Optimal measurements for symmetric quantum states with applications to optical communication

Abstract

The minimum probability of error (MPE) measurement discriminates between a set of candidate quantum states with the minimum average error probability allowed by quantum mechanics. Conditions for a measurement to be MPE were derived by Yuen, Kennedy and Lax (YKL) [1]. MPE measurements have been found for states that form a single orbit under a group action, i.e., there is a transitive group action on the states in the set. For such state sets, termed geometrically uniform (GU) in [2], it was shown that the 'pretty good measurement' (PGM) attains the MPE. Even so, evaluating the actual probability of error (and other performance metrics) attained by the PGM on a GU set involves inverting large matrices, and is not easy in general. Our first contribution is a formula for the MPE and conditional probabilities of GU sets, using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). MPE measurements for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the MPE measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many common group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs-which is what is needed to solve the YKL conditions numerically for arbitrary state sets. We show how to evaluate MPE measurements for CGU states for some examples relevant to quantum-limited classical optical communication.arXiv:1507.04737v1 [quant-ph] 16 Jul 2015

Structure of minimum error discrimination for linearly independent states

“…This shows that the PGM of Q is the optimal POVM for MED of P. In particular, in the case of pure states we have a nice property which is proved in [1,2]. 1 One can define a PGM for an arbitrary ensemble of states Q = {q i , σ i } m i=1 using equation (11). This is also true when supp σ is strictly smaller than H. In such cases, we restrict the space to span {supp σ i } m i=1 to define σ −1/2 .…”

“…Using the definition of Z given in (20), and equations (5) and (11) we see that p i ρ i should satisfy the following equation…”

Certified answers for ordered quantum discrimination problems

“…The light blue block corresponds to Z ME r and the darker one to Z ∆ r . The small black boxes show the elements of the minor of interest to obtain the bound(9).…”

Optimal quantum state discrimination via nested binary measurements

“…Triggered by the observation that non-orthogonal quantum states cannot be perfectly discriminated, this subject has stimulated much work, both from a theoretical and practical point of view: the seminal works of Helstrom [17], Holevo [18] and Yuen et al [19] formalized the problem, obtaining a set of conditions for the optimal measurement operators, which in turn provide the optimal success probability, then solved it for sets of states symmetric under a unitary transformation; more recently, acknowledging that a general analytical solution is hard to find, research focused on finding a solution for sets with more general symmetries [20][21][22], computing explicitly the optimal measurements for the most interesting sets of states [23][24][25][26][27] and studying the implementation of such measurements with available technology (see for example [28][29][30][31][32][33][34][35][36][37][38][39][40] for the case of two optical coherent states, the most relevant for optical communication). Also, the problem of discrimination has been identified as a convex optimization one, arguing that it can be solved efficiently with numerical optimization methods [41].…”