A bound on the mutual information, and properties of entropy reduction, for quantum channels with inefficient measurements

Abstract

The Holevo bound is a bound on the mutual information for a given quantum encoding. In 1996 Schumacher, Westmoreland and Wootters [Schumacher, Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] derived a bound which reduces to the Holevo bound for complete measurements, but which is tighter for incomplete measurements. The most general quantum operations may be both incomplete and inefficient. Here we show that the bound derived by SWW can be further extended to obtain one which is yet again tighter for inefficient measurements. This allows us in addition to obtain a generalization of a bound derived by Hall, and to show that the average reduction in the von Neumann entropy during a quantum operation is concave in the initial state, for all quantum operations. This is a quantum version of the concavity of the mutual information. We also show that both this average entropy reduction and the mutual information for pure state ensembles, are Schur-concave for unitarily covariant measurements; that is, for these measurements, information gain increases with initial uncertainty.

Integral quantum fluctuation theorems under measurement and feedback control

“…The last equality indicates that I gives the difference in the von Neumann entropy between the premeasurement and postmeasurement states, which is the information gain [30,31] with inefficient measurements [38]. In general, I can take on negative values for the following reason.…”

“…Then, the measurement operator M S k,a reduces to M S k , which is called an efficient measurement [31,38]. In this case, ρ S (k) = ρ S (k,a) holds and the information loss is zero: I loss = 0, which means that there is no loss of information in estimating the postmeasurement state.…”

“…We note that inequality (29) also holds without the assumption (22) which can be shown in a similar manner as in Eq. (38).…”

Entropy reduction of quantum measurements

“…By noting that ̺ = i∈I π i (ρ)̺ i we conclude from Lemma 6 in the Appendix that ̺ i = ̺ for all i. Since ̺ i = (π i (ρ)) −1 j,k λ j √ λ k [TrV i |j k|V * i ]|j k| and ̺ = k λ k |k k|, we obtain (11).…”

“…Roughly speaking, the entropy reduction characterizes a degree of purifying ("gain in purity") of a state in a measurement process. More details about the information sense of this value can be found in [11,15,21].…”