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arXiv:2505.15551 · Information

The Landau-Feynman Transiently Open Quantum System: Entanglement and Density Operators

Alain Deville, Yannick Deville

Published Jun 30, 2025• 0 citations
arXivHTMLPDF

Abstract

Users of quantum mechanics are familiar with the concept of a statistical mixture as introduced by von Neumann, and with the use of a density operator in that context. A density operator may also be used in another situation, introduced by Landau, with a transient coupling between the two parts of a quantum bipartite system. But more than fifty years after a clarifying work by Feynman on the subject, a confusion still persists about what we call the Landau-Feynman situation. In this paper we establish that, when facing that situation, the right concept to be used is not the one of a mixed state - be it qualified as proper or improper -, but the one of entanglement.

DOI
10.3390/info16070558

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Users of quantum mechanics are familiar with the concept of a statistical mixture as introduced by von Neumann, and with the use of a density operator in that context. A density operator may also be used in another situation, introduced by Landau, with a transient coupling between the two parts of a quantum bipartite system. But more than fifty years after a clarifying work by Feynman on the subject, a confusion still persists about what we call the Landau-Feynman situation. In this paper we establish that, when facing that situation, the right concept to be used is not the one of a mixed state - be it qualified as proper or improper -, but the one of entanglement.
Abstract In this chapter we discuss statistical mixtures of pure quantum states, resulting in so-called mixed states, and their representation using density operators. The properties of density operators for pure and mixed states are analysed and we examine their time evolution governed by the Liouville-von Neumann equation. As particular examples of interest we consider the density operators for thermal states, focusing on harmonic oscillators and the application of this model to calculate the average photon number for the modes of the electromagnetic field at thermal equilibrium. We then turn to two-level systems, or qubits, which are of particular relevance in quantum information theory and we inspect the geometry of the state space of density operators as a convex subspace of the Hilbert-Schmidt space. Finally, we take a closer look at density operators for bipartite systems which forms the mathematical basis for considering entanglement in later chapters.
Idea of quantum entanglement is discussed in the context of debate about the Einstein-Podolsky-Rosen thought experiment and some theoretical studies of quantum systems. It is noted that Schrödinger invented this idea in 1935 in order to fix some features of the quantum-mechanical description of two systems with temporary interaction. However, he did not grasp essence of these features really. In view of the concepts of mixture and statistical operator proposed by von Neumann and adopted by Schrödinger in 1936, it is argued that the idea of entanglement and related terminology are not necessary in quantum mechanics. One can use this idea and terms "entanglement" etc. as "visual" surrogates for the "mixture – statistical operator" pair. Deeper comparative analysis of several theoretical works by Schrödinger, von Neumann, and Landau showed that the modeling of non-trivial complex quantum systems as quasi-classical aggregates has been gradually overcome. Instead, wholeness of such quantum systems was actually recognized step by step. Thus, wholeness is immanent not only to quantum phenomena, as Niels Bohr had argued, but also to the quantum systems themselves, objectively. The pair "mixture – statistical operator" and especially the pair "mixed state – density matrix" similar to it appear to be adequate tools to comprehend and describe wholeness of diverse quantum reality. It is insisted, it is advisable to understand the surrogate idea of entanglement and relevant terminology in the same sense. In mature quantum paradigm, they are possible but not necessary theoretical tools to grasp wholeness of reality. Respectively, acceptable understanding of quantum entanglement must be based on recognition of quantum wholeness. Philosophically speaking, the idea of entanglement is understandable and conditionally acceptable in the view of contemporary rational holism, or holistic rationality. The clarified understanding of quantum entanglement, as well as Bohr's substantiation of wholeness of quantum phenomena, demonstrates irreducibility of the Universe to any quasi-classical aggregate. Moreover, all this supports the view of the Universe as real wholeness, which rational holism intends to grasp. It is concluded, further development and regular implementation of rational holism have the undoubted potential for revolutionary replacement of the hitherto widespread worldview in the spirit of Democritus and pure analytical methodology of knowledge.
Beyond state vectors We cannot always assign a definite state vector |ψ〉 to a quantum system Q. It may be that Q is part of a composite system RQ that is in an entangled state |Ψ (RQ) 〉. Or it may be that our knowledge of the preparation of Q is insufficient to determine a particular state |ψ〉. Consider, for instance, a qubit sent from Alice to Bob during the BB84 key distribution protocol from Section 4.4. The state of this qubit could be |0〉, |1〉, |+〉 or |−〉, each with equal likelihood. In either case – whether Q is a subsystem of an entangled system, or the state of Q is determined by a probabilistic process – we cannot specify a quantum state vector |ψ〉 for Q. Nevertheless, in either case we are in a position to make statistical predictions about the outcomes of measurements on Q. In this chapter we describe the mathematical machinery for doing this. Mixtures of states Suppose the state of Q arises by a random process, so that the state |ψ α 〉 is prepared with probability p α . The possible states |ψ α 〉 need not be orthogonal (as you can see in the BB84 example above). We call this situation a mixture of the states |ψ α 〉, or a mixed state for short. One way to interpret a mixed state is to return to the idea of an ensemble of systems, which we introduced in Section 3.3.
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 paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state ). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question. A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme. It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.