A Geometric Approach to Quantum State Separation

TL;DR

This paper introduces a geometric method for optimal probabilistic quantum state separation of two pure states, providing analytical solutions and a linear optics implementation, revealing a phase transition phenomenon.

Contribution

It formulates the quantum state separation problem geometrically and derives analytical solutions, including a phase transition analogy, with a practical optical implementation.

Findings

Analytical solutions for optimal state separation

Identification of a phase transition in the limit of perfect separation

Proposed linear optics implementation for quantum state separation

Abstract

Probabilistic quantum state transformations can be characterized by the degree of state separation they provide. This, in turn, sets limits on the success rate of these transformations. We consider optimum state separation of two known pure states in the general case where the known states have arbitrary a priori probabilities. The problem is formulated from a geometric perspective and shown to be equivalent to the problem of finding tangent curves within two families of conics that represent the unitarity constraints and the objective functions to be optimized, respectively. We present the corresponding analytical solutions in various forms. In the limit of perfect state separation, which is equivalent to unambiguous state discrimination, the solution exhibits a phenomenon analogous to a second order symmetry breaking phase transition. We also propose a linear optics implementation of separation which is based on the dual rail representation of qubits and single-photon multiport interferometry.