The aim of this work is to study two-prover quantum games (i.e., games with quantum inputs and outputs) from an operator-algebraic and operator-space point of view. We characterize several notions of the value of such games by formulating them in terms of tensor norms in the category of operator spaces. The main results of the paper concern the description of the so-called no-signalling value of these games, for which we not only provide a precise operator-space formulation, but also establish close connections between this study and some problems in operator algebras. In particular, we show how the recent counterexample to Grothendieck's theorem for operator spaces given in \cite{Ara} can be understood as a direct consequence of results in quantum information theory. We also obtain new upper bounds on the gap between the no-signalling value and the quantum value of two-prover quantum games, improving the best previously known estimates.
The aim of this work is to study two-prover quantum games (i.e., games with quantum inputs and outputs) from an operator-algebraic and operator-space point of view. We characterize several notions of the value of such games by formulating them in terms of tensor norms in the category of operator spaces. The main results of the paper concern the description of the so-called no-signalling value of these games, for which we not only provide a precise operator-space formulation, but also establish close connections between this study and some problems in operator algebras. In particular, we show how the recent counterexample to Grothendieck's theorem for operator spaces given in \cite{Ara} can be understood as a direct consequence of results in quantum information theory. We also obtain new upper bounds on the gap between the no-signalling value and the quantum value of two-prover quantum games, improving the best previously known estimates.
In the past few decades, the connections between non-local games arising in quantum information theory and the theory of operator algebras have undergone a phase of significant development. Operator algebras provide a particularly fruitful framework for approaching questions of non-locality in quantum systems, as the input-output behavior of measurements on bipartite quantum systems can be encoded through noncommutative operator algebras and their state spaces. This thesis is the compilation of a series of papers, written by the author in collaboration with Ivan G. Todorov, utilizing this framework by using the theory of operator algebras and completely positive maps to study questions involving non-local games. ☐ Using the simulation paradigm from information theory, we define a quantized version of homomorphisms and isomorphisms between classical hypergraphs, generalizing quantum homomorphisms and isomorphisms of graphs from the literature. We show that these quantum homomorphisms and isomorphisms constitute pre-orders and equivalence relations, respectively. We use quantum homomorphisms of hypergraphs to provide multiple examples of strict separation between correlation classes of varying types. Specializing to the case when our underlying hypergraphs arise from classical non-local games, we define quantum non-local game homomorphisms and isomorphisms. We also show how the existence of a homomorphism or isomorphism of some fixed type between games is reflected in a comparison of optimal and asymptotic game values with respect to this type. These non-local game homomorphisms are defined via a new class of non-signalling correlations which we introduce here. We develop the multivariate tensor product theory in the category of operator systems, which we then use to provide a characterization of game homomorphisms in terms of state spaces for various tensor products of canonically associated operator systems. We then apply these results to the study of synchronous non-local games, defining jointly synchronous correlations and showing how they correspond to tracial states of tensor products of C*-algebras canonically associated to each game party. ☐ We then move to a second level of quantization by proposing a "quantized" notion of hypergraphs (where "quantum" here is in the sense of a non-commutative analogue for a discrete combinatorial object), and introduce quantum homomorphisms between these quantum objects. We provide an initial foray into the properties quantum morphisms between quantum hypergraphs display, showing they satisfy analogous properties to quantum homomorphisms between classical hypergraphs. We also indicate initial connections these quantum homomorphisms between quantum hypergraphs have to the study of quantum input-output games. We end the thesis showing that homomorphisms of a local type between quantum hypergraphs is closely related, and in some cases identical, to the TRO equivalence of finite-dimensionally acting operator spaces canonically associated with each hypergraph. This suggests a quantum information theoretic approach to Morita equivalence in the category of operator spaces.
We develop a resource-theoretical approach that allows us to quantify values of two-player, one-round cooperative games with quantum inputs and outputs, as well as values of quantum probabilistic hypergraphs. We analyse the quantum game values arising from the type hierarchy of quantum no-signalling correlations, establishing tensor norm expressions for each of the correlation types. As a consequence, we provide metric characterisations of state convertibility via LOSR and LOCC.En route, we obtain an alternative description of the maximal tensor products of ternary rings of operators.
