State learning from pairs of states

TL;DR

This paper demonstrates that having pairs of identical qubits allows for unique quantum state learning, overcoming the ambiguity in state decomposition, with high accuracy achievable through practical measurement schemes.

Contribution

The authors show that pairs of qubits enable unique state decomposition and propose a feasible experiment for state estimation using existing technology.

Findings

High accuracy in state and probability inference from a few thousand qubit pairs

Simulation confirms effectiveness of symmetric informationally complete measurement

Proposed experiment can be implemented with current quantum technology

Abstract

Suppose you receive a sequence of qubits where each qubit is guaranteed to be in one of two pure states, but you do not know what those states are. Your task is to determine the states. This can be viewed as a kind of quantum state learning -- or quantum state estimation. A problem is that, without more information, all that can be determined is the density matrix of the sequence and, in general, density matrices can be decomposed into pure states in many different ways. To solve the problem, additional information, either classical or quantum, is required. We show that if an additional copy of each qubit is supplied -- that is, one receives pairs of qubits, both in the same state, rather than single qubits -- the task can be accomplished. This is possible because the mixed two-qubit state has a unique decomposition into pure product states. For illustration, we simulate numerically the symmetric, informationally complete measurement of a sequence of qubit pairs and show that the unknown states and their respective probabilities of occurrence can be inferred from the data with high accuracy. Finally, we propose an experiment that employs a product measurement and can be realized with existing technology, and we demonstrate how the data tell us the states and their probabilities. We find that it is enough to detect a few thousand qubit pairs.