Asymptotic Equivalence of Quantum Statistical Models

Cornell University, Ithaca NY

Investigators

Abstract

A fundamental insight of quantum mechanics is that randomness is a inherent feature of the physical world at the microscopic level. Any observation made on a quantum system such as an atom or a light pulse, results in a nondeterministic, stochastic outcome. The study of the direct map from the system?s state or preparation to the probability distribution of the measurement outcomes has been one of the core topics in traditional quantum theory. In many quantum protocols, the experimenter has incomplete knowledge and control of the system and its environment, or is interested in estimating an external field parameter which affects the system dynamics. In this case, one deals with a statistical inverse problem of inferring unknown state parameters from the measurement data obtained by probing a large number of individual quantum systems. The theory and practice arising from tackling such questions is shaping up into the field of quantum statistics, which lies at the intersection of quantum theory and statistical inference. The current project aims at a better understanding of statistical inference for infinite dimensional quantum systems, an area which can be seen as a quantum counterpart of nonparametric statistics. The ultimate goal is to develop a theory of comparison and convergence of quantum statistical models, thereby enabling efficient estimation techniques and establishing solid statistical methodology for computing reliable error bars. This project is well aligned with NSF's Quantum Leap Big Idea. The graduate student supported by this award will carry out research on quantum statistics, under the supervision of the PI. An area of particular interest is local asymptotic equivalence, with explicitly constructed quantum channels, of different quantum statistical models of pure states with a parameter in Hilbert space. Some elements of such a theory have already been put forward by a recent result of the proposer and collaborators, approximating a model of a large number of independent quantum systems by a Gaussian model of coherent states (shifted vacuum states). The goal will be to further develop these notions in cases of entanglement, focusing at first on quantum analogs of Gaussian stationary sequences. Among these, two cases are singled out: a stationary model of zero mean pure Gaussian states commonly used to model the squeezed vacuum in quantum optics, and gauge invariant stationary Gaussian states which appear to exhibit classical limiting behavior. In a related topic, results on sharp nonparametric risk asymptotics like the Pinsker bound, proved initially in classical white noise models, have motivated the development of equivalence theory. It is of interest to establish minimax risk bounds of this type for estimating pure quantum states, and their adaptive attainability. In the problem of symmetric quantum hypothesis testing, or discrimination between two quantum states, the proposer and collaborators solved the longstanding problem of the quantum Chernoff lower bound pertaining to the exponential rate of decay of the error probability. In that connection, several new problems appear, such as the optimal error exponent for discriminating composite hypotheses, and attainability of the bound by realizable receivers in quantum optics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.