Experimental test of quantum contextuality

Experimental test of quantum contextualityOtfried GühneG. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann,A. Cabello, R. Blatt, C. F. RoosInstitut für Quantenoptik und QuanteninformationÖsterreichische Akademie der Wissenschaften, Innsbruck

Outline1 The Kochen-Specker Theorem and the Mermin-Peres square2 An experimental test with trapped ions3 Extended inequalities for testing hidden variable models4 Conclusion & Outlook

Hidden variable modelsEinstein, Podolsky and Rosen’s questionIs quantum mechanics complete? Or is there some hidden variable (HV)theory behind it?A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 (1935).Bell’s answerIf a hidden variable model is local, it can be ruled out experimentally.J.S. Bell, Physics 1, 195 (1965); Aspect et al., Phys. Rev. Lett. 47, 460 (1981)Also other models for distant correlations can be ruled out.A. Leggett, Found. Phys. (2003); S. Groeblacher et al, Nature (2007); C. Branciard et al, Nat. Phys. (2008)But what if one has only one quantum system?

The Kochen-Specker theoremQuantum mechanics cannot be explainedby non-contextual hidden variable models.E. Specker, Dialectica 14, 239 (1960), J.S. Bell, RMP 38 (1966), S. Kochen, E. Specker, J. Math. Mech. 17 (1967).What does non-contextuality mean?

CompatibilityCompatibility of measurementsDefinition. Two measurements A and B are called compatible (A ∼ B) ifthey can be measured simultaneously or in any order without disturbance.If a sequence like A 1 B 2 A 3 B 4 A 5 A 6 B 7 ... is measured on a single state,then the values of A and B remain the same.Compatibility is experimentally testable.In QM: Commuting observables are compatible.This is a pairwise relation, but one can also talk about compatibilityof three or more observables.

ContextualityNon-contextual theoriesAssume that A ∼ B and A ∼ C. A theory is non-contextual, if it assigns toA a value v(A) independently whether B or C is measured jointly with A.This definition needs at least 3 measurements.B and C don’t have to be compatible.Is this a reasonable assumption about theories ?KS Theorem: Such models are in conflict with quantum theoryUsual proof: If |ψ i 〉 form an orthonormal basis, then P i = |ψ i 〉〈ψ i | arecompatible. Non-contextual models have to assign to each P i a value {0, 1}independent of the complete basis. On can write down sets of bases, suchthat this cannot be done.A. Cabello, Int. J. Quantum Inf. 4, 55 (2006)

The Mermin-Peres squareConsider a four level system (two qubits) and the observables:A = σ z ⊗½, B =½⊗σ z , C = σ z ⊗ σ z ,a =½⊗σ x , b = σ x ⊗½, c = σ x ⊗ σ x ,α = σ z ⊗ σ x , β = σ x ⊗ σ z , γ = σ y ⊗ σ y .The observables in each row (R i ) and column (C j ) commuteand are compatible.If we assign to each of them a value v = ±1 independently ofthe row or column, we have3∏R i C i = +1i=1In QM: C 3 = Ccγ = −½, hence ∏ 3i=1 R iC i = −1.A. Peres, Phys. Lett. A 151, 107 (1990); D. Mermin, Phys. Rev. Lett. 65, 3373 (1990).

A testable inequalityQuestionCan we translate this into an experimentally testable inequality?AnswerConsider sequences of measurements. Then, for non-contextual models〈X MP 〉 = 〈A 1 B 2 C 3 〉 + 〈a 1 b 2 c 3 〉 + 〈α 1 β 2 γ 3 〉+〈A 1 a 2 α 3 〉 + 〈B 1 b 2 β 3 〉 − 〈C 1 c 2 γ 3 〉= 〈R 1 〉 + 〈R 2 〉 + 〈R 3 〉 + 〈C 1 〉 + 〈C 2 〉 − 〈C 3 〉 ≤ 4.Here, 〈A 1 B 2 C 3 〉 means the product of the values, when the sequenceA 1 B 2 C 3 is measured on a single instance of a state.In QM, we have〈X MP 〉 = 6for any quantum state (in contrast to a Bell inequality violation).This can be tested in the lab!A. Cabello, PRL 101, 210401 (2008), for other experiments see Y.F. Huang et al, PRL 90, 250401 (2003),Y. Hasegawa et al., 97, 230401 (2006).

