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arXiv:2311.13474 · Physical Review A

Alternative robust ways of witnessing nonclassicality in the simplest scenario

Massy Khoshbin, Lorenzo Catani, Matthew Leifer

Published Mar 14, 2024• 9 citations
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Abstract

In this paper we relate notions of nonclassicality in what is known as the simplest nontrivial scenario (a prepare and measure scenario composed of four preparations and two binary-outcome tomographically complete measurements). Specifically, we relate the established method developed by Pusey [M. F. Pusey, Phys. Rev. A 98, 022112 (2018)] to witness a violation of preparation noncontextuality, that is not suitable in experiments where the operational equivalences to be tested are specified in advance, with an approach based on the notion of bounded ontological distinctness for preparations, defined by Chaturvedi and Saha [A. Chaturvedi and D. Saha, Quantum 4, 345 (2020)]. In our approach, we test bounded ontological distinctness for two particular preparations that are relevant in certain information processing tasks in that they are associated with the even and odd parity of the bits to communicate. When there exists an ontological model where this distance is preserved we talk of parity preservation. Our main result provides a noise threshold under which violating parity preservation (and so bounded ontological distinctness) agrees with the established method for witnessing preparation contextuality in the simplest nontrivial scenario. This is achieved by first relating the violation of parity preservation to the quantification of contextuality in terms of inaccessible information as developed by Marvian (I. Marvian, arXiv:2003.05984.), that we also show, given the way we quantify noise, to be more robust in witnessing contextuality than Pusey's noncontextuality inequality. As an application of our findings, we treat the case of two-bit parity-oblivious multiplexing in the presence of noise. In particular, given that we have a noise threshold below which preparation contextuality holds, we use it to establish a condition for which preparation contextuality is present in the case where the probability of success exceeds that achieved by any classical strategy.

