Behavior of the maximum likelihood in quantum state tomography

Abstract

Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d. They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good. Here, we consider the loglikelihood ratio. Because of the positivity constraint ρ ≥ 0, quantum state space does not generally satisfy local asymptotic normality, meaning the classical null theory for the loglikelihood ratio (the Wilks theorem) should not be used. Thus, understanding and quantifying how positivity affects the null behavior of this test statistic is necessary for its use in model selection for state tomography. We define a new generalization of local asymptotic normality, metric-projected local asymptotic normality, show that quantum state space satisfies it, and derive a replacement for the Wilks theorem. In addition to enabling reliable model selection, our results shed more light on the qualitative effects of the positivity constraint on state tomography.

A comparative study of estimation methods in quantum tomography

“…What should be the theory replacing asymptotic normality ? At the moment there isn't a complete answer to this question, but some important progress has been made in [66]. Following this work and [17], it is instructive to study an extended, 'non-physical' model in which the positivity requirement is dropped and (locally around ρ) the parameter space is taken to be that of selfadjoint matrices of trace-one M 1 sa (C d ).…”

“…As the left side is smaller than r, the integral needs to be smaller than rπ/2(d − r), which means that is close to 1 for r d. This agrees with the intuition that a large part of the eigenvalues of the lower block will be set to zero by projecting the LS onto states. Further details on finding an (approximate) solution to (28) can be found in [66]. In particular, we will approximate by the deterministic solution of equation (28) in which a = Tr(A) is set to zero; indeed, for large d this will have a negligible effect on but will allow us to compute it in terms of r and d deterministically.…”

“…For large sample size, the unconstrained estimator is asymptotically normal, while the constrained estimator is equal to its projection onto S d with respect to the metric defined by the classical Fisher information [66].…”

“…In this paper we focus on the asymptotic behaviour of the PLS estimator in the case of covariant measurements. Inspired by the techniques developed in [66] we show how the random matrix properties of the LS can be used to derive the asymptotic Frobenius and Bures risks of the PLS estimator for low rank states and large dimension d and samplesize N . In particular we uncover an interesting behaviour of the Bures risk which (unlike the Frobenius or the trace norm risks) increases steeply with rank for low rank states and then decreases towards full rank, cf.…”

Quantum gate teleportation between separated qubits in a trapped-ion processor

“…The general problem of deciding between models to fit data is called model selection. For a recent reference on the use of model selection in quantum tomography, see (37). A model M is a parametrized set of probability distributions.…”