Conference Agenda

Quantum information science attempts to use quantum phenomena as non-classical resources to perform new communication protocols and develop new computational paradigms. The theoretical advantages of quantum communication and quantum algorithms were proved in the 80-90’s and nowadays experimentalists are working on making that technology available. One of the quantum phenomena responsible for the speed up of quantum algorithms and the security of quantum communication is entanglement. A system of m-particules (a multipartite quantum state) is said to be entangled when the state of a particle of the system cannot be described independently of the others. Entanglement is a consequence of the superposition principle in quantum physics which mathematically translates to the fact that the Hilbert space of a composite system is the tensor product of the Hilbert space of each part. Algebraic geometry entered the study of entanglement of multipartite systems when it was both noticed in the early 2000s that the rank of tensors could be interpreted as a measure of entanglement and also that invariant theory could be used to distinguish different classes of entanglement. Since then a large amount of research has been produced in the mathematical-physics literature to classify and/or measure entanglement using techniques from classical invariant theory, representation theory, and geometric invariant theory. Because of the exponential growth of the dimension of the multipartite Hilbert spaces, when the number of factors increases, only a few examples of explicit classifications are known. Therefore to study entanglement in larger Hilbert spaces, techniques from tensor decomposition and asymptotic geometry of tensors have been recently introduced. These techniques establish new connections between entanglement and algebraic complexity theory.

This minisymposium on applications of algebraic geometry to quantum information will propose talks by mathematicians and physicists who have been studying entanglement from a geometrical perspective with classical and more recent techniques.

Despite considerable interest in recent years, understanding of quantum correlations in multipartite finite dimensional quantum systems is still incomplete. I will consider a simple scenario in which we have access to the results of all one-particle measurements of such system. The aim is to understand how much information about quantum correlations is encoded in this data. It turns out that mathematically consistent way of studying this problem involves methods that are used in classical mechanics to describe phase spaces with symmetries, symplectic geometry and geometric invariant theory. In this talk I will discuss these methods and show their usefulness to our problem. This is a joint work with Marek Kus, Tomasz Maciazek and Michal Oszmaniec.

Although the convex geometric questions of multipartite entanglement of mixed states are not directly linked to the algebraic-geometric questions of multipartite entanglement of pure states in general, there are some exotic exceptions, such as in the case of systems of three qubits. In this talk, we briefly review the lattice of the entanglement classification of three-qubit mixed states, based on the SLOCC classification of state vectors. The latter can be given in terms of a Freudenthal Triple System: state vectors of different SLOCC-class are in one-to-one correspondence with FTS-elements of different rank, which can be characterized by vanishing conditions of a set of LSL(2) (Local Special Linear) covariants. From these covariants we construct a larger set of LU(2) (Local Unitary) invariant polynomials on pure states, showing the proper vanishing conditions to be indicator functions for all the 21 partial separability classes for mixed states.

A great circle $T_1$ on a sphere is {sl non-displacable}, as any two great circles on a sphere $S_2=CP^1$ do intersect. This statement can be generalized for higher dimensions: Cho showed (2004) that any great torus $T_K$ embedded in $CP^K$ is non displacable for any integer $K$. Above fact implies that for any choice of two orthogonal basis in $N=K+1$ dimensional space there exists a vector mutually coherent with respect to both bases, so that the sum of the entropies characterizing measurements in both bases is maximal and equal to 2log N. Bounds for the sum of entropies obtained for more than two ortogonal measurements are also discussed.

A related result by Tamarkin (2008) states that real projective space $RP^K$ embedded in $CP^K$ is non-displacable. Making use of this statement for $K=3$ we show that for any two-qubit unitary gate U in U(4) there exists a mutually entangled pure quantum state, which is maximally entangled with respect to the standard computational product basis, $B={|00>, |01>, |10>, |11>}$, and also with respect to the rotated basis $UB$.

According to a recent idea bulk space-time is an emergent quantity coming from entanglement patterns of the boundary. By studying the space of geodesics in $AdS_3$, and quantizing a parametrized family of geodesic motion we show that scattering data is related to boundary entanglement of the $CFT_2$ vaccum. For the parametrized family of geodesics we calculate the Berry curvature living on the space of geomdesics. As a result we recover the Crofton form with a quantum coefficient related to the scattering energy. We argue that, by applying results coming from Algebraic Scattering Theory, this idea can be generalized for more general states and possibly for the general $AdS_{n+1}/CFT_n$ correspondence.