1. Quantum Information Technology Group in NUS Singapore Theoretical Section Experimental Section

2. QIT Group (Singapore)  Janet Anders  Chia Teck Chee  Chen Jingling  Chen Lai Keat  Choo Keng Wah  Du Jiangfeng  Berge Englert  Feng Xunli  Ajay Gopinathan  Darwin Gosal  Hor Wei Hann  D. Kaszlikowski  Christian Kurtsiefer  L.C. Kwek  C.H. Lai  Wayne Lawton  Lim Jenn Yang  Antia Lamas Linares Artur Ekert and C.H. Oh  Alex Ling  Looi Shiang Yong  Liu Xiongjun  Ivan Marcikic  Neelima Raitha  Kuldip Singh  Tey Meng Khoon  Tong Dianmin  Wang Zisheng  Wu Chunfeng  5-10 undergraduate students  www.quantumlah.org www.quantumlah.org

3. Speakers: Prof. A. Ekert (Oxford) Prof. C. Bennett (IBM) Prof. S. Popescu (Bristol) Prof. I. Chuang (MIT) First Workshop on Quantum Computation and Information held in 2001

4. Focus Quantum Cryptography; Quantum Algorithms; Quantum Games; Quantum Cloning; Quantum Channels; Geometric Phase Computation; Quantum Entanglement; Foundation of Quantum Mechanics; Bell Inequalities, Bures Fidelity

5. A New Bell Inequality with Improved Visibility for 3 qubits Chunfeng Wu, Jingling Chen, L. C. Kwek and C. H. Oh Physics Department National University of Singapore Presented at the International Conference on Recent Progress in Quantum Mechanics and Its Applications, Hong Kong, China December 13, 2005 to December 16, 2005 Quantum Information Technology Group – Quantumlah

6. Outline Introduction Bell inequalities A new Bell inequality for three qubits Quantum Information Technology Group – Quantumlah

7. Introduction In a 1935 Einstein, Podolsky, and Rosen (EPR) poised the question “can quantum mechanical description of physical reality be considered complete?” paper [1] Element of physical reality: “If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” (sufficient, not necessary condition to define an element of reality). Completeness: “In a complete theory there is an element corresponding to each element of reality.” Locality: “The real factual situation of the system A is independent of what is done with the system B, which is spatially separated from the former.” Quantum Information Technology Group – Quantumlah [1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935).

8.  Spooky action: the mysterious long-range correlations between the two widely separated particles.  Local hidden variables are suggested in order to restore locality and completeness to quantum mechanics. In a local hidden variable theory, measurement is fundamentally deterministic, but appears to be probabilistic because some degrees of freedom are not precisely known. EPR Paradox Quantum Information Technology Group – Quantumlah

9. Entanglement  Central to EPR paper is an entangled state.  The notion of entanglement [2] was introduced by Schrödinger to describe a situation in which “Maximal knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all…” [2] E. Schrodinger, The present situation in quantum mechanics. In J. Wheeler and W. Zurek, editors, it Quantum Theory and Measurement, P 152, Princeton University Press, 1983. Quantum Information Technology Group – Quantumlah

10. Entanglement  Understanding of quantum entanglement the information in a composite system resides more in the correlations than in properties of individuals. Quantum Information Technology Group – Quantumlah

11. The Bell Theorem  In 1964, Bell [3] showed that local realism imposes experimentally constraints on the statistical measurements of separated systems. These constraints, called Bell inequalities, can be violated by the predictions of quantum mechanics. J. Bell’s contribution: Consider the correlations predicted for three spin measurements not at right angles but at an arbitrary angle . He was able to prove that correlations predicted by quantum mechanics are greater than could be obtained from any local hidden variable theory.  Violation of Bell inequalities is one method to identify entanglement. [3] J. S. Bell, Physics, 1, 195 (1964). Quantum Information Technology Group – Quantumlah

12. The Bell Theorem  The original Bell inequalities are not suitable for realistic experimental verification. One of the most common form of Bell inequalities is Clauser-Horne-Shimony-Holt (CHSH) inequality [4] for two qubit system, Quantum Information Technology Group – Quantumlah [4] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).

