Entangled many-body states are an essential resource for quantum computing and interferometry. Determining the type of entanglement present in a system usually requires access to an exponential number of parameters. We show that in the case of pure, multiparticle quantum states, features of the global entanglement can already be extracted from local information alone. This is achieved by associating any given class of entanglement with an entanglement polytope-a geometric object that characterizes the single-particle states compatible with that class. Our results, applicable to systems of arbitrary size and statistics, give rise to local witnesses for global pure-state entanglement and can be generalized to states affected by low levels of noise.

Abstract:We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing methods involving Mumford's geometric invariant theory (GIT). We concentrate on actions of unipotent groups H, and define sets of stable points X s and semistable points X ss , often explicitly computable via the methods of reductive GIT, which reduce to the standard definitions due to Mumford in the case of reductive actions. We compare these with definitions in the literature. Results include (1) a geometric criterion determining whether or not a ring of invariants is finitely generated, (2) the existence of a geometric quotient of X s , and (3) the existence of a canonical "enveloping quotient" variety of X ss , denoted X//H, which (4) has a projective completion given by a reductive GIT quotient and (5) is itself projective and isomorphic to Proj(k[X] H ) when k [X] H is finitely generated.

Let U be a graded unipotent group over the complex numbers, in the sense that it has an extensionÛ by the multiplicative group such that the action of the multiplicative group by conjugation on the Lie algebra of U has all its weights strictly positive. Given any action of U on a projective variety X extending to an action ofÛ which is linear with respect to an ample line bundle on X, then provided that one is willing to replace the line bundle with a tensor power and to twist the linearisation of the action ofÛ by a suitable (rational) character, and provided an additional condition is satisfied which is the analogue of the condition in classical geometric invariant theory (GIT) that there should be no strictly semistable points for the action, we show that theÛ -invariants form a finitely generated graded algebra; moreover, the natural morphism from the semistable subset of X to the enveloping quotient is surjective and expresses the enveloping quotient as a geometric quotient of the semistable subset. Applying this result with X replaced by its product with the projective line gives us a projective variety which is a geometric quotient byÛ of an invariant open subset of the product of X with the affine line and contains as an open subset a geometric quotient of a U -invariant open subset of X by the action of U . Furthermore, these open subsets of X and its product with the affine line can be described using criteria similar to the Hilbert-Mumford criteria in classical GIT.

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Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.Comment: 51 pages, 7 figure