John Calsamiglia, Universitat Autònoma de Barcelona
Abstract
I will present three problems in quantum statistical inference that all stem from the difference between local and collective measurements on multipartite states.
The first one is that of hiding a secret direction in a multipartite state with the promise that the parties cannot learn the direction if they perform local operations and classical communication, but can accurately retrieve it if they perform collective measurements.
The second is that of giving the probability of discriminating between a large number of copies of two given quantum states. This leads to the identification of the Quantum Chernoff bound, which gives a natural operational distinguishability measure between quantum states. The question remains whether the bound can be attained by LOCC measurments or, on the contrary, collective operations on an assymptotically large number of copies are necessary. Our results indicate that the gap between collective and LOCC operations persists even in the asymptotic limit: in order for LOCC and collective protocols to achieve the same accuracy, the former requires up to twice the number of copies of the latter.
Finally, if time permits, I will present a quantum learning machine which classifies an unknown qubit after being trained with a number of already classified qubits. We prove that quantum memory is not required for optimal performance (i.e. quantum correlations between the training qubits and the classified qubit are of no aid here). In this sense, we show how to physically separate the protocol in two steps: training and discrimination, where the information gathered in the former is stored in a (finite) classical memory to be used for the latter. This 2-step functioning implies also the ability of having time-like separation between training and testing stages. Moreover, as the memory needed is classical, such a machine can be reused for the classification of any number of additional qubits without retraining.