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Martin B Plenio Koenraad MR Audenaert

Imperial College London. When are Correlations Quantum?: Verification and Quantification of Entanglement with simple measurements. Institute for Mathematical Sciences, 53 Princes Gate, Exhibition Road, Imperial College London, London SW7 2PG

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Martin B Plenio Koenraad MR Audenaert

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  1. Imperial College London When are Correlations Quantum?: Verification and Quantification of Entanglement with simple measurements Institute for Mathematical Sciences, 53 Princes Gate, Exhibition Road, Imperial College London, London SW7 2PG Quantum Optics and Laser Science Group, Blackett Laboratory, Prince Consort Road, Imperial College London, London SW7 2BW http://www.imperial.ac.uk/quantuminformation Martin B Plenio Koenraad MR Audenaert

  2. Simple correlations Measure: Observe: Optimist: Pessimist: Lots of literature: Horodecki et al, Phys. Lett A 1996, Buzek et al, JMO 1997, Horodecki et al, PRA 1999, Terhal, Phys. Lett. A 2000, Sancho & Huelga, PRA 2000, Hannover school, Guehne et al, JMO 2003, …

  3. The general principle Problem: Experiment does not have direct access to entanglementbut only a limited set of measurements data. Audenaert & Plenio, NJP 8, 266 (2006)

  4. The general principle Problem: Experiment does not have direct access to entanglementbut only a limited set of measurements data. General Question: Given the measurement data, what is the least amount of entanglement E that iscompatible with these data? Audenaert & Plenio, NJP 8, 266 (2006)

  5. The general principle Problem: Experiment does not have direct access to entanglementbut only a limited set of measurements data. General Question: Given the measurement data, what is the least amount of entanglement E that iscompatible with these data? Result: These measurement data have then proven that at least E units of entanglement are present. Audenaert & Plenio, NJP 8, 266 (2006)

  6. Correlations + Purity

  7. Correlations + Purity Unphysical

  8. Correlations + Purity Unphysical No entanglementmay be inferred

  9. Correlations + Purity Unphysical No entanglementmay be inferred Audenaert & Plenio, NJP 8, 266 (2006)

  10. Simple Correlations Consider Logarithmic Negativity: Minimal amount of entanglement compatible with and Audenaert & Plenio, NJP 8, 266 (2006)

  11. Simple Correlations Minimal amount of entanglement compatible with , , . Audenaert & Plenio, NJP 8, 266 (2006)

  12. Example Tsomokos, Hartmann, Huelga & Plenio, NJP 9, 2007

  13. Example Optimize measurement basis is the eigenbasis of Obtain from idealized model and then apply to experiment Tsomokos, Hartmann, Huelga & Plenio, NJP 9, 2007

  14. Being Economical Measure: Obtain as by-product: , … )

  15. Being economical with the data Measure: Obtain as by-product: , … ) & (

  16. Some Mathematics Audenaert & Plenio, NJP 8, 266 (2006)

  17. Some Mathematics Audenaert & Plenio, NJP 8, 266 (2006)

  18. Some Mathematics } Lagrange duality Audenaert & Plenio, NJP 8, 266 (2006) provides lower bound

  19. Some Mathematics = { = if > 0 Audenaert & Plenio, NJP 8, 266 (2006)

  20. Some Mathematics = { = if > 0 } Any choice yields lower bound & Clever guesses give analytical lower bounds Audenaert & Plenio, NJP 8, 266 (2006) Optimization is a semi-definite programme

  21. Applications and

  22. Another Application F = Flip operator

  23. The general principle Problem: Experiment does not have direct access to physical quantity P but only a limited set of measurements. General Question: Given the measurement data, what is the least amount q(P) of the physical quantity P that iscompatible with these data? Result: The measurement data have proven that at least q(P) of the physical quantity P are present but no more.

  24. Entanglement Theory Comparison to witnesses disentangled

  25. Entanglement Theory Comparison to witnesses tr[W r] < 0 1 disentangled Witness-operator W: tr[Wr] > 0 for all separable states tr[Wr] < 0 for some entangled states

  26. Entanglement Theory Comparison to witnesses W W 1 2 tr[W r] < 0 & tr[W r] < 0 1 1 disentangled Witness-operator W: tr[Wr] > 0 for all separable states tr[Wr] < 0 for some entangled states

  27. Entanglement Theory Comparison to witnesses Cleverly construct witness: Hard to measure directly measure locally and Then construct tr[Wr] . This approach discards much information that could be used to bound entanglement !

  28. Other approaches Jaynes principle: = l=0 Chose the state that reproduces the experimental data and maximizes the entropy. Buzek, Drobny, Adam, Derka, and Knight, J. Mod. Opt. 44, 2607 (1997) Question I: Do both approaches agree on entangled/disentangled both. Question II: Which approach is more reliable, i.e. closer to the truth ‘on average’

  29. Simple correlations Question I: Do both approaches agree on entangled/disentangled both. = entangled Horodecki, Horodecki, Horodecki, Phys. Rev. A 59, 1799 (1999) but = separable, is compatible with data

  30. Simple correlations Question II: Which approach is more reliable, i.e. closer to the truth ‘on average’ According to which distribution?

  31. Simple correlations Question II: Which approach is more reliable, i.e. closer to the truth ‘on average’ According to which distribution? Pure state distribution on doubled Hilbert space Chose mixed state, compute expectation of B and then • estimate state according to Jaynes principle • determine minimal entanglement compatible with data • Numerical results: Both approaches give very similar results

  32. References This talk was based on • Audenaert and Plenio, “When are correlations quantum?—Verification and quantification of entanglement by simple measurements”, NJP 8, 266 (2006) see also • Eisert, Audenaert and Brandao, NJP 9, 60 (2007) • Reimpell, Guehne & Werner, to appear PRL 2007 Sponsors of this work

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