We discuss a definition of maximally entangled states in terms of maximum uncertainty of corresponding measurements. We describe a method of construction of bases of maximally entangled states. The entangled states that can be obtained from the maximally entangled states by means of SLOCC (stochastic local operations assisted by classical communications) we consider as semistable vectors. We discuss a measure of entanglement expressed in terms of a geometric invariant.

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arXiv:1708.01645 · Annales Henri Poincaré

Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

Jim Bryan, Zinovy Reichstein, Mark Van Raamsdonk B

Published May 31, 2018• 16 citations

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DOI: 10.1007/s00023-018-0682-6

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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.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.

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{\cal {PT}}

symmetric unitary operations. If two parties share a maximally entangled state, then under local

{\cal {PT}}

symmetric unitary evolution the entropy of entanglement for pure bipartite states does not remain invariant. Furthermore, we show that if one of the party has access to

{\cal {PT}}

-symmetric quantum world, then a maximally entangled state in usual quantum theory appears as a non-maximally entangled states for the other party. This we call as the "entanglement mismatch" effect which can lead to the violation of the no-signaling condition.