The experimental implementation

Non-demolition measurementsProblemMeasurements like σ z ⊗ σ z cannot be implemented by measuring σ z oneach ion, as coherences (like |00〉 + |11〉) would be destroyed.SolutionWrite σ z ⊗ σ z = U nl [½⊗σ z ]U † nland read out only one ion.This needs 6 nonlocal gates for R 3 or C 3 .

Results for the singlet statea b c〈R 1〉 = 0.92(1) 〈R 2〉 = 0.93(1) 〈R 3〉 = 0.90(1)+1+1+1σ z ⊗σ z-1σ x ⊗σ x-1σ y ⊗σ y-1+1Ι ⊗ σ z-1 -1σ z ⊗ Ι+1+1σ x ⊗ Ι-1 -1Ι ⊗ σ x+1+1σ x ⊗ σ z-1 -1+1σ z ⊗ σ xd e f〈C 1〉 = 0.90(1) 〈C 2〉 = 0.89(1) 〈C 3〉 = -0.91(1)σ z ⊗σ x+1-1σ x ⊗σ z+1-1σ y ⊗σ y+1-1+1Ι ⊗ σ x-1 -1σ z ⊗ Ι+1+1σ x ⊗ Ι-1 -1Ι ⊗ σ z+1+1σ x ⊗ σ x-1 -1+1σ z ⊗ σ zIn summary a total value of 〈X MP 〉 = 5.46(4) > 4.

Permuting the observables10.950.90.850.80.75row 1 row 2 row 3 column 1 column 2 column 3The order does not really matter, on average 〈X MP 〉 = 5.38

State independenceViolations ranging from 5.23(5) to 5.46(4) for 10 different states.

Consequences and interpretation

What have we shown so far?Conclusion from the experimentNo HV theory which assigns (for a fixed HV λ) ±1 to the measurementsindependently of the row or column, can explain the experimental data.But...In the experiment, the observables are not perfectly compatibleSo does one need to assign fixed values ±1?Can we still conclude something about HV models based on some reasonableassumptions?Be careful: HV models without any assumptions can never be refuted!As one can always take the experimental data as a HV model or also Bohmian mechanics.

Generalizing non-contextualityConsider a HV model, with HV λ, and a state is effectively aprobability distribution p(λ). Then considerp[(B + 1 |B 1) and (B − 2 |A 1B 2 )]If A and B are compatible and the theory is non-contextualthis is zero.Otherwise, this term quantifies the disturbance of B by ameasurement of A.This probability is not experimentally accessible.Can we estimate it, if A and B are nearly compatible?

Generalized KS inequalityUsing this assumption, one can show that for a HV model〈X DHV 〉 := 〈A 1 B 2 〉 + 〈C 1 B 2 〉 + 〈A 1 D 2 〉 − 〈C 1 D 2 〉 − 2p err [B 1 A 2 B 3 ]− 2p err [B 1 C 2 B 3 ] − 2p err [D 1 A 2 D 3 ] − 2p err [D 1 C 2 D 3 ]≤ 2with p err [B 1 A 2 B 1 ] = p[B 1 + , B− 3 |B 1A 2 B 3 ] + p[B1 − , B+ 3 |B 1A 2 B 3 ].Experimental resultExperimentally, one finds〈X DHV 〉 = 2.23(5) > 2⇒ Also more general HV models can be ruled out.In principle, this can be generalized to the full MP square, but then theexperimental precision has to be improved.

ConclusionNon-contextuality is in conflict with quantum mechanics. Thisconflict is state independent.This contradiction has been investigated using trapped ions.Also some more general HV models have been ruled out.Are the correlations in KS experiments useful for some task?Reference:G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne,A. Cabello, R. Blatt, C. F. Roos, arXiv:0904.1655Funding