Multicopy programmable discriminators between two unknown qudit states with group-theoretic approach

Abstract

The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of $U(n)$ group, we construct the Jordan basis to derive the analytical results for both the optimal unambiguous discrimination and minimum-error discrimination. The POVM operators for unambiguous discrimination and orthogonal measurement operators for minimum-error discrimination are obtained. We find that the optimal failure probability and minimum-error probability for the discrimination between the mean input mixd states are dependent on the dimension of the unknown qudit states. We applied the approach to generalize the results of He and Bergou (2007) from qubit to qudit case, and we further solve the problem of programmable dicriminators with arbitrary copies of unknown states in both program and data systems.

The universal unambiguous discriminator between two unknown qudit states

“…Later, the cases where a certain number of copies in the registers are provided were discussed in refs. [23][24][25][26][27][28][29][30][31][32], which is the generalization of unambiguous discrimination discussed in refs. [22].…”

“…With the group-theoretic approach, the optimal probabilities of both the minimum-error discrimination and unambiguous discrimination have also been obtained [30], and the optimal success probability was further derived for the unambiguous discrimination [31], while only the minimum-error discrimination was discussed in ref. [32].…”

A generalized unknown qubit discriminator

“…[11], by means of the representation theory of U (n) group, Hayashi et al obtain the optimal UD solution for the averages of two uniformly distributed unknown qudit systems prepared with n copies of reference plus one copy of data, while in [12] we apply the Jordan basis method to the corresponding problem of two unknown qubit systems prepared with n copies of reference plus m copies of data which are prepared with arbitrary a prior probabilities. Other recent progress in unknown quantum state discriminations and their physical realization can be found in [14][15][16][17][18][19][20][21][22][23]. Many of these studies are about the optimal UD for the averages of the systems.…”