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arXiv:1309.4300 · Journal of Physics A Mathematical and Theoretical

Entanglement in fermionic Fock space

Gábor Sárosi, Péter Lévay

Published Feb 28, 2014• 25 citations
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Abstract

We propose a generalization of the usual stochastic local operations and classical communications (SLOCC) and local unitary (LU) classifications of entangled pure state fermionic systems based on the spin group. Our generalization relies on the fact that there is a representation of this group acting on the fermionic Fock space, which, when restricted to fixed particle number subspaces, naturally recovers the usual SLOCC transformations. The new ingredient is the occurrence of Bogoliubov transformations of the whole Fock space, which change the particle number. The classification scheme built on the spin group naturally prohibits entanglement between states containing even and odd numbers of fermions. In our scheme the problem of the classification of entanglement types boils down to the classification of spinors where totally separable states are represented by so-called pure spinors. We construct the basic invariants of the spin group and show how some of the known SLOCC invariants are just their special cases. As an example we present the classification of fermionic systems with a Fock space based on six single particle states; an intriguing duality between two different possibilities for embedding three-qubit systems inside the fermionic ones is revealed. This duality is elucidated via an interesting connection to the configurations of wrapped membranes reinterpreted as qubits.

DOI
10.1088/1751-8113/47/11/115304

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We propose a generalization of the usual stochastic local operations and classical communications (SLOCC) and local unitary (LU) classifications of entangled pure state fermionic systems based on the spin group. Our generalization relies on the fact that there is a representation of this group acting on the fermionic Fock space, which, when restricted to fixed particle number subspaces, naturally recovers the usual SLOCC transformations. The new ingredient is the occurrence of Bogoliubov transformations of the whole Fock space, which change the particle number. The classification scheme built on the spin group naturally prohibits entanglement between states containing even and odd numbers of fermions. In our scheme the problem of the classification of entanglement types boils down to the classification of spinors where totally separable states are represented by so-called pure spinors. We construct the basic invariants of the spin group and show how some of the known SLOCC invariants are just their special cases. As an example we present the classification of fermionic systems with a Fock space based on six single particle states; an intriguing duality between two different possibilities for embedding three-qubit systems inside the fermionic ones is revealed. This duality is elucidated via an interesting connection to the configurations of wrapped membranes reinterpreted as qubits.
We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physical sound meaning of some of the direct sum structures showing up in the context of the so-called black-hole/qubit correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self-conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of Spin(16,C)\text{Spin}(16,\mathbb{C})Spin(16,C) invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with eight modes. Here we find an algebraically independent generating set of the generalized stochastic local operations and classical communication invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group E8{E}_{8}E8​ is playing in these considerations.
Based on results well known in the mathematics literature but not yet common knowledge in the physics literature, we conduct a study on three-fermionic systems with six, seven, eight, and nine single-particle states. Via introducing special polynomial invariants playing the role of entanglement measures the structure of the stochastic local operations and classical communication (SLOCC) entanglement classes is investigated. The SLOCC classes of the six- and seven-dimensional cases can elegantly be described by special subconfigurations of the Fano plane. Some special embedded systems containing distinguishable constituents are arising naturally in our formalism, namely, three-qubits and three-qutrits. In particular, the three fundamental invariants I6{I}_{6}I6​, I9{I}_{9}I9​, and I12{I}_{12}I12​ of the three-qutrits system are shown to arise as special cases of the four fundamental invariants of three-fermions with nine single-particle states.
The embedding of the nnn-qubit space into the nnn-fermion space with 2n2n2n modes is a widely used method in studying various aspects of these systems. This simple mapping raises a crucial question: Does the embedding preserve the entanglement structure? It is known that the answer is affirmative for n=2n=2n=2 and n=3n=3n=3. That is, under either local unitary (LU) operations or with respect to stochastic local operations and classical communication (SLOCC), there is a one-to-one correspondence between the two- (or three)-qubit orbits and the two- (or three)-fermion orbits with four (or six) modes. However, these results do not generalize as the mapping from the nnn-qubit orbits to the nnn-fermion orbits with 2n2n2n modes is no longer surjective for n>3n>3n>3. Here we consider the case of n=4n=4n=4. We show that, surprisingly, the mapping of orbits from qubits to fermions is injective under SLOCC, and a similar result holds under LU for generic orbits. As a by-product, we obtain a complete answer to the problem of SLOCC equivalence of pure four-qubit states.
We consider special quantum systems containing both distinguishable and identical constituents. It is shown that for these systems the Freudenthal construction based on cubic Jordan algebras naturally defines entanglement measures invariant under the group of stochastic local operations and classical communication (SLOCC). For this type of multipartite entanglement the SLOCC classes can be explicitly given. These results enable further explicit constructions of multiqubit entanglement measures for distinguishable constituents by embedding them into identical fermionic ones. We also prove that the Plücker relations for the embedding system provide a sufficient and necessary condition for the separability of the embedded one. We argue that this embedding procedure can be regarded as a convenient representation for quantum systems of particles which are really indistinguishable but for some reason they are not in the same state of some inner degree of freedom.
In this paper the entanglement of pure 3-qubit states is discussed. The local unitary (LU) polynomial invariants that are closely related to the canonical forms are constructed and the relations of the coefficients of the canonical forms are given. Then the stochastic local operations and classical communication (SLOCC) classification of the states are discussed on the basis of the canonical forms, and the symmetric canonical form of the states without 3-tangle is discussed. Finally, we give the relation between the LU polynomial invariants and SLOCC classification.
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.