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.
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 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.
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,
I9, and
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 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.
The embedding of the
n-qubit space into the
n-fermion space with
2n 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=2 and
n=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
n-qubit orbits to the
n-fermion orbits with
2n modes is no longer surjective for
n>3. Here we consider the case of
n=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.
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-
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 consider a fermionic system for which there exist a single-reference configuration-interaction (CI) expansion of the ground-state wave function that converges, albeit not necessarily rapidly, with the increasing number of particle-hole excitations. For such systems, we show that, whenever the coefficients of Slater determinants (SD) with $l\ensuremath{\le}k$ excitations can be defined with a number of free parameters ${N}_{\ensuremath{\le}k}$ bounded polynomially in
k, the ground-state energy
E only depends on a small fraction of all the wave function parameters and is the solution of equations of the coupled-cluster (CC) form. This generalizes the standard CC method, for which ${N}_{\ensuremath{\le}k}$ is bounded by a constant. Based on that result and low-rank tensor decompositions (LRTD), we discuss two possible extensions of the CC approach for wave functions with general polynomial bound for ${N}_{\ensuremath{\le}k}$. The most straightforward of those extensions uses the LRTD to represent the amplitudes of the CC cluster operator
T which, unlike in the CC case, is not truncated with respect to number of excitations, and the energy and tensor parameters are given by a LRTD-adapted version of standard CC equations. The LRTD can also be used to directly parametrize the wave function coefficients, which involves different equations of the CC form. We derive those equations for the coefficients of SD's with $l\ensuremath{\le}4$ excitations, using the CC exponential wave function ansatz with a different type of excitation operator, and a representation of the Hamiltonian in terms of excited particle and hole operators. To complete the proposed computational methods, we construct compact tensor representations of coefficients, or
T amplitudes, using superpositions of tree tensor networks which take into account different possible types of entanglement between excited particles and holes. Finally, we discuss why the proposed CC extensions are theoretically applicable at larger coupling strengths than those treatable by the standard CC method.