Ortholattices and their automorphisms

Attempts to improve our understanding of quantum physics by means of an algebraic analysis of its basic model, the complex Hilbert space, are sometimes seen critically or are even considered as hopeless. Indeed, when observing the collection of lattice-theoretic properties by which, say, John Wilbur once characterised the ortholattice of closed subspaces, we might not feel anything else than having obtained the solution of a difficult mathematical problem. The procedure is somewhat far from physical reality; not even the composition of systems is easy to handle.

Nonetheless, we would like to defend the idea of reconstructing the complex Hilbert space by algebraic means. But we propose to get rid of cumbersome axioms that are, for instance, part of any purely lattice-theoretic approach. We rather propose to start from a conveniently simple structure and focus on postulating the existence of suitable automorphisms. As regards simplicity, probably the notion of an orthogonality space is unbeaten, which is nothing but a set endowed with a symmetric and irreflexive binary relation.

A characterisation of the complex Hilbert space is indeed possible this way. Unfortunately, though, it works well in infinite dimensions only. The usual proof method gives us little cause to hope for a modification to include the finite-dimensional case. But the construction of an orthomodular space is still feasible. The characterisation of the field of complex numbers among the division rings requires new methods, which to simplify is presently the challenge.

Towards a functorial quantum spectrum for noncommutative algebras

What kind of "quantum space" should play the role of the spectrum of a noncommutative algebra? Potential solutions to this problem might include a topological space, a sheaf of rings, or a noncommutative algebra of "discrete" functions. I will discuss various obstructions and partial progress with these approaches in both ring theory and operator algebra. Then I will report on work-in-progress toward coalgebras as a "quantum spectrum", yielding a potential solution to this problem in the restricted context of finitely generated noetherian algebras satisfying a polynomial identity.

Sinkhorn's theorem in quantum theory

Given a matrix with only positive entries, it can be scaled to a bistochastic matrix using two diagonal matrices. This innocent result by Sinkhorn and others has had a wealth of applications. In this talk, I'll start from the original Sinkhorn theorem conjectured sometimes in the 30s to discuss generalizations and related theorems useful for quantum (information) theory. In particular, we will see a version for unitary matrices and discuss its place among decompositions of unitary matrices which - as other talks in this workshop show - are increasingly useful for decompositions of quantum circuits. If time permits, I'll discuss other generalisations not limited to unitary matrices and how they find applications in quantum theory.

Effectus theory: an introduction, a reconstruction and possibilistic equivalences

Usually, in the (categorical) foundational study of quantum theory, the parallel composition of systems (tensor product) takes centre stage. Effectus theory takes a different and complementary approach: for us, the probabilistic disjunction (direct sum of algebras) of systems takes centre stage. This approach leads, out of necessity, to rather different notions and reasoning, than is usually encountered.

In this talk I will give an introduction to the structure of effectuses and motivate them (anachronistically) by a recent reconstruction of finite-dimensional quantum theory by van de Wetering. If time permits, I will discuss some new ideas which effectus theory inspired. The first is diamond-adjointness: two channels f,g in opposite direction are said to be diamond-adjoint if f(p) ≤ 1-q iff g(q) ≤ 1-p for all projections p,q. For pure channels, this is just adjointness of the underlying Krauss operators (up to a scalar), but in general it’s more interesting with a possibilistic flavour.

Choquet order and abelian subalgebras

An interplay between recent topos theoretic approach and standart convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of the posets of all abelian subalgebras of von Neumann algebras with classical Tomita's theorem from state space Choquet theory, we show that order isomorphism between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed the Choquet order are given by Jordan *-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated system of convex decompositions (discrete or continuous) of a fixed state. Joint work with J. Hamhalter.

Quantum functions and the Morita theory of quantum graph isomorphisms

I will describe a 2-categorical framework for noncommutative finite set and graph theory. In the spirit of noncommutative topology, finite-dimensional C*-algebras are regarded as noncommutative analogues of finite sets; the role of a function between finite sets A and B is then played by a quantum measurement with outcomes in B, controlled by elements of A. The resulting 2-category admits a simple string diagrammatic calculus and has close connections to the theory of quantum permutation groups in noncommutative topology.

