Abstract Sequences of compatible quantum measurements can be contextual and any simulation with a classical model conforming with the quantum predictions needs to use internal memory. Kleinmann et al (2011 New J. Phys . 13 113011) showed that simulating sequences from the Peres–Mermin scenario requires at least three different internal states in order not to contradict the deterministic predictions of quantum theory. We extend this analysis to probabilistic quantum predictions and ask how much memory is required to simulate correlations generated for sequences of compatible observables by any quantum state. We find that even in this comprehensive approach only three internal states are required for a perfect simulation of quantum correlations in the Peres–Mermin scenario.
Abstract Sequences of compatible quantum measurements can be contextual and any simulation with a classical model conforming with the quantum predictions needs to use internal memory. Kleinmann et al (2011 New J. Phys . 13 113011) showed that simulating sequences from the Peres–Mermin scenario requires at least three different internal states in order not to contradict the deterministic predictions of quantum theory. We extend this analysis to probabilistic quantum predictions and ask how much memory is required to simulate correlations generated for sequences of compatible observables by any quantum state. We find that even in this comprehensive approach only three internal states are required for a perfect simulation of quantum correlations in the Peres–Mermin scenario.
Classically simulating the quantum contextual correlations produced by sequences of ideal measurements of compatible observables requires the measured system to have an internal memory. Computing the minimum amount of memory needed is, in general, challenging. Here, building upon the work of Kleinmann et al. [New J. Phys. 13, 113011 (2011)], we prove that the memory cost for simulating the contextuality produced by the 10 three-qubit observables of Mermin's pentagram is only log2(5)≈2.32 bits, but the memory cost for simulating the contextuality produced by all 15 two-qubit Pauli observables is, at least, log2(6)≈2.58 bits, thus exceeding the classical capacity of the system on which the measurements are performed. We also add results on the memory for simulating some subsets of quantum predictions.
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≈4.585 bits, simulating a single qutrit tested with the Yu-Oh set requires, at least, 5.740 bits.
Abstract A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general.
We investigate temporal correlations in the simplest measurement scenario, i.e., that of a physical system on which the same measurement is performed at different times, producing a sequence of dichotomic outcomes. The resource for generating such sequences is the internal dimension, or memory , of the system. We characterize the minimum memory requirements for sequences to be obtained deterministically, and numerically investigate the probabilistic behavior below this memory threshold, in both classical and quantum scenarios. A particular class of sequences is found to offer an upper-bound for all other sequences, which suggests a nontrivial universal upper-bound of 1/e for the classical probability of realization of any sequence below this memory threshold. We further present evidence that no such nontrivial bound exists in the quantum case.
Quantum systems show contextuality.More precisely, it is impossible to reproduce the quantum-mechanical predictions using a non-contextual realist model, i.e., a model where the outcome of one measurement is independent of the choice of compatible measurements performed in the measurement context.There has been several attempts to quantify the amount of contextuality for specific quantum systems, for example, in the number of rays needed in a KS proof, or the number of terms in certain inequalities, or in the violation, noise sensitivity, and other measures.This paper is about another approach: to use a simple contextual model that reproduces the quantum-mechanical contextual behaviour, but not necessarily all quantum predictions.The amount of contextuality can then be quantified in terms of additional resources needed as compared with a similar model without contextuality.In this case the contextual model needs to keep track of the context used, so the appropriate measure would be memory.Another way to view this is as a memory requirement to be able to reproduce quantum contextuality in a realist model.The model we will use can be viewed as an extension of Spekkens' toy model [Phys.Rev. A 75, 032110 (2007)], and the relation is studied in some detail.To reproduce the quantum predictions for the Peres-Mermin square, the memory requirement is more than one bit in addition to the memory used for the individual outcomes in the corresponding noncontextual model.
Simulation of two spin-s singlet correlations for all s involving spin measurements
In a recent paper [A. Ahanj et al., quant-ph/0603053], we gave a classical protocol to simulate quantum correlations corresponding to the spin s singlet state for the infinite sequence of spins satisfying 2s+1=2n. In the present paper, we have generalized this result by giving a classical protocol to exactly simulate quantum correlations implied by the spin-s singlet state corresponding to all integer as well as half-integer spin values s. The class of measurements we consider here are only those corresponding to spin observables, as has been done in the above-mentioned paper. The required amount of communication is found to be ⌈log2(s+1)⌉ in the worst case scenario, where ⌈x⌉ is the least integer greater than or equal to x.