Quantum theory is not only successfully tested in laboratories every day but also constitutes a robust theoretical framework: small variations usually lead to implausible consequences, such as faster-than-light communication. It has even been argued that quantum theory may be special among possible theories. Here we report that, at the level of correlations among different systems, quantum theory is not so special. We define a set of correlations, dubbed 'almost quantum', and prove that it strictly contains the set of quantum correlations but satisfies all-but-one of the proposed principles to capture quantum correlations. We present numerical evidence that the remaining principle is satisfied too.

Quantum correlations exhibit behaviour that cannot be resolved with a local hidden variable picture of the world. In quantum information, they are also used as resources for information processing tasks, such as Measurement-based Quantum Computation (MQC). In MQC, universal quantum computation can be achieved via adaptive measurements on a suitable entangled resource state. In this paper, we look at a version of MQC in which we remove the adaptivity of measurements and aim to understand what computational abilities still remain in the resource. We show that there are explicit connections between this model of computation and the question of non-classicality in quantum correlations. We demonstrate this by focussing on deterministic computation of Boolean functions, in which natural generalisations of the Greenberger-Horne-Zeilinger (GHZ) paradox emerge; we then explore probabilistic computation, via which multipartite Bell Inequalities can be defined. We use this correspondence to define families of multi-party Bell inequalities, which we show to have a number of interesting contrasting properties. PACS numbers: I. INTRODUCTIONSpace-like separated measurements on entangled quantum systems can reveal correlations which would not appear if Nature were described by a local hidden variable (LHV) theory [1]. Such correlations can violate a Bell inequality, a condition expressing a restriction on the correlations permitted in any LHV theory. Violations of Bell inequalities provide a dramatic distinction between locally realistic classical physics and quantum mechanics.The field of quantum information processing has developed with the aim of exploiting the non-classical properties of quantum systems for practical tasks. Non-classical (i.e. Bell inequality violating) correlations are an important quantum resource in several areas of quantum information. For example, they can be used to guarantee the security of quantum key distribution [2] even if the devices used cannot be trusted; in distributed computational tasks, nonclassical correlations can reduce the communication required between parties [3,4]. One of the most renowned applications of quantum mechanics to information processing is quantum computation. We know that a quantum computer, if realised, would permit efficient solutions to problems for which no efficient classical algorithm is known [5]. Despite the importance of entanglement in quantum computation, and despite its evident non-classical nature, the link between Bell inequality violations and quantum computational speed-up remains unclear. Recently, however, it was shown [6,7] that Bell inequalities do have a concrete connection with a particular model of quantum computation, measurement-based quantum computation (MQC). MQC is a model of quantum computation, very different from the standard unitary circuit model. Computation progresses in two stages. First a many-particle entangled state is prepared, e.g. the cluster state [8], and then singlesite (qubit) measurements are performed. By an appropriate choi...

The verification of quantum devices is an important aspect of quantum information, especially with the emergence of more advanced experimental implementations of quantum computation and secure communication. Within this, the theory of device-independent robust self-testing via Bell tests has reached a level of maturity now that many quantum states and measurements can be verified without direct access to the quantum systems: interaction with the devices is solely classical. However, the requirements for this robust level of verification are daunting and require high levels of experimental accuracy. In this paper we discuss the possibility of self-testing where we only have direct access to one part of the quantum device. This motivates the study of self-testing via EPR-steering, an intermediate form of entanglement verification between full state tomography and Bell tests. Quantum non-locality implies EPR-steering so results in the former can apply in the latter, but we ask what advantages may be gleaned from the latter over the former given that one can do partial state tomography? We show that in the case of self-testing a maximally entangled two-qubit state, or ebit, EPR-steering allows for simpler analysis and better error tolerance than in the case of full device-independence. On the other hand, this improvement is only a constant improvement and (up to constants) is the best one can hope for. Finally, we indicate that the main advantage in self-testing based on EPR-steering could be in the case of self-testing multi-partite quantum states and measurements. For example, it may be easier to establish a tensor product structure for a particular party's Hilbert space even if we do not have access to their part of the global quantum system.

The emergence of quantum computers has challenged long-held beliefs about what is efficiently computable given our current physical theories. However, going back to the work of Abrams and Lloyd, changing one aspect of quantum theory can result in yet more dramatic increases in computational power, as well as violations of fundamental physical principles. Here we focus on efficient computation within a framework of general physical theories that make good operational sense. In prior work, Lee and Barrett showed that in any theory satisfying the principle of tomographic locality (roughly, local measurements suffice for tomography of multipartite states) the complexity bound on efficient computation is AWPP. This bound holds independently of whether the principle of causality (roughly, no signalling from the future) is satisfied. In this work we show that this bound is tight: there exists a theory satisfying both the principles of tomographic locality and causality which can efficiently decide everything in AWPP, and in particular can simulate any efficient quantum computation. Thus the class AWPP has a natural physical interpretation: it is precisely the class of problems that can be solved efficiently in tomographically-local theories. This theory is built upon a model of computing involving Turing machines with quasi-probabilities, to wit, machines with transition weights that can be negative but sum to unity over all branches. In analogy with the study of non-local quantum correlations, this leads us to question what physical principles recover the power of quantum computing. Along this line, we give some computational complexity evidence that quantum computation does not achieve the bound of AWPP.

The violation of certain Bell inequalities allows for device-independent information processing secure against nonsignaling eavesdroppers. However, this only holds for the Bell network, in which two or more agents perform local measurements on a single shared source of entanglement. To overcome the practical constraints that entangled systems can only be transmitted over relatively short distances, large-scale multisource networks have been employed. Do there exist analogs of Bell inequalities for such networks, whose violation is a resource for device independence? In this Letter, the violation of recently derived polynomial Bell inequalities will be shown to allow for device independence on multisource networks, secure against nonsignaling eavesdroppers.

