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Journal of Mathematical Physics

Density Matrices Arising from Incomplete Measurements

Eyvind H. Wichmann

Published Jul 1, 1963• 77 citations
Publisher

Abstract

The problem of how to associate a statistical ensemble of a quantum mechanical system with an incomplete set of measured ensemble averages is discussed. The view adhered to in this note is that the most chaotic ensemble consistent with the measured ensemble averages is a reasonable choice of ensemble. The properties of this ensemble are studied rigorously for the case when the quantum mechanical system is associated with a finite-dimensional Hilbert space. The results are extended heuristically to some particular ensembles of many-boson and many-fermion systems.

DOI
10.1063/1.1704014

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The problem of how to associate a statistical ensemble of a quantum mechanical system with an incomplete set of measured ensemble averages is discussed. The view adhered to in this note is that the most chaotic ensemble consistent with the measured ensemble averages is a reasonable choice of ensemble. The properties of this ensemble are studied rigorously for the case when the quantum mechanical system is associated with a finite-dimensional Hilbert space. The results are extended heuristically to some particular ensembles of many-boson and many-fermion systems.
In quantum mechanics, states are described by density matrices. Though their probabilistic interpretation is rooted in ensemble theory, density matrices embody a known shortcoming. They do not completely express the physical realization of an ensemble. Conveniently, the outcome statistics of projective and positive operator-valued measurements do not depend on the ensemble realization, only on the density matrix. Here, we show how the geometric approach to quantum mechanics tracks ensemble realizations. We do so in two concrete cases of a finite-dimensional quantum system interacting with another one with (i) finite-dimensional Hilbert space, relevant for quantum thermodynamics, and (ii) infinite-dimensional Hilbert space, relevant for state-manipulation protocols.
We show that two chosen ensembles of spin states, which are differently prepared but are described by the same density matrix in quantum mechanics, do not fully share the same measurable characteristics. One characteristic in which they differ is shown to be the variance of the spin along a given direction. We conclude that the statistical description of an ensemble of states as given by its density matrix, although sufficient in many cases, should be considered incomplete, as it does not fully describe the measurable characteristics of the ensemble. A discussion a posteriori on the problem is provided.
A quantum mechanical system that is described by a state vector can be equally described by a density operator, although the inverse is not true. When information about a system is incomplete the only way to describe the system is by the density operator formalism. Incompleteness arises in two general cases: a mixed ensemble of physical systems and a system that interacts with its environment. We shed light on some subtle points that are usually misunderstood. We show explicitly that in the first case the averaged state vector of the system is zero, and in the second case no state vector can be assigned to the system in general.
Abstract This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out on the ensemble, it will split into a number of sub-ensembles, each sub-ensemble being conditioned on a particular outcome of the measurements. The mathematical description of the collection of sub-ensembles thus created leads to probability distributions on projective Hilbert space. The chapter develops an appropriate mathematical framework which enables the general formulation of such a distribution, and leads to the concepts of stochastic state vectors and stochastic density matrices. These concepts are required in later chapters to construct appropriate stochastic differential equations describing the continuous monitoring of open quantum systems.
We investigate here the concept of generalized complementarity introduced by Bohr. It is shown that it is not possible to ascertain the microscopic state of a system without, for the purpose of measurement, imposing upon it physical conditions so stringent that they preclude the application of the statistical description which was appropriate before the measurement. One may conclude that the statistical averages which appear in the formation of an ensemble should in quantum statistics be given an interpretation somewhat different from that of classical statistics. These results lead naturally to a critical survey of the foundations of quantum statistical mechanics. We discuss the physical interpretation of statistical matrices which are the quantum equivalent of the classical ensemble. By a straightforward generalization of classical methods one obtains a statistical representation of a system if a set of measurements are given. Let $\ensuremath{\rho}$ designate the statistical matrix which represents the system; the method consists in making the diagonal sum of $\ensuremath{\rho} log \ensuremath{\rho}$ a minimum, while the results of the performed measurements appear as conditions of the minimum. As an application we treat the composition of two separate systems and derive from this the general proof of the HHH theorem which was first given by Klein.