## Statistical Science

- Statist. Sci.
- Volume 8, Number 4 (1993), 433-457.

### Quantum Statistical Inference

James D. Malley and John Hornstein

#### Abstract

The three main points of this article are: 1. Quantum mechanical data differ from conventional data: for example, joint distributions usually cannot be defined conventionally; 2. rigorous methods have been developed for analyzing such data; the methods often use quantum-consistent analogs of classical statistical procedures; 3. with these procedures, statisticians, both data-analytic and more theoretically oriented, can become active participants in many new and emerging areas of science and biotechnology. In the physical realm described by quantum mechanics, many conventional statistical and probabilistic assumptions no longer hold. Probabilistic ideas are central to quantum theory but the standard Kolmogorov axioms are not uniformly applicable. Studying such phenomena requires an altered model for sample spaces, for random variables and for inference and decision making. The appropriate decision theory has been in development since the mid-1960s. It is both mathematically and statistically rigorous and conforms to the requirements of the known physical results. This article provides a tour of the structure and current applications of quantum-consistent statistical inference and decision theory. It presents examples, outlines the theory and considers applications and open problems. Certain central concepts of quantum theory are more clearly apprehended in terms of the quantum-consistent statistical decision theory. For example, the Heisenberg uncertainty principle can be obtained as a consequence of the quantum version of the Cramer-Rao inequality. This places concepts of statistical estimation and decision theory, and thus the statistician, at the center of the quantum measurement process. Quantum statistical inference offers considerable scope for participation by the statistical community, in both applications and foundational questions.

#### Article information

**Source**

Statist. Sci., Volume 8, Number 4 (1993), 433-457.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.ss/1177010787

**Digital Object Identifier**

doi:10.1214/ss/1177010787

**Mathematical Reviews number (MathSciNet)**

MR1250150

**Zentralblatt MATH identifier**

0955.81500

**JSTOR**

links.jstor.org

**Keywords**

Quantum mechanics Heisenberg uncertainty joint distribution Hilbert space self-adjoint operator spectral measure probability-operator measure decision theory Bayesian inference de Finetti representation theorem Cramer-Rao inequality

#### Citation

Malley, James D.; Hornstein, John. Quantum Statistical Inference. Statist. Sci. 8 (1993), no. 4, 433--457. doi:10.1214/ss/1177010787. https://projecteuclid.org/euclid.ss/1177010787