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arXiv:1310.2863

Spin entanglement and nonlocality of multifermion systems

M. R. Arifullin, Vitaly L. Berdinskiy

Published Oct 10, 2013• 0 citations
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Abstract

Spin density matrices of the system, containing arbitrary even number N of indistinguishable fermions with spin S = 1/2, described by antisymmetric wave function, have been calculated. The indistinguishability and the Pauli principles are proved to determine uniquely spin states, spin correlations and entanglement of fermion spin states. Increase of the particle number in the multifermion system reduces the spin correlation in any pair of fermions. The fully entangled system of N electrons are shown to be composed by pairs with nonentangled spin states that is the incoherent superposition of the singlet and triplet states. Any large system of N fermions, such as electrons with spin S = 1/2, the spin state of any particle are shown to be entangled with the other part of the system containing N-1 particle. However, the spin state of this electron is not entangled with any other particle, and spin state of any electron pair is not entangled. These properties of spin states manifest in the Einstein-Podolsky-Rosen as confirmation or violation of the Bell inequalities indicating the presence of non-local quantum spin correlations.

DOI
10.48550/arxiv.1310.2863

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Spin density matrices of the system, containing arbitrary even number N of indistinguishable fermions with spin S = 1/2, described by antisymmetric wave function, have been calculated. The indistinguishability and the Pauli principles are proved to determine uniquely spin states, spin correlations and entanglement of fermion spin states. Increase of the particle number in the multifermion system reduces the spin correlation in any pair of fermions. The fully entangled system of N electrons are shown to be composed by pairs with nonentangled spin states that is the incoherent superposition of the singlet and triplet states. Any large system of N fermions, such as electrons with spin S = 1/2, the spin state of any particle are shown to be entangled with the other part of the system containing N-1 particle. However, the spin state of this electron is not entangled with any other particle, and spin state of any electron pair is not entangled. These properties of spin states manifest in the Einstein-Podolsky-Rosen as confirmation or violation of the Bell inequalities indicating the presence of non-local quantum spin correlations.
Spin density matrices of the system, containing arbitrary even number N of indistinguishable fermions with spin S = 1/2, described by antisymmetric wave function, have been calculated. The indistinguishability and the Pauli principles are proved to determine uniquely spin states, spin correlations and entanglement of fermion spin states. Increase of the particle number in the multifermion system reduces the spin correlation in any pair of fermions. The fully entangled system of N electrons are shown to be composed by pairs with nonentangled spin states that is the incoherent superposition of the singlet and triplet states.
Any large system of N fermions, such as electrons with spin S = 1/2, the spin state of any particle are shown to be entangled with the other part of the system containing N-1 particle. However, the spin state of this electron is not entangled with any other particle, and spin state of any electron pair is not entangled. These properties of spin states manifest in the Einstein-Podolsky-Rosen as confirmation or violation of the Bell inequalities indicating the presence of non-local quantum spin correlations.
Entanglement, the Einstein-Podolsky-Rosen (EPR) paradox and Bell’s failure of local-hiddenvariable (LHV) theories are three historically famous forms of “quantum nonlocality”. We give experimental criteria for these three forms of nonlocality in multi-particle systems, with the aim of better understanding the transition from microscopic to macroscopic nonlocality. We examine the nonlocality of N separated spin J systems. First, we obtain multipartite Bell inequalities that address the correlation between spin values measured at each site, and then we review spin squeezing inequalities that address the degree of reduction in the variance of collective spins. The latter have been particularly useful as a tool for investigating entanglement in Bose-Einstein condensates (BEC). We present solutions for two topical quantum states: multi-qubit Greenberger-Horne-Zeilinger (GHZ) states, and the ground state of a two-well BEC.
The characterization of entanglement is a fundamental issue for Quantum Information Theory. But the definition of entanglement depends on the notion of locality, and thus on the tensor product structure of the state space of the composite system. This notion is affected by the presence of superselection rules that restrict the accessible Hilbert space to a direct sum of subspaces. Indistinguishability of particles imposes one such restriction, namely to totally symmetric or totally antisymmetric states. The entanglement can in this case be defined with respect to partitions of modes in the second quantization formalism. For fermionic systems the Fock space of m modes is isomorphic to the space of m qubits, but the action of creation and annihilation operators is not local, due to their anticommutation. Conservation of the parity of fermion number imposes another relevant superselection rule. It requires that local physical observables commute with the local parity operator. Taking into account the considerations above, it is possible to define the set of separable states or equivalently the concept of entanglement for fermionic systems in a number of ways. Here we analyze systematically these possibilities and the relation among the various sets of separable states. We also discuss the behavior of the different classes when taking several copies of the state, as well as the characterization of the sets in terms of the usual criteria regarding the tensor product.
The original formula of Bell inequality (BI) in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the state-density operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by PBP_{B}PB​ is always less than or at most equal to one for the local realistic model (PBlc≤1P_{B}^{lc}\leq1PBlc​≤1) regardless of the specific superposition coefficients of entangled state. Including the non-local quantum interference the maximum violation of BI is found as PBmax⁡P_{B}^{\max}PBmax​ =2=2=2, which, however depends on state parameters and three measuring directions as well. Our result is suitable for entangled photon pairs.