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arXiv:1801.03508 · Quantum

Locally Maximally Entangled States of Multipart Quantum Systems

Jim Bryan, Samuel Leutheusser, Zinovy Reichstein, Mark Van Raamsdonk B

Published Jan 6, 2019• 19 citations
arXivHTMLPDF

Abstract

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to~[J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491-2511. MR3830220], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions (d1,d2,…,dn) , and computes the dimension of the space SLME/K of LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions (2,A,B) and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points in SLME/K correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup S⊂SL⁡(d1,C)×⋯×SL⁡(dn,C) for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

DOI
10.22331/q-2019-01-06-115

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For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to~[J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491-2511. MR3830220], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions (d1,d2,…,dn) , and computes the dimension of the space SLME/K of LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions (2,A,B) and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points in SLME/K correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup S⊂SL⁡(d1,C)×⋯×SL⁡(dn,C) for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.
Kruszynska and Kraus [Phys. Rev. A 79, 052304 (2009)] have recently introduced the so-called locally maximally entangleable (LME) states of nnn qubits which can be maximally entangled with local auxiliary qubits using controlled operations. We characterize the local entangleability of hypergraph states and WWW states using the approach of Kruszynska and Kraus. We show that (i) all hypergraph states are LME; (ii) hypergraph states and LME states are not equivalent under local unitaries; (iii) a WWW state of nnn qubits is not LME; and (iv) no hypergraph state of nnn qubits can be converted into the WWW state under local unitary transformations. Moreover, we also present an approach for encoding weighted hypergraphs into LME states.
We investigate the relationship between locally maximally entangleable (LME) states and W-type states which are equivalent to a W state under stochastic local operations and classical communication (SLOCC). We prove that (i) some special W-type states of three qubits are LME; and (ii) no W-type state of four and more qubits is LME. Our results show that the W state of four and more qubits cannot be prepared by using LME states under SLOCC. However, for three qubits some special LME states can be used to prepare the W state under SLOCC.
Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for three and generic four qubit states. Here, we consider the non-generic four qubit states and analyze their properties regarding local transformations. As already the most coarse grained classification, due to Stochastic LOCC (SLOCC), of four qubit states is much richer than in case of three qubits, the investigation of possible LOCC transformations is correspondingly more difficult. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify here three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Although the possibility of transforming states into local-unitary inequivalent states by LOCC turns out to be very rare, we identify those states (with exception of the latter class) which are in the MES and those, which can be obtained (transformed) non-trivially from (into) other states respectively. These investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states.
In quantum information theory, maximally entangled states, specifically locally maximally entangled (LME) states, are essential for quantum protocols. While many focus on bipartite entanglement, applications such as quantum error correction and multiparty secret sharing rely on multipartite entanglement. These LME states naturally appear in the invariant subspaces of tensor products of irreducible representations of the symmetric group SnS_nSn​, called Kronecker subspaces, whose dimensions are the Kronecker coefficients. A Kronecker subspace is a space of multipartite LME states that entangle high-dimensional Hilbert spaces. Although these states can be derived from Clebsch-Gordan coefficients of SnS_nSn​, known methods are inefficient even for small nnn. A quantum-information-based alternative comes from entanglement concentration protocols, where Kronecker subspaces arise in the isotypic decomposition of multiple copies of entangled states. Closed forms have been found for the multiqubit WWW-class states, but not in general. This thesis extends that approach to any multiqubit system. We first propose a graphical construction called W-state Stitching, where multiqubit entangled states are represented as tensor networks built from WWW states. By analyzing the isotypic decomposition of copies of these graph states, corresponding graph Kronecker states can be constructed. In particular, graph states of generic multiqubit systems can generate any Kronecker subspace. We explicitly construct bases for three- and four-qubit systems and show that the W-stitching technique also serves as a valuable tool for multiqubit entanglement classification. These results may open new directions in multipartite entanglement resource theories, with bipartite and tripartite WWW states as foundational elements, and asymptotic analysis based on Kronecker states.
Abstract The classification of multipartite entanglement is essential as it serves as a resource for various quantum information processing tasks. This study concerns a particular class of highly entangled multipartite states, the so-called absolutely maximally entangled (AME) states. These are characterized by maximal entanglement across all possible bipartitions. In particular we analyze the local unitary (LU) equivalence among AME states using invariants. One of our main findings is that the existence of special irredundant orthogonal arrays implies the existence of an infinite number of equivalence classes of AME states constructed from these. In particular, we show that there are infinitely many LU inequivalent three-party AME states for local dimension d > 2 and five-party AME states for d ⩾ 2 .
Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis, the entanglement of symmetric multipartite states is categorized, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalization of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n-qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterization of entanglement equivalence classes such as stochastic local operations and classical communication (SLOCC) and the degeneracy configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Möbius transformations of complex analysis, which allows for a clear visualization of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to five qubits are derived. Well-known entanglement classification schemes such as the four qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.