Non-locality and steering are both non-classical phenomena witnessed in nature as a result of quantum entanglement. It is now well-established that one can study non-locality independently of the formalism of quantum mechanics, in the so-called device-independent framework. With regards to steering, although one cannot study it completely independently of the quantum formalism, 'post-quantum steering' has been described, which is steering that cannot be reproduced by measurements on entangled states but does not lead to superluminal signalling. In this work we present a framework based on the study of quantum channels in which one can study steering (and non-locality) in quantum theory and beyond. In this framework, we show that kinds of steering, whether quantum or post-quantum, are directly related to particular families of quantum channels that have been previously introduced by Beckman et al (2001 Phys. Rev. A 64 052309). Utilizing this connection we also demonstrate new analytical examples of post-quantum steering, give a quantum channel interpretation of almost quantum non-locality and steering, easily recover and generalize the celebrated Gisin-Hughston-Jozsa-Wootters theorem, and initiate the study of post-quantum Buscemi non-locality and non-classical teleportation. In this way, we see post-quantum non-locality and steering as just two aspects of a more general phenomenon.Entanglement is one of the most striking non-classical features of quantum mechanics. Given appropriately chosen measurements certain, but not all, entangled states can exhibit a violation of local realism (local causality), called 'non-locality' [1]. Apart from its fundamental interest, non-locality has also turned into a key resource for certain information-theoretic tasks, such as key distribution [2] or certified quantum randomness generation [3], and has been witnessed experimentally in a loophole-free manner [4][5][6].The non-classical implications of entanglement also manifest as a phenomenon called 'Einstein-Podolsky-Rosen steering', henceforth referred to as solely 'steering'. There, one party, Alice, by performing appropriately chosen measurements on one half of an entangled state, remotely 'steers' the states held by a distant party, Bob, in a way which has no local explanation [7]. A modern approach to steering describes it as a way to certify entanglement in cryptographic situations where some devices in the protocol are not characterized [8]. Steering hence allows for a 'one-sided device-independent' implementation of several information-theoretic tasks, such as quantum key distribution [9], randomness certification [10,11], measurement incompatibility certification [12][13][14], and self-testing of quantum states [15,16].Even though these phenomena arise naturally within quantum mechanics, they are not restricted to it. Non-local correlations and steering beyond what quantum theory allows are conceivable while still complying with natural physical assumptions, such as relativistic causality [17,18]. By 'post-quantum' we m...

We develop a unified approach to classical, quantum and post-quantum steering. The framework is based on uncharacterised (black-box) parties performing quantum measurements on their share of a (possibly unphysical) quantum state, and its starting point is the characterisation of general no-signalling assemblages via non-positive local hidden-state models, which will be defined in this work. By developing a connection to entanglement witnesses, this formalism allows for new definitions of families of assemblages, in particular via (i) non-decomposable positive maps and (ii) unextendible product bases. The former proves to be useful for constructing post-quantum assemblages with the built-in feature of yielding only quantum correlations in Bell experiments, while the latter always gives certifiably post-quantum assemblages. Finally, our framework is equipped with an inherent quantifier of post-quantum steering, which we call the negativity of post-quantum steering. We postulate that post-quantum steering should not increase under one-way quantum operations from the steered parties to the steering parties, and we show that, in this sense, the negativity of post-quantum steering is a convex post-quantum-steering monotone.The concept of steering was first introduced by Schrödinger in 1935 [1] in response to the Einstein et al paradox [2]. It refers to the phenomenon where one party, Alice, by performing measurements on one part of a shared system, seemingly remotely 'steers' the state of the system held by a distant party, Bob, in a way which has no explanation in terms of local causal influences. Steering has only recently been formally defined in a quantum information-theoretic setting [3], as a way of certifying the entanglement of quantum systems without the need to trust one of the parties, or when one of the parties is using uncharacterised devices. In this setting, the uncharacterised party convinces the other party that they shared entanglement by demonstrating steering. Furthermore, if all parties are uncharacterised (or untrusted) then one recovers the device-independent setting of a standard Bell test. Steering thus may be seen as one in a family of non-classical phenomena, closely related to entanglement and Bell non-locality [4]. Indeed, Bell nonlocality implies steering, and steering implies entanglement, however all three concepts are inequivalent [3,5].It is well-known that, in spite of demonstrating non-locality, local measurements on entangled quantum systems cannot be used to communicate superluminally. That is, correlations that are generated by varying the choice of local measurements on space-like separated quantum subsystem-which we define to be quantum correlations-satisfy the principle of no-signalling. We will call no-signalling colleations all correlations that do not permit signalling. One can conceive of no-signalling correlations that cannot be realised by local measurements on quantum states, hence called post-quantum correlations; this possibility was first pointed out in a seminal work...

Measurement-based quantum computation (MBQC) is a model of quantum computation, in which computation proceeds via adaptive single qubit measurements on a multiqubit quantum state. It is computationally equivalent to the circuit model. Unlike the circuit model, however, its classical analog is little studied. Here we present a classical analog of MBQC whose computational complexity presents a rich structure. To do so, we identify uniform families of quantum computations [refining the circuits introduced by Bremner Proc. R. Soc. A 467, 459 (2010)] whose output is likely hard to exactly simulate (sample) classically. We demonstrate that these circuit families can be efficiently implemented in the MBQC model without adaptive measurement and, thus, can be achieved in a classical analog of MBQC whose resource state is a probability distribution which has been created quantum mechanically. Such states (by definition) violate no Bell inequality, but, if widely held beliefs about computational complexity are true, they, nevertheless, exhibit nonclassicality when used as a computational resource—an imprint of their quantum origin.