Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1607.07357 · Physical Review A

Constructing entanglement measures for fermions

Markus Johansson, Zahra Raissi

Published Oct 13, 2016• 4 citations
arXivHTMLPDF

Abstract

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-$\frac{1}{2}$ 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.

DOI
10.1103/physreva.94.042319

Related papers

Entanglement in fermionic Fock space

Gábor Sárosi, Péter Lévay

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.

arXiv:1309.4300•Journal of Physics A Mathematical and Theoretical•Feb 28, 2014•25 citations

Entanglement classification of three fermions with up to nine single-particle states

Gábor Sárosi, Péter Lévay

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 ${I}_{6}$, ${I}_{9}$, and ${I}_{12}$ 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.

arXiv:1312.2786•Physical Review A•Apr 11, 2014•19 citations

Special entangled quantum systems and the Freudenthal construction

Péter Vrana, Péter Lévay

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.

arXiv:0902.2269•Journal of Physics A Mathematical and Theoretical•Jun 19, 2009•32 citations

Maximal entanglement of two delocalized spin- 12 particles

Markus Johansson

We describe the entanglement of two indistinguishable delocalized spin-$\frac{1}{2}$ particles in the simplest spatial configuration of three spatial modes with the constraint that at most one particle occupy each mode. It is shown that this is the only number of modes for which maximally entangled states exist in such a system. The set of entangled states, including the set of maximally entangled states, is described and different types of entanglement in terms of Bell-nonlocal correlations for different partitions of the system are identified. In particular we focus on the entangled states that are Bell local for a tripartition of the system and cannot be described as a superposition of bipartite entangled pairs of localized particles. Two entanglement invariants are constructed and it is shown that all entanglement monotones are functions of these. Furthermore, the system has a generic nontrivial local unitary symmetry with a corresponding $2\ensuremath{\pi}/3$ fractional topological phase. In addition to this a necessary and sufficient condition for the existence of maximally entangled states in systems of arbitrary numbers of delocalized particles with arbitrary spin, where at most one particle can occupy each mode, is derived.

arXiv:1509.02037•Physical Review A•Feb 23, 2016•4 citations

Multipartite entanglement in fermionic systems via a geometric measure

Behzad Lari, P. Durganandini, Pramod S. Joag

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.

arXiv:1007.2908•Physical Review A•Dec 2, 2010•5 citations

The coupled cluster method and entanglement in three fermion systems

Péter Lévay, Szilvia Nagy, Jānos Pipek +1 more

The Coupled Cluster (CC) and full CI expansions are studied for three fermions with six and seven modes. Surprisingly the CC expansion is tailor made to characterize the usual stochastic local operations and classical communication (SLOCC) entanglement classes. It means that the notion of a SLOCC transformation shows up quite naturally as a one relating the CC and CI expansions, and going from the CI expansion to the CC one is equivalent to obtaining a form for the state where the structure of the entanglement classes is transparent. In this picture, entanglement is characterized by the parameters of the cluster operators describing transitions from occupied states to singles, doubles, and triples of non-occupied ones. Using the CC parametrization of states in the seven-mode case, we give a simple formula for the unique SLOCC invariant J. Then we consider a perturbation problem featuring a state from the unique SLOCC class characterized by J≠0. For this state with entanglement generated by doubles, we investigate the phenomenon of changing the entanglement type due to the perturbing effect of triples. We show that there are states with real amplitudes such that their entanglement encoded into configurations of clusters of doubles is protected from errors generated by triples. Finally we put forward a proposal to use the parameters of the cluster operator describing transitions to doubles for entanglement characterization. Compared to the usual SLOCC classes, this provides a coarse grained approach to fermionic entanglement.

Journal of Mathematical Physics•Jan 1, 2017•2 citations
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 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.
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.
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.
We describe the entanglement of two indistinguishable delocalized spin-12\frac{1}{2}21​ particles in the simplest spatial configuration of three spatial modes with the constraint that at most one particle occupy each mode. It is shown that this is the only number of modes for which maximally entangled states exist in such a system. The set of entangled states, including the set of maximally entangled states, is described and different types of entanglement in terms of Bell-nonlocal correlations for different partitions of the system are identified. In particular we focus on the entangled states that are Bell local for a tripartition of the system and cannot be described as a superposition of bipartite entangled pairs of localized particles. Two entanglement invariants are constructed and it is shown that all entanglement monotones are functions of these. Furthermore, the system has a generic nontrivial local unitary symmetry with a corresponding $2\ensuremath{\pi}/3$ fractional topological phase. In addition to this a necessary and sufficient condition for the existence of maximally entangled states in systems of arbitrary numbers of delocalized particles with arbitrary spin, where at most one particle can occupy each mode, is derived.
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 $m\ensuremath{\geqslant}2$ subsets. Thus this measure applies to $m\ensuremath{\leqslant}2L$ 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 Coupled Cluster (CC) and full CI expansions are studied for three fermions with six and seven modes. Surprisingly the CC expansion is tailor made to characterize the usual stochastic local operations and classical communication (SLOCC) entanglement classes. It means that the notion of a SLOCC transformation shows up quite naturally as a one relating the CC and CI expansions, and going from the CI expansion to the CC one is equivalent to obtaining a form for the state where the structure of the entanglement classes is transparent. In this picture, entanglement is characterized by the parameters of the cluster operators describing transitions from occupied states to singles, doubles, and triples of non-occupied ones. Using the CC parametrization of states in the seven-mode case, we give a simple formula for the unique SLOCC invariant J. Then we consider a perturbation problem featuring a state from the unique SLOCC class characterized by J≠0. For this state with entanglement generated by doubles, we investigate the phenomenon of changing the entanglement type due to the perturbing effect of triples. We show that there are states with real amplitudes such that their entanglement encoded into configurations of clusters of doubles is protected from errors generated by triples. Finally we put forward a proposal to use the parameters of the cluster operator describing transitions to doubles for entanglement characterization. Compared to the usual SLOCC classes, this provides a coarse grained approach to fermionic entanglement.