Geometric bloch vector solution to minimum-error discriminations of mixed qubit states

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 $$\Gamma $$ Γ , we develop a four-step structured instruction to find $$\Gamma $$ 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.

Protocol for Nonlinear State Discrimination in Rotating Condensate

“…We refer to this idealization as a single‐input discriminator to distinguish it from the more familiar minimum error and unambiguous discriminators based on linear CPTP channels. [ 44–47 ] Of course, any actual atomtronic realization is likely to suffer from all such errors and may fail to give the correct answer or return an answer at all. Although the theoretically achievable performance depends sensitively on the system and device details, and is beyond the scope of this work, we note that combining torsion with non‐CP dissipation is predicted to implement an autonomous discriminator, [ 36 ] whose control sequence and operation is (mostly) independent of $$| a \rangle$$ and $$| b \rangle$$.…”

“…As an application, we consider the problem of quantum state discrimination, [ 44–47 ] a basic task in quantum information science. In the two‐state variant considered here, a quantum state $$ | \psi \rangle \in \lbrace | a \rangle, | b \rangle \rbrace $$ is input to a processor, which knows the values of $$| a \rangle$$ and $$| b \rangle$$ ahead of time and tries to determine which was provided (with a bounded failure probability).…”