Experimental Implementation of a Kochen-Specker Set of Quantum Tests

The conflict between classical and quantum physics can be identified through a series of yes-no tests on quantum systems, without it being necessary that these systems be in special quantum states. Kochen-Specker (KS) sets of yes-no tests have this property and provide a quantum-versus-classical advantage that is free of the initialization problem that affects some quantum computers. Here, we report the first experimental implementation of a complete KS set that consists of 18 yes-no tests on four-dimensional quantum systems and show how to use the KS set to obtain a state-independent quantum advantage. We first demonstrate the unique power of this KS set for solving a task while avoiding the problem of state initialization. Such a demonstration is done by showing that, for 28 different quantum states encoded in the orbital-angular-momentum and polarization degrees of freedom of single photons, the KS set provides an impossible-to-beat solution. In a second experiment, we generate maximally contextual quantum correlations by performing compatible sequential measurements of the polarization and path of single photons. In this case, state independence is demonstrated for 15 different initial states. Maximum contextuality and state independence follow from the fact that the sequences of measurements project any initial quantum state onto one of the KS set’s eigenstates. Our results show that KS sets can be used for quantum-information processing and quantum computation and pave the way for future developments.

Popular Summary

Does a moving car have a definitive position before we make a measurement of it? Anyone who answers this question with a “no” will be considered delusional: That the act of observation of an object and the object’s “properties” are independent is philosophically the most fundamental tenet of classical physics as we know it. When one replaces the moving car with a quantum particle, however, the answer is indeed a mind-boggling “no.” According to quantum mechanics, “unperformed experiments have no conceivable results.” Why that is so is still one of the ongoing and most fundamental debates about quantum mechanics.

Some of the contestants for a fundamental explanation of what physical “reality” is in the quantum realm have been the so-called hidden-variable theories, built on the assumption that more fundamental mechanisms that are consistent with our notion of classical physical reality are behind the apparent quantum-mechanical puzzles but are just not observable. Such hidden-variable theories have been challenged by many renowned scientists from many different angles for decades. In the 1960s, Simon Kochen and Ernst Specker proved a theorem, known as the KS theorem, that excludes a type of hidden-variable theory known as noncontextual in which the results of measurements reveal preexisting properties and are independent of other compatible measurements. So far, however, the KS theorem has remained a purely theoretical and abstract construct. In this paper, we report the first implementation of the KS theorem in two different single-photon experiments, each also illustrating one possible application in quantum-information processing.

The proof of the KS theorem has one basic ingredient: a set of measurements or tests that return only either a yes or no answer on their own but have the peculiarity that there is no way for them to return yes or no answers in a way that is in agreement with the assumptions of noncontextual hidden-variable theories and the foundations of quantum mechanics. Our experiments have realized a complete set of KS tests by employing the polarization, orbital angular momentum, and path of single photons. But, what may also be promising is the potential of the KS tests for quantum computing and information processing: We have demonstrated that the KS tests can be used for solving an algorithmic task with an unbeatable quantum advantage (compared to the classical protocol) and for producing correlations that, independent of the initial state of the system, are maximally contextual, as they lead to the maximum possible violation of an inequality satisfied by any noncontextual hidden-variable theory.

Figure 1

(a) The 18-test KS set. Each vertex represents a yes-no test (associated in QM with a projector ${\Pi }_i=|v_i⟩⟨v_i|$, where $⟨v_i|$ are the unit vectors displayed in the Figure; normalization factors are omitted to simplify the notation), and adjacent vertices correspond to exclusive tests (i.e., they cannot both have the answer yes on the same system; in QM, they are associated with orthogonal projectors). This vector representation is the one that is adopted in our experiments. (b) Optimal strategy for the task that is described in the text using classical resources. The system is a ball that can be placed in one out of 18 boxes, and “1, 2, 11, 16” denotes the following yes-no test: “Is the ball in box 1 or in box 2 or in box 11 or in box 16?.” The set of classical tests in (b) results in the maximum probability of obtaining yes by using classical resources (see the Supplemental Material [39]). (c) Propositions tested in the noncontextuality inequality (5) that are used to obtain state-independent maximally contextual quantum correlations. Each vertex represents a proposition $abc|xyz$ that denotes that “the result of measuring $x$ is $a$, the result of measuring $y$ is $b$, and the result of measuring $z$ is $c$.” When the measurements are those measurements in (4), then each of these sequences of measurements and results projects any initial state onto the corresponding state in (a).

Figure 2

Experimental setup for the measurement of the probabilities $P({\Pi }_i=1)$ on different states encoded in the space of polarization and orbital angular momentum. In the upper left corner, the single-photon source is represented. The four schemes we use for the experiment are presented in the right part of the figure. Each state is prepared by one of the two setups of the column labeled “Generation”: setup (a) for separable states (quantum transferrer $\pi \to o_2$ [45]) and setup (b) for entangled ones (an “entangler” based on a QP and wave plates). The column labeled “Analysis” shows the setups for the projection onto the desired states: setup (c) for separable states, a deterministic transferrer $o_2\to \pi$, and setup (d) for entangled states, where a QP is needed to have a deterministic detection.

Figure 3

Experimental results for $\Sigma$ for 28 quantum states. The solid (dashed) blue line refers to the (corrected) classical upper bound for $\Sigma$. The red area represents the range $[{\Sigma }_{\min },{\Sigma }_{\max }]$ in which we theoretically expect to find all experimental values of $\Sigma$. The first 18 states correspond to the ones in Fig. 1a. States 19–28 are defined in Table .

Figure 4

Experimental setups for the observables in (4). For the measurement of observable 0, it is only necessary to distinguish between the paths $r$ and $t$. To measure observable 4, a polarization-independent beam splitter is used to distinguish the eigenstates through interference. The measurements of observables 1 and 3 are standard polarization measurements using PBSs and HWPs. Observables 2, 5, and 8 are Bell-state measurements, and so are the measurements of 6 and 7, but in the latter group the Bell measurement is preceded by a rotation of the polarization to guarantee compatibility with observable 8. To measure the probabilities that appear in inequality (5), these measurement devices are arranged in a cascaded manner [30, 33].