Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories

Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamiolkowski isomorphism, one and the same --- namely the homogeneous self-dual cones of positive semi-definite linear endomorphisms on finite dimensional complex Hilbert spaces. From the perspective of category theory, these cones are the sets of morphisms in finite dimensional quantum theory as a dagger compact closed category. Understanding the intricate geometry of these cones and charting the wider landscape for their host category is imperative for foundational physics. In Part I of this thesis, we study the shape of finite dimensional quantum theory in terms of quantum information. We introduce novel geometric structures inscribed within quantum cones: conical t-designs. Conical t-designs are a natural, strictly inclusive generalization of complex projective t-designs. We prove that symmetric informationally complete measurements of arbitrary rank (SIMs), and full sets of mutually unbiased measurements of arbitrary rank (MUMs) are conical 2-designs. SIMs and MUMs correspond to highly symmetric polytopes within the Bloch body. The same holds for the entire class of homogeneous conical 2-designs; moreover, we establish necessary and sufficient conditions for a Bloch polytope to represent a homogeneous conical 2-design. Furthermore, we show that infinite families of such designs exist in all finite dimensions. It turns out that conical 2-designs are naturally adapted to a geometric description of bipartite entanglement. We prove that a quantum measurement is a conical 2-design if and only if there exists a (regular) entanglement monotone whose restriction to pure states is a function of the norm of the probability vector over the outcomes of the bipartite measurement formed from its tensor products. In that case the concurrence is such a monotone. In addition to monotones, we formulate entanglement witnesses in terms of geometric conditions on the aforementioned conical 2-design probabilities. In Part II of this thesis, we move beyond quantum theory within the vein of Euclidean Jordan algebras (EJAs). In light of the Koecher-Vinberg theorem, the positive cones of EJAs are the only homogeneous self-dual cones to be found in a finite dimensional setting. We consider physical theories based on EJAs subject to nonsignaling axioms regarding their compositional structure. We prove that any such Jordanic composite is a Jordan ideal of Hanche-Olsen's universal tensor product. Consequently, no Jordanic composite exists having the exceptional Jordan algebra as a direct summand, nor does any such composite exist if either factor is exceptional. So we focus on special EJAs of self-adjoint matrices over the real, complex, and quaternionic division rings. We demonstrate that these can be organized in a natural way as a symmetric monoidal category, albeit one that is not compact closed. We then construct a related category InvQM of embedded EJAs, having fewer objects but more morphisms, that is dagger compact closed. This category unifies finite dimensional real, complex and quaternionic quantum theories, except that the composite of two complex quantum systems comes with an extra classical bit. Our notion of composite requires neither tomographic locality, nor preservation of purity under monoidal products. The categories we construct include examples in which both of these conditions fail. Our unification cannot be extended to include any finite dimensional spin factors (save the rebit, qubit, and quabit) without destroying compact closure

