Maximum Likelihood Estimator of Quantum Probabilities

Vikesh Siddhu

Using a Bayesian methodology, we introduce the maximum a posteriori probability (MAP) estimator for quantum state and process tomography. We show that the maximum likelihood, the hedged maximum likelihood, and the maximum likelihood-maximum entropy estimator, and estimators of this general type, can be viewed as special cases of the MAP estimator. The MAP, like the Bayes mean estimator, includes prior knowledge. For cases of interest to tomography MAP can take advantage of convex optimization tools, making it numerically tractable. We show how the MAP and other Bayesian quantum state estimators can be corrected for noise produced if the quantum state passes through a noisy quantum channel prior to measurement. Numerical simulations on a single qubit indicate that, on average, including these corrections significantly improves the estimate even when the measurement data are modestly large.

Mirko Navara, Jan Ševic

Classical probability theory is based on assumptions which are often violated in practice. Therefore quantum probability is a proposed alternative not only in quantum physics, but also in other sciences. However, so far it mostly criticizes the classical approach, but does not suggest a working alternative. Maximum likelihood estimators were given very low attention in this context. We show that they can be correctly defined and their computation in closed form is feasible at least in some cases.

Hideo Mabuchi, J. K. Chin, Andrew C. Doherty +6 more

Z. Hradil, Jaroslav Řeháček, Jaromı́r Fiurášek +1 more

Demosthenes Ellinas, Peter Jarvis, Matthew Pearce

Abstract The parametric maximum likelihood estimation problem is addressed in the context of quantum walk theory for quantum walks on the lattice of integers. A coin action is presented, with the real parameter θ to be estimated identified with the angular argument of an orthogonal reshuffling matrix. We provide analytic results for the probability distribution for a quantum walker to be displaced by d units from its initial position after k steps. For k large, we show that the likelihood is sharply peaked at a displacement determined by the ratio d/k which is correlated with the reshuffling parameter θ. We suggest that this ‘reluctant walker’ behaviour provides the framework for maximum likelihood estimation analysis, allowing for robust parameter estimation of θ via return probabilities of closed evolution loops and quantum measurements of the position of quantum walker with ‘reluctance index’ r = d/k.