1. 1 Realization of the Contextuality- Nonlocality Tradeoff with a Qutrit-Qubit Photon Pair Peng Xue the Department of Physics, Southeast University gnep.eux@gmail.com

2. 2  Brief review on nonlocality and contextuality  Idea of monogamy relation on contextualityvs nonlocality  Realization of contextuality-nonlocalitytradeoff with a qubit-qutrit photon pair Outlines

3. Einstein-Poldosky-Rosen Elements of Reality... 3

4. Einstein-Poldosky-Rosen Elements of Reality 4 Einstein, Podolsky and Rosen’s question Is quantum mechanics complete? Or is there some hidden variable theory behind it? Bell’s answer Both a local realistic picture and quantum mechanics can explain the perfect correlations observed. If a hidden variable model is local, it can be ruled out experimentally. Bell inequality states that certain statistical correlations predicted by quantum mechanics for measurements on two-particle ensembles cannot be understood within a realistic picture based on local properties of each individual particle. J. S. Bell, Physics 1, 195 (1964).

5. CHSH inequality and violation of local realism 5 Generalized Bell inequality — CHSH inequality (easier for experimental test), Quantum Mechanics prediction: Local Reality prediction: J. Clauser et al., Phys. Rev. Lett. 23, 880 (1969). If E=1, perfect correlation. If E=-1, perfect anticorrelation. Corresponding experimental tests support the necessity of quantum mechanics by violating CHSH inequality.

6. 6 Contextuality Q: What if one has only one quantum system? A: The Kochen-Specker theorem states that noncontextual theories are incompatible with quantum mechanics. Noncontextuality means that the value for an observable predicted by such a theory does not depend on the experimental context which other co-measurable （ compatible ） observables are measured simultaneously.

7. Definition of compatibility: two measurements A and B are called compatible if they can be measured simultaneously or in any order without disturbance. KCBS inequality: Alice randomly chooses two compatible measurements from five measurements {A i } (i=1,…,5) and performs them on her system. Each two of A i and A (i+1)mod5 are compatible. Non-contextual hidden variable model: 7 Contextuality （ KCBS inequality ）

8. 8 KCBS inequality Quantum mechanical prediction KCBS inequality is violated based on QM prediction!

9. 9 Q ： Are two realities independent ？ A ： There is a monogamy relation on contextuality versus nonlocality, i.e., in the same quantum system, two inequalities can not be violated at the same time. Two spatial separated systems: Quantum theory local reality One quantum system: Quantum theory noncontextual reality Monogamy relation Ref: P. Kurzyński et al., PRL 106 180402 (2014)

10. 10 The ND principle is a generalization of the no-signaling principlethat refers to compatible observables instead of spacelikeseparated observables Based on the ND principle, the probabilities of outcomes of themeasurement A i do not depend on if A i is measured with A (i+1)mod5. The ND principle imposes a nontrivial tradeoff between theviolation of CHSH and KCBS inequalities. Quantum theory is one kind of the ND principle. No-disturbance (ND) principle Ref: P. Kurzyński et al., PRL 106 180402 (2014)

11. 11 The monogamy relation holds in any theory satisfying the NDprinciple such as quantum theory. Quantum theory imposes a more stringent monogamy relationbetween quantum contextual and nonlocal correlations. Consider Alice and Bob share a qutrit-qubit system. Alice’smeasurements are, with In particular, the state is assumed to be Bob’s observables are chosen to be two Pauli operators B 1 =Z and B 2 =X. Stronger monogamy relation imposed by quantum theory Ref: P. Kurzyński et al., PRL 106 180402 (2014)

12. 12 The more stringentmonogamy relation makesthe quantum regionsmaller than that imposedby the ND principle. Arbitrary quantum statecorresponds a point insidethe quantum region. The boundary of thequantum region can beproduced by the statestaking the form. Stronger monogamy relation imposed by quantum theory =-2.08, =-2.92 Ref: P. Kurzyński et al., PRL 106 180402 (2014)

13. 13 PX, X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B. C. Sanders, and PX, Phys. Rev. Lett. 116, 090401 (2016) Realization of the Contextuality-Nonlocality Tradeoff with a Qutrit-Qubit Photon Pair

14. 14 Measurements Alice randomly chooses two compatiblemeasurements from five measurements {A i } (i=1,..,5) and performs on her system. Each twoof five measurements are compatible. Bob chooses one of two incompatiblemeasurements, B 1 and B 2 and performs on his system.

15. 15 Experimental demonstration Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016) Entangled photon source 15 A qutrit-qubit state preparation: entangle photons aregenerated via type-I spontaneous parameteric down-conversion (SPDC) and the initial state is prepared in thisform

16. 16 Experimental demonstration Bob’s measurements: setting H b =0, he performs projective measurement along the z axis; settingH b =22.5, he performs projective measurement along the x axis. D h clicks, the result of the measurement is +1; D v clicks, the result of the measurement is -1. Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016)

17. 17 Experimental demonstration Alice’s measurements via three steps Step1: performs A i via BD1-2 and HWP2-6 Step2: recreates the eigenstate of A i via BD3 and BD6, HWP7- 10, HWP15-17 Step3: performs A i+1 via BD4-5, BD7-8, HWP11-14, HWP18- 21. Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016) 17

18. 18 Results Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016) Direct experimental evidence of a tradeoff between locally contextual correlations and spatially separated correlations In the same quantum system, two inequalities can not be violated at the same time.

