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arXiv:quant-ph/0512253 · Physical Review A

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

Marcin Pawłowski, Marek Czachor

Published Apr 21, 2006• 24 citations
arXivHTMLPDF

Abstract

We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\ensuremath{\infty}$-representation and show in two ways that in two-mode states the modes are maximally entangled with the vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with the vacuum is present even for single-mode states. Still, the degree of entanglement here is difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.

DOI
10.1103/physreva.73.042111

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We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the \ensuremath∞\ensuremath{\infty}\ensuremath∞-representation and show in two ways that in two-mode states the modes are maximally entangled with the vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with the vacuum is present even for single-mode states. Still, the degree of entanglement here is difficult to estimate, mainly because there are N+2N+2N+2 subsystems, with NNN unspecified and large.
Quantum Field Theory (QFT) represents a vast generalization of Quantum Mechanics (QM), as it deals with systems that have an infinite number of degrees of freedom. The Stone-von Neumann theorem, which establishes the equivalence of irreducible representations of the canonical commutation relations (CCR) in QM, does not extend to QFT. Consequently, QFT admits multiple inequivalent irreducible representations, leading to a much richer algebraic structure. This essay aims to explore the physics of QFT from the operator algebra perspective, particularly focusing on entanglement entropy. We discuss the role of von Neumann algebras of different types in QFT, describe the local operator algebra approach to QFT, and explain how entanglement entropy can be defined in terms of the algebra of observables. Additionally, we explore the benefits of this approach in concrete applications, specifically in quantum field theory on curved spacetime.
The entanglement properties of the system of two two-level atoms interacting with a single-mode vacuum field are explored. The quantum entanglement between two two-level atoms and a single-mode vacuum field is investigated by using the quantum reduced entropy; the quantum entanglement between two two-level atoms and that between a single two-level atom and a single-mode vacuum field are studied in terms of the quantum relative entropy. The influences of the atomic dipole–dipole interaction on the quantum entanglement of the system are also discussed. Our results show that three entangled states of two atoms–field, atom–atom and atom–field can be prepared via two two-level atoms interacting with a single-mode vacuum field.
The so-called entanglement with vacuum is not a property of the Fock space, but of some rather pathological representations of CCR/CAR algebras. In some other Fock space representations the notion simply does not exist. We have checked all the main Gedanken experiments where the notion of entanglement with vacuum was used, and found that all the calculations could be performed at a representation-independent level. In particular any such experiment can be formulated in a Fock-space representation where the notion of entanglement with vacuum is meaningless. So, for the moment there is no single experiment where the notion is needed, and probably it is simply unphysical.
In this paper, we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogs of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations.
This thesis can be regarded as a light, but thorough, introduction to the algebraic approach to quantum statistical mechanics and a subsequent test of this framework in the form of an application to Bose-Einstein condensates. The success of the algebraic approach to quantum statistical mechanics hinges upon the remarkable properties of special operator algebras known as C^*-algebras. These algebras have unique characterization properties which allows one to readily identify the mathematical counterparts of concepts in physics while at the same time maintaining mathematical rigour and clarity. In the first half of this thesis, we focus on abstract C^*-algebras known as the canonical commutation relation algebras (CCR algebras) which are generated by elements satisfying specific commutation relations. The main result in this section is the proof of a certain kind of algebraic uniqueness of these algebras. The main idea of the proof is to utilise the underlying common structure of any of the CCR algebras and explicitly construct an isomorphism between the generators of these algebras. The construction of this isomorphism involves the use of abstract Fourier analysis on groups and various arguments concerning bounded operators. The second half of the thesis concerns the rigorous set-up of the formation of Bose-Einstein condensation. First, one defines the Gibbs grand canonical equilibrium states, and then we specialize to studying the taking of the thermodynamic limit of these systems in various contexts. The main result of this section involves two main elements. The first is that by fixing the temperature and density of the system while varying its activity and volume, there exists a limiting state corresponding to the taking of the thermodynamic limit. The second element concerns the existence of a critical density after which the limiting state begins to show the physical characteristics of Bose-Einstein condensation. The mathematical issues one faces with Bose-Einstein condensation are mainly related to the unboundedness of the creation and annihilation operators and the definition of the algebra that we are working on. The first issue is relevant to all areas of mathematical physics, and one deals with it in the standard ways. The second issue is more nuanced and is a direct result of the first issue we mentioned. In particular, we would like to define the states on an algebra which contains the operators that we are interested in. The problem is that these operators are unbounded, and, as a result, one must instead use the CCR algebra and show by extension that we can, in fact, also use the unbounded operators in this state.