associated with entropy. Finally, we note as in Part I that when using either the von Neumann approach or Shannon’s entropy one must know which probabilities to use. For a system with energy levels en, P(n)=P(en/T) is an immediate candidate, but there may be more probabilities in the system related to a particle jumping from one ei to another. In such a case, if one uses these probabilities and then sums over some of them leaving only P(en/T), the form of entropy changes from that of Shannon’s form. (Originally, however, Shannon’s form is used before summing.)