As an application, I will focus on noncommutative graph theory and show how the quantum graph homomorphisms and isomorphisms recently introduced in the study of quantum pseudo-telepathy can be understood in our framework. In particular, I will discuss how graphs which are quantum isomorphic to a given graph G can be classified in terms of certain algebras in the monoidal category of quantum graph automorphisms of G.

This is joint work with Ben Musto and Dominic Verdon and is based on arXiv:1711.07945 and arXiv:1801.09705.

ZXZ decompositions of quantum and classical reversible circuits

Regardless of how the commodity quantum computer will look like, its architecture will likely be based on a decomposition of a unitary evolution matrix into a library of simpler quantum gates. This viewpoint is very much rooted in how classical computations are decomposed into a concatenation of simpler (and universal) classical gates, like the AND, OR and NOT. However, despite the common language, it is not straightforward how classical computations are embedded within the Hilbert space of quantum computations, partly due to lack of reversibility of the former. The (unitary) group structure of quantum circuits is lost at the level of classical circuits. However, classical reversible systems come with a finite group representation, and allow for a clear-cut connection with quantum circuits.

In this presentation, I will show how the generalized Birkhoff theorem on the decomposition of arbitrary classical reversible circuits into elementary controlled CNOT gates is a corrolary of the block-ZXZ decomposition of an arbitary unitary matrix of dimension 2^w x 2^w for w a positive integer (nr qubits). The latter is a generalization of the ZXZ Sinkhorn scaling of a unitary matrix, discussed by Martin Idel in this workshop. Joint work with Alex De Vos.

AF C*-algebras, Many-valued Logics, and Effect Algebras

Effect algebras arose in the 1990's as part of the abstract theory of quantum measurement. Earlier, in the 1980's, D. Mundici found deep and surprising connections of AF (Approximately Finite) C*-algebras with algebras of Lukasiewicz's many-valued logics, called MV algebras. MV algebras turn out to be non-trivially related to both effect algebras and the logic of infinite quantum systems, as well as to several areas of contemporary mathematics. In work with M. Lawson, we gave a coordinatization theory of MV algebras (via effect algebras) using Boolean inverse monoids. We survey recent results in the area.

Posets of commutative C*-subalgebras

The main subject of this talk is the set C(A) of all commutative unital C*-subalgebras of some given unital C*-algebra A, which we order by inclusion. Physically, we are motivated to study this partially ordered set since it represents the classical contexts of a quantum system represented by A. Our mathematical motivation lies in the fact that Gelfand duality provides us a very good understanding of commutative C*-algebras, therefore the study of the C*-algebra A via C(A) is a way of studying non-commutative C*-algebras by expoiting Gelfand duality as much as possible. Indeed, it turns out that C(A) is a powerful invariant for C*-algebras. We will show that we can reconstruct the orthomodular poset Proj(A) of projections in A from C(A), which has some major implications, namely we can detect whether or not A is an AW*-algebra or even a von Neumann algebra, and we can even determine its type in those cases. Moreover, we show that type I AW*-algebras are determined up to isomorphism by C(A).

Tensor topology

In categories of Hilbert bundles, Hilbert modules, and sheaves over a topological space, the open subsets of the base space can be algebraically recovered via so-called idempotent subunits. These subunits provide any braided monoidal category with a built-in notion of space, and form a meet-semilattice that corresponds to different intuitions in different categories (e.g. truth values in the context of logic and side-effect-free observations in the context of order theory). We introduce a notion of support that captures 'where morphisms happen' and that is characterised by a universal property. Finally, braided monoidal categories come with notions of localisation (to a region of space) and of causal structure. All together, these concepts can be used, for instance, to categorically model relativistic quantum information protocols, and we believe that they offer the possibility of other applications and of new interesting directions of investigation. Joint work with Chris Heunen and Sean Tull.

Quantum computation in the hall of mirrors

The successes of the theories of quantum information and quantum computation, demonstrate the power of abstraction in describing what is possible in principle in the real world. However, as a mathematical formalism, it is sometimes presented in isolation even from closely related subjects such as randomised computation; and at the same time, there are unanswered questions about what limits there are to the power of quantum computation. By further abstraction from the standard quantum computational model, we can describe models of computation which are funhouse-mirror images of quantum computation, and whose computational power we can exactly characterise relative to classical computation. This opens up the question of whether we can somehow find the power of quantum computation in this hall of mirrors by accounting for the distortions in those funhouse image models.

