We consider a group covariant quantum state set with a group in which each element corresponds to a unitary or antiunitary operator. This type of quantum state set can express a broad class of quantum state sets, including geometrically uniform state sets and self-symmetric state sets. We derive that for any quantum measurement for a group-covariant state set (both with and without a certain fraction of inconclusive results), a group covariant quantum measurement exists with respect to the same group with the same performance, that is, the same probability of correct detection and the same rate of inconclusive results. We then show that a group covariant optimal measurement exists for any group-covariant state set. In some cases, we derive an optimal measurement for a group-covariant state set.
In general, for a bipartite quantum system
Cd⊗Cd and an integer
k such that
4≤k≤d,there are few necessary and sufficient conditions for local discrimination of sets of
k generalized Bell states (GBSs) and it is difficult to locally distinguish
k-GBS this http URL this paper, we consider the local discrimination of GBS sets and the purpose is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems,specifically, we show some necessary and sufficient conditions for local discrimination of GBS sets by which the local discrimination of GBS sets can be quickly determined.Firstly some sufficient conditions are given, these sufficient conditions are practical and effective.Fan
,s and Wang et al.
,s results (Phys Rev Lett 92:177905, 2004: Phys Rev A 99:022307, 2019) can be deduced as special cases of these conditions.Secondly in
C4⊗C4, a necessary and sufficient condition for local discrimination of GBS sets is provided,all locally indistinguishable 4-GBS sets are found,and then we can quickly determine the local discriminability of an arbitrary GBS this http URL
C5⊗C5, a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case
d=5 of the problem proposed by Wang et al. (Phys Rev A 99:022307, 2019).
"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".