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arXiv:quant-ph/0105120 · Physical Review A

Spin-space entanglement transfer and quantum statistics

Yasser Omar, Nikola Paunković, Sougato Bose, Vlatko Vedral

Published May 30, 2002• 34 citations
arXivHTMLPDF

Abstract

Both the topics of entanglement and particle statistics have aroused enormous research interest since the advent of quantum mechanics. Using two pairs of entangled particles we show that indistinguishability enforces a transfer of entanglement from the internal to the spatial degrees of freedom without any interaction between these degrees of freedom. Moreover, subensembles selected by local measurements of the path will, in general, have different amounts of entanglement in the internal degrees of freedom depending on the statistics (either fermionic or bosonic) of the particles involved.

DOI
10.1103/physreva.65.062305

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Both the topics of entanglement and particle statistics have aroused enormous research interest since the advent of quantum mechanics. Using two pairs of entangled particles we show that indistinguishability enforces a transfer of entanglement from the internal to the spatial degrees of freedom without any interaction between these degrees of freedom. Moreover, subensembles selected by local measurements of the path will, in general, have different amounts of entanglement in the internal degrees of freedom depending on the statistics (either fermionic or bosonic) of the particles involved.
We reconsider the effect of indistinguishability on the reduced density operator of the internal degrees of freedom (tracing out the spatial degrees of freedom) for a quantum system composed of identical particles located in different spatial regions. We explicitly show that if the spin measurements are performed in disjoint spatial regions then there are no constraints on the structure of the reduced state of the system. This implies that the statistics of identical particles has no role from the point of view of separability and entanglement when the measurements are spatially separated. We extend the treatment to the case of n particles and show the connection with some recent criteria for separability based on subalgebras of observables.
We define statistics of quantum mechanical particles, obtaining the Bose–Einstein and Fermi–Dirac varieties of indistinguishable particles. After finding the rotation and Lorentz matrices in different Lorentz representations, we describe the spin-statistics theorem, relating fermions with half-integer spin and bosons with integer spin. We explain two simple proofs and say some words on two others. We end by discussing symmetries in more generality, and we discuss the fact that internal symmetries must commute with spacetime ones, due to the Coleman–Mandula theorem.
We study properties of entangled systems in the (mainly non-relativistic) second quantization formalism. This is then applied to interacting and non-interacting bosons and fermions and the differences between the two are discussed. We present a general formalism to show how entanglement changes with the change of modes of the system. This is illustrated with examples such as the Bose condensation and the Unruh effect. It is then shown that a non-interacting collection of fermions at zero temperature can be entangled in spin providing that their distances do not exceed the inverse Fermi wavenumber. Beyond this distance all bipartite entanglement vanishes, although classical correlations still persist. We compute the entanglement of formation as well as the mutual information for two spin-correlated electrons as a function of their distance. The analogous non-interacting collection of bosons displays no entanglement in the internal degrees of freedom. We show how to generalize our analysis of the entanglement in the internal degrees of freedom to an arbitrary number of particles.
The predictions of quantum mechanics often differ from our everyday experiences. In the classical case, the state of a system can be predicted precisely, assuming the exact knowledge of all parameters. This is not possible in quantum mechanics. For example, the spin of one particle can be simultaneously in two different states, until a measurement defines the actual state. If multiple particles are in such superposition states, their spins can be coupled with each other such that the measurement of one particle changes the physical state of the other particles. This coupling effect is a fundamental part of quantum mechanics and is called entanglement. It is a central requirement for applications in the fields of quantum communication, quantum cryptography and quantum computing. One possibility to create entanglement are spin-changing collisions in a spinor Bose-Einstein condensate. These collisions are described by a nonlinear process, which creates non-classical correlations between all atoms of the Bose-Einstein condensate. Within this thesis, 87Rb spinor Bose-Einstein condensates are used to demonstrate Einstein-Podolsky-Rosen entanglement by the generation and analysis of a two-mode entangled state in spin space [A1]. These states are applied in an interferometric measurement with a resolution of 2.05+0.34 -0.37 dB beyond the standard quantum limit [A2]. Depending on the dynamics, the process can also be analogous to the dynamical Casimir effect [A3]. The entanglement between the particles in these publications is strongly connected with their indistinguishability. It is thus possible to make the particles distinguishable again and recover the entanglement between the now separated particles? This question is approached by cutting an entangled state of indistinguishable particles into two spatially separated atomic modes - creating a two-mode entangled state in real space. As shown in Ref. [A4], the entanglement between the two separated clouds of a Bose-Einstein condensate can be measured directly. In the future, the created state can be employed for a test of quantum nonlocality.
A general and an arbitrarily efficient scheme for entangling the spins (or any spinlike degree of freedom) of two independent uncorrelated identical particles by a combination of two particle interferometry and which way detection is formulated. It is shown that the same setup could be used to identify the quantum statistics of the incident particles from either the sign or the magnitude of measured spin correlations. Our setup also exhibits a curious complementarity between particle distinguishability and the amount of generated entanglement.
A generalized framework for particle statistics is developed within which it becomes clear how bosons can combine to form fermions without violating the normal connection between spin and statistics. In the situation where internal and external particle degrees of freedom are separable, the possible statistical types are found and their combining rules derived. Here the statistical "interaction" between two particles is characterized either by the familiar sign or by an integer ("twist number"). However, further possibilities arise if the separability of external from internal degrees of freedom fails, as it can in gauge theories. In spatial dimensions other than three the "fermion from boson" phenomenon does not occur.