Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory We study a question which has natural interpretations both in quantum mechanics and in geometry. Let $$V_{1},\cdots , V_{n}$$ V 1 , ⋯ , V n be complex vector spaces of dimension $$d_{1},\ldots ,d_{n}$$ d 1 , … , d n and let $$G= {\text {SL}}_{d_{1}} \times \cdots \times {\text {SL}}_{d_{n}}$$ G = SL d 1 × ⋯ × SL d n . Geometrically, we ask: Given $$(d_{1},\ldots ,d_{n})$$ ( d 1 , … , d n ) , when is the geometric invariant theory quotient $$\mathbb {P}(V_{1}\otimes \cdots \otimes V_{n})/\!/G$$ P ( V 1 ⊗ ⋯ ⊗ V n ) / / G non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space $$V_{1}\otimes \cdots \otimes V_{n}$$ V 1 ⊗ ⋯ ⊗ V n has a locally maximally entangled state, i.e., 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 $$R(d_{1},\cdots ,d_{n})\geqslant 0$$ R ( d 1 , ⋯ , d n ) ⩾ 0 where $$\begin{aligned} R(d_{1},\cdots ,d_{n}) = \prod _{i}d_{i} +\sum _{k=1}^{n} (-1)^{k}\sum _{1\leqslant i_{1}<\cdots <i_{k}\leqslant n} \left( \gcd (d_{i_{1}},\cdots ,d_{i_{k}}) \right) ^{2}. \end{aligned}$$ R ( d 1 , ⋯ , d n ) = ∏ i d i + ∑ k = 1 n ( - 1 ) k ∑ 1 ⩽ i 1 < ⋯ < i k ⩽ n gcd ( d i 1 , ⋯ , d i k ) 2 . We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases. 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

We study a question which has natural interpretations both in quantum mechanics and in geometry. Let $$V_{1},\cdots , V_{n}$$ V 1 , ⋯ , V n be complex vector spaces of dimension $$d_{1},\ldots ,d_{n}$$ d 1 , … , d n and let $$G= {\text {SL}}_{d_{1}} \times \cdots \times {\text {SL}}_{d_{n}}$$ G = SL d 1 × ⋯ × SL d n . Geometrically, we ask: Given $$(d_{1},\ldots ,d_{n})$$ ( d 1 , … , d n ) , when is the geometric invariant theory quotient $$\mathbb {P}(V_{1}\otimes \cdots \otimes V_{n})/\!/G$$ P ( V 1 ⊗ ⋯ ⊗ V n ) / / G non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space $$V_{1}\otimes \cdots \otimes V_{n}$$ V 1 ⊗ ⋯ ⊗ V n has a locally maximally entangled state, i.e., 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 $$R(d_{1},\cdots ,d_{n})\geqslant 0$$ R ( d 1 , ⋯ , d n ) ⩾ 0 where $$\begin{aligned} R(d_{1},\cdots ,d_{n}) = \prod _{i}d_{i} +\sum _{k=1}^{n} (-1)^{k}\sum _{1\leqslant i_{1}<\cdots <i_{k}\leqslant n} \left( \gcd (d_{i_{1}},\cdots ,d_{i_{k}}) \right) ^{2}. \end{aligned}$$ R ( d 1 , ⋯ , d n ) = ∏ i d i + ∑ k = 1 n ( - 1 ) k ∑ 1 ⩽ i 1 < ⋯ < i k ⩽ n gcd ( d i 1 , ⋯ , d i k ) 2 . We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases.

Journal

Annales Henri PoincaréSpringer Journals

Published: May 31, 2018

References

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