Quantum Communications and Measurement pp 411-418 | Cite as

# Probabilities for Observing Mixed Quantum States Given Limited Prior Information

## Abstract

The original development of the formalism of quantum mechanics involved the study of isolated quantum systems in pure states. Such systems fail to capture important aspects of the warm, wet, and noisy physical world which can better be modelled by quantum statistical mechanics and local quantum field theory using mixed states of continuous systems. In this context, we need to be able to compute quantum probabilities given only partial information. Specifically, suppose that *β* is a set of operators. This set need not be a von Neumann algebra. Simple axioms are proposed which allow us to identify a function which can be interpreted as the probability, per unit trial of the information specified by *β*,of observing the (mixed) state of the world restricted to *β* to be *σ* when we are given *ρ* — the restriction to *β* of a prior state. This probability generalizes the idea of a mixed state (*ρ*) as being a sum of terms (*σ*) weighted by probabilities. The unique function satisfying the axioms can be defined in terms of the relative entropy. The analogous inference problem in classical probability would be a situation where we have some information about the prior distribution, but not enough to determine it uniquely. In such a situation in quantum theory, because only what we observe should be taken to be specified, it is not appropriate to assume the existence of a fixed, definite, unknown prior state, beyond the set *β* about which we have information. The theory was developed for the purposes of a fairly radical attack on the interpretation of quantum theory, involving many-worlds ideas and the abstract characterization of observers as finite information-processing structures, but deals with quantum inference problems of broad generality.

### Keywords

quantum measurement theory relative entropy.## Preview

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### References

- [1]M.J. Donald, “Quantum theory and the brain”.
*Proc. R. Soc. Lond. A***427**, 43–93 (1990).ADSMATHCrossRefGoogle Scholar - [2]M.J. Donald, “A priori probability and localized observers”.
*Found. Phys.***22**, 1111–1172 (1992).MathSciNetADSCrossRefGoogle Scholar - [3]M.J. Donald, “A mathematical characterization of the physical structure of observers”.
*Found. Phys.*(to appear).Google Scholar - [4]M.J. Donald, “On the relative entropy”.
*Commun. Math. Phys.***105**, 13–34 (1986).MathSciNetADSMATHCrossRefGoogle Scholar - [5]M.J. Donald, “Further results on the relative entropy”.
*Math. Proc. Camb. Phil. Soc.***101**, 363–373 (1987).MathSciNetCrossRefGoogle Scholar - [6]I.N. Sanov, “On the probability of large deviations of random variables”.
*Mat. Sbornik*,**42***11*-44 (1957) and*Selected Translations in Math. Stat. and Prob.***1**, 213–244 (1961).MathSciNetMATHGoogle Scholar - [7]W.H. Zurek, “Decoherence and the transition from quantum to classical”.
*Physics Today*36–44 (October, 1991 ).Google Scholar