Matteo Rosati

We study the problem of estimating distance measures between unknown quantum states, given M and N copies of each type. For 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 a more precise 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 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 measure cannot be expressed in terms of a single distance. With the swap test, one can estimate the Hilbert-Schmidt distance and the overlap, but it is not possible to obtain other very relevant distance measures. 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 and it can be used to estimate a wider class of properties and distinguishability measures. In particular we show how to measure the Holevo quantity of a an ensemble constructed with the states. We discuss how to get bounds on the trace distance using these estimates, and apply these bounds to closeness and remoteness testing, improving sample complexity with respect to the swap test.

Bit commitment is a fundamental cryptographic primitive and a cornerstone for numerous two- party cryptographic protocols, including zero-knowledge proofs. However, it has been proven that unconditionally secure bit commitment, both classical and quantum, is impossible. In this work, we demonstrate that imposing a restriction on the committing party to perform only separable operations enables secure quantum bit commitment schemes. Specifically, we prove that in any perfectly hiding bit commitment protocol, an honestly-committing party limited to separable operations will be detected with high probability if they attempt to alter their commitment. To illustrate our findings, we present an example protocol.

Quantum metrology and cryptography can be combined in a distributed and/or remote sensing setting, where distant end-users with limited quantum capabilities can employ quantum states, transmitted by a quantum-powerful provider via a quantum network, to perform quantum-enhanced parameter estimation in a private fashion. Previous works on the subject have been limited by restricted assumptions on the capabilities of a potential eavesdropper and the use of abort-based protocols that prevent a simple practical realization. Here we introduce, theoretically analyze, and experimentally demonstrate single- and two-way protocols for distributed sensing combining several unique and desirable features: (i) a safety-threshold mechanism that allows the protocol to proceed in low-noise cases and quantifying the potential tampering with respect to the ideal estimation procedure, effectively paving the way for wide-spread practical realizations; (ii) equivalence of entanglement-based and mutually-unbiased-bases-based formulations; (iii) robustness against collective attacks via a LOCC-de-Finetti theorem, for the first time to our knowledge. Finally, we demonstrate our protocols in a photonic-based implementation, observing that the possibility of guaranteeing a safety threshold may come at a significant price in terms of the estimation bias, potentially overestimating the effect of tampering in practical settings.