A∞-persistence

This work is a mixture of two different worlds: Persistent Homology and A∞-(co)algebras. The former is a new tool in Topological Data Analysis, whilst the latter is a classical concept in Homological Perturbation Theory. Persistent Homology (in the sense of [1,2,3]) is a topological technique which has proved to be useful in areas such as digital imaging [4], sensor networks [5], molecular modelling [6], dynamical systems [7,8] and speech pattern analysis [9]. It allows one to extract global structural information about data sets (specially high dimensional ones) which may contain noise. In this dissertation, we enhance the power of Persistent Homology through the use of A∞-(co)algebras [10]. These are structures with which we can endow the (co)homology of a topological space in order to obtain information of its homotopy type beyond its homology groups and its cohomology algebra. Throughout this abstract, the word “homology” will refer to "homology with coefficients in some field". If we want to study a data set, we can build a point cloud (i.e., a finite metric space) to represent it so that the details we find about the structure of the cloud can be translated into patterns in the data. Specifically, if we view the point cloud “P” as a reasonable sample of points of an unknown metric space “M”, the shape of M will tell us about the structure of P. In a situation like this, persistent homology (that we recall in Chapter 1) allows us to approximate the Betti numbers of M. We work with A∞-(co)algebras (that we recall in Chapter 2) with the aim of providing more information about the space M. To do so, we start by showing how powerful A∞-(co)algebras are (Section 2.4) and choose a part of every A∞-(co)algebra to focus on (Section 3.1), e.g., given an A∞-coalgebra "{Δ_n}_n" on the homology of a topological space "X", we can look at the kernel of each of the linear maps Δ_n restricted to some homology degree. Any such kernel is a vector subspace of the corresponding homology group but its dimension is not a homotopy invariant of X (Section 3.2.1). Despite this, we show situations in which this dimension can give information on the homotopy type of X (Sections 3.2.2 and 3.2.3). Next step is adding persistence to the scene. The Fundamental Theorem of Persistent Homology (of which we give a new proof in Section 1.2) claims that given a family of nested topological spaces with finite dimensional homology groups, the sequence formed by the homology of each of the spaces, with all maps induced by the corresponding inclusions, can be decomposed as a “barcode” in a unique way (up to reordering of bars). The point is to find analogues of this classical result to capture the information of the vector subspaces we are focusing on, like the kernels of the maps Δ_n mentioned above. We first prove a decomposition result of this kind (Section 3.3) for nested topological spaces and A∞-coalgebra structures on their homology satisfying the following condition, that we call C: the maps induced in homology by the inclusions send elements in the kernel of Δ_n to elements in the kernel of the corresponding Δ_n. We then provide a construction that guarantees that for certain families of nested topological spaces, we can choose an A∞-coalgebra on the homology of each term so that condition C holds (Section 3.4). Notice that when the chosen A∞-coalgebra structures do not satisfy C, an interesting intermittent behavior takes place (Section 3.5), but in spite of this, we can still prove a decomposition result by using zigzag persistence techniques (Section 3.6). On the computational side, we modify an algorithm by P. Real et al. [11] to compute A∞-coalgebra structures on the homology of finite CW complex by building a discrete vector field (Section 2.3), and we provide an abstract algorithm to compute the barcode decomposition that encapsulates the information of the kernels of the maps Δ_n in an A∞-coalgebra (Section 3.7). To make this work as self-contained as possible, we have added an Appendix with a collection of tools of Rational Homotopy Theory that we use. REFERENCES [1] G. Carlsson and A. Zomorodian. Computing persistent homology. Discrete Comput. Geom., 33(2):249-274, 2005. [2] H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simpli_cation. Discrete Comput. Geom., 28:511-533, 2002. [3] A. Zomorodian. Computing and Comprehending Topology: Persistence and Hierarchical Morse Complexes. PhD thesis, University of Illinois at Urbana-Champaign, 2001. [4] V. Robins, P. J. Wood, and A. P. Sheppard. Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Analysis and Machine Intelligence, 33(8):1646-1658, 2011. [5] V. de Silva and R. Ghrist. Coverage in sensor networks via persistent homology. Alg. & Geom. Top., 7:339-358, 2007. [6] P. K. Agarwal, H. Edelsbrunner, J. Harer, and Y. Wang. Extreme elevation on a 2-manifold. Discrete Comput. Geom., 36(4):553-572, 2006. [7] H. Edelsbrunner, G. Jablonski, and M. Mrozek. The Persistent Homology of a Self-Map. Found. Comput. Math., 15(5):1213-1244, 2015. [8] V. Robins. Towards computing homology from finite approximations. In Proceedings of the 14th Summer Conference on General Topology and its Applications (Brookville, NY, 1999), volume 24, pages 503-532 (2001), 1999. [9] K. A. Brown and K. P. Knudson. Nonlinear statistics of human speech data. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19(7):2307-2319, 2009. [10] J. D. Stasheff. Homotopy associativity of H-spaces. I, II. Trans. Amer. Math. Soc., 108:275-312, 1963. [11] H. Molina-Abril and P. Real. Cell AT-models for digital volumes. In Graph-Based Representations in Pattern Recogn. (7th IAPR-TC-15 International Workshop, GbRPR 2009), volume 5534 of Lecture Notes in Computer Science, pages 314-323. Springer, 2009

