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arXiv:1309.0791 · Physical Review A

Four-qubit pure states as fermionic states

Lin Chen, Dragomir Ž. Ðoković, Markus Grassl, Bei Zeng

Published Nov 11, 2013• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.88.052309

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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 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.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.
We discuss the entanglement properties of symmetric states of n qubits. The Majorana representation maps a generic such state into a system of n points on a sphere. Entanglement invariants, either under local unitaries (LU) or stochastic local operations and classical communication (SLOCC), can then be addressed in terms of the relative positions of the Majorana points. In the LU case, an overcomplete set of invariants can be built from the inner product of the radial vectors pointing to these points; this is detailed for the well-documented three-qubits case. In the SLOCC case, a cross ratio of related Möbius transformations are shown to play a central role, exemplified here for four qubits. Finally, as a side result, we also analyze the manifold of maximally entangled 3 qubit state, both in the symmetric and generic case.
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.
L. Lamata et al. use an inductive approach to classify the entangled pure states of four qubits under stochastic local operations and classical communication (SLOCC) [Phys. Rev. A 75, 022318 (2007)]. The inductive method yields a priori ten entanglement superclasses, of which they discard three as empty. One of the remaining superclasses is split into two, resulting in eight superclasses of genuine four-qubit entanglement. Here, we show that two of the three discarded superclasses are in fact nonempty and should have been retained. We give explicit expressions for the canonical states of those superclasses, up to SLOCC and qubit permutations. Furthermore, we confirm that the third discarded superclass is indeed empty, yielding a total of ten superclasses of genuine four-qubit entanglement under the inductive classification scheme.