Quantum Foundations

Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model and the dimension of the underlying Hilbert spaces, various notions of sets of quantum correlations can be defined. This talk is devoted to the separations of some of these sets via simple ideas in quantum information theory, namely self-testing and entanglement embezzlement.

"Quantum computing requires the ability to manipulate large nonclassical quantum systems. As we are far from any useful quantum computing advantage, certifying this ability is an important benchmark to assess progress toward this goal. This can be done using the nonlocal nature of quantum correlations, which allows to certify a non-trusted experimental apparatus from its input/output behaviour in a device independent way. It first requires to introduce the concept of Genuine Multipartite Nonlocality (GMNL) of size n, which designate systems which nonlocality cannot be understood an obtained from many states composed of n − 1 (or less) constituents. The first historical definition of GMNL, proposed by Svetlichny, is ill-defined when used to assess the large nonclassical nature of quantum systems, as it predicts that maximal GMNL states can be obtain from bipartite sources only. A more appropriate re-definition of that concept, called LOSR-GMNL, was proposed recently [arXiv:2105.09381]. However, it is not satisfactory in all experimental situations, as it cannot (by design) capture potential communications between the systems which could occur in some realistic experimental systems (e.g., many-body systems) – which Svetlichny definition captures in a naïve way. In this talk, I will propose a new alternative re-definition solving this issue, called Communication-Genuine Multipartite Nonlocality of length t (C-GMNL). It is based on a model inspired from synchronous distributed computing, that involves t communications steps along a graph. I will show that (i) the GHZ state is maximally nonlocal according to this C-GMNL definition, (ii) the cluster state is trivial in this C-GMNL definition but that (iii) the cluster state is maximally difficult in the LOSR-GMNL definition. Hence, some complicated LOSR-GMNL states become trivial when a small amount of communication is allowed. Based on a joint work in preparation with Xavier Coiteux-Roy, Owidiusz Makuta, Fionnuala Curran, Remigiusz Augusiak."

When some variables in a directed acyclic graph (DAG) are hidden, a notoriously complicated set of constraints on the distribution of observed variables is implied. In this talk, we present inequality constraints implied by graphical criteria in hidden variable DAGs. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entropy. For DAGs that exhibit e-separation relations, we present entropic inequality constraints and we show how they can be used to learn about the true causal model from an observed data distribution (arXiv:2107.07087).

"If one only performs experiments involving passive observations, in general there are multiple causal structures that can explain the same set of distributions over the observed variables. In this case, we say that these causal structures are observationally equivalent. In this work, we explore all the known techniques for proving observational equivalence or inequivalence, as well as some original ones. Even if the existing rules are not enough to achieve the full classification of the causal structures with four observed variables, our results get close to such classification and show that admitting inequality constraints is a generic feature among structures with four observed variables."

"Imsets, introduced by Studený (see Studený, 2005 for details), are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice when applied to directed acyclic graph (DAG) models. In particular, the standard imset for a DAG is in one-to-one correspondence with the independence model it induces, and hence is a label for its Markov equivalence class. We present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, which have directed and bidirected edges, using the parameterizing set representation of Hu and Evans (2020). By construction, our imset also represents the Markov equivalence class of the MAG. We show that for many such graphs our proposed imset defines the model, though there is a subclass of graphs for which the representation does not. We prove that it does work for MAGs that include models with no adjacent bidirected edges, as well as for a large class of purely bidirected models. If there is time, we will also discuss applications of imsets to structure learning in MAGs. This is joint work with Zhongyi Hu (Oxford). References Z. Hu and R.J. Evans, Faster algorithms for Markov equivalence, In Proceedings for the 36th Conference on Uncertainty in Artificial Intelligence (UAI-2020), 2020. M. Studený, Probabilistic Conditional Independence Structures, Springer-Verlag, 2005."

There are numerous properties of quantum states that one might be interested in characterizing, including statistical moments of observables such as expectations or variances, or more generally purities, entropies, probabilities, eigenvalues, symmetries, marginals, etc. Given a fixed collection of properties, the realizability problem aims to determine which value-assignments to those properties are jointly exhibited by at least one quantum state. In addition to the decision problem of realizability, one might also be interested in quantifying what proportion of quantum states possesses those property values. 

Any property of a quantum state can always, at least in principle, be estimated empirically by suitably measuring an ensemble of many independently and identically prepared copies of that quantum state. The particular sequence of positive operator valued measures which estimates a given property is known as a property estimation scheme. The purpose of this talk is to discuss a strategy for tackling realizability problems by studying the large deviation behaviour of property estimation schemes.

The key idea of this approach is the following:
A given collection of properties is realized by a quantum state if and only if a random quantum state occasionally produces that collection of properties as estimates.
Under suitable conditions, this observation leads to a complete hierarchy of necessary conditions for realizability.

Zoom link:  https://pitp.zoom.us/j/91945653382?pwd=QTZqSnpjYjlxYndqaHZwN2lES1h1Zz09

The theory of polynomial optimisation considers a polynomial objective function subject to countable many polynomial constraints. In a seminal contribution Navascués, Pironio and Acín (NPA) generalised a previous result from Lassere, allowing for its application in quantum information theory by considering its non-commutative variant. Non-commutative variables are represented as bounded operators on potentially infinite dimensional Hilbert spaces. These infinite-dimensional non-commutative polynomials optimisation (NPO) problems are recast as a complete hierarchy of semidefinite programming (SDP) relaxations by a suitable partitioning of the underlying spaces.

The reformulation into convex optimisation problems allows for numerical analysis. We focus on an operator theoretical approach to the NPA hierarchy and show its equiv-
alence to the original NPA hierarchy. To do so, we introduce the necessary mathematical preliminaries from operator algebra theory and semidefinite programming. We conclude by showing how certain relations on operators translate to SDP relaxations yielding drastically reduced problem sizes.

Zoom Link: https://pitp.zoom.us/j/98583295694?pwd=SlcvNG90RzFrODBKSHNaUi84bG9DZz09

Abstract: TBD

Zoom Link: https://pitp.zoom.us/j/95212522185?pwd=eWx4R3o3cmZISmtPY0xwMmdxc2EzZz09