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Small violations of Bell inequalities by random states
Raphael C. Drumond
(in collaboration with R. I. Oliveira,
to appear in Phys. Rev. A)
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Question
• For a fixed quantum systemand a propertyP of interest:
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Question
• For a fixed quantum systemand a propertyP of interest:
“How many” quantum states satisfyP?
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Examples
• Entanglemente.g.: P. Hayden, D. W. Leung, A. Winter,Comm. Math. Phys. 265(1) 95, (2006).
• Ergodicity:e.g.: N. Linden, S. Popescu, A.J. Short, A. Winter, Phys. Rev. E 79, 061103
(2009).
• (1-way) Quantum computatione.g.: J. Bremner, C. Mora, A. Winter,Phys. Rev. Lett. 102, 190502 (2009).
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“How many”
• Measure (or Volume or Probability Measure)
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“How many”
• Measure (or Volume or Probability Measure)
• For pure states of a finite dimensional Hilbert space we think them as high dimensional spheres:
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What about non-locality?
• For pure states is generic: non-local iff entangled
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What about non-locality?
• For pure states is generic: non-local iff entangled
• Mixed states: ?
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What about non-locality?
• For pure states is generic: non-local iff entangled
• Mixed states: ?
•What is the typical degree?
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Question’
• What is the measure (probability) of the set (event) constitued by the pure states that violate some Bell inequality by at least v?
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WWZB inequalities
• Consider N systems where, on each of them, one can measure two observables A0,j
and A1,j, with outcomes +1 and -1. We have, for LHV models:
M. Zukowski and C. Brukner, Phys. Rev. Lett. 88, 210401 (2002).
R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001).
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Violations by pure states
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Violations by pure states
• For appropriate states and observables:
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Question’’
• If
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Question’’
• If
• What is
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Our result:• For systemof N parts, each withd-dimensional
Hilbert space:
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Our result:• For systemof N parts, each withd-dimensional
Hilbert space:
• For d=2:
• For d>2
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Idea of the proofWe want to estimate the measure of:
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Idea of the proofWe want to estimate the measure of:
So:
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Idea of the proofWe want to estimate the measure of:
So:To sum up:
non-linear inequality+epsilon-net+Lévy’s lemma=bound
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Noiseandthecase d=2
• For white noise:
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Noiseandthecase d=2
• For white noise:
• For any
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Further work:
• Generalization to an arbitrary scenario
(n,m,N)
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Further work:
• Generalization to an arbitrary scenario
(n,m,N)
• Result valid as long as d>>n,m
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Further work:
• Generalization to an arbitrary scenario
(n,m,N)
• Result valid as long as d>>n,m
• What happens if d<<n,m?
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Thank You!