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Abstract
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Minimum error discrimination (MED) and Unambiguous discrimination
(UD) are two common strategies for quantum state discrimination that can
be modified by imposing a finite error margin on the error probability. Error
margins 0 and 1 correspond to two common strategies. In this paper, for an
arbitrary error margin m, the discrimination problem of equiprobable quantum
symmetric states is analytically solved for four distinct cases. A generating set
of irreducible and reducible representations of a subgroup of a unitary group
are considered, separately, as unitary operators that produce one set of the
symmetric states. In the irreducible case, for N d mixed and pure qudit states,
one critical m which divides the parameter space into two domains is obtained.
The number of critical values m in the reducible case is two, for both N mixed
and pure qubit states. The reason for this difference between numbers of critical
values m is explained. The optimal set of measurements and corresponding
maximum success probability in fully analytical form are determined for all
values of the error margin. The relationship between the amount of error that is
imposed on error probability and geometrical situation of states with changes
in rank of element corresponding to inconclusive result is determined. The
behaviors of elements of measurement are explained geometrically in order
to decrease the error probability in each domain. Furthermore, the problem
of the discrimination with error margin among elements of two different sets
of symmetric quantum states is studied. The number of critical values m is equivalent to one set in both reducible and irreducible cases. In addition,
optimal measurements in each domain are obtained.
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