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arXiv:1204.1071

Underlining some limitations of the statistical formalism in quantum mechanics: Reply to the Comment of Bodor and Diósi

F. Fratini, Armen G. Hayrapetyan

Published Apr 4, 2012• 1 citation
arXivHTMLPDF

Abstract

In a paper of us, it is showed that Density Matrices do not provide a complete description of ensembles of states in quantum mechanics, since they lack measurable information concerning the preparation of the ensembles. Bodor and Diósi have later posted a comment on that article, which agrees on some points of it but disagrees on some others. This reply is intended to clarify the discussion.

DOI
10.48550/arxiv.1204.1071

Related papers

Underlining some limitations of the statistical formalism in quantum mechanics

F. Fratini, A G Hayrapetyan

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.

arXiv:1108.6249•Physica Scripta•Aug 31, 2011•8 citations

Comment on `Underlining some limitations of the statistical formalism in quantum mechanics' by Fratini and Hayrapetyan

András Bodor, Diósi, Lajos

We point out that Fratini and Hayrapetyan ignored the randomness of mixing, a basic request to prepare a statistical ensemble rather than an ensemble in general. Their analysis is irrelevant for standard statistical ensembles, their conclusions about the limitations of standard theory become unjustified.

arXiv:1110.4549•Oct 20, 2011•1 citation

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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.

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Comments

Bernard d’Espagnat

This chapter discusses some comments, and the rules of quantum mechanics. The quantum mechanics is universally valid for all the ensembles to which the more general formalism just alluded to is applicable. Any ensemble of systems the specification of which involves no reference to the results of future measurements that are to be performed on these systems obeys the quantum rules. One of the views most often expressed by the supporters of the "Copenhagen interpretation" is that quantum mechanics is a complete theory. The main motivation for the assumption is that the splitting of ensemble and subensembles cannot be made in any straightforward manner. However, this assumption implies the existence of particular, hitherto unknown forces, and it is therefore quite generally considered as unattractive. Thus the probabilities, or statistical distributions, that conventional quantum mechanics introduces into physics are inherently different from those found in classical mechanics, for example, in the kinetic theory of gases.

CRC Press eBooks•Mar 5, 2018•0 citations

Individuals, Ensembles, States

Bernard d’Espagnat

Most of the predictions of quantum mechanics are of a statistical nature and therefore make sense only for ensembles. The axiomatics based on the quantum calculus of propositions might open the way toward a general definition of the quantum state of an individual system. It was soon recognized that this leads to separability and thereby to violation of some of the quantum-mechanical predictions. In contemporary literature, the concept of state is not wholly unambiguous. Some authors associate it with the notion of an individual system and say, for instance that a given system is in a given state. Others associate it exclusively with ensembles. The situation is complicated by the fact that for some theorists the concept of state refers only to ensembles that can be described by some state vector. For other authors it refers just as well to ensembles that cannot be associated with any state vector and for which only a density matrix description is therefore acceptable.

CRC Press eBooks•Mar 5, 2018•0 citations
In a paper of us, it is showed that Density Matrices do not provide a complete description of ensembles of states in quantum mechanics, since they lack measurable information concerning the preparation of the ensembles. Bodor and Diósi have later posted a comment on that article, which agrees on some points of it but disagrees on some others. This reply is intended to clarify the discussion.
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.
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.
As a consequence of the place ascribed to measurements in the postulates of quantum mechanics, if two differently prepared systems are described with the same density operator \r{ho}, they are said to be in the same quantum state. For more than fifty years, there has been a lack of consensus about this postulate. In a 2011 paper, considering variances of spin components, Fratini and Hayrapetyan tried to show that this postulate is unjustified. The aim of the present paper is to discuss major points in this 2011 article, and in their reply to a 2012 paper by Bodor and Diosi claiming that their analysis was irrelevant. Facing some ambiguities or inconsistencies in the 2011 paper and in the reply, we first try to guess their aim, then establish results useful in this context, and finally discuss the use or misuse of several concepts implied in this debate.
This chapter discusses some comments, and the rules of quantum mechanics. The quantum mechanics is universally valid for all the ensembles to which the more general formalism just alluded to is applicable. Any ensemble of systems the specification of which involves no reference to the results of future measurements that are to be performed on these systems obeys the quantum rules. One of the views most often expressed by the supporters of the "Copenhagen interpretation" is that quantum mechanics is a complete theory. The main motivation for the assumption is that the splitting of ensemble and subensembles cannot be made in any straightforward manner. However, this assumption implies the existence of particular, hitherto unknown forces, and it is therefore quite generally considered as unattractive. Thus the probabilities, or statistical distributions, that conventional quantum mechanics introduces into physics are inherently different from those found in classical mechanics, for example, in the kinetic theory of gases.
Most of the predictions of quantum mechanics are of a statistical nature and therefore make sense only for ensembles. The axiomatics based on the quantum calculus of propositions might open the way toward a general definition of the quantum state of an individual system. It was soon recognized that this leads to separability and thereby to violation of some of the quantum-mechanical predictions. In contemporary literature, the concept of state is not wholly unambiguous. Some authors associate it with the notion of an individual system and say, for instance that a given system is in a given state. Others associate it exclusively with ensembles. The situation is complicated by the fact that for some theorists the concept of state refers only to ensembles that can be described by some state vector. For other authors it refers just as well to ensembles that cannot be associated with any state vector and for which only a density matrix description is therefore acceptable.