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arXiv:1306.0742

Locally maximally entangleable states and W-type states

Ri Qu, Juan Wang, Yan-ru Bao, Zong-shang Li

Published Jun 4, 2013• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1306.0742

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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.
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Absolutely maximally entangled (AME) states are maximally entangled for every bipartition of the system. They are crucial resources for various quantum information protocols. We present techniques for verifying that two AME states are equivalent concerning stochastic local operations and classical communication (SLOCC). The conjecture that for a given multipartite quantum system all AME states are SLOCC equivalent is proven false. We also show that the existence of AME states with minimal support of six or more particles results in the existence of infinitely many such non-SLOCC-equivalent states. Moreover, we present AME states which are not SLOCC equivalent to the existing AME states with minimal support.
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.
We show there are at least 28 distinct true stochastic local operations and classical communication (SLOCC) entanglement classes for four qubits by means of SLOCC invariant and semi-invariants and derive the number of degenerate SLOCC classes for nnn qubits.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.