Conference Agenda

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Session Overview
Session
MS169, part 1: Applications of Algebraic geometry to quantum information
Time:
Friday, 12/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-111
30 seats, 56m^2

Presentations
10:00am - 12:00pm

Applications of Algebraic geometry to quantum information

Chair(s): Frédéric Holweck (University of Bourgogne Franche-Comté)

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.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Tensor rank, border rank, multiplicativity and entanglement

Fulvio Gesmundo
University of Copenhagen

Matrix rank has several different generalizations to the setting of tensors which are natural measures of the entanglement of the quantum state described by the tensor. Some recent results show that, unlike the classical matrix rank, these generalizations are not multiplicative under the operation of tensor product. We describe this phenomenon and some of its consequences in a general geometric framework, which allows for further generalizations.

 

Hyperdeterminants from $E_8$

Luke Oeding
Auburn University

Projective duality can be used to study singularities. A matrix is singular precisely when its determinant vanishes, or equivalently, when it belongs to the projective dual to rank-one matrices, the Segre variety. A higher order tensor is singular when its hyperdeterminant vanishes, i.e. when it belongs to the dual of a higher order Segre product. Efficient expressions for hyperdeterminants are mostly unknown and they are difficult to compute. We describe a connection to the exceptional Lie algebra $E_8$. This gives an interpretation of certain hyperdeterminants (of formats $2times 2times 2times 2$ and $3times 3times 3$) and certain discriminants (of the Grassmannians $Gr(3,9)$ and $Gr(4,8)$) as sparse $E_8$-discriminants. We give expressions of these high degree invariants in terms of lower degree fundamental invariants, which allow evaluation, and may be useful for Quantum Information Theory as measures of entanglement. This is joint work with Frédéric Holweck.

 

Tensor network representations from the geometry of entangled states

Matthias Christandl
University of Copenhagen

Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.

 

Tensor scaling, quantum marginals, and moment polytopes

Michael Walter
University of Amsterdam

Given a collection of quantum states of individual particles, are they compatible with a global quantum state? I will give an introduction to the mathematics of this "quantum marginal problem" (which has applications from entanglement theory to quantum chemistry), explain its connection to geometric invariant theory, and present an efficient algorithmic solution. Our numerical algorithm applies more generally to the problems of deciding semistability and computing moment polytopes.