All CHSH polytopes

The correlations that admit a local hidden-variable model are described by a family of polytopes, whose facets are the Bell inequalities. The CHSH inequality is the simplest such Bell inequality and is a facet of every Bell polytope. We investigate for which Bell polytopes the CHSH inequality is also the unique (non-trivial) facet. We prove that the CHSH inequality is the unique facet for all bipartite polytopes where at least one party has a binary choice of dichotomic measurements, irrespective of the number of measurement settings and outcomes for the other party. Based on numerical results, we conjecture that it is also the unique facet for all bipartite polytopes involving two measurements per party where at least one measurement is dichotomic. Finally, we remark that these two situations can be the only ones for which the CHSH inequality is the unique facet, i.e., any polytope that does not correspond to one of these two cases necessarily has facets that are not of the CHSH form. As a byproduct of our approach, we derive a new family of facet inequalities

Random 'choices' and the locality loophole

It has been claimed that to close the locality loophole in a Bell experiment, random numbers of quantum origin should be used for selecting the measurement settings. This is how it has been implemented in all recent Bell experiment addressing this loophole. I point out in this note that quantum random number generators are unnecessary for such experiments and that a Bell experiment with a pseudo-random (but otherwise completely deterministic) mechanism for selecting the measurement settings, such as taking a hash function of the latest million tweets with the hashtag #quantum, would be as convincing, or even more, than one using quantum random number generators.Comment: This note is based on a talk I gave at the GISIN'14 workshop in September 2014 and at the Randomness in Quantum Physics and Beyond conference in May 201

Effects of preparation and measurement misalignments on the security of the BB84 quantum key distribution protocol

The ideal Bennett-Brassard 1984 (BB84) quantum key distribution protocol is based on the preparation and measurement of qubits in two alternative bases differing by an angle of pi/2. Any real implementation of the protocol, though, will inevitably introduce misalignments in the preparation of the states and in the alignment of the measurement bases with respect to this ideal situation. Various security proofs take into account (at least partially) such errors, i.e., show how Alice and Bob can still distil a secure key in the presence of these imperfections. Here, we consider the complementary problem: how can Eve exploit misalignments to obtain more information about the key than would be possible in an ideal implementation? Specifically, we investigate the effects of misalignment errors on the security of the BB84 protocol in the case of individual attacks, where necessary and sufficient conditions for security are known. Though the effects of these errors are small for expected deviations from the perfect situation, our results nevertheless show that Alice and Bob can incorrectly conclude that they have established a secure key if the inevitable experimental errors in the state preparation and in the alignment of the measurements are not taken into account. This gives further weight to the idea that the formulation and security analysis of any quantum cryptography protocol should be based on realistic assumptions about the properties of the apparatus used. Additionally, we note that BB84 seems more robust against alignment imperfections if both the x and z bases are used to generate the key

Popescu-Rohrlich correlations as a unit of nonlocality

A set of nonlocal correlations that have come to be known as a PR box suggest themselves as a natural unit of nonlocality, much as a singlet is a natural unit of entanglement. We present two results relevant to this idea. One is that a wide class of multipartite correlations can be simulated using local operations on PR boxes only. We show this with an explicit scheme, which has the interesting feature that the number of PR boxes required is related to the computational resources necessary to represent a function defining the multipartite box. The second result is that there are quantum multipartite correlations, arising from measurements on a cluster state, that cannot be simulated with n PR boxes, for any n.Comment: 5 pages, no figures. v2: minor modification

Maximally Non-Local and Monogamous Quantum Correlations

We introduce a version of the chained Bell inequality for an arbitrary number of measurement outcomes, and use it to give a simple proof that the maximally entangled state of two d dimensional quantum systems has no local component. That is, if we write its quantum correlations as a mixture of local correlations and general (not necessarily quantum) correlations, the coefficient of the local correlations must be zero. This suggests an experimental programme to obtain as good an upper bound as possible on the fraction of local states, and provides a lower bound on the amount of classical communication needed to simulate a maximally entangled state in dxd dimensions. We also prove that the quantum correlations violating the inequality are monogamous among non-signalling correlations, and hence can be used for quantum key distribution secure against post-quantum (but non-signalling) eavesdroppers.Comment: 5 pages, no figure

