Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory Ann. Henri Poincar´e Online First 2018 Springer International Publishing AG, part of Springer Nature Annales Henri Poincar´ e https://doi.org/10.1007/s00023-018-0682-6 Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory Jim Bryan, Zinovy Reichstein and Mark Van Raamsdonk Abstract. We study a question which has natural interpretations both in quantum mechanics and in geometry. Let V ,...,V be complex vector 1 n spaces of dimension d ,...,d and let G =SL ×··· × SL . Geometri- 1 n d d 1 n cally, we ask: Given (d ,...,d ), when is the geometric invariant theory 1 n quotient P(V ⊗· · ·⊗ V )//G non-empty? This is equivalent to the quan- 1 n tum mechanical question of whether the multipart quantum system with Hilbert space V ⊗· · ·⊗ V has a locally maximally entangled state, i.e., 1 n a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if R(d ,...,d )  0where 1 n k 2 R(d ,...,d )= d + (−1) (gcd(d ,...,d )) . 1 n i i i i k=1 1i <···<i n http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Henri Poincaré Springer Journals

Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Physics; Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Elementary Particles, Quantum Field Theory
ISSN
1424-0637
eISSN
1424-0661
D.O.I.
10.1007/s00023-018-0682-6
Publisher site
See Article on Publisher Site

Abstract

Ann. Henri Poincar´e Online First 2018 Springer International Publishing AG, part of Springer Nature Annales Henri Poincar´ e https://doi.org/10.1007/s00023-018-0682-6 Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory Jim Bryan, Zinovy Reichstein and Mark Van Raamsdonk Abstract. We study a question which has natural interpretations both in quantum mechanics and in geometry. Let V ,...,V be complex vector 1 n spaces of dimension d ,...,d and let G =SL ×··· × SL . Geometri- 1 n d d 1 n cally, we ask: Given (d ,...,d ), when is the geometric invariant theory 1 n quotient P(V ⊗· · ·⊗ V )//G non-empty? This is equivalent to the quan- 1 n tum mechanical question of whether the multipart quantum system with Hilbert space V ⊗· · ·⊗ V has a locally maximally entangled state, i.e., 1 n a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if R(d ,...,d )  0where 1 n k 2 R(d ,...,d )= d + (−1) (gcd(d ,...,d )) . 1 n i i i i k=1 1i <···<i n

Journal

Annales Henri PoincaréSpringer Journals

Published: May 31, 2018

References

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