Efficient Optimal Minimum Error Discrimination of Symmetric Quantum States

TL;DR

This paper presents a new, more efficient method for optimal quantum state discrimination that leverages symmetry and semidefinite programming, enabling analytical solutions and improved performance in high-dimensional quantum systems.

Contribution

It introduces a novel dual formulation for symmetric quantum state discrimination, simplifying the optimization and enabling analytical solutions and reduced computational complexity.

Findings

Derived a dual problem formulation for symmetric quantum states

Provided closed-form analytical solutions for specific cases

Enabled scalable numerical solutions for high-dimensional systems

Abstract

This paper deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator $\rho_0$. It is well-known that the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator $\Pi_0$, and maximizing $\textrm{Tr}(\rho_0 \Pi_0)$. In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a semidefinite positive operator $X$ commuting with the symmetry operator and such that $X >= \rho_0$. The new formulation gives a deeper insight into the optimization problem and allows to obtain closed-form analytical solutions, as shown by a simple but not trivial explanatory example. Besides the theoretical interest, the result leads to semidefinite programming solutions of reduced complexity, allowing to extend the numerical performance evaluation to quantum communication systems modeled in Hilbert spaces of large dimension.