Temporal quantum correlations and hidden variable models

This thesis is devoted to the investigation of the differences between the predictions of classical and quantum theory. More precisely, we shall analyze such differences starting from their consequences on quantities with a clear empirical meaning, such as probabilities, or relative frequencies, that can be directly observed in experiments. Different kind of classical probability theories, or hidden variable theories, corresponding to different physical constraints imposed on the measurement scenario are discussed, namely, locality,noncontextuality and macroscopic realism. Each of these theories predicts bounds on the strength of correlations among different variables, and quantum mechanical predictions violate such bounds, thus revealing a stark contrast with our classical intuition. Our work starts with the investigation of the set of classical probabilities by means of the correlation polytope approach, which provides a minimal and optimal set of bounds for classical correlations. In order to overcome some of the computational difficulties associated with it, we develop an alternative method that avoid the direct computation of the polytope and we apply it to Bell and noncontextuality scenarios showing its advantages both for analytical and numerical computations. A different notion of optimality is then discussed for noncontextuality scenarios that provide a state-independent violation: Optimal expression are those maximizing the ratio between the quantum and the classical value. We show that this problem can be formulated as a linear program and solved with standard numerical techniques. Moreover, optimal inequalities for the cases analyzed are also proven to be part of the minimal set described above. Subsequently, we provide a general method to analyze quantum correlations in the sequential measurement scenario, which allows us to compute the maximal correlations. Such a method has a direct application for computation of maximal quantum violations of Leggett-Garg inequalities, i.e., the bounds for correlation in a macroscopic realist theories, and it is relevant in the analysis of noncontextuality tests, where sequential measurements are usually employed. Finally, we discuss a possible application of the above results for the construction of dimension witnesses, i.e., as a certification of the minimal dimension of the Hilbert spaces needed to explain the arising of certain quantum correlations.to Bell and noncontextuality scenarios showing its advantages both for analytical and numerical computations.Diese Doktorarbeit befasst sich mit der Untersuchung der unterschiedlichen Vorhersagen von klassischen Theorien und Quantenmechanik. Es werden verschiedene klassische Wahrscheinlichkeitstheorien oder Theorien, die auf der Existenz versteckter Variablen basieren, diskutiert und besonders auf ihre Vorhersagen bezüglich der möglichen Stärke der Korrelationen zwischen verschiedenen Variablen eingegangen. Die klassischen Theorien machen dabei unterschiedliche physikalischen Annahmen wie Lokalität, Nichtkontextualität oder makroskopischer Realismus. Für jede dieser Theorien sagt die Quantenmeachnik stärkere Korrelationen voraus, die die klassischen Schranken verletzen und damit im Widerspruch zu unserer klassisch geprägten Intuition stehen. Unsere Arbeit beginnt mit der Untersuchung der Menge von klassischen Wahrscheinlichkeiten mittels des Korrelations-Polytop-Verfahrens, welches einen minimalen und optimalen Satz an Grenzen für klassische Korrelationen liefert. Um einige der mit diesem Verfahren verbundenen rechnerischen Schwierigkeiten zu überwinden, entwickeln wir eine alternative Methode, die die direkte Berechnung des Polytops umgeht. Angewendet auf Bell- und Kontextualitätsszenarien zeigen wir die Vorteile unserer Methode, sowohl bezüglich analytischer, als auch numerischer Berechnungen. Danach wird eine andere Möglichkeit betrachtet, Optimalität für Nichtkontextualitätsungleichungen zu definieren, die eine zustandsunabhängige Verletzung aufweisen: Optimale Ungleichungen sind solche, die das Verhältnis zwischen quantenmachanischem und klassischem Wert maximieren. Wir zeigen, dass dieses Problem als lineares Programm formuliert und mit standardmäßigen, numerischen Methoden gelöst werden kann. Darüber hinaus beweisen wir, dass die optimalen Ungleichungen für die betrachteten Fälle jene sind, die Teil des oben beschriebenen minimalen Satzes von Grenzflächen sind. Anschließend stellen wir eine allgemeine Methode vor mit der man Quantenkorrelationen bei sequentiellen Messungen analysieren kann und die maximalen Korrelationen berechnen kann. Ein solches Verfahren hat als direkte Anwendung die Berechnung maximaler Quantenverletzung von Leggett-Garg Ungleichungen, d.h. der Grenzen für Korrelationen in Theorien, die auf der Annahme des makroskopischem Realismus basieren. Zudem ist diese Methode relevant in der analytischen Betrachtung von Kontextualitätstests, in denen üblicherweise sequentielle Messungen verwendet werden. Abschließend diskutieren wir für die obigen Resultate Anwendungen bei der Konstruktion von Zeugenoperatoren für die Dimension von Quantensystemen. Damit ist es möglich, die minimale Dimension des Hilbertraums zu zertifizieren, die nötig ist, um das Auftreten von gegebenen Quantenkorrelationen zu erklären

