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Realization of the ContextualityNonlocality Tradeoff with a QutritQubit Photon Pair Peng Xue the Department of Physics, Southeast University [email protected] 1 Outlines Brief review on nonlocality and contextuality Idea of monogamy relation on contextuality vs nonlocality Realization of contextuality-nonlocality tradeoff with a qubit-qutrit photon pair 2 Einstein-Poldosky-Rosen Elements of Reality ... 3 Einstein-Poldosky-Rosen Elements of Reality 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). 4 CHSH inequality and violation of local realism Generalized Bell inequality — CHSH inequality (easier for experimental test), S E a, b E a, b ' E a ', b E a ', b ' If E=1, perfect correlation. If E=-1, perfect anticorrelation. Local Reality prediction: S MAX 2 Quantum Mechanics prediction: S MAX 2 2 Corresponding experimental tests support the necessity of quantum mechanics by violating CHSH inequality. J. Clauser et al., Phys. Rev. Lett. 23, 880 (1969). 5 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 comeasurable (compatible) observables are measured simultaneously. 6 Contextuality (KCBS inequality) 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 {Ai} (i=1,…,5) and performs them on her system. Each two of Ai and A(i+1)mod5 are compatible. Non-contextual hidden variable model: 7 KCBS inequality Quantum mechanical prediction KCBS inequality is violated based on QM prediction! 8 Monogamy relation • Two spatial separated systems: Quantum theory local reality • One quantum system: Quantum theory noncontextual reality • 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. Ref: P. Kurzyński et al., PRL 106 180402 (2014) 9 No-disturbance (ND) principle • The ND principle is a generalization of the no-signaling principle that refers to compatible observables instead of spacelike separated observables • Based on the ND principle, the probabilities of outcomes of the measurement Ai do not depend on if Ai is measured with A(i+1)mod5. • The ND principle imposes a nontrivial tradeoff between the violation of CHSH and KCBS inequalities. • Quantum theory is one kind of the ND principle. Ref: P. Kurzyński et al., PRL 106 180402 (2014) 10 Stronger monogamy relation imposed by quantum theory • The monogamy relation holds in any theory satisfying the ND principle such as quantum theory. • Quantum theory imposes a more stringent monogamy relation between quantum contextual and nonlocal correlations. • Consider Alice and Bob share a qutrit-qubit system. Alice’s measurements are , with • In particular, the state is assumed to be • Bob’s observables are chosen to be two Pauli operators B1=Z and B2=X. Ref: P. Kurzyński et al., PRL 106 180402 (2014) 11 Stronger monogamy relation imposed by quantum theory • The more stringent monogamy relation makes the quantum region smaller than that imposed by the ND principle. • Arbitrary quantum state corresponds a point inside the quantum region. • The boundary of the quantum region can be produced by the states taking the form. <CHSH>=-2.08, <KCBS>=-2.92 Ref: P. Kurzyński et al., PRL 106 180402 (2014) 12 Realization of the Contextuality-Nonlocality Tradeoff with a Qutrit-Qubit Photon Pair X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B. C. Sanders, and PX, Phys. Rev. Lett. 116, 090401 (2016) 13 Measurements • Alice randomly chooses two compatible measurements from five measurements {Ai} (i=1,..,5) and performs on her system. Each two of five measurements are compatible. • Bob chooses one of two incompatible measurements, B1 and B2 and performs on his system. 14 Experimental demonstration A qutrit-qubit state preparation: entangle photons are generated via type-I spontaneous parameteric downconversion (SPDC) and the initial state is prepared in this form Entangled photon source 15 Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016) 15 Experimental demonstration Bob’s measurements: setting Hb=0, he performs projective measurement along the z axis; setting Hb=22.5, he performs projective measurement along the x axis. Dh clicks, the result of the measurement is +1; Dv 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) 16 Experimental demonstration • Alice’s measurements via three steps • Step1: performs Ai via BD1-2 and HWP2-6 • Step2: recreates the eigenstate of Ai via BD3 and BD6, HWP7-10, HWP15-17 • Step3: performs Ai+1 via BD4-5, BD7-8, HWP11-14, HWP18-21. 17 Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016) 17 Results 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. Ref: X. Zhan, X. Zhang, J. Li, Y. S. Zhang, B.C. Sanders and PX, PRL 116 040901 (2016) 18 Meaning • 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. 19 Meaning • 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. 20 Violation of a generalized non-contextuality inequality with single-photon qubit • 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) 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 Violation of a generalized non-contextuality inequality with single-photon qubit • A violation of LSW inequality is interesting because (1) more stringent than that set by the KCBS (or KS) noncontextuality---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 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 Vertices---measurement, edges---jointly measurable contextuality. 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; 23 G+- generalized noncontextuality inequality Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted 24 Realization of unsharp measurements • State being measured • LSW inequality concerns average probability of anticorrelations (η is sharpness ) • 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 sitedependent coin rotations Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted 25 Realization of joint POVM • 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 Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted 26 Joint POVMs Single photona qubit G- - G+G-+ G++ sandwich-type QWP-HWP-QWP sets HWP • 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 Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted 27 Violation of generalized noncontextual inequality with a smallest quantum system 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. Ref: X. Zhan, J. Li, H. Qin, Z. H. Bian, Y. S. Zhang, and PX, submitted 28 Conclusion Realization of contextualitynonlocality tradeoff with a qubitqutrit photon pair Violation of a generalized noncontextuality inequality 30 Collaborators: Xiang Zhan, Zhihao Bian, Kunkun Wang, Xin Zhang, Jian Li @ Southeast Univ.; Barry C. Sanders @ USTC & Univ. of Calgary Yongsheng Zhang @ USTC 31 31 Thank you for your attention… 32 32 Implementation of POVM {E1,…,En}, where E i =λi |ψi>< ψi| 1 initialize the walker at x=0 with coin state corresponding to the qubit state to be measured |φo> 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 33 Implementation of POVM {E1,…,En}, where E i =λi |ψi>< ψi| 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 |ψi><ψi| (one of the elements of POVM E i=λi |ψi><ψi| ), i.e. mapping the state of the horizontal photons to E i |φo> 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 λiTr(|ψi><ψi|φo><φo|) , where |φo> is the state to be measured. 34