We develop a method for the transfer of perfect strategies between various classes of two-player, one round cooperative non-local games with quantum inputs and outputs using the simulation paradigm in quantum information theory. We show that such a transfer is possible when canonically associated operator spaces for each game are quantum homomorphic or isomorphic, as defined in the joint work of H. and Todorov (2024). We examine a new class of QNS correlations, needed for the transfer of strategies between games, and characterize them in terms of states on tensor products of canonical operator systems. We define jointly tracial correlations and show they correspond to traces acting on tensor products of canonical C∗-algebras associated with individual game parties. We then make an inquiry into the initial application of such results to the study of concurrent quantum games.
Quantum non-local games with quantum inputs or outputs have been formulated in several different languages, including rank-one and quantum XOR games, support maps between projection lattices, probabilistic quantum hypergraphs, and Frobenius-algebraic rule operators on finite quantum sets. We give a unified operator-algebraic comparison of these models by assigning to each rule its winning transformation space: the operator space of transformations accepted with certainty. We introduce R-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra R. Their winning transformation spaces are precisely operator spaces with a natural bimodule structure, equivalently rectangular quantum relations. We compare this formalism with projection-lattice games, hypergraph quantum games, and the graphical rule-operator definition. Projection-lattice games capture exactly the reflexive winning bimodules. Hypergraph quantum games admit value-preserving projection-test realisations, and, after passing to perfect transformations, describe the same reflexive part as projection-lattice games. Using Daws' technique, we also translate rule operators to projections in tensor products of finite-dimensional von Neumann algebras. This identification places graphical rules in the same operator-bimodule framework and preserves both values and perfectness. Finally, we compare concurrency with the synchronicity conditions of Goldberg and of Bochniak--Kasprzak--Sołtan. The framework is illustrated by classical, rank-one, quantum XOR, colouring, and quantum graph homomorphism and isomorphism games.
A cryptographic compiler introduced by Kalai et al. (STOC'23) converts any nonlocal game into an interactive protocol with a single computationally bounded prover. Although the compiler is known to be sound in the case of classical provers and complete in the quantum case, quantum soundness has so far only been established for special classes of games. In this work, we establish a quantum soundness result for all compiled two-player nonlocal games. In particular, we prove that the quantum commuting operator value of the underlying nonlocal game is an upper bound on the quantum value of the compiled game. Our result employs techniques from operator algebras in a computational and cryptographic setting to establish information-theoretic objects in the asymptotic limit of the security parameter. It further relies on a sequential characterization of quantum commuting operator correlations which may be of independent interest.
This thesis studies one-turn quantum refereed games, which are abstract zero-sum games with two competing computationally unbounded quantum provers and a computationally bounded quantum referee. The provers send quantum states to the referee, who plugs the two states into his quantum circuit, measures the output of the circuit in the standard basis, and declares one of the two players as the winner depending on the outcome of the measurement. The complexity class QRG(1) comprises of those promise problems for which there exists a one-turn quantum refereed game such that one of the players wins with high probability for the yes-instances, and the other player wins with high probability for the no-instances, irrespective of the opponent’s strategy. QRG(1) is a generalization of QMA (or co-QMA), and can informally be viewed as QMA with a no-prover (or co-QMA with a yes-prover). \n \nWe have given a full characterization of QRG(1), starting with appropriate definitions and known results, and building on to two new results about this class. Previously, the best known upper bound on QRG(1) was PSPACE. We have proved that if one of the provers is completely classical, sending a classical probability distribution instead of a quantum state, the new class, which we name CQRG(1), is contained in Ǝ · PP (non- deterministic polynomial-time operator applied to the class PP). We have also defined another restricted version of QRG(1) where both provers send quantum states, but the referee measures one of the quantum states first, and plugs the classical outcome into the measurement, along with the other prover’s quantum state, into a quantum circuit, before measuring the output of the quantum circuit in the standard basis. The new class, which we name MQRG(1), is contained in P · PP (the probabilistic polynomial time operator applied to PP). Ǝ · PP is contained in P · PP, which is, in turn, contained in PSPACE. Hence, our results give better containments than PSPACE for restricted versions of QRG(1).