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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Quantum Information and Computation

J. Math. Phys. 53, 022201 (2012); http://dx.doi.org/10.1063/1.3681905 (30 pages)

Local quasi hidden variable modelling and violations of Bell-type inequalities by a multipartite quantum state

Elena R. Loubenets

Applied Mathematics Department, Moscow State Institute of Electronics and Mathematics, Moscow 109028, Russia

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(Received 30 June 2011; accepted 30 December 2011; published online 15 February 2012; publisher error corrected 21 February 2012)

We introduce for a general correlation scenario a new simulation model, a local quasi hidden variable (LqHV) model, where locality and the measure-theoretic construction inherent to a local hidden variable (LHV) model are preserved but positivity of a simulation measure is dropped. We specify a necessary and sufficient condition for LqHV modelling and, based on this, prove that every quantum correlation scenario admits a LqHV simulation. Via the LqHV approach, we construct analogs of Bell-type inequalities for an N-partite quantum state and find a new analytical upper bound on the maximal violation by an N-partite quantum state of S1 × ⋯ × SN-setting Bell-type inequalities – either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type, discrete or continuous. This general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an S1 × ⋯ × SN-setting LHV description but also leads to the new exact numerical upper estimates on the maximal Bell violations for concrete N-partite quantum states used in quantum information processing and for an arbitrary N-partite quantum state. We, in particular, prove that violation by an N-partite quantum state of an arbitrary Bell-type inequality (either on correlation functions or on joint probabilities) for S settings per site cannot exceed (2S − 1)N − 1 even in case of an infinite dimensional quantum state and infinitely many outcomes.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. PRELIMINARIES: SOURCE OPERATORS, TENSOR POSITIVITY, THE COVERING NORM
  3. LqHV MODELLING OF A GENERAL CORRELATION SCENARIO
  4. BELL-TYPE INEQUALITIES
  5. LqHV MODELLING OF A QUANTUM CORRELATION SCENARIO
  6. QUANTUM VIOLATIONS OF BELL-TYPE INEQUALITIES
    1. Numerical estimates
    2. Discussion
  7. CONCLUSIONS

EDITORIALLY RELATED

  1. Publisher's Note: “Local quasi hidden variable modelling and violations of Bell-type inequalities by a multipartite quantum state” [J. Math. Phys. 53, 022201 (2012)]
    Elena R. Loubenets
    J. Math. Phys. 53, 049901 (2012)JMAPAQ000053000004049901000001

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KEYWORDS and PACS

Keywords

quantum theory, statistical analysis, Bell theorem

PACS

  • 03.65.Ta

    Foundations of quantum mechanics; measurement theory

  • 03.65.Ud

    Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

  • 03.67.-a

    Quantum information

  • 02.50.Cw

    Probability theory

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3681905

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
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    A. Fine, Phys. Rev. Lett. 48, 291 (1982).

    R. F. Werner, Phys. Rev. A 40, 4277 (1989).

    M. Terhal, A. C. Doherty, and D. Schwab, Phys. Rev. Lett. 90, 157903 (2003).

    A. Acin, I. Masanes, and N. Gisin, Phys. Rev. Lett. 97, 120405 (2006).

    H. Buhrman, R. Cleve, S. Massar, and R. De Wolf, Rev. Mod. Phys. 82, 665 (2010).

    A. N. Acin, N. Gisin, and B. Toner, Phys. Rev. A 73, 062105 (2006).



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