Beyond the Swap Test: Optimal Estimation of Quantum State Overlap.

We study the estimation of the overlap between two unknown pure quantum states of a finite-dimensional system, given M and N copies of each type. This 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 squared overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies. 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 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. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.

Improved Sensing with a Single Qubit.

We consider quantum metrology with arbitrary prior knowledge of the parameter. We demonstrate that a single sensing two-level system can act as a virtual multilevel system that offers increased sensitivity in a Bayesian single-shot metrology scenario, and that allows one to estimate (arbitrary) large parameter values by avoiding phase wraps. This is achieved by making use of additional degrees of freedom or auxiliary systems not participating in the sensing process. The joint system is manipulated by intermediate control operations in such a way that an effective Hamiltonian, with an arbitrary spectrum, is generated that mimics the spectrum of a multisystem interacting with the field. We show how to use additional internal degrees of freedom of a single trapped ion to achieve a high-sensitivity magnetic field sensor for fields with arbitrary prior knowledge.

Deterministic superreplication of one-parameter unitary transformations.

We show that one can deterministically generate, out of N copies of an unknown unitary operation, up to N^{2} almost perfect copies. The result holds for all operations generated by a Hamiltonian with an unknown interaction strength. This generalizes a similar result in the context of phase-covariant cloning where, however, superreplication comes at the price of an exponentially reduced probability of success. We also show that multiple copies of unitary operations can be emulated by operations acting on a much smaller space, e.g., a magnetic field acting on a single n-level system allows one to emulate the action of the field on n^{2} qubits.