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Physical Review A

Optimal discrimination of geometrically uniform qubit and qutrit states

Xin Lü

Published Feb 2, 2023• 5 citations
Publisher

Abstract

We consider discrimination of geometrically uniform states, and analytically solve this problem for qubits. For qutrits, we obtain the exact solution to the optimal measurement when the defining unitary matrix for the geometrically uniform states is degenerate. We also show that if the unitary is nondegenerate then the optimal measurement for discriminating geometrically uniform qubits or qutrits can always be expressed as a rank-1 operator, which converts the original discrimination problem into an optimization of a sum of trigonometric functions with two real variables in the case of qutrits. Additionally, a geometrical interpretation of the optimal measurement for discriminating geometrically uniform qubits is given via the Bloch sphere representation.

DOI
10.1103/physreva.107.022401

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Abstract We investigate minimum-error (ME) discrimination for mixed qubit states using a geometric approach. By analyzing positive operator-valued measure (POVM) solutions and introducing Lagrange operator Γ\Gamma Γ Γ , we develop a four-step structured instruction to find Γ\Gamma Γ Γ for N mixed qubit states. Our method covers optimal solutions for two, three, and four mixed qubit states, including a novel result for four qubit states. We introduce geometric-based POVM classes and non-decomposable subsets for constructing optimal solutions, enabling us to find all possible answers for the general problem of minimum-error discrimination for N mixed qubit states with arbitrary a priori probabilities.