Generalized Satisfiability Problems via Operator Assignments

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For the case of constraints given by a system of linear equations over the two-element field, this relaxation has received considerable attention in the foundations of quantum mechanics, where such constructions as the Mermin-Peres magic square show that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. We obtain a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments. To establish our main result, we adapt the notion of primitive-positive definability (pp-definability) to our setting, a notion that has been used extensively in the study of constraint satisfaction problems. Here, we show that pp-definability gives rise to gadget reductions that preserve satisfiability gaps. We also present several additional applications of this method. In particular and perhaps surprisingly, we show that the relaxed notion of pp-definability in which the quantified variables are allowed to range over operator assignments gives no additional expressive power in defining Boolean relations

Uncertainty relations for the realisation of macroscopic quantum superpositions and EPR paradoxes

We present a unified approach, based on the use of quantum uncertainty relations, for arriving at criteria for the demonstration of the EPR paradox and macroscopic superpositions. We suggest to view each criterion as a means to demonstrate an EPR-type paradox, where there is an inconsistency between the assumptions of a form of realism, either macroscopic realism (MR) or local realism (LR), and the completeness of quantum mechanics.Comment: 9 pages, 2 figures, to appear Journ Mod Optics work presented at PQE 2007 conferenc

Halfvortices in flat nanomagnets

We discuss a new type of topological defect in XY systems where the O(2) symmetry is broken in the presence of a boundary. Of particular interest is the appearance of such defects in nanomagnets with a planar geometry. They are manifested as kinks of magnetization along the edge and can be viewed as halfvortices with winding numbers \pm 1/2. We argue that halfvortices play a role equally important to that of ordinary vortices in the statics and dynamics of flat nanomagnets. Domain walls found in experiments and numerical simulations are composite objects containing two or more of these elementary defects. We also discuss a closely related system: the two-dimensional smectic liquid crystal films with planar boundary condition.Comment: 7 pages, 8 figures, To appear as a chapter in Les Houches summer school on Quantum Magnetis

On the solution of trivalent decision problems by quantum state identification

The trivalent functions of a trit can be grouped into equipartitions of three elements. We discuss the separation of the corresponding functional classes by quantum state identifications

Realization of GHZ States and the GHZ Test via Cavity QED

In this article we discuss the realization of atomic GHZ states involving three-level atoms and we show explicitly how to use this state to perform the GHZ test in which it is possible to decide between local realism theories and quantum mechanics. The experimental realizations proposed makes use of the interaction of Rydberg atoms with a cavity prepared in a coherent state.Comment: 16 pages and 3 figures. submitted to J. Mod. Op

Drawing quantum contextuality with 'dessins d'enfants'

The Frontiers Collection: "It from Bit orBit from It", ed. by A. Aguirre, B. Foster, Z. Meralli (Springer, 2014), pp 37-50International audienceIn the standard formulation of quantum mechanics, there exists an inherent feedback of the measurement setting on the elementary object under scrutiny. Thus one cannot assume that an 'element of reality' prexists to the measurement and, it is even more intriguing that unperformed/counterfactual observables enter the game. This is called quantum contextuality. Simple finite projective geometries are a good way to picture the commutation relations of quantum observables entering the context, at least for systems with two or three parties. In the essay, it is further discovered a mathematical mechanism for 'drawing' the contexts. The so-called 'dessins d'enfants' of the celebrated mathematician Alexandre Grothendieck feature group, graph, topological, geometric and algebraic properties of the quantum contexts that would otherwise have been 'hidden' in the apparent randomness of measurement outcomes

Generalised Compositional Theories and Diagrammatic Reasoning

This chapter provides an introduction to the use of diagrammatic language, or perhaps more accurately, diagrammatic calculus, in quantum information and quantum foundations. We illustrate the use of diagrammatic calculus in one particular case, namely the study of complementarity and non-locality, two fundamental concepts of quantum theory whose relationship we explore in later part of this chapter. The diagrammatic calculus that we are concerned with here is not merely an illustrative tool, but it has both (i) a conceptual physical backbone, which allows it to act as a foundation for diverse physical theories, and (ii) a genuine mathematical underpinning, permitting one to relate it to standard mathematical structures.Comment: To appear as a Springer book chapter chapter, edited by G. Chirabella, R. Spekken

All-Versus-Nothing Proof of Einstein-Podolsky-Rosen Steering

Einstein-Podolsky-Rosen steering is a form of quantum nonlocality intermediate between entanglement and Bell nonlocality. Although Schr\"odinger already mooted the idea in 1935, steering still defies a complete understanding. In analogy to "all-versus-nothing" proofs of Bell nonlocality, here we present a proof of steering without inequalities rendering the detection of correlations leading to a violation of steering inequalities unnecessary. We show that, given any two-qubit entangled state, the existence of certain projective measurement by Alice so that Bob's normalized conditional states can be regarded as two different pure states provides a criterion for Alice-to-Bob steerability. A steering inequality equivalent to the all-versus-nothing proof is also obtained. Our result clearly demonstrates that there exist many quantum states which do not violate any previously known steering inequality but are indeed steerable. Our method offers advantages over the existing methods for experimentally testing steerability, and sheds new light on the asymmetric steering problem.Comment: 7 pages, 2 figures. Accepted in Sci. Re

Topological Schr\"odinger cats: Non-local quantum superpositions of topological defects

Topological defects (such as monopoles, vortex lines, or domain walls) mark locations where disparate choices of a broken symmetry vacuum elsewhere in the system lead to irreconcilable differences. They are energetically costly (the energy density in their core reaches that of the prior symmetric vacuum) but topologically stable (the whole manifold would have to be rearranged to get rid of the defect). We show how, in a paradigmatic model of a quantum phase transition, a topological defect can be put in a non-local superposition, so that - in a region large compared to the size of its core - the order parameter of the system is "undecided" by being in a quantum superposition of conflicting choices of the broken symmetry. We demonstrate how to exhibit such a "Schr\"odinger kink" by devising a version of a double-slit experiment suitable for topological defects. Coherence detectable in such experiments will be suppressed as a consequence of interaction with the environment. We analyze environment-induced decoherence and discuss its role in symmetry breaking.Comment: 7 pages, 4 figure

Measurements on the reality of the wavefunction

Quantum mechanics is an outstandingly successful description of nature, underpinning fields from biology through chemistry to physics. At its heart is the quantum wavefunction, the central tool for describing quantum systems. Yet it is still unclear what the wavefunction actually is: does it merely represent our limited knowledge of a system, or is it an element of reality? Recent no-go theorems argued that if there was any underlying reality to start with, the wavefunction must be real. However, that conclusion relied on debatable assumptions, without which a partial knowledge interpretation can be maintained to some extent. A different approach is to impose bounds on the degree to which knowledge interpretations can explain quantum phenomena, such as why we cannot perfectly distinguish non-orthogonal quantum states. Here we experimentally test this approach with single photons. We find that no knowledge interpretation can fully explain the indistinguishability of non-orthogonal quantum states in three and four dimensions. Assuming that some underlying reality exists, our results strengthen the view that the entire wavefunction should be real. The only alternative is to adopt more unorthodox concepts such as backwards-in-time causation, or to completely abandon any notion of objective reality.Comment: 7 pages, 4 figure