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arXiv:1901.08847 · Journal of Physics A Mathematical and Theoretical

Characterizing multipartite entanglement classes via higher-dimensional embeddings

Christina Ritz, Cornelia Spee, Otfried Gühne

Published Jul 4, 2019• 39 citations
arXivHTMLPDF

Abstract

Abstract Witness operators are a central tool to detect entanglement or to distinguish among the different entanglement classes of multiparticle systems, which can be defined using stochastic local operations and classical communication (SLOCC). We show a one-to-one correspondence between general SLOCC witnesses and a class of entanglement witnesses in an extended Hilbert space. This relation can be used to derive SLOCC witnesses from criteria for full separability of quantum states; moreover, given SLOCC witnesses can be viewed as entanglement witnesses. As applications of this relation we discuss the calculation of overlaps between different SLOCC classes and the SLOCC classification in -dimensional systems.

DOI
10.1088/1751-8121/ab2f54

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Abstract Witness operators are a central tool to detect entanglement or to distinguish among the different entanglement classes of multiparticle systems, which can be defined using stochastic local operations and classical communication (SLOCC). We show a one-to-one correspondence between general SLOCC witnesses and a class of entanglement witnesses in an extended Hilbert space. This relation can be used to derive SLOCC witnesses from criteria for full separability of quantum states; moreover, given SLOCC witnesses can be viewed as entanglement witnesses. As applications of this relation we discuss the calculation of overlaps between different SLOCC classes and the SLOCC classification in -dimensional systems.
We introduce an operational entanglement classification of symmetric mixed states for an arbitrary number of qubits based on stochastic local operations assisted with classical communication (SLOCC operations). We define families of SLOCC entanglement classes successively embedded into each other, we prove that they are of nonzero measure, and we construct witness operators to distinguish them. Moreover, we discuss how arbitrary symmetric mixed states can be realized in the laboratory via a one-to-one correspondence between well-defined sets of controllable parameters and the corresponding entanglement families.
The coefficient matrix is an efficient tool in entanglement classification under stochastic local operation and classical communication (SLOCC). In this work, we take all the ranks of the coefficient matrices into account in the method of entanglement classification, and give the entanglement classification procedure for arbitrary-dimensional multipartite pure states under SLOCC. As a main application, we study the entanglement classification for quantum systems in the Hilbert space H = C2 ⊗ C2 ⊗ C2 ⊗ Cd in terms of the ranks of the coefficient matrices.
We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.
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 develop a novel method in classifying the multipartite entanglement state of 2×N ×N under stochastic local operations and classical communication (SLOCC). In this method, all inequivalent classes of a genuine entangled state can be assorted directly without knowing the classification information of lower dimension ones for any given dimension N. It also gives a nature explanation for the non-local parameters remaining in the entanglement classes when N ≥ 4. Entanglement is at the heart of the quantum information theory (QIT) and is now thought as a physical resource to realize quantum information tasks, such as quantum cryptography [1, 2], superdense coding [3, 4], and quantum computation [5]. The study of entanglement improves our further understanding of quantum non-locality [6] and also is of particular interest in QIT. One such case is the classification of multipartite entanglement. Two quantum states can be employed to implement the same tasks in QIT and are thought to be equivalent while they are mutually convertible by Stochastic Local Operations and Classical Communication (SLOCC) [7]. Nevertheless, in practice the classification of multipartite entanglement and in high dimensions in the Hilbert space seems to be a formidable task [8]. Matrix decomposition keeps to be a useful tool as in the two-partite case [9, 10]. A widely adopted philosophy in dealing with this issue is first to know the classification of the state in lower dimension (or less partite) and then extend to the higher dimension case [11, 12] (or more partite [13, 14]) in an inductive way. However, nontrivial aspect emerges as the dimension increases, i.e. some non-local parameters may nest in the entangled states [15, 16]. In literature, in
We study multipartite entanglement under stochastic local operations and classical communication (SLOCC) and propose the entanglement classification under SLOCC for arbitrary-dimensional multipartite (nnn-qudit) pure states via the rank of coefficient matrix, together with the permutation of qudits. The ranks of the coefficient matrices have been proved to be entanglement monotones. The entanglement classification of the 2⊗2⊗2⊗42 \otimes 2 \otimes 2 \otimes 42⊗2⊗2⊗4 system is discussed in terms of the generalized method, and 22 different SLOCC families are found.