Local unambiguous discrimination of symmetric ternary states

Abstract

We investigate unambiguous discrimination between given quantum states with a sequential measurement, which is restricted to local measurements and one-way classical communication. If the given states are binary or those each of whose individual systems is two-dimensional, then it is in some cases known whether a sequential measurement achieves a globally optimal unambiguous measurement. In contrast, for more than two states each of whose individual systems is more than two-dimensional, the problem becomes extremely complicated. This paper focuses on symmetric ternary pure states each of whose individual systems is three-dimensional, which include phase shift keyed (PSK) optical coherent states and a lifted version of "double trine" states. We provide a necessary and sufficient condition for an optimal sequential measurement to be globally optimal for the bipartite case. A sufficient condition of global optimality for multipartite states is also presented. One can easily judge whether these conditions hold for given states. Some examples are given, which demonstrate that, despite the restriction to local measurements and one-way classical communication, a sequential measurement can be globally optimal in quite a few cases.

Online identification of symmetric pure states

“…Rather surprisingly, the performance of online sequential strategies and their comparison with the global optimal values for zero-error identification have hardly been explored. Although even the simplest case of three symmetric states (TSS) is quite a big challenge, as discussed in [42], the constraints imposed by the zero-error requirement provide more chances to obtain analytical results. Here we will mainly focus our attention in the TSS case, and also address some straightforward generalizations.…”

“…For complex values of the overlap and for n = 2 copies, it is possible to find a region with a "Mitsubishi-logo" shape where a sequence of two local measurements yields the same success probability as the global measurement [42]. However, the strategy proposed in [42] is not online, since it requires knowing the horizon.…”

“…For complex values of the overlap and for n = 2 copies, it is possible to find a region with a "Mitsubishi-logo" shape where a sequence of two local measurements yields the same success probability as the global measurement [42]. However, the strategy proposed in [42] is not online, since it requires knowing the horizon. Indeed, it sacrifices optimality in the first step (by not putting the maximum possible weight on the POVM elements F 0,1,2 ) in order to match global performance at the second step.…”

“…When trying to discriminate between two states, it is known that online strategies attain optimal global performance, regardless of whether one considers minimum error discrimination [22] or unambiguous identification [19,21], the two usual approaches to state discrimination. Discriminating more than two hypotheses is a much harder problem: optimal protocols are only known for certain special cases [4,[34][35][36][37][38][39], and results on local distinguishability are even more scarce [24,25,[40][41][42]. Here we tackle the problem of unambiguous (zero error) identification of three symmetric pure quantum states with constant (but arbitrary) overlap c when n copies are provided, characterizing for which parameter ranges do online schemes attain global performance.…”

Online identification of symmetric pure states

“…When trying to discriminate between two states, it is known that online strategies attain optimal global performance, regardless of whether one considers minimum error discrimination [22] or unambiguous identification [19,21], the two usual approaches to state discrimination. Discriminating more than two hypotheses is a much harder problem: optimal protocols are only known for certain special cases [34,4,35,36,37,38,39], and results on local distinguishability are even more scarce [24,25,40,41,42]. Here we tackle the problem of unambiguous (zero error) identification of three symmetric pure quantum states with constant (but arbitrary) overlap c when n copies are provided, characterizing for which parameter ranges do online schemes attain global performance.…”