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arXiv:1304.6275 · Physical Review A

Relationship among locally maximally entangleable states, W states, and hypergraph states under local unitary transformations

Ri Qu, Yi-Ping Ma, Bo Wang, Yan-ru Bao

Published May 29, 2013• 15 citations
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Abstract

Kruszynska and Kraus [Phys. Rev. A 79, 052304 (2009)] have recently introduced the so-called locally maximally entangleable (LME) states of $n$ qubits which can be maximally entangled with local auxiliary qubits using controlled operations. We characterize the local entangleability of hypergraph states and $W$ 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 $W$ state of $n$ qubits is not LME; and (iv) no hypergraph state of $n$ qubits can be converted into the $W$ state under local unitary transformations. Moreover, we also present an approach for encoding weighted hypergraphs into LME states.

DOI
10.1103/physreva.87.052331

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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.
Several entanglement measures are used to define equivalence classes in the set of hypergraph states of three qubits. Our classifications reveal that (i) under local unitary transformations, hypergraph states of three qubits are split into six classes and only one of them is not equivalent to any graph state; (ii) under stochastic local operations with classical communication, for the single-copy case hypergraph states of three qubits, partitioned into five classes which cannot be converted into a WWW state, are equivalent to graph states; and (iii) when bipartite entanglement in three qubits is considered, hypergraph states of three qubits are split into five classes and only one of them has the same entangled graph as the WWW state.
Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well--established notion of graph states. Mathematically, they can conveniently be described by a hypergraph that indicates a possible generation procedure of these states; alternatively, they can also be phrased in terms of a non-local stabilizer formalism. In this paper, we explore the entanglement properties and nonclassical features of hypergraph states. First, we identify the equivalence classes under local unitary transformations for up to four qubits, as well as important classes of five- and six-qubit states, and determine various entanglement properties of these classes. Second, we present general conditions under which the local unitary equivalence of hypergraph states can simply be decided by considering a finite set of transformations with a clear graph-theoretical interpretation. Finally, we consider the question whether hypergraph states and their correlations can be used to reveal contradictions with classical hidden variable theories. We demonstrate that various noncontextuality inequalities and Bell inequalities can be derived for hypergraph states.
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.
In this paper we investigate the entanglement properties of the class of $\ensuremath{\pi}$--locally-maximally-entangleable $(\ensuremath{\pi}$-LME) states, which are also known as the real equally weighted states or the hypergraph states. The $\ensuremath{\pi}$-LME states comprise well-studied classes of quantum states (e.g., graph states) and exhibit a large degree of symmetry. Motivated by the structure of LME states, we show that the capacity to (efficiently) determine if a $\ensuremath{\pi}$-LME state is entangled would imply an efficient solution to the Boolean satisfiability problem. More concretely, we show that this particular problem of entanglement detection, phrased as a decision problem, is NP\mathsf{NP}NP-complete. The restricted setting we consider yields a technically uninvolved proof, and illustrates that entanglement detection, even when quantum states under consideration are highly restricted, still remains difficult.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.