Constraints on Multiparticle Entanglement

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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.