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arXiv:1007.2908 · Physical Review A

Multipartite entanglement in fermionic systems via a geometric measure

Behzad Lari, P. Durganandini, Pramod S. Joag

Published Dec 2, 2010• 5 citations
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Abstract

We study multipartite entanglement in a system consisting of indistinguishable fermions. Specifically, we have proposed a geometric entanglement measure for $N$ spin-$\frac{1}{2}$ fermions distributed over $2L$ modes (single-particle states). The measure is defined on the $2L$ qubit space isomorphic to the Fock space for $2L$ single-particle states. This entanglement measure is defined for a given partition of $2L$ modes containing $m\ensuremath{\geqslant}2$ subsets. Thus this measure applies to $m\ensuremath{\leqslant}2L$ partite fermionic systems where $L$ is any finite number, giving the number of sites. The Hilbert spaces associated with these subsets may have different dimensions. Further, we have defined the local quantum operations with respect to a given partition of modes. This definition is generic and unifies different ways of dividing a fermionic system into subsystems. We have shown, using a representative case, that the geometric measure is invariant under local unitary operators corresponding to a given partition. We explicitly demonstrate the use of the measure to calculate multipartite entanglement in some correlated electron systems.

DOI
10.1103/physreva.82.062302

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We study multipartite entanglement in a system consisting of indistinguishable fermions. Specifically, we have proposed a geometric entanglement measure for NNN spin-12\frac{1}{2}21​ fermions distributed over 2L2L2L modes (single-particle states). The measure is defined on the 2L2L2L qubit space isomorphic to the Fock space for 2L2L2L single-particle states. This entanglement measure is defined for a given partition of 2L2L2L modes containing m\ensuremath⩾2m\ensuremath{\geqslant}2m\ensuremath⩾2 subsets. Thus this measure applies to m\ensuremath⩽2Lm\ensuremath{\leqslant}2Lm\ensuremath⩽2L partite fermionic systems where LLL is any finite number, giving the number of sites. The Hilbert spaces associated with these subsets may have different dimensions. Further, we have defined the local quantum operations with respect to a given partition of modes. This definition is generic and unifies different ways of dividing a fermionic system into subsystems. We have shown, using a representative case, that the geometric measure is invariant under local unitary operators corresponding to a given partition. We explicitly demonstrate the use of the measure to calculate multipartite entanglement in some correlated electron systems.
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.
We discuss the relation between fermion entanglement and bipartite entanglement. We first show that an exact correspondence between them arises when the states are constrained to have a definite local number parity. Moreover, for arbitrary states in a four-dimensional single-particle Hilbert space, the fermion entanglement is shown to measure the entanglement between two distinguishable qubits defined by a suitable partition of this space. Such entanglement can be used as a resource for tasks like quantum teleportation. On the other hand, this fermionic entanglement provides a lower bound to the entanglement of an arbitrary bipartition, although in this case the local states involved will generally have different number parities. Finally, the fermionic implementation of the teleportation and superdense coding protocols based on qubits with odd and even number parity is discussed, together with the role of the previous types of entanglement.
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with a fixed number parity yet not necessarily a fixed particle number. The mode entanglement between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space H\mathcal{H}H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of H\mathcal{H}H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined by J. Schliemann et al. [Phys. Rev. A 64, 022303 (2001)], implying that particle entanglement may be seen as minimum mode entanglement. It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that in the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy is evaluated analytically, extending the results in previous reference to general states.
We introduce geometric measures of entanglement for indistinguishable particles, which apply to mixed states, multipartite systems, and arbitrary dimensions. They are based on generalized (i.e., not necessarily finite) norms on the set of quantum states and lead to the first necessary and sufficient computational separability criterion in this general setting. The coherent approach developed in the paper allows us to compare, in particular, entanglement for fermionic and distinguishable particles: The entanglement measure for fermionic particles coincides with the corresponding entanglement measure for distinguishable particles up to a factor of k!k!k! where kkk is the number of particles involved. By this result the amount of entanglement emerging from fermi statistics alone is clearly separated from the overall amount of entanglement. Finally, our techniques are applied to entanglement related to Schmidt and Slater numbers.
In this paper we describe a method for finding polynomial invariants under stochastic local operations and classical communication (SLOCC) for a system of delocalized fermions shared between different parties, with global particle-number conservation as the only constraint. These invariants can be used to construct entanglement measures for different types of entanglement in such a system. It is shown that the invariants, and the measures constructed from them, take a nonzero value only if the state of the system allows for the observation of Bell-nonlocal correlations. Invariants of this kind are constructed for systems of two and three spin-12\frac{1}{2}21​ fermions and examples of maximally entangled states are given that illustrate the different types of entanglement distinguished by the invariants. A general condition for the existence of SLOCC invariants and their associated measures is given as a relation between the number of fermions, their spin, and the number of spatial modes of the system. In addition, the effect of further constraints on the system, including the localization of a subset of the fermions, is discussed. Finally, a hybrid Ising-Hubbard Hamiltonian is constructed for which the ground state of a three-site chain exhibits a high degree of entanglement at the transition between a regime dominated by on-site interaction and a regime dominated by Ising interaction. This entanglement is well described by a measure constructed by the introduced method.
We investigate the behaviour of multipartite entanglement in finite-size quantum spin systems, resorting to a hierarchy of geometric measures of multipartite entanglement recently introduced in the literature. In particular, we investigate the ground-state entanglement in the XY model defined on finite chains of N sites with periodic boundary conditions. We analyse the behaviour of the geometric measures of (N 1)-partite and (N/2)-partite entanglement and compare them with the Wei‐Goldbart geometric measure of global entanglement.