# Measuring distances in Hilbert space by many-particle interference

## TL;DR

This paper introduces an interferometric method using multiparticle interference to measure quantum state distances directly, potentially surpassing traditional quantum state tomography in efficiency.

## Contribution

It proposes a novel experimental approach to estimate quantum distances between unknown two-qubit states using multiparticle interference, avoiding full state reconstruction.

## Key findings

- Demonstrates measurement of Uhlmann-Jozsa fidelity, Hilbert-Schmidt, and trace distances
- Uses bounds of fidelity (superfidelity and subfidelity) for estimation
- Employs multiparticle interference between qubit copies

## Abstract

The measures of distances between points in a Hilbert space are one of the basic theoretical concepts used to characterize properties of a quantum system with respect to some etalon state. These are not only used in studying fidelity of signal transmission and basic quantum phenomena but also applied in measuring quantum correlations, and also in quantum machine learning. The values of quantum distance measures are very difficult to determine without completely reconstructing the state. Here we demonstrate an interferometric approach to measuring distances between quantum states that in some cases can outperform quantum state tomography. We propose a direct experimental method to estimate such distance measures between two unknown two-qubit mixed states as Uhlmann-Jozsa fidelity (or the Bures distance), the Hilbert-Schmidt distance, and the trace distance. The fidelity is estimated via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method is based on the multiparticle interactions (i.e., interference) between copies of the unknown pairs of qubits.