DOI
10.1103/physreva.109.032212

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In this paper we relate notions of nonclassicality in what is known as the simplest nontrivial scenario (a prepare and measure scenario composed of four preparations and two binary-outcome tomographically complete measurements). Specifically, we relate the established method developed by Pusey [M. F. Pusey, Phys. Rev. A 98, 022112 (2018)] to witness a violation of preparation noncontextuality, that is not suitable in experiments where the operational equivalences to be tested are specified in advance, with an approach based on the notion of bounded ontological distinctness for preparations, defined by Chaturvedi and Saha [A. Chaturvedi and D. Saha, Quantum 4, 345 (2020)]. In our approach, we test bounded ontological distinctness for two particular preparations that are relevant in certain information processing tasks in that they are associated with the even and odd parity of the bits to communicate. When there exists an ontological model where this distance is preserved we talk of parity preservation. Our main result provides a noise threshold under which violating parity preservation (and so bounded ontological distinctness) agrees with the established method for witnessing preparation contextuality in the simplest nontrivial scenario. This is achieved by first relating the violation of parity preservation to the quantification of contextuality in terms of inaccessible information as developed by Marvian (I. Marvian, arXiv:2003.05984.), that we also show, given the way we quantify noise, to be more robust in witnessing contextuality than Pusey's noncontextuality inequality. As an application of our findings, we treat the case of two-bit parity-oblivious multiplexing in the presence of noise. In particular, given that we have a noise threshold below which preparation contextuality holds, we use it to establish a condition for which preparation contextuality is present in the case where the probability of success exceeds that achieved by any classical strategy.
In this work, we experimentally investigate the classical-light emulation of different notions of nonclassicality in the simplest scenario. We implement this prepare-and-measure scenario involving four preparations and two binary-outcome measurements using two distinct experimental setups that exploit different degrees of freedom of light: polarization and first-order Hermite-Gaussian transverse modes. We additionally model experimental noise through an all-optical setup that reproduces the operational effect of a depolarizing channel. Our experimental results are consistent with the findings of Khoshbin et al. [Phys. Rev. A 109, 032212 (2024)]: under the assumption that the two measurements performed form a tomographically complete set, the observed statistics violate their noise-robust inequalities, indicating inconsistencies with preparation noncontextuality and bounded ontological distinctness for preparations. Although our implementation uses classical light, it reproduces the statistics predicted for the simplest scenario. Since the states and measurements of this scenario underpin computational advantages in tasks such as two-bit quantum random access codes -- among the simplest communication primitives enabling semi-device-independent certification of nonclassicality -- our implementation is directly relevant for such applications.
Contextuality is the leading notion of nonclassicality for a single system. However, an experimental demonstration requires finding procedures that are operationally equivalent, which might seem impossible to achieve exactly. Here I focus on the simplest nontrivial case, four preparations and two tomographically complete binary measurements. Exploiting a subtle connection to the Clauser-Horne-Shimony-Holt scenario gives eight nonlinear inequalities which are together necessary and sufficient for the experimental statistics to admit a preparation noncontextual model in such a scenario. No fixed operational equivalences are required, removing a key difficulty with experimental tests of older preparation noncontextuality inequalities.
The quantum ontological feature of contextuality apart from being central to foundations of quantum theory forms the basis of quantum advantage in a multitude of information processing tasks. In particular, the contextuality of preparation procedures was shown to power a particular two-party information processing task “parity oblivious multiplexing” (Spekkens et al. Phys Rev Lett 102:010401 (2009). Specifically, it was shown that there exists a limit to how well any preparation noncontextual theory can perform in this task. This limit constitutes a noncontextuality inequality. Moreover, the authors demonstrated quantum violation of this inequality along with preparation contextuality associated with the ontic description underlying two-level completely mixed quantum state. In this work, we extend these arguments to apply to arbitrary dimensions by introducing a class of two-party information processing tasks, namely d-level parity oblivious random access codes. We analytically obtain classical (or equivalently preparation noncontextual) bounds on the success probability for these tasks for arbitrary d. For each value of d, this bound constitutes a unique noncontextuality inequality. Remarkably, these bounds are independent of the amount of communication. Furthermore, we find a classical protocol utilizing a dc=dd_\mathrm{c}=ddc​=d -dimensional classical message which saturates this bound. In order to establish nontriviality of these inequalities, we provide evidence of significant quantum violations. Specifically, by numerical techniques, we show that for d=3,…,10d=3,\ldots ,10d=3,…,10 , the noncontextuality bound is violated by quantum theory. (1) We provide explicit quantum protocols which violate the associated noncontextuality inequality for d=3,4,5d=3,4,5d=3,4,5 employing dq=dd_\mathrm{q}=ddq​=d -leveled quantum systems. (2) Using see-saw semi-definite programming (SDP) technique, we find evidence (lower bounds) of significant quantum violation of these inequalities for d=3,…,10d=3,\ldots ,10d=3,…,10 . (3) With the help of state-of-the-art (NPA-hierarchy like) SDP technique, we provide upper bounds (independent of the dimension of the involved quantum systems) on quantum violation for d=3,…,10d=3,\ldots ,10d=3,…,10 . The introduced class of information tasks, thus, provides for operational depiction of preparation contextuality of the ontic description underlying mixed higher dimensional quantum systems.
Preparation contextuality is often manifested operationally through a quantitative advantage over preparation-noncontextual models in information-processing tasks, yet quantum theory typically falls short of perfect success. Here we introduce a parity-oblivious Hidden Matching task that instead yields an all-vs-nothing manifestation of preparation contextuality. Alice encodes an nnn-bit string so that her message reveals no information about any input parity other than the two-bit parities. Bob, given a perfect matching on the input positions, must output an edge of the matching together with the parity of its two bits. A quantum protocol using a single ⌈log⁡2n⌉\lceil\log_2 n\rceil⌈log2​n⌉-qubit message satisfies the parity-obliviousness constraint and succeeds with certainty. We prove that no preparation-noncontextual model achieves perfect success for any even n≥6n\ge6n≥6. The result holds for arbitrary ontic state spaces and assumes neither outcome determinism nor measurement noncontextuality. For n=6n=6n=6 and n=8n=8n=8, the optimal noncontextual success probabilities are 4/54/54/5 and 3/43/43/4, respectively, while asymptotically the optimal noncontextual success probability is 1/2+Θ(1/n)1/2+Θ(1/\sqrt n)1/2+Θ(1/n​). Our asymptotic upper bound follows from a Fourier-analytic sum rule derived via hypercontractivity on the Boolean hypercube. The resulting separation is qualitative rather than merely quantitative: quantum theory achieves perfect success, whereas preparation-noncontextual models cannot. We aslo show that at n=8n=8n=8 the noncontextual bound remains 3/43/43/4 even when Bob is restricted to just suitable 444 of the 105105105 possible perfect matchings, bringing the effect within experimental reach.
Contextuality is the leading notion of non-classicality for a single system. However, experimental demonstrations of contextuality have generally been limited by conceptual difficulties. Here I focus on the simplest non-trivial case, four preparations and two tomographically complete binary measurements. Exploiting a subtle connection to the CHSH scenario gives eight non-linear inequalities which are together necessary and sufficient for the experimental statistics to admit a preparation noncontextual model. No fixed operational equivalences are required. Violating such an inequality is therefore arguably the most direct route to an experimental refutation of noncontextuality.
Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, called unit separability , is formulated as the relevant classicality criterion – the name is inspired by the usual notion of quantum state separability. Using this criterion, we derive a new upper bound on the cardinality of the ontic space. Then, we recast the unit separability criterion as a (possibly infinite) set of linear constraints, from which we obtain two separate hierarchies of algorithmic tests to witness the non-classicality or certify the classicality of a scenario. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories.