Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate "measurements" in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.

Original languageEnglish
Article number042105
JournalPhysical Review A
Volume77
Issue number4
DOIs
Publication statusPublished - Apr 9 2008

Fingerprint

Hilbert space
bells
real numbers
constrictions
proportion

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Physics and Astronomy(all)

Cite this

Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities. / Pál, K.; Vértesi, T.

In: Physical Review A, Vol. 77, No. 4, 042105, 09.04.2008.

Research output: Contribution to journalArticle

@article{dd5154a9ce324866b8e9e7b54f47d918,
title = "Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities",
abstract = "We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate {"}measurements{"} in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.",
author = "K. P{\'a}l and T. V{\'e}rtesi",
year = "2008",
month = "4",
day = "9",
doi = "10.1103/PhysRevA.77.042105",
language = "English",
volume = "77",
journal = "Physical Review A",
issn = "2469-9926",
publisher = "American Physical Society",
number = "4",

}

TY - JOUR

T1 - Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities

AU - Pál, K.

AU - Vértesi, T.

PY - 2008/4/9

Y1 - 2008/4/9

N2 - We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate "measurements" in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.

AB - We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate "measurements" in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.

UR - http://www.scopus.com/inward/record.url?scp=42149143610&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42149143610&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.77.042105

DO - 10.1103/PhysRevA.77.042105

M3 - Article

AN - SCOPUS:42149143610

VL - 77

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 4

M1 - 042105

ER -