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arXiv:2303.05827

Use and misuse of variances for quantum systems in pure or mixed states

Alain Deville, Yannick Deville

Published Mar 10, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2303.05827

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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.
Quantum information mobilizes the description of quantum systems, their states, and their behavior. Since a measurement postulate introduced by von Neumann in 1932, if a quantum system has been prepared in two different mixed states represented by the same density operator ρ, these preparations are said to have led to the same mixture. For more than 50 years, there has been a lack of consensus about this postulate. In a 2011 article, 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, establish results useful in this context, and finally discuss the use or misuse of several concepts implied in this debate.
The distinction between pure states and mixed states is a kernel ingredient of what is considered to be the standard formulation of quantum mechanics and plays today a kernel role in foundational debates about the meaning of quantum probability, the separability of quantum systems, the definition and measure of entanglement, etc. In this work we attempt to expose the many inconsistencies introduced by this distinction and the serious consequences this has for many ongoing research programs within quantum physics which apply these notions uncritically.
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.
It is well known, that the causal Schrödinger evolution of a quantum state is not compatible with the reduction postulate, even when decoherence is taken into account. The violation of the causal evolution, introduced by the standard reduction postulate distinguishes certain systems (as the measurement devices), whose states are very close to statistical mixtures (as the ones resulting from the process of decoherence). In these systems, this violation has not any observable effect. In arbitrary quantum systems, the transition from the initial density matrix to a diagonal matrix predicted by the standard reduction postulate, would lead to a complete breakdown of the Schrödinger evolution, and consequently would destroy all the predictive power of quantum mechanics. What we show here, is that there is a modified version of the postulate that allows to treat all the quantum mechanical systems on equal footing. The standard reduction postulate may be considered as a good approximation, useful for practical purposes, of this modified version which is consistent with the Schrödinger evolution and via decoherence with the experimental results. Within this approach, the physical role played by the reduction postulate is as a tool for the computation of relative probabilities and consequently for the determination of the probabilities of consistent histories.