We illustrate how quantum information theory and free (i.e. noncommutative) semialgebraic geometry often study similar objects from different perspectives. We give examples in the context of positivity and separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the benefits of combining the two perspectives. This paper is an invitation to consider the intersection of the two fields, and should be accessible for researchers from either field.
We illustrate how quantum information theory and free (i.e. noncommutative) semialgebraic geometry often study similar objects from different perspectives. We give examples in the context of positivity and separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the benefits of combining the two perspectives. This paper is an invitation to consider the intersection of the two fields, and should be accessible for researchers from either field.
This paper explores the intersection of quantum algebraic geometry and quantum information theory, focusing on how noncommutative algebraic structures provide a powerful framework for understanding and manipulating quantum systems. We investigate the role of noncommutative rings and schemes in encoding quantum states, quantum operations, and entanglement. Furthermore, we examine the application of tools from algebraic geometry, such as sheaves and cohomology, to characterize the complexity and properties of quantum information processing tasks. Specifically, we delve into the use of noncommutative tori and their classifications, alongside explorations of quantum metrics and preordered semialgebras, to build a bridge between abstract mathematical structures and concrete applications in quantum technologies. Our goal is to highlight the potential of quantum algebraic geometry to offer new insights and techniques for advancing the field of quantum information.
This thesis explores various perspectives on quantum phenomena, and how our understanding of these phenomena is informed by the study of general probabilistic theories. Particular attention is given to quantum nonlocality, and its interaction with areas of physical and mathematical interest such as entropy, reversible dynamics, information-based games and the idea of negative probability. We begin with a review of non-signaling distributions and convex operational theories, including “black box” descriptions of experiments and the mathematics of convex vector spaces. In Chapter 3 we derive various classical and quantum-like quasiprobabilistic representations of arbitrary non-signaling distributions. Previously, results in which the density operator is allowed to become non-positive [1] have proved useful in derivations of quantum theory from physical requirements [2]; we derive a dual result in which the measurement operators instead are allowed to become non-positive, and show that the generation of any non-signaling distribution is possible using a fixed separable state with negligible correlation. We also derive two distinct “quasi-local” models of non-signaling correlations. Chapter 4 investigates non-local games, in particular the game known as Information Causality. By analysing the probability of success in this game, we prove the conjectured tightness of a bound given in [3] concerning how well entanglement allows us to perform the task of random access coding, and introduce a quadratic bias bound which seems to capture a great deal of information about the set of quantum-achievable correlations. By reformulating Information Causality in terms of entropies, we find that a sensible measure of entropy precludes many general probabilistic theories whose non-locality is stronger than that of quantum theory. Chapter 5 explores the role that reversible transitivity (the principle that any two pure states are joined by a reversible transformation) plays as a characteristic feature of quantum theory. It has previously been shown that in Boxworld, the theory allowing for the full set of non-signaling correlations, any reversible transformation on a restricted class of composite systems is merely a composition of relabellings of measurement choices and outcomes, and permutations of subsystems [4]. We develop a tabular description of Boxworld states and effects first introduced in [5], and use this to extend this reversibility result to any composite Boxworld system in which none of the subsystems are classical.
The algebraic approach to physical theories provides a general framework encompassing both classical and quantum mechanics. Accordingly, by looking at the behaviour of the relevant algebras of observables one can investigate structural and conceptual differences between the theories. Interesting foundational questions can be formulated algebraically and their answers are then given in a mathematically compelling way. My dissertation focuses on some philosophical issues concerning entanglement and quantum information as they arise in the algebraic. These two concepts are connected in that one can exploit the non-local character of quantum theory to construct protocols of information theory which are not realized in the classical world. I first introduce the basic concepts of the algebraic formalism, by reviewing von Neumann's work on the mathematical foundations of quantum theories. After presenting the reasons why von Neumann abandoned the standard Hilbert space formulation in favour of the algebraic approach, I show how his axiomatic program remained a mathematical "utopia" in mathematical physics. The Bayesian interpretation of quantum mechanics is grounded in information-theoretical considerations. I take on some specific problems concerning the extension of Bayesian statistical inference in infinite dimensional Hilbert space. I demonstrate that the failure of a stability condition, formulated as a rationality constraint for classical Bayesian conditional probabilities, does not undermine the Bayesian interpretation of quantum probabilities. I then provide a solution to the problem of Bayesian noncommutative statistical inference in general probability spaces. Furthermore, I propose a derivation of the a priori probability state in quantum mechanics from symmetry considerations. Finally, Algebraic Quantum Field Theory offers a rigorous axiomatization of quantum field theory, namely the synthesis of quantum mechanics and special relativity. In such a framework one can raise the question of whether or not quantum correlations are made stronger by adding relativistic constraints. I argue that entanglement is more robust in the relativistic context than in ordinary quantum theory. In particular, I show how to generalize the claim that entanglement across space-like separated regions of Minkowski spacetime would persist, no matter how one acts locally.
Combining physics and philosophy, this is a uniquely interdisciplinary examination of quantum information science which provides an up-to-date examination of developments in this field. The authors provide coherent definitions and theories of information, taking clearly defined approaches to considering information in connection with quantum mechanics, probability, and correlations. Concepts addressed include entanglement of quantum states, the relation of quantum correlations to quantum information, and the meaning of the informational approach for the foundations of quantum mechanics. Furthermore, the mathematical concept of information in the communicational context, and the notion of pragmatic information are considered. Suitable as both a discussion of the conceptual and philosophical problems of this field and a comprehensive stand-alone introduction, this book will benefit both experienced and new researchers in quantum information and the philosophy of physics.
Quantum entanglement, and the resulting peculiar non-classical correlations are one of the most counter-intuitive aspects of quantum mechanics.
The formalism of interactive games from computational complexity theory provides a useful framework in which to understand the power of entanglement. In an interactive game, a verifier interacts with a number of infinitely powerful provers who are allowed to share quantum entanglement but otherwise can't communicate. The ability of the verifier to extract useful information from the provers, whom he does not trust, provides an interesting measure of the ability of the provers to coordinate using their shared entanglement. Two particular interactive games we'll look at are Magic Square and 3SAT.
The Magic Square game is the iconic example of a game where two classical provers cannot perfectly coordinate their strategies using shared randomness, but quantum provers with shared entanglement can win with probability 1. We show that by adding an extra prover, we disallow perfect cheating. For 3SAT with three provers we also show that perfect cheating is not possible.
We then generalize the results for Magic Square and 3SAT by looking at non-commuting provers, a superset of entangled provers (communication is allowed, but operators applied by different provers must commute). Using this method, we obtain a generalized Tsirelson inequality that we apply to the Magic Square. Hence, we are able to give provably optimal strategies for the general Magic Square with n players. We also recover a similar result for 3SAT as with entangled provers, and we improve on it by showing that the gap is inverse exponential in the input size.
The no-signaling principle, which forbids faster than light communication, is a fundamental constraint on the non-local correlations resulting from quantum entanglement. However, no-signaling allows distributions that are more general than those arising from quantum mechanics. In particular, using a specific unit of general non-local correlation (the Popescu-Rohlich box), we show that there are classes that are equal to NEXP classically, but collapse to AM once such correlations are allowed. We also show that MIP where the verifier only looks at the XOR of the answers collapses to PSPACE. For the second approach, we show that by writing general non-local correlations as linear constraints, MIP is included in EXP under such correlations (vs NEXP classically).
We can extend these results to entangled quantum provers, by formulating an artificial MIP-like class built on a promise problem, that classically is equal to NEXP, but that also collapses to AM when quantum correlations are present.
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors.
This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions.
We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.