We introduce a new two-player nonlocal game, called the (G, H)-isomorphism game, where classical players can win with certainty if and only if the graphs G and H are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the (G, H)-isomorphism game, respectively. First, we prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to determine quantum isomorphic graphs that are not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our proof techniques are related to the Feige, Goldwasser, Lovász, Safra, and Szegedy reduction from the inapproximability literature [

Prepare-and-measure (P&M) quantum networks are the basic building blocks of quantum communication and cryptography. These networks crucially rely on non-orthogonal quantum encodings to distribute quantum correlations, thus enabling superior communication rates and informationtheoretic security. Here, we present a computational toolbox that is able to efficiently characterise the set of input-output probability distributions for any discrete-variable P&M quantum network, assuming only the inner-product information of the quantum encodings. Our toolbox is thus highly versatile and can be used to analyse a wide range of quantum network protocols, including those that employ infinite-dimensional quantum code states. To demonstrate the feasibility and efficacy of our toolbox, we use it to reveal new results in multipartite quantum distributed computing and quantum cryptography. Taken together, these findings suggest that our method may have implications for quantum network information theory and the development of new quantum technologies. arXiv:1803.04796v2 [quant-ph]

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the non-convexity of certain restricted-dimensional quantum correlations.In what ranks as one of the most important achievements of modern physics, it was shown by John Bell in 1964 that some correlations generated within the framework of quantum mechanics can be nonlocal, in the sense that the statistics generated by quantum mechanics cannot always be reproduced by a local hidden-variable model [1,2]. Over the last 40 years, there have been significant efforts in trying to verify this fact experimentally. The first such experimental data [3] was published in 1972 and this remains an active area of research [4]. Moreover, as a central concept in quantum physics and quantum information theory, fully understanding quantum entanglement and nonlocality still remains a very interesting and important problem with far-reaching applications. Indeed, profound relationships between quantum nonlocality and other fundamental quantum concepts or phenomena such as entanglement measures [5,6], entanglement distillation [7,8], and teleportation [9] have been identified. Meanwhile, for many tasks, e.g. in cryptography [10,11], it has been realized that due to quantum nonlocality, quantum strategies enjoy remarkable advantages over their classical counterparts.However, even though quantum nonlocal effects can lead to interesting and often surprising advantages in some applications, this does not paint the full picture. After all, for practical applications, it is just as important to understand the amount of quantum resources required for these advantages to manifest. For instance, if there is an exponential blowup in the amount of resources required, then whatever advantage gained by employing quantum mechanics may not be useful in practice. Quantifying the amount of quantum resources needed to perform a certain task is the central focus of this Letter.We study quantum nonlocality from the viewpoint of two-party quantum correlations that arise from a Bell experiment. A two-party Bell experiment is performed between two parties, Alice and Bob, whose labs are set up in separate locations. Alice (resp. Bob) has in her possession a measurement apparatus whose possible settings are labelled by the elements of a finite set X (resp. Y ) and the possible measurement outcomes are labelled by a finite set A (resp. B). After repeat...