Optimal Classical Simulation of State-Independent Quantum Contextuality

Simulating quantum contextuality with classical systems requires memory. A fundamental yet open question is what is the minimum memory needed and, therefore, the precise sense in which quantum systems outperform classical ones. Here, we make rigorous the notion of classically simulating quantum state-independent contextuality (QSIC) in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite QSIC set. We obtain the minimum memory needed to simulate arbitrary QSIC sets via classical systems under the assumption that the simulation should not contain any oracular information. In particular, we show that, while classically simulating two qubits tested with the Peres-Mermin set requires ${\log }_{2}24\approx 4.585$ bits, simulating a single qutrit tested with the Yu-Oh set requires, at least, 5.740 bits.

Figure 1

Contextuality experiment on a recycled system. Buttons represent possible measurements. Light bulbs represent possible outcomes. We consider experiments in which there are as many buttons as elements of the QSIC set, and all of them have two outcomes.

Figure 2

Observables in the Peres-Mermin set. $z_1$ denotes the quantum observable represented by the operator ${\sigma }_z^{(1)}\otimes {\mathbb{1}}^{(2)}$. Similarly, $zx$ denotes ${\sigma }_z^{(1)}\otimes {\sigma }_x^{(2)}$. Observables in each row or column are mutually compatible and their corresponding operators have four common eigenstates. In the figure, these eigenstates are represented by straight lines numbered from 1 to 24. For example, quantum state $|2⟩$ is the one satisfying ${\sigma }_z^{(1)}\otimes {\mathbb{1}}^{(2)}|2⟩=|2⟩$, ${\mathbb{1}}^{(1)}\otimes {\sigma }_z^{(2)}|2⟩=-|2⟩$, and ${\sigma }_z^{(1)}\otimes {\sigma }_z^{(2)}|2⟩=-|2⟩$.

Figure 3

Assuming that the experiment starts with a qutrit in one of the 13 quantum states of the Yu-Oh set, represented by the dots over a semisphere in (a), the successive figures show the possible postmeasurement quantum states after one (b), three (c), five (d), and seven measurements (e). The number of possible postmeasurement states is 25, 265, 3649, and 50 293, respectively. All the states lie in one of the 13 semicircles corresponding to the states with real components orthogonal to the 13 states of the Yu-Oh set. However, not all of them occur with the same probability. To illustrate this, the volume of each point in (e) is proportional to the probability with which the corresponding state appears. Collectively, these figures exemplify the typical behavior of QSIC sets (e.g., [32, 33]).

Figure 4

Classical memory in bits needed to simulate sequential Yu-Oh and Peres-Mermin experiments, as given by Eq. (1), as a function of the number of steps, as defined in the caption of Fig. 3 for the case of Yu-Oh (and, similarly, for the case of Peres-Mermin). Values are obtained from considering all possible measurement sequences of a given number of steps and then analytically calculating the corresponding results and postmeasurement states.