Measuring distances in Hilbert space by many-particle interference

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.

Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance

“…Some examples of widely used distance measures are the trace distance (d tr ), Hilbert-Schmidt distance (d HS ), Bures distance (d B ), and Hellinger distance (d H ). For given two density matrices ρ 1 , ρ 2 , these are defined respectively as [34][35][36][53][54][55][56][57][58][59][60] d tr = tr |ρ 1 − ρ 2 |,…”

Experimental Measurement of the Hilbert-Schmidt Distance between Two-Qubit States as a Means for Reducing the Complexity of Machine Learning

“…In quantum communications theory one can quantify the accuracy of a signal transmission by measuring the distance in Hilbert space between the transmitted and received states. The most prominent distance measures include Uhlmann-Jozsa fidelity (Bures metrics), trace distance, and Hilbert-Schmidt distance (HSD) (for overviews see, e.g., [18][19][20]).…”

“…This is because the HSD can be expressed by first-order overlaps O(ρ i ,ρ j ) as described in Ref. [20,27,28]…”

“…[29][30][31]. Each overlap or other functions of overlaps can be measured directly by utilizing multi-particle interactions between copies of the investigated states [20,27,30,[32][33][34][35][36][37]. In contrast by applying full quantum tomography (see, e.g., [ can become even grater in case of larger values of D. The number of the required observables to measure in each of the 3 steps for a multi-qubit overlap depends linearly on the number of qubits forming the density matrix, i.e., n = log 2 (D) as O(n) = O[log 2 ( √ N + 1)] (see, e.g., Refs.…”

“…[29][30][31]. Each overlap or other functions of overlaps can be measured directly by utilizing multi-particle interactions between copies of the investigated states [20,27,30,[32][33][34][35][36][37]. In contrast by applying full quantum tomography (see, e.g., [24]) 4 qubits.…”