1. When are Correlations Quantum?: Verification and Quantification of Entanglement with simple measurements Imperial College London Martin B Plenio Koenraad MR Audenaert 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

2. Kish Island, 8 th September 2007 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. Kish Island, 8 th September 2007 The general principle Problem: Experiment does not have direct access to entanglement  but only a limited set of measurements data. Audenaert & Plenio, NJP 8, 266 (2006)

4. Kish Island, 8 th September 2007 The general principle Problem: Experiment does not have direct access to entanglement  but only a limited set of measurements data. General Question: Given the measurement data, what is the least amount of entanglement E that is  compatible with these data? Audenaert & Plenio, NJP 8, 266 (2006)

5. Kish Island, 8 th September 2007 The general principle Problem: Experiment does not have direct access to entanglement  but only a limited set of measurements data. General Question: Given the measurement data, what is the least amount of entanglement E that is  compatible 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. Kish Island, 8 th September 2007 Correlations + Purity

7. Kish Island, 8 th September 2007 Correlations + Purity Unphysical

8. Kish Island, 8 th September 2007 Correlations + Purity Unphysical No entanglement may be inferred

9. Kish Island, 8 th September 2007 Correlations + Purity Unphysical No entanglement may be inferred Audenaert & Plenio, NJP 8, 266 (2006)

10. Kish Island, 8 th September 2007 Simple Correlations Consider Logarithmic Negativity: Minimal amount of entanglement compatible with and Audenaert & Plenio, NJP 8, 266 (2006)

11. Kish Island, 8 th September 2007 Simple Correlations Minimal amount of entanglement compatible with,,. Audenaert & Plenio, NJP 8, 266 (2006)

12. Kish Island, 8 th September 2007 Example Tsomokos, Hartmann, Huelga & Plenio, NJP 9, 2007

13. Kish Island, 8 th September 2007 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. Kish Island, 8 th September 2007 Being Economical Measure: Obtain as by-product:, … )

15. Kish Island, 8 th September 2007 Being economical with the data Measure: Obtain as by-product:, … ) ( &

16. Kish Island, 8 th September 2007 Some Mathematics Audenaert & Plenio, NJP 8, 266 (2006)

17. Kish Island, 8 th September 2007 Some Mathematics Audenaert & Plenio, NJP 8, 266 (2006)

18. Kish Island, 8 th September 2007 Some Mathematics Lagrange duality provides lower bound Audenaert & Plenio, NJP 8, 266 (2006) }

19. Kish Island, 8 th September 2007 Some Mathematics = = if > 0 { Audenaert & Plenio, NJP 8, 266 (2006)

20. Kish Island, 8 th September 2007 Some Mathematics = = if > 0 { Any choice & yields lower bound Clever guesses give analytical lower bounds Optimization is a semi-definite programme Audenaert & Plenio, NJP 8, 266 (2006) }

21. Kish Island, 8 th September 2007 Applications and

22. Kish Island, 8 th September 2007 Another Application F = Flip operator

23. Kish Island, 8 th September 2007 The general principle Problem: Experiment does not have direct access to physical quantity  but only a limited set of measurements. General Question: Given the measurement data, what is the least amount q(  ) of the physical quantity  that is  compatible with these data? Result: The measurement data have proven that at least q(  ) of the physical quantity  are present but no more.

24. Kish Island, 8 th September 2007 Entanglement Theory Comparison to witnesses disentangled

25. Kish Island, 8 th September 2007 Entanglement Theory Comparison to witnesses disentangled tr[ W  ] > 0 for all separable states tr[ W  ] < 0 for some entangled states Witness-operator W : 1 tr[ W  ] < 0

26. Kish Island, 8 th September 2007 Entanglement Theory Comparison to witnesses disentangled tr[ W  ] > 0 for all separable states tr[ W  ] < 0 for some entangled states Witness-operator W : W W 1 2 tr[ W  ] < 0 & 1 tr[ W  ] < 0 1

27. Kish Island, 8 th September 2007 Entanglement Theory Comparison to witnesses Cleverly construct witness: Hard to measure directly measure locally and Then construct tr[W  This approach discards much information that could be used to bound entanglement !

28. Kish Island, 8 th September 2007 Other approaches Jaynes principle:  = Chose the state that reproduces the experimental data and maximizes the entropy. 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’ Buzek, Drobny, Adam, Derka, and Knight, J. Mod. Opt. 44, 2607 (1997)

29. Kish Island, 8 th September 2007 Simple correlations Question I: Do both approaches agree on entangled/disentangled both. Horodecki, Horodecki, Horodecki, Phys. Rev. A 59, 1799 (1999) = entangled but = separable, is compatible with data

30. Kish Island, 8 th September 2007 Simple correlations Question II: Which approach is more reliable, i.e. closer to the truth ‘on average’ According to which distribution?

31. Kish Island, 8 th September 2007 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  Numerical results: Both approaches give very similar results Chose mixed state, compute expectation of B and then estimate state according to Jaynes principle determine minimal entanglement compatible with data

32. Kish Island, 8 th September 2007 Sponsors of this work 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)  Eisert, Audenaert and Brandao, NJP 9, 60 (2007) see also  Reimpell, Guehne & Werner, to appear PRL 2007