Testing Contextuality on Quantum Ensembles with One Clean Qubit

Abstract

We present a protocol to evaluate the expectation value of the correlations of measurement outcomes for ensembles of quantum systems, and use it to experimentally demonstrate-under an assumption of fair sampling-the violation of an inequality that is satisfied by any non-contextual hidden-variables (NCHV) theory. The experiment is performed on an ensemble of molecular nuclear spins in the solid state, using established Nuclear Magnetic Resonance (NMR) techniques for quantum information processing (QIP).The Bell-Kochen-Specker (BKS) theorem [1,2,3,4] states that no noncontextual hidden-variables (NCHV) theory can reproduce the predictions of quantum mechanics for correlations between measurement outcomes of some sets of observables. Any such set of observables constitutes a proof of the theorem. Recently, Cabello [5] and others [6] used BKS proofs to derive a set of inequalities that are satisfied by any NCHV theory but are violated by quantum mechanics for any quantum state. These inequalities bound certain linear combinations of ensemble averages of correlations between measurement outcomes of compatible observables; thus creating a separation between the predicted outcomes of quantum mechanics, and the bound that is satisfied by NCHV theories.This provides an opportunity to test noncontextuality with finite-precision experiments -which has been the subject of contention for many years [7,8,9]-and without the need for the creation of special quantum states [10,11,12]. Already, two experiments, on a pair of trapped 40 Ca + ions [13], and with single photons [14], have demonstrated this state-independent conflict with noncontextuality. In this letter, we examine testing contextuality on quantum ensembles. This manuscript is organized as follows. First, we sketch the arguments leading to one of the inequalities derived in [5]. Then we present an algorithm to estimate the expectation value of the correlations of measurement outcomes for ensembles of quantum systems. And lastly, we report and discuss the result of experimentally implementing the algorithm on a 3-qubit ensemble of molecular nuclear spins in the solid state.Inequality -For a quantum system prepared according to some state, ρ, one can assign simultaneous outcomes {ν(S k )} of measurements of a set {S k } of coobservables (i.e. comeasureable; mutually compatible; commuting). In this case, the correlation between the measurement outcomes is given byirrespective of the product ordering. Repeating the preparation and measurement many times, and averaging over the outcomes, one obtains an estimate of the ensemble average of the correlation π {S k } ρ = k ν(S k ) ρ . For the case where the coobservables {S k } are also dichotomic, with possible outcomes {ν(S k ) = ±1}, the correlation (1) also takes on the possible values ±1, and the ensemble average satisfies −1 ≤ π {S k } ρ ≤ +1. Note, that in this case, these operators are Hermitian and unitary (also known as Quantum Boolean Functions).Consider any set of observables with possible outcomes ±1 arranged in ...

Proposed test of macroscopic quantum contextuality

“…For example, preserving R 2 and C 1 , we obtain the proof in Table II [15,16], which leads to the inequality tested in [6][7][8].…”

“…The quantum prediction (assuming an ideal experiment without imperfections) maximally violates inequality (1), that is, χ QM = N . In practice, the experimental values are in the range χ expt = 5.2-5.5 [6][7][8], instead of χ QM = 6, since the experimental values for the correlations are | C i | = 1 − ǫ i , with 0 < ǫ i ≪ 2, instead of the quantum predictions for an ideal experiment,…”

“…Alternatively, it can be easily seen that b = 2s − N , where s is the maximum number of quantum predictions which can be simultaneously satisfied by a noncontextual HV model. For instance, in the inequality tested in [6][7][8], there are N = 6 predictions and a noncontextual HV model can only satisfy s = 5 of them at most. Therefore, the bound is b = 4.…”

“…The inequality tested in [6][7][8] involves correlations between sequential measurements of three observables on two-qubit systems and tolerates an error per correlation of ε = 1/3 ≈ 0.33. However, none of these experiments on SIQC is free of the compatibility loophole [11].…”

“…Indeed, state-independent quantum contextuality (SIQC) is experimentally testable [4,5]. So far, it has only been observed in two-qubit systems: pairs of ions with two internal levels [6], polarization and path of single photons [7], and two-qubit nuclear magnetic resonance systems [8]. Related quantum contextuality experiments for specific states have been performed using also two-qubit systems: spin and path of single neutrons [9], and polarization of two-photon systems [10].…”