Random Variables Recorded under Mutually Exclusive Conditions: Contextuality-by-Default

We present general principles underlying analysis of the dependence of random variables (outputs) on deterministic conditions (inputs). Random outputs recorded under mutually exclusive input values are labeled by these values and considered stochastically unrelated, possessing no joint distribution. An input that does not directly influence an output creates a context for the latter. Any constraint imposed on the dependence of random outputs on inputs can be characterized by considering all possible couplings (joint distributions) imposed on stochastically unrelated outputs. The target application of these principles is a quantum mechanical system of entangled particles, with directions of spin measurements chosen for each particle being inputs and the spins recorded outputs. The sphere of applicability, however, spans systems across physical, biological, and behavioral sciences.Comment: In H. Liljenstr\"om (Ed.) Advances in Cognitive Neurodynamics IV (pp. 405-410) (2015

Testing axioms for Quantum Mechanics on Probabilistic toy-theories

In Ref. [1] one of the authors proposed postulates for axiomatizing Quantum Mechanics as a "fair operational framework", namely regarding the theory as a set of rules that allow the experimenter to predict future events on the basis of suitable tests, having local control and low experimental complexity. In addition to causality, the following postulates have been considered: PFAITH (existence of a pure preparationally faithful state), and FAITHE (existence of a faithful effect). These postulates have exhibited an unexpected theoretical power, excluding all known nonquantum probabilistic theories. Later in Ref. [2] in addition to causality and PFAITH, postulate LDISCR (local discriminability) and PURIFY (purifiability of all states) have been considered, narrowing the probabilistic theory to something very close to Quantum Mechanics. In the present paper we test the above postulates on some nonquantum probabilistic models. The first model, "the two-box world" is an extension of the Popescu-Rohrlich model, which achieves the greatest violation of the CHSH inequality compatible with the no-signaling principle. The second model "the two-clock world" is actually a full class of models, all having a disk as convex set of states for the local system. One of them corresponds to the "the two-rebit world", namely qubits with real Hilbert space. The third model--"the spin-factor"--is a sort of n-dimensional generalization of the clock. Finally the last model is "the classical probabilistic theory". We see how each model violates some of the proposed postulates, when and how teleportation can be achieved, and we analyze other interesting connections between these postulate violations, along with deep relations between the local and the non-local structures of the probabilistic theory.Comment: Submitted to QIP Special Issue on Foundations of Quantum Informatio

Three-slit experiments and quantum nonlocality

An interesting link between two very different physical aspects of quantum mechanics is revealed; these are the absence of third-order interference and Tsirelson's bound for the nonlocal correlations. Considering multiple-slit experiments - not only the traditional configuration with two slits, but also configurations with three and more slits - Sorkin detected that third-order (and higher-order) interference is not possible in quantum mechanics. The EPR experiments show that quantum mechanics involves nonlocal correlations which are demonstrated in a violation of the Bell or CHSH inequality, but are still limited by a bound discovered by Tsirelson. It now turns out that Tsirelson's bound holds in a broad class of probabilistic theories provided that they rule out third-order interference. A major characteristic of this class is the existence of a reasonable calculus of conditional probability or, phrased more physically, of a reasonable model for the quantum measurement process.Comment: 9 pages, no figur

PR-box correlations have no classical limit

One of Yakir Aharonov's endlessly captivating physics ideas is the conjecture that two axioms, namely relativistic causality ("no superluminal signalling") and nonlocality, so nearly contradict each other that a unique theory - quantum mechanics - reconciles them. But superquantum (or "PR-box") correlations imply that quantum mechanics is not the most nonlocal theory (in the sense of nonlocal correlations) consistent with relativistic causality. Let us consider supplementing these two axioms with a minimal third axiom: there exists a classical limit in which macroscopic observables commute. That is, just as quantum mechanics has a classical limit, so must any generalization of quantum mechanics. In this classical limit, PR-box correlations violate relativistic causality. Generalized to all stronger-than-quantum bipartite correlations, this result is a derivation of Tsirelson's bound without assuming quantum mechanics.Comment: for a video of this talk at the Aharonov-80 Conference in 2012 at Chapman University, see quantum.chapman.edu/talk-10, published in Quantum Theory: A Two-Time Success Story (Yakir Aharonov Festschrift), eds. D. C. Struppa and J. M. Tollaksen (New York: Springer), 2013, pp. 205-21

