Violating the modified Helstrom bound with nonprojective measurements

Abstract

We consider the discrimination of two pure quantum states with three allowed outcomes: a correct guess, an incorrect guess, and a non-guess. To find an optimum measurement procedure, we define a tunable cost that penalizes the incorrect guess and non-guess outcomes. Minimizing this cost over all projective measurements produces a rigorous cost bound that includes the usual Helstrom discrimination bound as a special case. We then show that nonprojective measurements can outperform this modified Helstrom bound for certain choices of cost function. The Ivanovic-Dieks-Peres unambiguous state discrimination protocol is recovered as a special case of this improvement. Notably, while the cost advantage of the latter protocol is destroyed with the introduction of any amount of experimental noise, other choices of cost function have optima for which nonprojective measurements robustly show an appreciable, and thus experimentally measurable, cost advantage. Such an experiment would be an unambiguous demonstration of a benefit from nonprojective measurements.A fundamental consequence of quantum mechanics is the inability to perfectly distinguish between two nonorthogonal quantum states. Any attempt to guess which state is which after making a measurement will have an unavoidable probability of error that is bounded from below, as shown originally by Helstrom [1, 2] and in related work by Holevo [3]. This lower bound, known as the Helstrom bound (HB), grows with the overlap of the two states being discriminated.The HB can be circumvented, however, if a third option is added to the guessing game. Ivanovic, Dieks, and Peres showed that if one can also decline to guess after a measurement, then it is possible to reduce the probability of error to zero while still retaining a significant chance of guessing correctly [4][5][6]. Intriguingly, to maximize the correct guess probability in such "Unambiguous State Discrimination" (USD), it is not sufficient to use standard projective measurements; instead, one must use generalized (nonprojective) measurements [7,8].This advantage of nonprojective measurements in state discrimination is so surprising that it has become a featured example in modern quantum information textbooks (e.g., [9,10]), and has led to considerable research, both in theory [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] and experiment [28][29][30][31][32][33][34][35][36][37] (reviewed, e.g., in [38,39]). Most of this work has focused on the extreme cases of zero declining (as with the HB) or zero error (as with USD), with fewer papers considering intermediate cases that minimize the declining probability given a fixed nonzero error rate [23][24][25][26][27]. Moreover, to our knowledge all but one [27] of these few works have neglected the effect that experimental imperfections will have upon the accessible minima. We are thus not aware of any paper that discusses a rigorous bound suitable to experimentally demonstrate that nonprojective measurements have a definitive advantage over projective...

Cost of postselection in decision theory

“…In this limit there is an alternative protocol which is equally optimal: always report "reject". Finally we show how our results can be connected to previous approaches where there is a tradeoff between the rejection probablity and the error probablity [25][26][27][28][29][30][31]. Our analysis adheres to the desiderata suggested in Ref.…”

“…Other studies of inconclusive state discrimination [27,28,30,31] concern themselves with the probabilities of error and reporting the "reject" result. This avoids the question of what to do given the outcome of some measurement.…”

“…So far we have focused on the decision function and the loss function. In this section we connect our approach to the previous approaches which focus on tradeoffs between reject and error probabilities [31], and rejection thresholds [30].…”

“…The USD measurement is optimized in the sense that it has minimal probability of reporting the inconclusive result "reject". Prior work on inconclusive state discrimination has focused on exploring and optimizing schemes which interpolate between minimum error probability and minimum inconclusive result probability [25][26][27][28][29][30][31]. Typically in USD and its generalizations [32] there is no explicit penalty for reporting "reject".…”

Optimal measurements for symmetric quantum states with applications to optical communication

“…In [20], the YKL conditions for the optimal measurement was were written in terms of equalities (rather than involving an inequality, which makes it harder to check). Recently, in [21], a generalization of the optimal measurement to non-projective measurements has been considered.…”