13. The Bell Theorem  The function is the correlation of measurements between and for the two systems.  Classically, thus, It is easy to check that the above inequality is always valid under classical theory. Since, either or. Quantum Information Technology Group – Quantumlah

14. The Bell Theorem  Quantum mechanically, For maximally entangled state For appropriate angles, Quantum Information Technology Group – Quantumlah

15. The Bell Theorem Thus, CHSH inequality is violated. Quantum Information Technology Group – Quantumlah

16. Bell Inequalities 1982 Aspect experiment: two detectors were placed 13m apart and a container of excited calcium atoms midway between them; spin states of two entangled photons. 1997 N Gisin: two detectors were placed 11km apart Rule out local hidden variables

17. Bell Inequalities  Are Bell inequalities violated by all pure entangled states?  Recent developments. (1) Gisin’s theorem [5]: every pure bipartite entangled state in two dimensions violates the CHSH inequality. Quantum Information Technology Group – Quantumlah [5] N. Gisin, Phys. Lett. A 154, 201 (1991); N. Gisin and A. Peres, Phys. Lett. A 162, 15 (1992).

18. Gisin’s Theorem (1991) Phys. Lett. A, 154, 201 (1991) All entangled pure states violate Bell inequalities. Quantum Information Technology Group – Quantumlah

19. Bell Inequalities (2) Mermin-Ardehali-Belinskii-Klyshko inequalities for N qubits [6]. Quantum Information Technology Group – Quantumlah Denoting the expectation value of the products of the results of the experiments where denote the same expression but with all the and exchanged. [6] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990); M. Ardehali, Phys. Rev. A 46, 5375 (1992); A. V. Belinskii and D. N. Klyshko, Phys. Usp. 36, 653 (1993).

20. Bell Inequalities Take for example that the three-qubit MABK inequality is given as Quantum Information Technology Group – Quantumlah If we use Q to describe correlation function, the above inequality can also be written as here we take where

21. Bell Inequalities (3) Scarani and Gisin [7] noticed that there exist pure states of N qubits which do not violate MABK inequalities. For, the states do not violate the MABK inequalities. Quantum Information Technology Group – Quantumlah [7] V. Scarani and N. Gisin, J. Phys. A 34, 6043 (2001). They consider to find maximal quantum violation, with a MABK Bell operator. For example, where

22. Bell Inequalities Quantum Information Technology Group – Quantumlah [8] M. Zukowski and C. Brukner, Phys. Rev. Lett. 88, 210401 (2002). [9] R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001). (4) Zukowski [8] and Werner [9] independently found the most general Bell inequalities for N qubits (called Zukowski-Brukner inequalities here). To violate the MABK inequalities, it is required This results So for, the generalized GHZ states do not violate the MABK inequalities.

23. Bell Inequalities The assumption of local realism implies the existence of two numbers and each taking values +1 or -1 which describe the result of a measurement by the jth observer of the observables defined, by Consider N observers and allow each of them to choose between two dichotomic observables, determined by local parameters denoted and. The correlation function, in the case of a local realistic theory, is the average over many runs of the experiment Quantum Information Technology Group – Quantumlah Zukowski-Brukner(ZB) inequalities for N qubits

24. Bell Inequalities After averaging the expression over the ensemble of the runs of the experiment, the following set of Bell inequality is obtained Quantum Information Technology Group – Quantumlah These are the ZB inequalities.

25. Bell Inequalities (5) Zukowski et al [10] showed that For N=even, the generalized GHZ states violate the Zukowski-Brukner inequalities; For N=odd and, the correlations between measurements on qubits in the generalized GHZ states satisfy the ZB inequalities for correlations. Quantum Information Technology Group – Quantumlah [10] M. Zukowski, C. Brukner, W. Laskowski and M. Wiesniak, Phys. Rev. Lett. 88, 210402 (2002). [11] J. L. Chen, C. F. Wu, L. C. Kwek and C. H. Oh, Phys. Rev. Lett. 93, 140407 (2004). (6) We constructed Bell inequalities [11] for three qubits in terms of correlation functions. These inequalities are violated by all pure entangled states.

26. Bell Inequalities  Bell inequalities are sensitive to the presence of noise and above a certain amount of noise, the Bell inequalities will cease to be violated by QM. Quantum Information Technology Group – Quantumlah In the case of considering violation strength, quantum system is described by and The strength of violation or visibility (V ) is considered as the minimal amount V of the given entangled state that one has to add to pure noise so that the resulting state still violates local realism.

