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Workshop on Physics and Computation (PhysComp)

The Two Extremes Of Information In Quantum Mechanics

William K. Wootters

Published Aug 24, 2005• 1 citation
Publisher

Abstract

In the problem of estimating the amount of information one can extract from a quantum system, the entropy S of the system is known to play a special role: it is an upper bound on the accessible information. This paper calls attention to another property of quantum systems, called Q, which appears to play an analogous role in defining a lower bound on the accessible information. Some parallels between S and Q are noted.

DOI
10.1109/phycmp.1992.615526

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In the problem of estimating the amount of information one can extract from a quantum system, the entropy S of the system is known to play a special role: it is an upper bound on the accessible information. This paper calls attention to another property of quantum systems, called Q, which appears to play an analogous role in defining a lower bound on the accessible information. Some parallels between S and Q are noted.
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.
We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.
Quantum entropy and skew information play important roles in quantum information science. They are defined by the trace of the positive operators so that the trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this paper, we study some properties for information quantities in quantum system through trace inequalities. Especially, we give upper bounds and lower bounds of Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum system. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.