Optimal universal learning machines for quantum state discrimination

TL;DR

This paper investigates the optimal quantum machine learning approach for classifying qubits into two states with limited prior information, deriving strategies and finite-size corrections for various scenarios.

Contribution

It introduces a comprehensive framework for quantum state discrimination with partial prior knowledge and training data, including finite-size corrections and extensions to higher-dimensional systems.

Findings

Optimal classification strategies derived for different prior information scenarios.

Finite-size corrections to asymptotic limits obtained.

Extension of results to d-level quantum systems for pure states.

Abstract

We consider the problem of correctly classifying a given quantum two-level system (qubit) which is known to be in one of two equally probable quantum states. We assume that this task should be performed by a quantum machine which does not have at its disposal a complete classical description of the two template states, but can only have partial prior information about their level of purity and mutual overlap. Moreover, similarly to the classical supervised learning paradigm, we assume that the machine can be trained by $n$ qubits prepared in the first template state and by $n$ more qubits prepared in the second template state. In this situation we are interested in the optimal process which correctly classifies the input qubit with the largest probability allowed by quantum mechanics. The problem is studied in its full generality for a number of different prior information scenarios and for an arbitrary size $n$ of the training data. Finite size corrections around the asymptotic limit $n\rightarrow \infty$ are also derived. When the states are assumed to be pure, with known overlap, the problem is also solved in the case of d-level systems.