19. 19 The contextuality-nonlocality monogamy suggests the existence of a quantum resource of which entanglement is just a particular form. That is, to violate the locality inequality costs entanglement as a resource, while to violate the noncontextuality inequality costs contextuality as a resource. In a quantum system, only one of the two inequalities can be violated because nothing is left to violate the other one. The resource required to violate the noncontextuality inequality and that required to violate the locality inequality are fungible through entanglement.Meaning

20. 20 Nonlocality and contextuality are both just different manifestations of a more fundamental concept, the assumption of realism. The reason for the nonlocality-contextuality tradeoff arises from the fact that both properties have the same root: the assumption of realism, which is the assumption that the physical world exists independent of our observations, and that the act of observation does not change it. Since nonlocality and contextuality can be thought of as two different manifestations of the basic assumption of realism, then one of them can be transformed into the other, but both cannot exist at the same time because they are essentially the same thing.Meaning

21. A qutrit and five projectors are required for a proof of KS-contextuality (in a state-dependent manner) A qutrit and thirteen projectors are required for a proof of KS-contextuality (in a state-independent manner) A qubit (smallest quantum system) and three unsharp binary qubit measurements are enough to violate a generalized non-contexutality---Liang, Spekkens, Wiseman (LSW) inequality (state-dependent) Violation of a generalized non-contextuality inequality with single-photon qubit Ref: Y. C. Liang, R. W. Spekkens, and H. M. Wiseman, Physics Reports 506 1-39 (2011); Ref: R. Kunjwal and S. Ghosh, Phys. Rev. A 89, 042118 (2014) 21

22. A violation of LSW inequality is interesting because (1) more stringent than that set by the KCBS (or KS) non- contextuality---larger upper bound (1-η/3 v.s. 2/3 with 0<η<1) to rule out the non-contextual models; (2) less requirements---a smallest quantum system (a qubit) and three unsharp measurements. The key point: how to realize unsharp measurements Violation of a generalized non-contextuality inequality with single-photon qubit Ref: Y. C. Liang, R. W. Spekkens, and H. M. Wiseman, Physics Reports 506 1-39 (2011); Ref: R. Kunjwal and S. Ghosh, Phys. Rev. A 89, 042118 (2014) 22

23. 23 KCBS contextuality scenario: A qutrit and five projectors; Upper bound 2/3 of the probability of anticorrelations LSW contextuality scenario: A qubit and three unsharp measurements; Upper bound 1-η/3 (0<η<1) of the probability of anticorrelations; Vertices---measurement, edges---jointly measurable contextuality.

24. 24 G +- Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted generalized noncontextuality inequality

25. 25 Realization of unsharp measurements 3 unsharp measurements can be constructed and realized by the joint POVMs, each of which has four elements Joint POVM via a 5-step quantum walk with site- dependent coin rotations Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted LSW inequality concerns average probability of anticorrelations (η is sharpness ) State being measured

26. 26 The key point is to construct and realize the joint POVMs each of which has four elements Joint POVM via a 5-step QW with site-dependent coin rotations Realization of joint POVM 26 Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted

27. 27 The probabilities of the clicks on the detectors D4, D3, D2, D1 correspond to those of the joint POVMs elements on the polarization state of single photons Joint POVMs Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted G +- G -+ G ++ G- -G- - HWP sandwich-type QWP-HWP-QWP sets Single photon- a qubit

28. Violation of generalized noncontextual inequality with a smallest quantum system Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted The measured average probability of anticorrelations violates the boundary ( ) and is in a good agreement with quantum prediction 0.8075. Support the necessity of quantum machines. Quantum machines is proven to be complete even with a smallest quantum system---a single qubit. 28

29. 29 Applications of unsharp measurements  unsharp measurement does not strongly disturb the state dynamics but still obtain some information about the system  estimating an unknown single pure qubit state from repeated unsharp measurements  monitoring the Rabi oscillations of a single qubit in a driving field  understanding the relation between information gain and disturbance

30. 30  Realization of contextuality- nonlocality tradeoff with a qubit-qutrit photon pair  Violation of a generalized noncontextuality inequality Conclusion

31. 31 Collaborators: Xiang Zhan, Zhihao Bian, Kunkun Wang, Xin Zhang, Jian Li @ Southeast Univ.; Barry C. Sanders @ USTC & Univ. of Calgary Yongsheng Zhang @ USTC 31

32. 32 Thank you for your attention…

33. 33 Implementation of POVM {E 1,…,E n }, where E i =λ i |ψ i >< ψ i | 1 initialize the walker at x=0 with coin state corresponding to the qubit state to be measured |φ o > 2 set i:=1 3 while i<n (n is the number of the elements of the POVM) do (a)For each odd step, apply coin operation at position x=0 and identity elsewhere and then apply position shift operation (b)For each even step, apply coin operation at x =1, NOT gate at x=-1 and identity elsewhere and then position shift operation (c) i:=i+1, next round

34. 34 Implementation of POVM {E 1,…,E n }, where E i =λ i |ψ i >< ψ i | coin operation C (1) i is chosen to guarantee that after the step 3(a) the unitary operation at position x=1 on the ‘initial’ state is proportional to |ψ i > coin operation C (2) i is chosen to guarantee after the step 3(b) the probability of click at x=2 is the probability of the element of POVM E i applied on the system of interest λ i Tr(|ψ i > is the state to be measured.