Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

Bryan Jim; Reichstein Zinovy; Van Raamsdonk Mark

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Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

机译：通过几何不变理论存在局部最大缠绕量子状态

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We study a question which has natural interpretations both in quantum mechanics and in geometry. Let V1,..., Vn be complex vector spaces of dimension d1,..., dn and let G = SLd1 x center dot center dot center dot xSLdn. Geometrically, we ask: Given (d1,..., dn), when is the geometric invariant theory quotient P(V1. center dot center dot center dot. Vn)// G non- empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space V1. center dot center dot center dot. Vn 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(d1,..., dn) 0 where R(d1,..., dn) = i di + n k= 1 (- 1) k 1 i1 center dot center dot center dot ik n (gcd(di1,..., dik)) 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.

机译：我们研究了一个问题，它在量子力学和几何中具有自然解释。让V1，...，VN是维度D1，...，DN的复杂向量空间，...，DN和设法G = SLD1 x中心点中心点中心点XSLDN。几何上，我们问：给定（d1，...，dn），什么时候是几何不变理论商P（V1。中心点中心点中心点。VN）// g非空？这相当于Quantum机械问题，用于HILBERT SPACE V1的多部分量子系统。中心DOT中心点中心点。 Vn具有局部最大纠缠的状态，i。 e，这样的状态，使得每个基本系统的密度矩阵是身份的倍数。我们表明，如果r（d1，...，dn）= i di + nk = 1（ - 1）k 1 i1＆lt ;中心点中心点中心点＆ IK N（GCD（DI1，...，DIK））2。我们还提供了一种简单的递归算法，它决定了问题的答案，并在非空案件中计算所得商的维度。

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《Annales Henri Poincare》 |2018年第8期|共21页
作者
Bryan Jim; Reichstein Zinovy; Van Raamsdonk Mark;
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Univ British Columbia Dept Math Room 121 1984 Math Rd Vancouver BC V6T 1Z2 Canada;

Univ British Columbia Dept Math Room 121 1984 Math Rd Vancouver BC V6T 1Z2 Canada;

Univ British Columbia Dept Phys &

Astron 6224 Agr Rd Vancouver BC V6N 3T8 Canada;

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中图分类理论物理学;
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1. Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory [J] . Bryan Jim, Reichstein Zinovy, Van Raamsdonk Mark Annales Henri Poincare . 2018,第8期

机译：通过几何不变理论存在局部最大缠绕量子状态
2. On the non-existence of the micro-causal sector of Lorentz non-invariant quantum field theory on a continuum [J] . Rashidi R, Salehi H Physics Letters, B. Nuclear Physics and High Energy Physics . 2007,第5a6期

机译：关于连续论上的洛伦兹非不变量子场论的微观因果扇区的不存在
3. Editor's preface for the topical issue on "Geometric invariants and quantum field theory" [J] . Bruzzo U. Journal of geometry and physics . 2011,第7期

机译：关于“几何不变量与量子场论”的主题的编者序
4. A Quantum TITO Diversity Transmission Scheme with Quantum Teleportation of Non-maximally Entangled Bell State [C] . Shi Jinjing, Shi Ronghua, Guo Ying, 11th IEEE International Conference on Trust, Security And Privacy In Computing And Communications, 11th IEEE International Conference On Ubiquitous Computing and Communications . 2012

机译：非最大纠缠贝尔态的量子隐形传态的量子TITO分集传输方案
5. Topics in ergodic theory: Existence of invariant elements and ergodic decompositions of Banach lattices. [D] . Prabaharan, Kanagarajah. 1991

机译：遍历理论的主题：Banach格的不变元素的存在和遍历分解。
6. General existence of locally distinguishable maximally entangled states only with two-way classical communication [O] . Guojing Tian, Xia Wu, Ya Cao, -1

机译：仅存在双向经典通信的局部可区分最大纠缠态的一般存在
7. Existence of locally maximally entangled quantum states via geometric invariant theory [O] . Bryan, Jim, Reichstein, Zinovy, Van Raamsdonk, Mark 2017

机译：局部最大纠缠量子态的几何存在性不变理论

Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

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