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A unified formulation for entanglement of distinguishable and indistinguishable particles

Swapnamay Mondal

Published Jan 8, 2015• 0 citations
OpenAlex

Abstract

Entanglement is well understood concept for distinguishable particles. However fundamental particles are inherently indistinguishable and when they occupy orthogonal modes, i.e. have orthogonal wave functions, they are effectively distinguishable. Thus there should exist a formulation for entanglement which works in the generic case, i.e. even when wave functions overlap. To achieve this, we consider R\`{e}nyi entropies which are measurable quantities. Outcomes of such measurements for a generic case are conjectured in terms of R\`{e}nyi entropies of spatial regions in a field theory. To leading order naive quantum mechanical answers are reproduced. Corrections depend on overlap of wave functions and exhibit novel features. In a generic case the Von Neumann entropy computed from these R\`{e}nyi entropies is argued to be a measure of entanglement of pure states and a formula for this is conjectured in terms of Von Neumann entropies of spatial regions in field theory. It seems, due to the corrections it is possible for the entanglement to exceed maximal quantum mechanical value, e.g. $log 2$ for a single qubit.

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Entanglement is well understood concept for distinguishable particles. However fundamental particles are inherently indistinguishable and when they occupy orthogonal modes, i.e. have orthogonal wave functions, they are effectively distinguishable. Thus there should exist a formulation for entanglement which works in the generic case, i.e. even when wave functions overlap. To achieve this, we consider R`{e}nyi entropies which are measurable quantities. Outcomes of such measurements for a generic case are conjectured in terms of R`{e}nyi entropies of spatial regions in a field theory. To leading order naive quantum mechanical answers are reproduced. Corrections depend on overlap of wave functions and exhibit novel features. In a generic case the Von Neumann entropy computed from these R`{e}nyi entropies is argued to be a measure of entanglement of pure states and a formula for this is conjectured in terms of Von Neumann entropies of spatial regions in field theory. It seems, due to the corrections it is possible for the entanglement to exceed maximal quantum mechanical value, e.g. log2log 2log2 for a single qubit.
Consider an arbitrary local quantum field theory with a gap or an arbitrary gapless free theory. We consider states in such a theory, that describe two entangled particles localized in disjoint regions of space. We show that in such a state, to leading order, Rényi entropies of spatial regions, containing only one of the particles are same as their quantum mechanical counterparts, after subtraction of vacuum contribution. Subleading corrections depend on overlap of wave functions. These results suggest that Von Neumann entropy of a spatial region, after subtraction of vacuum contribution, can serve as a measure of entanglement of indistinguishable particles in pure states.
We study the entanglement R\'enyi $\ensuremath{\alpha}$ entropy $(\mathrm{ER}\ensuremath{\alpha}\mathrm{E})$ as the measure of entanglement. Instead of a single quantity in standard entanglement quantification for a quantum state by using the von Neumann entropy for the well-accepted entanglement of formation (EoF), the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ gives a continuous spectrum parametrized by variable $\ensuremath{\alpha}$ as the entanglement measure, and it reduces to the standard EoF in the special case $\ensuremath{\alpha}\ensuremath{\rightarrow}1$. The $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ provides more information in entanglement quantification and can be used, for example, to determine the convertibility of entangled states by local operations and classical communication. A series of results is obtained: (i) we show that the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ of two states, which can be mixed or pure, may be incomparable, in contrast to the fact that there always exists an order for the EoF of two states; (ii) similar to the case of EoF, we study in a fully analytical way the $\mathrm{ER}\ensuremath{\alpha}\mathrm{E}$ for arbitrary two-qubit states, the Werner states, and isotropic states in the general ddd dimension; and (iii) we provide a proof of the previous conjecture for the analytical functional form of the EoF of isotropic states in the arbitrary ddd dimension.
Classical information theory, built on Shannon entropy, and quantum information theory, built on von Neumann entropy, quantify uncertainty and compression properties of probability distributions and density operators. Both are powerful, but both treat information as a property of states of individual systems or ensembles. In this manuscript we argue that entanglement—the pattern of non-classical correlations between systems—is more fundamental than either Shannon or von Neumann information. We motivate this claim with a condensate thought experiment in which many bosons occupy the same single-particle state: Shannon and von Neumann entropies assign them identical information content, yet they can be distinguished by their distinct entanglement histories with external systems. Formally, states with identical marginal density operators but different purifications represent different global information, accessible only through their entanglement structure. Entropic quantities are then seen as coarse-grained summaries of this underlying relational information. We connect this perspective to an entanglement-geometric view of physics, including Entanglement Manifold Cosmology (EMC), in which spacetime and physical “truth” emerge from the geometry of correlations. The goal is an information-theoretic narrative that is rigorous enough for physics and information theory while readable to a non-specialist who is comfortable with basic linear algebra and quantum mechanics.
While entanglement in the case of distinguishable particles is clearly understood, in the case of indistinguishable particles, there is still a wide debate regarding basic conceptual issues. Here, I will address two such issues. First, the debate almost always rests on a differentiated treatment of the cases of distinguishability and indistinguishability, even with respect to the definition of the concept of entanglement. Second, the fact that symmetrized and antisymmetrized states of indistinguishable particles are non-factorizable leads to certain perplexities about the very nature of entanglement. In the present work, both issues will be addressed from the perspective of an ontology of properties, which finds its natural expression in the algebraic formalism of quantum mechanics. The final goal will be to show that the proposed ontology allows a unified treatment of entanglement in the distinguishability and indistinguishability cases and offers a straightforward way out to perplexities. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.
It has been known for some years that entanglement entropy obtained from partial trace does not provide the correct entanglement measure when applied to systems of identical particles. Several criteria have been proposed that have the drawback of being different according to whether one is dealing with fermions, bosons, or distinguishable particles. In this Letter, we give a precise and mathematically natural answer to this problem. Our approach is based on the use of the more general idea of the restriction of states to subalgebras. It leads to a novel approach to entanglement, which is suitable to be used in general quantum systems and especially in systems of identical particles. This settles some recent controversy regarding entanglement for identical particles. The prospects for applications of our criteria are wide ranging, from spin chains in condensed matter to entropy of black holes.
Ryu and Takayanagi conjectured a formula for the entanglement (von Neumann) entropy of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called R\'enyi entropies. Using Euclidean gravity, Fursaev computed the entanglement R\'enyi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the RT formula. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional conformal field theories (CFTs), considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the mutual information between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large NNN symmetric product theories have the same EREs as holographic ones.