2,726 research outputs found

    Quantum Entanglement and Projective Ring Geometry

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    The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15×\times15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of nn copies of the Galois field GF(2), with nn = 2, 3 and 4.Comment: 13 pages, 6 figures Fig. 3 improved, typos corrected; Version 4: Final Version Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

    Commutativity, comeasurability, and contextuality in the Kochen-Specker arguments

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    If noncontextuality is defined as the robustness of a system's response to a measurement against other simultaneous measurements, then the Kochen-Specker arguments do not provide an algebraic proof for quantum contextuality. Namely, for the argument to be effective, (i) each operator must be uniquely associated with a measurement and (ii) commuting operators must represent simultaneous measurements. However, in all Kochen-Specker arguments discussed in the literature either (i) or (ii) is not met. Arguments meeting (i) contain at least one subset of mutually commuting operators which do not represent simultaneous measurements and hence fail to physically justify the functional composition principle. Arguments meeting (ii) associate some operators with more than one measurement and hence need to invoke an extra assumption different from noncontextuality.Comment: 27 pages, 1 figur

    Quantum Probability Theory

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    The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.Comment: 28 pages, LaTeX, typos removed and some minor modifications for clarity and accuracy made. This is the version to appear in Studies in the History and Philosophy of Modern Physic

    Selective and Efficient Quantum Process Tomography

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    In this paper we describe in detail and generalize a method for quantum process tomography that was presented in [A. Bendersky, F. Pastawski, J. P. Paz, Physical Review Letters 100, 190403 (2008)]. The method enables the efficient estimation of any element of the χ\chi--matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required to implement it scale polynomically with the number of qubits of the system. The estimation of all diagonal elements of the χ\chi--matrix can be efficiently done without any ancillary qubits. In turn, the estimation of all the off-diagonal elements requires an extra clean qubit. The key ideas of the method, that is based on efficient estimation by random sampling over a set of states forming a 2--design, are described in detail. Efficient methods for preparing and detecting such states are explicitly shown.Comment: 9 pages, 5 figure

    Partially Unbiased Entangled Bases

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    In this contribution we group the operator basis for d^2 dimensional Hilbert space in a way that enables us to relate bases of entangled states with single particle mutually unbiased state bases (MUB), each in dimensionality d. We utilize these sets of operators to show that an arbitrary density matrix for this d^2 dimensional Hilbert space system is analyzed by via d^2+d+1 measurements, d^2-d of which involve those entangled states that we associate with MUB of the d-dimensional single particle constituents. The number d2+d+1d^2+d+1 lies in the middle of the number of measurements needed for bipartite state reconstruction with two-particle MUB (d^2+1) and those needed by single-particle MUB [(d^2+1)^2].Comment: 5 page
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