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Reduced Density Matrix of Identical Particles from Three Aspects: the First Quantization, Exterior Products, and GNS representation

Seungbeom Chin, Joonsuk Huh

Published Jun 3, 2019• 0 citations
OpenAlex

Abstract

We probe the theoretical connection among three different approaches to analyze the entanglement of identical particles, i.e., the first quantization language (1QL), exterior and elementary-symmetric products (which has the mathematical equivalence to no-labeling approaches), and the algebraic approach based on the GNS construction. Among several methods to quantify entanglement of identical particles, we focus on the computation of reduced density matrix, which can be achieved by the concept of \emph{symmetrized partial trace} defined in 1QL. We show that the symmetrized partial trace corresponds to the interior product between exterior (or elementary symmetric) state vectors, which also corresponds to the subalgebra restriction in the algebraic approach based on GNS representation. By expressing the state of identical particles as exterior and elementary symmetric state vectors, we can also prove that GHJW theorem holds for identical particles.

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We probe the theoretical connection among three different approaches to analyze the entanglement of identical particles, i.e., the first quantization language (1QL), exterior and elementary-symmetric products (which has the mathematical equivalence to no-labeling approaches), and the algebraic approach based on the GNS construction. Among several methods to quantify entanglement of identical particles, we focus on the computation of reduced density matrix, which can be achieved by the concept of \emph{symmetrized partial trace} defined in 1QL. We show that the symmetrized partial trace corresponds to the interior product between exterior (or elementary symmetric) state vectors, which also corresponds to the subalgebra restriction in the algebraic approach based on GNS representation. By expressing the state of identical particles as exterior and elementary symmetric state vectors, we can also prove that GHJW theorem holds for identical particles.
We probe the theoretical connection among three different approaches to analyze the entanglement of identical particles, i.e., the first quantization language (1QL), elementary-symmetric/exterior products (which has the mathematical equivalence to no-labeling approaches), and the algebraic approach based on the GNS construction. Among several methods to quantify the entanglement of identical particles, we focus on the computation of reduced density matrices, which can be achieved by the concept of \emph{symmetrized partial trace} defined in 1QL. We show that the symmetrized partial trace corresponds to the interior product in symmetric and exterior algebra (SEA), which also corresponds to the subalgebra restriction in the algebraic approach based on GNS representation. Our research bridges different viewpoints for understanding the quantum correlation of identical particles in a consistent manner.
We suggest a formalism to illustrate the entanglement of identical particles in the first quantization language (1QL). Our 1QL formalism enables one to exploit all the well-established quantum information tools to understand the indistinguishable ones, including the reduced density matrix and familiar entanglement measures. The rigorous quantitative relation between the amount of entanglement and the spatial coherence of particles is possible in this formalism. Our entanglement detection process is a generalization of the entanglement extraction protocol for identical particles with mode splitting proposed by Killoran et al. [Phys. Rev. Lett. 112, 150501 (2014)].
We consider entanglement in a system with a fixed number of identical particles. Since any operation should be symmetrized over all the identical particles and there is the precondition that the spatial wave functions overlap, the meaning of identical-particle entanglement is fundamentally different from that of distinguishable particles. The identical-particle counterpart of the Schmidt basis is shown to be the single-particle basis in which the one-particle reduced density matrix is diagonal. But it does not play a special role in the issue of entanglement, which depends on the single-particle basis chosen. The nonfactorization due to (anti)symmetrization is naturally excluded by using the (anti)symmetrized basis or, equivalently, the particle number representation. The natural degrees of freedom in quantifying the identical-particle entanglement in a chosen single-particle basis are occupation numbers of different single-particle basis states. The entanglement between effectively distinguishable spins is shown to be a special case of the occupation-number entanglement.
We present a general approach to quantum entanglement and entropy that is based on algebras of observables and states thereon. In contrast to more standard treatments, Hilbert space is an emergent concept, appearing as a representation space of the observable algebra, once a state is chosen. In this approach, which is based on the Gel'fand-Naimark-Segal construction, the study of subsystems becomes particularly clear. We explicitly show how the problems associated with the partial trace for the study of entanglement of identical particles are readily overcome. In particular, a suitable entanglement measure is proposed that can be applied to systems of particles obeying Fermi, Bose, para-, and even braid-group statistics. The generality of the method is also illustrated by study of the time evolution of subsystems emerging from restriction to subalgebras. Problems related to anomalies and quantum epistemology are also discussed.
Here, we discuss a particle-based approach to deal with systems of many identical quantum objects (particles) that never employs labels to mark them. We show that it avoids both methodological problems and drawbacks in the study of quantum correlations associated with the standard quantum mechanical treatment of identical particles. The core of this approach is represented by the multiparticle probability amplitude, whose structure in terms of single-particle amplitudes we derive here by first principles. To characterize entanglement among the identical particles, this new method uses the same notions, such as partial trace, adopted for non-identical ones. We highlight the connection between our approach and second quantization. We also define spin-exchanged multipartite states which contain a generalization of W states to identical particles. We prove that particle spatial overlap plays a role in the distributed entanglement within multipartite systems and is responsible for the appearance of non-local quantum correlations.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.
Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle.