Asymptotic Methods in Quantum Statistics

Cornell University, Ithaca NY

Investigators

Abstract

The processing of quantum information is emerging as a challenging new field for statisticians. While the concept of inherent randomness is central to quantum mechanics, it cannot be described in terms of traditional probability alone, i.e. notions such as observed random variables, sample spaces etc. are not sufficient. On the simplest level, finite probability laws have to be replaced by states, which are defined as complex positive definite Hermitian matrices of trace one. A very general framework is provided by the theory of operator algebras. In quantum statistical decision theory, families of states generalize families of probability measures (statistical experiments), and the tensor product of many states replaces the classical simple random sample. One problem which is already known in the classical context is the risk asymptotics for symmetric hypothesis testing, or Bayesian discrimination of two states with equal prior weights (the Chernoff bound on the exponential rate of decay of the error probability). In the recent solution of this problem on the quantum level by the investigator and coauthors, a new method has been developed to reduce the quantum risk to a classical one, via associating a pair of probability distributions to a pair of states by measurement on a purification. The present project aims at exploring further this new method, with regard to wider applicability in quantum testing, estimation and possibly in approximation of quantum statistical experiments. Further subjects of study are quantum statistical applications of information theoretic concepts like channel capacity, Kolmogorov complexity, and rate distortion. Natural phenomena on the very small (subatomic) level are governed by quantum theory, where physical laws and cause-effect relationships are thoroughly different from the world of ordinary human experience. Physicists and computer scientists realized some time ago that these phenomena might be harnessed to build computers of extraordinary speed, and also allow rapid advances in cryptography such as breaking all presently known secret codes or constructing new unbreakable ones. While quantum computers have long remained an abstract idea and have only been built at an embryonic stage so far, the theoretical groundwork for far-reaching applications is already being laid in the interdisciplinary field of Quantum Information Theory. In the need for finding benchmarks for optimal performance, researchers in this field have recently begun to exploit some well developed theories of signal processing and statistics. The present project is situated precisely at this new frontier between traditional statistics and quantum theory. The aim is to achieve a better mathematical and statistical understanding of quantum computing and communication, areas which promise to be of great technological impact once they reach an applied stage.