Constraints on Multiparticle Entanglement

由量子粒子（如电子）组成的系统可以具有称为“纠缠”的特性。纠缠的一个基本结果是，对不同粒子的测量可以以经典粒子不可能的方式相关联。纠缠的技术后果包括建立安全的量子通信系统的可能性，以及量子计算机将比经典计算机更有效地解决某些问题。但纠缠有一些局限性。例如，如果两个粒子完全纠缠在一起，那么它们中的任何一个都不能与其他粒子纠缠。 这种现象被称为“纠缠的一夫一妻制”，并对某些固体的基态（最低能量）有影响。关于纠缠的这一基本观点有助于解释某些材料的技术价值行为，例如反铁磁体，它们可用作磁场传感器。该项目的目标是更全面地了解量子系统中多个粒子之间纠缠的限制。除了对量子固体基态的影响外，这些结果还可能揭示量子通信网络容量或量子计算机计算能力的根本限制。该项目将涉及研究科学家和研究生的培训，并将与PI在UC San Diego教授的量子信息和计算研究生课程相协调。三粒子系统中的一夫一妻制是线性一夫一妻制不等式的结果，该不等式表明粒子A和B之间的纠缠加上粒子A和C之间的纠缠不大于A和粒子对BC之间的纠缠。这个不等式可以通过量子比特（具有二维内部自由度的粒子，如自旋1/2粒子）之间的纠缠的几种度量来满足：例如，平方并发度和平方负性。然而，在后一种情况下，这并不是完整的故事：平方负性满足一个更具限制性的非线性不等式。该项目将开发方法（一些来自计算代数几何和线性矩阵不等式）来确定这种新的非线性纠缠约束。这些还将包括对纠缠的对称集合的约束，例如，在A和B，B和C，C和A之间，这将是新的，不同于原始的一夫一妻制约束。将开发相关的方法来导出对更高维粒子的非线性约束（例如，“qutrits”而不是qubit），超过三个粒子，以及超过两个粒子之间的纠缠度，如3-tangle。

Systems that are composed of quantum particles, such as electrons, can have a property called "entanglement". A fundamental consequence of entanglement is that the measurements on distinct particles can be correlated in ways that would be impossible for classical particles. Technological consequences of entanglement include the possibility of building secure quantum communication systems, and quantum computers which will solve certain problems more efficiently than classical computers. But entanglement has some limitations. For example, if two particles are completely entangled with each other, neither can be entangled at all with any other particle. This phenomenon is called "monogamy of entanglement" and has implications for the ground (lowest energy) state of some solids. This fundamental point about entanglement helps explain the technologically valuable behavior of some materials, such as antiferromagnets, which are useful as magnetic field sensors. The goal of this project is to more fully understand the limitations on entanglements between multiple particles in quantum systems. In addition to implications for ground states of quantum solids, the results may reveal fundamental limits on the capacity of quantum communication networks, or the computational power of quantum computers. The project will involve training of research scientists and graduate students, and will be coordinated with a graduate course on quantum information and computation taught at UC San Diego by the PI. Monogamy in three particle systems is a consequence of a linear monogamy inequality stating that the entanglement between particles A and B, plus the entanglement between particles A and C is no greater than the entanglement between A and the pair of particles BC. This inequality is satisfied by several measures of entanglement between qubits (particles with a 2 dimensional internal degree of freedom, like spin 1/2 particles): the squared concurrence and the squared negativity, for example. This is not the complete story in the latter case, however: a more restrictive nonlinear inequality is satisfied by the squared negativity. This project will develop methods (some from computational algebraic geometry and linear matrix inequalities) to determine such new, nonlinear constraints on entanglement. These will also include constraints on symmetrical sets of entanglements, e.g., between A and B, B and C, C and A, which will be new, and distinct from the original monogamy constraints. Related methods will be developed to derive nonlinear constraints on higher dimensional particles (e.g., "qutrits" rather than qubits), more than three particles, and also measures of entanglement among more than two particles, like the 3-tangle.