A statistical interpretation of Grothendieck's inequality and its relation to the size of non-locality of quantum mechanics

In 1953 A. Grothendieck proved a theorem that he called The Fundamental Theorem on the Metric Theory of Tensor Products. This result is known today as Grothendieck's inequality (or Grothendieck's theorem). Originally, it is recognised as one of the major results of Banach space theory.

Grothendieck formulated his deep result in the language of tensor norms on tensor products of Banach spaces. To this end he described how to generate new tensor norms from known ones and unfolded a powerful duality theory between tensor norms. Only in 1968, thanks to J. Lindenstrauss and A. Pelczynski Grothendieck's inequality was decoded and equivalently rewritten - in matrix form - which lead to its global breakthrough.

However, since the appearance of Grothendieck's paper in 1953 there exists the (still) open problem to determine the smallest possible constant - called the Grothendieck constant - which can be used in Grothendieck's inequality.

In addition to its multifaceted representations in functional analysis the Grothendieck inequality admits further equivalent formulations - each one of them reflecting deep and surprising links with different scientific branches, such as semidefinite programming in convex optimisation, NP-hard combinatorial optimisation, graph theory, communication complexity, private data analysis, geomathematics and - due to the pioneering work of B. S. Tsirelson in 1985 - even foundations and philosophy of quantum mechanics.

Based on matrix analysis and a few techniques from multivariate statistics we will present a further equivalent representation of Grothendieck's inequality (over the reals) which reveals also its deep underlying statistical nature. Using this representation, we will revisit Tsirelson's approach and sketch how Grothendieck's inequality is intertwined with the violation of a Bell inequality, based on Tsirelson's observation that the Grothendieck constant - which is strictly larger than 1 - gives an upper bound of the deviation of the quantum mechanical correlation model from the ``classic'' Kolmogorovian statistical correlation model.

Categories of physical processes

I will discuss a symmetric monoidal functor GNS: Phys -> *Mod, based on the Gelfand-Naimark-Segal construction. This functor and its minor variants capture a fairly complete picture of quantum theory in a categorical formalism. I will show how to derive the link between symmetries and group representations, the probabilistic aspects of quantum theory (including Markov processes), and the link between the Heisenberg and Schrodinger pictures in this formalism. I also briefly touch upon the differential geometry of GNS, which captures infinitesimal symmetries and a correctly typed classical limit.

Quanta and qualia

Adding to the well known idea that abelian subalgebras of a C*-algebra A represent classical contexts of a quantum system, each such context is naturally equipped with a rich collection of symmetries induced by A. These are described by spatial pseudogroups or, equivalently, by topological étale groupoids. In particular I examine this in the case of C*-algebras of (possibly twisted) locally compact groupoids, and as a result obtain a notion of diagonal of a C*-algebra which consists of a locale embedded in the quantale Max A satisfying suitable properties related to the quantale structure. Whereas this first part of the talk is purely mathematical, in the second part I shall look at the structures obtained from the point of view of a tentative physical interpretation derived from the notion of quale (plural qualia), which in philosophy and psychology is used in order to refer to an individual instance of subjective, conscious experience.

Measurement functors

Given a physical system, one can assign to each set X (or suitable type of space) the collection of all measurements taking values in X. Postprocessing along a function X --> Y gives these assignments the structure of a functor to Set. So how much does this functor know about the physical system? I will discuss this question and prove that the answer is 'surprisingly much': one can reconstruct the entire piecewise C*-algebra of observables from the functor. Finally, I will present some partial results towards a complete reconstruction and reaxiomatization of quantum theory by postulating a form of Noether's theorem for a measurement functor. Based on arXiv:1512.01669.

From symmetric pattern-matching to quantum control

From a programmer's perspective, quantum algorithms are simply classical algorithms having access a special kind of memory featuring particular operations: quantum operations. They therefore feature two kinds of control flow. One is purely conventional and is concerned in the classical part of the algorithm. The other -- dubbed quantum control -- is more elusive and still subject to debate: if the notion of quantum test is reasonably consensual, the quantum counterpart of loops is still not believed to be meaningful.

