One and only one ray is assigned to value 1 among all the rays in a complete orthonormal basis. The first condition reflects the non-contextuality and the second condition arises from the requirement that the algebraic structure of compatible observables be preserved. For a Hilbert space of a dimension greater than 2 there always exists a finite set of rays to which the KS value assignment is impossible. Svozil in and pointed out by Bub schu. The best KS proof known so far is given by Conway and Kochen 31 with 31 rays.
To warm up let us present a state-independent proof of KS theorem for qutrit using only 13 rays. Consider the following 13 rays. Obviously a given set of rays determines uniquely the orthogonality graph and usually not vice versa. In fact without loss of generality we can choose z k as in Eq. The diagonal unitary transformation taking h 0 to 1 , 1 , 1 leaves z k unchanged so that the standard form of 13 rays in Eq. The KS value assignments to the ray set are possible, i.
Suppose that this is not true, i. In the reasonings above we have taken into account of condition 2 which demands that linked rays not be assigned simultaneously to value 1 and in a triangle one and only one ray be assigned to value 1.
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A set of eight rays in each case considered above, e. As a result the following inequality. This proves the original KS theorem: the non-contextual HV model satisfying conditions 1 and 2 cannot reproduce all the predictions of QM on non-sequential measurements. Usually the KS theorem is proved by finding a set of rays to which the KS value assignment does not exist so that we need not to check other predictions of QM.
Our proof here is a set of 13 rays, to which all possible KS value assignments, which do exist, fail to reproduce a certain prediction of QM. Notably the inequality Eq. In an experimental test of the inequality Eq. It turns out that the requirement of preserving the algebraic structure, i.
However the product rule is not necessary either. Instead we need only to require that the quantum correlations of compatible observables, the quantum mechanical predictions on sequential measurements, be reproduced. This imposes no additional constraints on the non-contextual HV models since they must reproduce all the predictions of QM in the first place.
As noticed earlier, the toy model by Kochen and Specker did not take into account of the quantum correlations of compatible observables. To generate measurements in arbitrary directions from M 0 , we apply unitary rotations before and after measurement Fig. Each measurement procedure starts by sending microwave pulses to the qutrit to rotate the desired measurement basis defined by one of the KCBS states ; see Fig.
After a readout pulse and delay of ns for cavity ring-down, further microwave pulses return the qutrit to its initial reference frame, necessary to allow subsequent measurements to be implemented independently. The rightmost pulse in a product is applied first in time. The trajectory of the state under transformation U 4 is shown as an example.
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The unitaries rotate the measurement axis into one of the states of the KCBS pentagram. Measurement of the M 0 observable is implemented with a cavity probe signal and the qutrit rotations are constructed with microwave pulses applied at the qutrit transition frequencies. Preserving coherence in the subspace orthogonal to the measurement direction is also crucial for ensuring the compatibility of context-independent sequential measurements.
Since noncontextuality tests aim to falsify noncontextuality using the assumptions of noncontextual realism, which contain no notion of compatibility, it is important to ask why test protocols only consider compatible measurements. It is well established that individual outcome probabilities for incompatible observables will depend on the order in which they are measured, but this overt contextuality does not reveal any further insight into the nature of reality.
However, restricting attention to compatible measurements allows a study of whether context dependence still remains when this overt contextuality is absent. In practice, experimental imperfections make the actual measurement procedures only approximately compatible. This loophole can be addressed by an extended KCBS inequality Here the order of the observables in the two-outcome correlations A i A j corresponds to the timing order for two corresponding sequential measurements, and ij are the operational bounds for the incompatibility of these measurement procedures.
A bound on incompatibility 19 :. In the final protocol, we measure the five combinations , , , and , and their reverse-order variants, followed by calibration blocks to detect phase drifts of the cavity signal. As the qutrit is operated in a dilution refrigerator at 20 mK, its thermal state is close to the ground state. A further delay of ns allows the cavity to ring-down before the measurement sequence begins. The outcomes were recorded to the hard drive and were later used to calculate expectation values A i and correlations A i A j.