27. Bell Inequalities  The Bell inequalities given by us [11] are not good enough to the resistance of noise. For the GHZ state, threshold visibility is 0.77 (it is 0.5 for 3-qubit Zukowski-Brukner inequality).  Our recent research shows that there is one new Bell inequality for three qubits with improved visibility. [11] J. L. Chen, C. F. Wu, L. C. Kwek and C. H. Oh, Phys. Rev. Lett. 93, 140407 (2004). Quantum Information Technology Group – Quantumlah

28. Bell Inequalities for 3 qubits (1)3-qubit Zukowski-Brukner inequality [8] Quantum Information Technology Group – Quantumlah (2) Our previous 3-qubit Bell inequality [11] where are three-particle correlation functions defined as after many runs of experiments. Similar definition for two-particle correlation functions

29. A new Bell Inequality for 3 qubits (3) Bell inequality with improved visibility: Quantum Information Technology Group – Quantumlah (+)

30. A new Bell Inequality for 3 qubits  Quantum mechanically, Quantum Information Technology Group – Quantumlah  Correlation functions in quantum mechanics

31. A new Bell Inequality for 3 qubits Quantum Information Technology Group – Quantumlah [13] W. Dur, G. Vidal and J. I. Cirac, Phys. Rev. A 62, 062314 (2000). [14] D.M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 69. [15] A. Zeilinger, M. A. Horne, and D.M. Greenberger, in Workshop on Squeezed States and Uncertainty Relations, edited by D. Han et al., NASA Conference Publication No. 3135 (NASA, Washington, DC, 1992), p. 73. For three qubits, there are only two classes of genuinely three particle entangled states which are inequivalent[13]. (2) The second by the so-called W state [15]. (1) The first class is represented by the GHZ state [14],

32. A new Bell Inequality for 3 qubits Quantum Information Technology Group – Quantumlah (1) Numerical results for Generalized GHZ states which violate the inequality except. For the GHZ state 4.404

33. Quantum Information Technology Group – Quantumlah A new Bell Inequality for 3 qubits For the W state (2) Numerical results for Generalized W states which violate the inequality except the cases with and.

34. Quantum Information Technology Group – Quantumlah A new Bell Inequality for 3 qubits To show that the new Bell inequality is more resistant to noise than our previous one. We rewrite them as follows,

35. Quantum Information Technology Group – Quantumlah A new Bell Inequality for 3 qubits Usually, the left hand side of Bell inequality can be described by a quantity, called Bell quantity. For the two Bell inequalities, we write separately

36. Quantum Information Technology Group – Quantumlah A new Bell Inequality for 3 qubits Curve A is for our previous inequality and curve B is for the new inequality. Quantum violation in the figure is the quantum prediction for Bell quantity. It is clear that the new inequality is more resistant to noise than the previous inequality.

37. Generalized Bell Inequalities  Bell inequalities for M qubits (M>3)  Bell inequalities for M qudits (M>3) M-qudit: M particles in d-dimensional Hilbert space. Quantum Information Technology Group – Quantumlah

38. Summary  A new Bell inequality in terms of correlation functions with improved visibility for 3 qubits is constructed.  However, the threshold visibility has not reached the optimal value 0.5 as exhibited by the maximal violation of the ZB inequality by the GHZ state. To construct such a Bell inequality for three qubits is an open problem.  Generalization of Gisin’s theorem for N qubit (N>3, odd numbers) is still unsolved at this stage. Quantum Information Technology Group – Quantumlah

41. Bell Inequalities for 3 qubits  Our previous inequality is violated by all pure entangled states of three qubits.  The quantum violation strength of the GHZ state of the inequality is not as strong as that of ZB inequality as seen in (1) and (2) below. (1)The threshold visibility of the inequality for the GHZ state is 0.77. Quantum Information Technology Group – Quantumlah For the GHZ state, namely, when, the maximal violation of the inequality is. So the threshold visibility is.

42. Bell Inequalities for 3 qubits (2)The threshold visibility of the ZB inequality for the GHZ state is 0.5. Quantum Information Technology Group – Quantumlah For the GHZ state, namely, when, the maximal violation of the ZB inequality is 4. So the threshold visibility is 2/4=0.5.

43. Generalized Bell Inequalities  Bell inequalities for M qubits (M>3)  Bell inequalities for M qudits (M>3) M-qudit: M particles in d-dimensional Hilbert space. Quantum Information Technology Group – Quantumlah