Fermion Sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states

Fermionic Linear Optics (FLO) is a restricted model of quantum computation which in its original form is known to be efficiently classically simulable. We show that, when initialized with suitable input states, FLO circuits can be used to demonstrate quantum computational advantage with strong hardness guarantees. Based on this, we propose a quantum advantage scheme which is a fermionic analogue of Boson Sampling: Fermion Sampling with magic input states. We consider in parallel two classes of circuits: particle-number conserving (passive) FLO and active FLO that preserves only fermionic parity and is closely related to Matchgate circuits introduced by Valiant. Mathematically, these classes of circuits can be understood as fermionic representations of the Lie groups $U(d)$ and $SO(2d)$. This observation allows us to prove our main technical results. We first show anticoncentration for probabilities in random FLO circuits of both kind. Moreover, we prove robust average-case hardness of computation of probabilities. To achieve this, we adapt the worst-to-average-case reduction based on Cayley transform, introduced recently by Movassagh, to representations of low-dimensional Lie groups. Taken together, these findings provide hardness guarantees comparable to the paradigm of Random Circuit Sampling. Importantly, our scheme has also a potential for experimental realization. Both passive and active FLO circuits are relevant for quantum chemistry and many-body physics and have been already implemented in proof-of-principle experiments with superconducting qubit architectures. Preparation of the desired quantum input states can be obtained by a simple quantum circuit acting independently on disjoint blocks of four qubits and using 3 entangling gates per block. We also argue that due to the structured nature of FLO circuits, they can be efficiently certified.Comment: 65 pages, 13 figures, 1 table, v2: improved discussion and narrative, numerics about anticoncentration added, references updated, comments and suggestions are welcom

The bracket geometry of statistics

In this thesis we build a geometric theory of Hamiltonian Monte Carlo, with an emphasis on symmetries and its bracket generalisations, construct the canonical geometry of smooth measures and Stein operators, and derive the complete recipe of measure-constraints preserving dynamics and diffusions on arbitrary manifolds. Specifically, we will explain the central role played by mechanics with symmetries to obtain efficient numerical integrators, and provide a general method to construct explicit integrators for HMC on geodesic orbit manifolds via symplectic reduction. Following ideas developed by Maxwell, Volterra, Poincaré, de Rham, Koszul, Dufour, Weinstein, and others, we will then show that any smooth distribution generates considerable geometric content, including ``musical" isomorphisms between multi-vector fields and twisted differential forms, and a boundary operator - the rotationnel, which, in particular, engenders the canonical Stein operator. We then introduce the ``bracket formalism" and its induced mechanics, a generalisation of Poisson mechanics and gradient flows that provides a general mechanism to associate unnormalised probability densities to flows depending on the score pointwise. Most importantly, we will characterise all measure-constraints preserving flows on arbitrary manifolds, showing the intimate relation between measure-preserving Nambu mechanics and closed twisted forms. Our results are canonical. As a special case we obtain the characterisation of measure-preserving bracket mechanical systems and measure-preserving diffusions, thus explaining and extending to manifolds the complete recipe of SGMCMC. We will discuss the geometry of Stein operators and extend the density approach by showing these are simply a reformulation of the exterior derivative on twisted forms satisfying Stokes' theorem. Combining the canonical Stein operator with brackets allows us to naturally recover the Riemannian and diffusion Stein operators as special cases. Finally, we shall introduce the minimum Stein discrepancy estimators, which provide a unifying perspective of parameter inference based on score matching, contrastive divergence, and minimum probability flow.Open Acces