The results used to test inequality 3 and its reverse-order counterpart are presented in Table 1. For all pairs, the first measurement yields expectation values very close to the ideal value of 0. Correlations contribute to the left side of equation 3. We also provide for the equation with the reversed order of measurements. Single expectation values and are used to evaluate bounds ij on compatibility contributing to the right side of equation 3.
Inspired by the extended inequality derived in ref. The null hypothesis that the experiment is described by a noncontextual HV model with compatibility 0. Our analysis requires only the assumption that the devices perform the same in every single trial and the no-memory assumption without any additional assumptions on compatibility of the measurements or on the measurement contrast. Our results strongly contradict the predictions of noncontextual HV models, closing two common loopholes: the detection loophole, via high-fidelity, deterministic single-shot readout and the individual-existence loophole 18 , using separate, sequential measurements.
The compatibility loophole was treated by violating an extended inequality 19 and, independently, by formulating the problem in the form of a hypothesis test without any assumptions on compatibility and bounding the incompatibility of the measurements in a separate hypothesis test. As a key ingredient in addressing these loopholes, we implemented sequential dichotomic qutrit measurements that project out one target state without disturbing the information stored in the remaining two-dimensional subspace.
This allows a classical result from the first measurement to be obtained before implementing the setting to be used for the second measurement. Our results demonstrate that quantum mechanics departs from predictions of noncontextual realism, without reliance on nonlocality or entanglement correlations, and provide evidence of the contextuality resource in superconducting circuits. While we used the simpler state-dependent inequality for demonstration of contextual nature of the superconducting circuits, the state-independent test will be the straightforward extension of our experiment.
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One key point that differentiates our noncontextuality analysis for an indivisible system from a similar analysis for Bell inequalities with locality constraints is that it is difficult to avoid the need for additional i. Since quantum contextuality can be simulated by a classical system with memory 27 , these loopholes will most likely remain for any Kochen—Specker tests without nonlocality.
In another case, for the finite-precision loophole 28 , 29 , debate continues about whether this loophole can be closed in principle 30 , 31 , It remains an important open challenge to identify a clear, general prescription for how to implement a noncontextuality test with minimal assumptions. The qutrit was fabricated on an intrinsic Si substrate in a single step of electron beam lithography followed by shadow evaporation of two Al layers with an oxidation step between the depositions. The design of the circuit is identical to the one in ref.
Magnetic flux supplied by a superconducting coil attached to the copper cavity is used to control the transition frequencies of the qutrit. The qutrit was incorporated into a three-dimensional microwave copper cavity attached to the cold stage of a dilution cryostat Fig. To measure transmission, a signal from a microwave generator RF was applied to the input port of the cavity.
Microwaves transmitted through the cavity were amplified by a Josephson parametric amplifier, high-electron-mobility transistor amplifier at 4 K and a chain of room-temperature amplifiers. The sample at 20 mK was isolated from the higher-temperature fridge stages by three circulators C in series. The amplified transmission signal was down-converted to an intermediate frequency of 25 MHz in an IQ mixer driven by a dedicated local oscillator LO , and digitized by an analogue-to-digital converter for data analysis.
State-independent proof of Kochen-Specker theorem with 13 rays
To implement single-shot readout, we used a Josephson parametric dimer amplifier 23 JPDA as a preamplifier of the signal. The Josephson parametric dimer amplifier consists of two coupled non-linear resonators and can be operated in the non-degenerate mode if a pump tone frequency is set between resonance frequencies of the resonators. In our experiment, the pump tone was set at 7. Two circulators installed between the readout cavity and JPDA, combined with the readout cavity itself, eliminated any effect of the pump tone on the qutrit.
Experimental tests of HV models can be formulated as a hypothesis test, where the null hypothesis to be rejected is that the measurement statistics can be modelled using HVs Specifically, we test an i. This limit is then tested separately. An -incompatible model assumes that. For the KCBS inequality, a trial is won if the two outcomes of a context are not equal. The total number of wins is recorded over the whole experimental run of n trials. The P -value is then the probability that the game could have been won at least that many times given a noncontextual HV model with incompatibility.
This may be because the snippet appears in a figure legend, contains special characters or spans different sections of the article. Nat Commun. Published online Oct 4. PMID: Langford , 4, 6 and Arkady Fedorov a, 1, 2.