In this talk, we argue that, under the right circumstances, a reasonable notion of quantum loop is possible. To this aim, we propose a typed, reversible language with lists and fixpoints, extensible to linear combinations of terms. The language admits a reduction strategy in the spirit of algebraic lambda-calculi. Under the restriction of structurally recursive fixpoints, it is shown to capture unitary operations. It is expressive enough to represent several of the unitaries one might be interested in expressing.

Two linearities for quantum computing in the lambda calculus

I will discuss a recent joint work with Gilles Dowek [LNCS 10687:281-293, 2017] on lambda calculus with quantum control. We proposed a way to add a measurement operator to a lambda calculus with quantum control. In order to do so, we provided the calculus with a type system which is is linear on superposition, while allows cloning (canonical) base vectors. In addition, we provided an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis. Finally, I will give some clues on an ongoing work with Octavio Malherbe in which we are studying a categorical semantics for such a calculus.

Non-commutative Stone dualities and étale groupoids

I shall describe the way in which classical Stone dualities can be generalized from a commutative to a non-commutative setting. These non-commutative dualities are important in the theory of C*-algebras, group theory and multiple-valued logic.

Contextuality as a resource

Contexts of compatible observables provide multiple partial, classical perspectives on a quantum system. Any two contexts fit nicely together, but they cannot all be pasted consistently into a global perspective. This gap between local consistency and global inconsistency is what constitutes contextuality, a concept that is elegantly expressed in the language of sheaf theory.

A fundamental phenomenon of quantum mechanics setting it apart from classical physical theories, recent results have also established its rôle as a source of advantage in informatic tasks, including the additional computational power in some specific schemes for quantum computation.

We consider contextuality from the point of view of resource theory, with an emphasis on operations that combine contextual systems into larger ones and control their use as resources, and on quantum advantages afforded by the presence of contextuality.

Dagger limits

A dagger category is a category equipped with a dagger: a contravariant involutive identity-on-objects endofunctor. Such categories are used to model quantum computing and reversible computing, amongst others. The philosophy when working with dagger categories is that all structure in sight should cooperate with the dagger. This causes dagger category theory to differ in many ways from ordinary category theory. Standard theorems have dagger analogues once one figures out what "cooperation with the dagger" means for each concept, but often this is not just an application of formal 2-categorical machinery or a passage to (co)free dagger categories.

We discuss limits in dagger categories. To cooperate with the dagger, limits in dagger categories should be defined up to an unique unitary (instead of only up to iso), that is, an isomorphism whose inverse is its dagger. We rework an initial attempt to a more elegant and general theory. Moreover, we consider the problem of building dagger limits from smaller building blocks, exhibit deep connections to polar decomposition and time permitting, discuss how to phrase dagger limits in terms of adjunctions.

Collective quantum games

Quantum Games have opened new routes for a better understanding of quantum information and quantum computation. Their inception has allowed to connect quantum mechanics, which determines the behaviour of systems at microscopic scales, with game theory, which is a cornerstone in Economics. Since the pioneering work by Eisert, Wilkens and Lewenstein (EWL) in 1999, several authors have studied different games using quantum information tools, e.g., the Prisoner’s Dilemma, the Haw-Dove, the Samaritan’s Dilemma, or the Battle of Sexes. In all these games, two players (A and B) compete following different strategies in order to maximize their own rewards. The use of quantum mechanical laws boosts the number of different states of the players as they are no longer limited to the classical ones but can also be in linear superpositions of them, which certainly outperforms the classical results.

The outline of this seminar is as follows. First, a short account of classical, i.e., non quantum, games will be given, then the EWL quantum approach in game theory will be described, paying special attention to the new Nash equilibria emerging in this context. Then, collective games involving NxN players, half of type A and half of type B, will come into consideration. Each player occupies a site in a two-dimensional lattice or a node of a fully random network in such a way that every player is connected to four partners and four mates. The collective games are played in a CA manner, i.e., with uniform, local and synchronous interactions. In this way, every player competes with his four adjacent partners, and imitates his best paid mate. Last but not least, collective correlated classical games will come also under scrutiny if time permits.