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 [H. P. Yuen et al., IEEE Trans. Info. Theory IT21, 125134 (1975)]. 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) previously, it was shown that the "pretty good measurement" attains the MPE. Even so, evaluating the actual probability of error (and other performance metrics) attained by the pretty good measurement 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 representationtheoretic 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 semidefinite programs—which is what is needed to solve the YuenKennedyLax conditions numerically for arbitrary state sets. We show how to evaluate MPE measurements for CGU states in some examples relevant to quantumlimited classical optical communication.
 Publication:

Physical Review A
 Pub Date:
 December 2015
 DOI:
 10.1103/PhysRevA.92.062333
 arXiv:
 arXiv:1507.04737
 Bibcode:
 2015PhRvA..92f2333K
 Keywords:

 03.67.Hk;
 03.65.Ta;
 03.65.Aa;
 Quantum communication;
 Foundations of quantum mechanics;
 measurement theory;
 Quantum Physics
 EPrint:
 11 pages, 3 figures