Matteo Rosati (UAdB): Beyond the swap test: efficient estimation of distances between quantum states
Channel: Centrum Fizyki Teoretycznej PAN
Date: 30/04/2021

Matteo Rosati (Universitat Autonoma de Barcelona).
Beyond the swap test: efficient estimation of distances between quantum states
We study the problem of estimating distance measures between unknown quantum states, given M and N copies of each type. In the case of pure states, any distance measure can be expressed in terms of the overlap between the states. Overlap estimation is a fundamental primitive in quantum information processing that is commonly accomplished from the outcomes of N swap-tests, a joint measurement on one copy of each type whose outcome probability is a linear function of the overlap. We show that an optimal estimate can be obtained by allowing for general collective measurements on all copies, in particular using weak Schur sampling. We derive the statistics of the optimal measurement and compute the optimal mean square error in the asymptotic pointwise and finite Bayesian estimation settings. Besides, we consider two strategies relying on the estimation of one or both the states, and show that, although they are suboptimal, they outperform the swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. We also show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states. In the case of mixed states, distance measures cannot be expressed in terms of a single quantity. With the swap test, one can estimate the Hilbert-Schmidt distance and the overlap. We show that simultaneous weak Schur sampling can instead estimate the spectra of each of the states and of their convex combination, with weight given by M/(M+N) and N/(M+N). This estimate is much more informative than the outcome of swap tests, as it can be used to estimate also the Holevo quantity of an ensemble constructed with the states, which is also a proper measure of distinguishability. We discuss how to obtain bounds on the trace distance using these estimates, and apply these bounds to distance testing, improving sample complexity with respect to the swap test. 
CTP Quantum Information Days (online), 22-24 February 2021.
Center for Theoretical Physics PAS (Warsaw, Poland)