Strong Kochen-Specker theorem and incomputability of quantum randomness

Abbott, Alastair A.; Calude, Cristian S.; Conder, Jonathan; Svozil, Karl

doi:10.1103/physreva.86.062109

Cited by 73 publications

(155 citation statements)

References 55 publications

(63 reference statements)

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“…In this way, confining the contextuality serves to simplify its experimental observation. Such a simplification not only raises interesting foundational questions [26], but may also suggest future quantum information processing applications [3,6]. Additionally two single intensities with one or the other beam blocked are recorded (I z± ), and again background measurements with orthogonal preparation and postselection states are performed and subtracted from the signal.…”

Section: Discussionmentioning

confidence: 99%

“…This phenomenon is one of the most counterintuitive aspects of quantum mechanics, and finds itself at the heart of recent quantum information processing applications [3][4][5][6][7][8]. The BKS theorem is proved by exhibiting a BKS-set of observables [9,10] that contains geometrically related and mutually commuting subsets (or measurement contexts) that result in a logical incompatibility: Any noncontextual hidden variable theory (NCHVT) that pre-assigns eigenvalues globally to the entire BKS-set (i.e., noncontextually) results in a contradiction with the predictions of quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

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Confined contextuality in neutron interferometry: Observing the quantum pigeonhole effect

et al. 2017

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Previous experimental tests of quantum contextuality based on the Bell-Kochen-Specker (BKS) theorem have demonstrated that not all observables among a given set can be assigned noncontextual eigenvalue predictions, but have never identified which specific observables must fail such assignment. We now remedy this shortcoming by showing that BKS contextuality can be confined to particular observables by pre-and postselection, resulting in anomalous weak values that we measure using modern neutron interferometry. We construct a confined contextuality witness from weak values, which we measure experimentally to obtain a 5σ average violation of the noncontextual bound, with one contributing term violating an independent bound by more than 99σ. This weakly measured confined BKS contextuality also confirms the quantum pigeonhole effect, wherein eigenvalue assignments to contextual observables apparently violate the classical pigeonhole principle.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Confined contextuality in neutron interferometry: Observing the quantum pigeonhole effect

et al. 2017

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“…In addition, the Bell-type certification schemes can be regarded as random expanders rather than generators due to the requirement of "a small private random seed" to operate [6,9,10]. Finally, the random number generators certified by Bell inequalities utilize no-signaling assumption and is, therefore, inherently a non-local device which is challenging to use for practical applications.To address this problem, a different approach to QRNG certification based on the Kochen-Specker theorem and contextual measurements has recently been proposed [9]. It does not allow certification of the data in a device-independent fashion like the CHSH inequality, but yields a rigorous theoretical proof of measurements outcomes being value-indefinite even in the presence of experimental imperfections.…”

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confidence: 99%

Realization of a Quantum Random Generator Certified with the Kochen-Specker Theorem

et al. 2017

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Random numbers are required for a variety of applications from secure communications to MonteCarlo simulation. Yet randomness is an asymptotic property and no output string generated by a physical device can be strictly proven to be random. We report an experimental realization of a quantum random number generator (QRNG) with randomness certified by quantum contextuality and the Kochen-Specker theorem. The certification is not performed in a device-independent way but through a rigorous theoretical proof of each outcome being value-indefinite even in the presence of experimental imperfections. The analysis of the generated data confirms the incomputable nature of our QRNG.While we can consider a mathematical abstraction of a true random number generator and examine its properties, in the physical world we are confined to performing finite statistical tests on the output strings. By applying sets of such tests (like NIST [1] or diehard [2]) we can verify with arbitrarily high probability that the generator is NOT random (if it has failed at least one test), but cannot prove its randomness in the opposite case. As an example, one may construct a pseudo-random number generator which passes all above-mentioned tests while the produced sequence is deterministic and even computable [3]. The impossibility of a rigorous proof of randomness for a finite string generated by a physical device motivates the consideration of more fundamental arguments to support a RNG's randomness. From this point of view, no classical RNG may be truly random as it is deterministic by the laws of classical mechanics, and may in principle be predicted. A natural foundation to build a RNG would be quantum theory, as it is intrinsically random.However, although quantum mechanics obeys probabilistic rules, the possibility of separating intrinsic randomness from apparent randomness arising from a lack of control or from experimental noise is still under debate [4]. Moreover, while quantum mechanics for a two-level system is described by the same intrinsically-probabilistic measurement rules, one may not strictly prove valueindefiniteness, and hence indeterminism, of its results [5].These considerations led to the next advance in quantum number generation: the protocols certified by violation of certain Bell-type inequalities [6][7][8]. More specifically, through violation of the CHSH inequality one may certify that the observed outputs are not entirely predetermined and write a lower bound on the generating process entropy. Unfortunately, this approach does not allow one to close the gap between this lower bound and true randomness. In addition, the Bell-type certification schemes can be regarded as random expanders rather than generators due to the requirement of "a small private random seed" to operate [6,9,10]. Finally, the random number generators certified by Bell inequalities utilize no-signaling assumption and is, therefore, inherently a non-local device which is challenging to use for practical applications.To address this problem, a diff...

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“…As a matter of fact, randomness, in all existing physical and computational theories, may be understood as unpredictability w.r.to the intended theory. It is thus a form of (in time) undecidability w.r.to the given (more or less) formal frame (see Calude and Longo 2015;Abbott et al 2012;Gács et al 2011 for analyses in relation to algorithmic randomness). In other words, the (joint) analysis of (algorithmic, physical and biological) randomness crucially helps to go beyond formal deductions and computations, as given by conventional theories.…”

Section: Introductionmentioning

confidence: 99%

The Unconventionality of Nature: Biology, from Noise to Functional Randomness

Bravi

Longo

2015

Lecture Notes in Computer Science

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In biology, phenotypes' variability stems from stochastic gene expression as well as from intrinsic and extrinsic fluctuations that are largely based on the contingency of evolutionary and developmental paths and on ecosystemic changes. Both forms of randomness constructively contribute to biological robustness, as resilience, far away from conventional computable dynamics, where elaboration and transmission of information are robust when they resist to noise. We first survey how fluctuations may be inserted in biochemical equations as probabilistic terms, in conjunction to diffusion or path integrals, and treated by statistical approaches to physics. Further work allows to better grasp the role of biological "resonance" (interactions between different levels of organization) and plasticity, in a highly unconventional frame that seems more suitable for biological processes. In contrast to physical conservation properties, thus symmetries, symmetry breaking is particularly relevant in biology; it provides another key component of biological historicity and of randomness as a source of diversity and, thus, of onto-phylogenetic stability and organization as these are also based on variation and adaptativity.

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Strong Kochen-Specker theorem and incomputability of quantum randomness

Cited by 73 publications

References 55 publications

Confined contextuality in neutron interferometry: Observing the quantum pigeonhole effect

Confined contextuality in neutron interferometry: Observing the quantum pigeonhole effect

Realization of a Quantum Random Generator Certified with the Kochen-Specker Theorem

The Unconventionality of Nature: Biology, from Noise to Functional Randomness

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