The Holevo bound and Landauer’s principle

Landauer’s principle [1] states that the erasure of classical information leads to an increase in the entropy of the environment by at least the same amount. The importance of this principle is the insight that it is not the act of obtaining information that necessarily generates heat and therefore entropy, but the erasure of information. This profound insight has led to the resolution of the problem of Maxwell’s demon by Bennett [2]. Here the missing entropy is generated when the demon’s memory is erased. Clearly Landauer’s principle provides a bridge that links classical information theory to thermodynamics. Recently, however, it has been shown that one can also use Landauer’s principle to connect the entropy of erasure and the efficiency of entanglement purification [3]. Here we demonstrate that there is also a connection between Landauer’s principle and the Holevo bound which limits the classical capacity of a quantum channel. In classical communication via quantum states, we encode classical signals in possibly mixed, non-orthogonal quantum states and send those to a receiver. The receiver then has the task to deduce from those quantum states the original classical message. The maximal information that the receiver can obtain is limited by the Holevo bound [4]. It has also been shown that this bound can be achieved asymptotically [5]. The purpose of this paper is to understand the Holevo bound in terms of Landauer’s principle. This approach is more intuitive than the unavoidably very technical, rigorous mathematical proofs. The hope is that this new approach may stimulate ideas that may lead to a better understanding of the more intricate problem of the quantum capacity of a quantum channel, a problem which is not yet fully understood (see however [6]). This paper is organized as follows. In the next section we will present a general method for the erasure of information which was introduced by Lubkin [7] and later generalized by Vedral [3] to non-commuting variables. In section III we will first briefly introduce the Holevo bound and then proceed to show how the Holevo bound can be obtained from Landauer’s principle. II. HOW TO ERASURE INFORMATION?