### Probabilities for Observing Mixed Quantum States given Limited Prior
Information.

#### From “Quantum Communications and Measurement”, pages 411 -
418, edited by V.P. Belavkin, O. Hirota, and R.L. Hudson,
Plenum (1995)

** 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 B 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 B, of observing the (mixed) state
of the world restricted to B to be σ when we are given ρ --
the restriction to B 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 B
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.

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#### Matthew J. Donald

#### The Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, Great
Britain.

#### e-mail :
mjd1014@cam.ac.uk

#### home page:
http://people.bss.phy.cam.ac.uk/~mjd1014