Two qubits of a W state violate Bell's inequality beyond Cirel'son's bound

It is shown that the correlations between two qubits selected from a trio prepared in a W state violate the Clauser-Horne-Shimony-Holt inequality more than the correlations between two qubits in any quantum state. Such a violation beyond Cirel'son's bound is smaller than the one achieved by two qubits selected from a trio in a Greenberger-Horne-Zeilinger state [A. Cabello, Phys. Rev. Lett. 88, 060403 (2002)]. However, it has the advantage that all local observers can know from their own measurements whether their qubits belongs or not to the selected pair.Comment: REVTeX4, 5 page

Bell's inequalities for states with positive partial transpose

We study violations of n particle Bell inequalities (as developed by Mermin and Klyshko) under the assumption that suitable partial transposes of the density operator are positive. If all transposes with respect to a partition of the system into p subsystems are positive, the best upper bound on the violation is 2^((n-p)/2). In particular, if the partial transposes with respect to all subsystems are positive, the inequalities are satisfied. This is supporting evidence for a recent conjecture by Peres that positivity of partial transposes could be equivalent to existence of local classical models.Comment: 4 pages, REVTe

Quantum Key Distribution between N partners: optimal eavesdropping and Bell's inequalities

Quantum secret-sharing protocols involving N partners (NQSS) are key distribution protocols in which Alice encodes her key into $N-1$ qubits, in such a way that all the other partners must cooperate in order to retrieve the key. On these protocols, several eavesdropping scenarios are possible: some partners may want to reconstruct the key without the help of the other ones, and consequently collaborate with an Eve that eavesdrops on the other partners' channels. For each of these scenarios, we give the optimal individual attack that the Eve can perform. In case of such an optimal attack, the authorized partners have a higher information on the key than the unauthorized ones if and only if they can violate a Bell's inequality.Comment: 14 pages, 1 figur

Greenberger-Horne-Zeilinger nonlocality for continuous variable systems

As a development of our previous work, this paper is concerned with the Greenberger-Horne-Zeilinger (GHZ) nonlocality for continuous variable cases. The discussion is based on the introduction of a pseudospin operator, which has the same algebra as the Pauli operator, for each of the $N$ modes of a light field. Then the Bell-CHSH (Clauser, Horne, Shimony and Holt) inequality is presented for the $N$ modes, each of which has a continuous degree of freedom. Following Mermin's argument, it is demonstrated that for $N$-mode parity-entangled GHZ states (in an infinite-dimensional Hilbert space) of the light field, the contradictions between quantum mechanics and local realism grow exponentially with $N$, similarly to the usual $N$-spin cases.Comment: RevTEX; comments are welcomed; new version with minor change

Monogamy of Correlations vs. Monogamy of Entanglement

A fruitful way of studying physical theories is via the question whether the possible physical states and different kinds of correlations in each theory can be shared to different parties. Over the past few years it has become clear that both quantum entanglement and non-locality (i.e., correlations that violate Bell-type inequalities) have limited shareability properties and can sometimes even be monogamous. We give a self-contained review of these results as well as present new results on the shareability of different kinds of correlations, including local, quantum and no-signalling correlations. This includes an alternative simpler proof of the Toner-Verstraete monogamy inequality for quantum correlations, as well as a strengthening thereof. Further, the relationship between sharing non-local quantum correlations and sharing mixed entangled states is investigated, and already for the simplest case of bi-partite correlations and qubits this is shown to be non-trivial. Also, a recently proposed new interpretation of Bell's theorem by Schumacher in terms of shareability of correlations is critically assessed. Finally, the relevance of monogamy of non-local correlations for secure quantum key distribution is pointed out, although, and importantly, it is stressed that not all non-local correlations are monogamous.Comment: 12 pages, 2 figures. Invited submission to a special issue of Quantum Information Processing. v2: Published version. Open acces

Effects of decoherence and errors on Bell-inequality violation

We study optimal conditions for violation of the Clauser-Horne-Shimony-Holt form of the Bell inequality in the presence of decoherence and measurement errors. We obtain all detector configurations providing the maximal Bell inequality violation for a general (pure or mixed) state. We consider local decoherence which includes energy relaxation at the zero temperature and arbitrary dephasing. Conditions for the maximal Bell-inequality violation in the presence of decoherence are analyzed both analytically and numerically for the general case and for a number of important special cases. Combined effects of measurement errors and decoherence are also discussed.Comment: 18 pages, 5 figure