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arXiv:1108.6249 · Physica Scripta

Underlining some limitations of the statistical formalism in quantum mechanics

F. Fratini, A G Hayrapetyan

Published Aug 31, 2011• 8 citations
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Abstract

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.

DOI
10.1088/0031-8949/84/03/035008

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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.
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.
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.
Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.