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Minimal Distance to Approximating Noncontextual System as a Measure of Contextuality

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Abstract

Let random vectors \(R^{c}=\{R_{p}^{c}:p\in P_{c}\}\) represent joint measurements of certain subsets \(P_{c}\subset P\) of properties \(p\in P\) in different contexts \(c\in C\). Such a system is traditionally called noncontextual if there exists a jointly distributed set \(\{Q_{p}:p\in P\}\) of random variables such that \(R^{c}\) has the same distribution as \(\{Q_{p}:p\in P_{c}\}\) for all \(c\in C.\) A trivial necessary condition for noncontextuality and a precondition for many measures of contextuality is that the system is consistently connected, i.e., all \(R_{p}^{c},R_{p}^{c^{\prime }},\dots \) measuring the same property \(p\in P\) have the same distribution. The contextuality-by-default (CbD) approach allows defining more general measures of contextuality that apply to inconsistently connected systems as well, but at the price of a higher computational cost. In this paper we propose a novel measure of contextuality that shares the generality of the CbD approach and the computational benefits of the previously proposed negative probability (NP) approach. The present approach differs from CbD in that instead of considering all possible joints of the double-indexed random variables \(R_{p}^{c}\), it considers all possible approximating single-indexed systems \(\{Q_{p}:p\in P\}\). The degree of contextuality is defined based on the minimum possible probabilistic distance of the actual measurements \(R^{c}\) from \(\{Q_{p}:p\in P_{c}\}\). We show that this measure, called the optimal approximation (OA) measure, agrees with a certain measure of contextuality of the CbD approach for all systems where each property enters in exactly two contexts. The OA measure can be calculated far more efficiently than the CbD measure and even more efficiently than the NP measure for sufficiently large systems. We also define a variant, the OA-NP measure of contextuality that agrees with the NP measure for consistently connected (non-signaling) systems while extending it to inconsistently connected systems.

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Notes

  1. Contextuality has been studied in various settings and the probabilistic formulation here is not necessarily the tradional one in all applications (see, e.g., [3, 15] for a traditional account on quantum contextuality and [19] for another theory-independent approach to contextuality).

  2. This property has also been referred to as non-signaling and marginal selectivity in other contexts. Note however, that in our terminology non-signaling generalizes differently than the other two terms to more complex systems in Sect. 3.

  3. The maximal couplings \((\hat{X}_{1},\ldots ,\hat{X}_{n})\) of discrete random variables \(X_{1},\ldots ,X_{n}\) satisfy

    $$\begin{aligned} \Pr \left[ \hat{X}_{1}=\ldots =\hat{X}_{n}=x\right] =\min _{i\in \{1,\ldots ,n\}}\Pr [X_{i}=x] \end{aligned}$$

    for all x (see Thorisson [21], Chap. 1, Theorem 4.2) and so the maximum coupling probability is given by the sum of the above probability over all x. The same result generalizes to densities of arbitrary random variables (see [21], Chap. 3, Sect. 7).

  4. A measure of contextuality is said to be consistent with a given definition of noncontextuality if it applies to the same class of systems and yields zero values for systems that are noncontextual according to the definition and positive values for systems that are contextual according to the definition.

  5. The precise argument here is that for \(i\in \{1,2\}\) and any joint of \(\left( \hat{A}_{i},\hat{A}_{i1},\hat{A}_{i2}\right) \) having the marginal expectations \(\left\langle A_{i}\right\rangle ,\left\langle A_{i1}\right\rangle ,\left\langle A_{i2}\right\rangle \), Lemma 1 implies that \(\frac{1}{2}\left| \left\langle A_{i1}\right\rangle -\left\langle A_{i2}\right\rangle \right| \le \Pr \left[ \hat{A}_{i1}\ne \hat{A}_{i}\right] +\Pr \left[ \hat{A}_{i2}\ne \hat{A}_{i}\right] \) with equality for some joint and analogously for \(\left( \hat{B}_{i},\hat{B}_{1j},\hat{B}_{2j}\right) \) for \(j\in \{1,2\}\). Given joints for which the equality holds, we obtain a \(\Delta \) equaling \(\Delta _{0}\) with the couplings \(\left( \hat{A}_{i},\hat{A}_{ij}\right) \) and \(\left( \hat{B}_{j},\hat{B}_{ij}\right) \) defined as marginals of the 3-joints with \((A_{1},A_{2},B_{1},B_{2})\) given as an arbitrary joint with the same marginals as \(\hat{A}_{1},\hat{A}_{2},\hat{B}_{1},\hat{B}_{2}\). Conversely, any \(A_{1},A_{2},B_{1},B_{2}\) together with couplings \(\left( \hat{A}_{i},\hat{A}_{ij}\right) \) and \(\left( \hat{B}_{j},\hat{B}_{ij}\right) \) for \(i,j\in \{1,2\}\) can be used to define joints for \(\left( \hat{A}_{i},\hat{A}_{i1},\hat{A}_{i2}\right) \) and \(\left( \hat{B}_{i},\hat{B}_{1j},\hat{B}_{2j}\right) \) whose 2-marginals agree with \(\left( \hat{A}_{i},\hat{A}_{ij}\right) \) and \(\left( \hat{B}_{j},\hat{B}_{ij}\right) \) and so it follows \(\Delta \ge \Delta _{0}\) implying that no value smaller than \(\Delta _{0}\) can be obtained.

  6. It is easy to show that \(\Delta \) given by (10) and Definition 14 for couplings \(\left( \hat{R}^{c},\hat{Q}^{c}\right) \) of \(\left( R^{c},Q^{c}\right) \) that minimize (10) is indeed a metric when taken as a function of the two sets of bunches \(R=\{R^{c}:c\in C\}\) and \(Q=\{Q^{c}:c\in C\}\).

  7. In the latest developments of the CbD theory, a modified “version 2.0” definition of CbD contextuality has been proposed [12, 13]. Comparison of the present approach to CbD 2.0 is a matter of ongoing research.

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Acknowledgements

The author is grateful to Ehtibar Dzhafarov, Acacio de Barros, and Gary Oas for discussions related to this work, and to Ehtibar Dzhafarov and an anonymous reviewer for many helpful comments.

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  1. Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Janne V. Kujala

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Kujala, J.V. Minimal Distance to Approximating Noncontextual System as a Measure of Contextuality. Found Phys 47, 911–932 (2017). https://doi.org/10.1007/s10701-017-0094-3

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