Robustness of Measurement, Discrimination Games, and Accessible Information

Abstract

We introduce a way of quantifying how informative a quantum measurement is, starting from a resourcetheoretic perspective. This quantifier, which we call the robustness of measurement, describes how much 'noise' must be added to a measurement before it becomes completely uninformative. We show that this geometric quantifier has operational significance in terms of the advantage the measurement provides over guessing at random in an suitably chosen state discrimination game. We further show that it is the single-shot generalisation of the accessible information of a certain quantum-to-classical channel. Using this insight, we also show that the recently-introduced robustness of coherence is the single-shot generalisation of the accessible information of an ensemble. Finally we discuss more generally the connection between robustness-based measures, discrimination problems and single-shot information theory.

Incompatibility robustness of quantum measurements: a unified framework

“…, which is a strictly feasible point of the dual for sufficiently large δ. Thus, the theorem applies and justifies the equality between the two problems in equation (23). Similar arguments apply to all pairs of primal-dual SDPs that we discuss in this work.…”

“…In fact, in appendix A, using an explicit counterexample, we show that none of the measures studied in this paper are concave. It is common to use the measure t 1 1 * * h = instead of * h because it is easy to prove its convexity, and it also has the appealing property that it vanishes for every A B JM , Î ( ) (a property referred to as faithfulness in [23]-also note that in [24], faithfulness, post-processing monotonicity and convexity were postulated as natural properties of any measure of incompatibility). Moreover, whenever * h is monotonic under pre-or post-processings, then so is t* (with opposite relation in the inequality defining monotonicity).…”

“…In this appendix we show that these relations can be strengthened, which leads to strict separations between some of the measures. In order to tighten the inequality between d h and jm h , we take the optimal point for the primal for d h in equation (23), and construct from it a feasible point for the primal for jm h in equation (45). Specifically, for a pair of measurements A B , ( ) we subtract some fraction of the original POVM element from the noise reaching the optimum in the primal for d h in equation (23), such that the remaining noise is jointly measurable and can thus serve as a feasible point for the primal for jm h in equation (45):…”

“…In order to tighten the inequality (B1) between d h and g h , we take the optimal point for the primal for d h in equation (23), and construct from it a feasible point for the primal for g h in equation (54). Specifically, we use…”

“…Then, this constitutes a feasible point for the dual given in equation (32) and we obtain Recall that we consider pairs of rank-one projective measurements A B , ( ) whose d i first outcomes live in the first d i dimensions of the total d f -dimensional space, and whose d f −d i remaining outcomes live in the remaining space. For this structure, an ansatz for the dual for d h given in equation (23) has been presented in equation (C14). However, this ansatz does not satisfy the additional constraints present in the dual for p h given in equation (40), namely, X tr a  x and Y tr b  u.…”