Quantifying the randomness of copies of noisy Popescu-Rohrlich correlations

In a no-signaling world, the outputs of a nonlocal box cannot be completely predetermined, a feature that is exploited in many quantum information protocols exploiting non-locality, such as device-independent randomness generation and quantum key distribution. This relation between non-locality and randomness can be formally quantified through the min-entropy, a measure of the unpredictability of the outputs that holds conditioned on the knowledge of any adversary that is limited only by the no-signaling principle. This quantity can easily be computed for the noisy Popescu-Rohrlich (PR) box, the paradigmatic example of non-locality. In this paper, we consider the min-entropy associated to several copies of noisy PR boxes. In the case where n noisy PR-boxes are implemented using n non-communicating pairs of devices, it is known that each PR-box behaves as an independent biased coin: the min-entropy per PR-box is constant with the number of copies. We show that this doesn't hold in more general scenarios where several noisy PR-boxes are implemented from a single pair of devices, either used sequentially n times or producing n outcome bits in a single run. In this case, the min-entropy per PR-box is smaller than the min-entropy of a single PR-box, and it decreases as the number of copies increases.Comment: 14 pages + 8 figures. Mathematica files attached. Comments welcom

Bounding the set of quantum correlations

We introduce a hierarchy of conditions necessarily satisfied by any distribution P(ab) representing the probabilities for two separate observers to obtain outcomes a and b when making local measurements on a shared quantum state. Each condition in this hierarchy is formulated as a semidefinite program. Our approach can be used to obtain upper-bounds on the quantum violation of an arbitrary Bell inequality. It yields, for instance, tight bounds for the violations of the Collins et al. inequalities.Comment: 5 pages, no figures. v2: minor modification

Proposal for Implementing Device-Independent Quantum Key Distribution based on a Heralded Qubit Amplification

In device-independent quantum key distribution (DIQKD), the violation of a Bell inequality is exploited to establish a shared key that is secure independently of the internal workings of the QKD devices. An experimental implementation of DIQKD, however, is still awaited, since hitherto all optical Bell tests are subject to the detection loophole, making the protocol unsecured. In particular, photon losses in the quantum channel represent a fundamental limitation for DIQKD. Here, we introduce a heralded qubit amplifier based on single-photon sources and linear optics that provides a realistic solution to overcome the problem of channel losses in Bell tests.Comment: 5 pages, 4 figures, 6 page appendi

Hyperdense coding and superadditivity of classical capacities in hypersphere theories

In quantum superdense coding, two parties previously sharing entanglement can communicate a two bit message by sending a single qubit. We study this feature in the broader framework of general probabilistic theories. We consider a particular class of theories in which the local state space of the communicating parties corresponds to Euclidean hyperballs of dimension n (the case n = 3 corresponds to the Bloch ball of quantum theory). We show that a single n-ball can encode at most one bit of information, independently of n. We introduce a bipartite extension of such theories for which there exist dense coding protocols such that log_2 (n+1) bits are communicated if entanglement is previously shared by the communicating parties. For n > 3, these protocols are more powerful than the quantum one, because more than two bits are communicated by transmission of a system that locally encodes at most one bit. We call this phenomenon hyperdense coding. Our hyperdense coding protocols imply superadditive classical capacities: two entangled systems can encode log_2 (n+1) > 2 bits, even though each system individually encodes at most one bit. In our examples, hyperdense coding and superadditivity of classical capacities come at the expense of violating tomographic locality or dynamical continuous reversibility.Comment: Expanded discussion in response to referee comments. Accepted for publication in New Journal of Physic