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Lower Bound on Locally Accessible Information: Local Subentropy

Apratim De, Sen U., Maciej Lewenstein

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

We find a universal lower bound on locally accessible information for arbitrary bipartite ensembles, when one of the parties is two-dimensional. In higher dimensions and in higher number of parties, the lower bound is on accessible information by separable operations. We evaluate this bound for several ensembles, for which the exact locally accessible information is not known. We show that for any given density matrix (of arbitrary number of parties and dimensions), there exists an ensemble, the so-called ``Scrooge ensemble'', which averages to the given density matrix and whose locally accessible information saturates the lower bound.

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We find a universal lower bound on locally accessible information for arbitrary bipartite ensembles, when one of the parties is two-dimensional. In higher dimensions and in higher number of parties, the lower bound is on accessible information by separable operations. We evaluate this bound for several ensembles, for which the exact locally accessible information is not known. We show that for any given density matrix (of arbitrary number of parties and dimensions), there exists an ensemble, the so-called ``Scrooge ensemble'', which averages to the given density matrix and whose locally accessible information saturates the lower bound.
We find a universal lower bound on locally accessible information for arbitrary bipartite quantum ensembles, when one of the parties is two dimensional. In higher dimensions and in a higher number of parties, the bound is on accessible information by separable operations. We show that for any given density matrix (of an arbitrary number of parties and dimensions), there exists an ensemble, which averages to the given density matrix and whose locally accessible information saturates the lower bound. Moreover, we give a general method to obtain bounds on the yield of singlets in distillation protocols that involves local distinguishing, by using lower and upper bounds on locally accessible information. We then illustrate it by using our lower bound, along with a previously obtained upper bound, on locally accessible information.
It has long been known that the von Neumann entropy S is an upper bound on the information one can extract from a quantum system in an unknown pure state. In this paper we define the ``subentropy'' Q, which we prove to be a lower bound on this information. Moreover, just as the von Neumann entropy is the best upper bound that depends only on the density matrix, we show that Q is the best lower bound that depends only on the density matrix. Other parallels between S and Q are also demonstrated.
Subentropy is an entropy-like quantity that arises in quantum information theory; for example, it provides a tight lower bound on the accessible information for pure state ensembles, dual to the von Neumann entropy upper bound in Holevo's theorem. Here we establish a series of properties of subentropy, paralleling the well-developed analogous theory for von Neumann entropy. Further, we show that subentropy is a lower bound for min-entropy. We introduce a notion of conditional subentropy and show that it can be used to provide an upper bound for the guessing probability of any classical-quantum state of two qubits; we conjecture that the bound applies also in higher dimensions. Finally, we give an operational interpretation of subentropy within classical information theory.