Group-theoretic approach for multi-copy programmable discriminators between two unknown qudit states

TL;DR

This paper introduces a group-theoretic method for designing optimal programmable discriminators for unknown qudit states, extending previous qubit results to higher dimensions and multiple copies.

Contribution

It develops a group-theoretic framework to analytically derive optimal discrimination strategies for unknown qudit states with multiple copies, generalizing prior qubit-based results.

Findings

Derived analytical formulas for optimal unambiguous discrimination

Obtained minimum-error discrimination operators and probabilities

Extended results from qubits to qudits with arbitrary copies

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 (Phys. Rev. A {\bf 75}, 032316 (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.