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Temporal correlations and device-independent randomness

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Abstract

Leggett–Garg inequalities (LGI) are constraints on certain combinations of temporal correlations obtained by measuring one and the same system at two different instants of time. The usual derivations of LGI assume macroscopic realism per se and noninvasive measurability. We derive these inequalities under a different set of assumptions, namely the assumptions of predictability and no signaling in time (NSIT). As a novel implication of this derivation, we find application of LGI in randomness certification. It turns out that randomness can be certified from temporal correlations, even without knowing the details of the experimental devices, provided the observed correlations violate LGI but satisfy NSIT.

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Notes

  1. In the present context, this postulate says that properties of an ensemble are determined exclusively by initial conditions and cannot be affected by final conditions.

  2. In Bell scenario, similar kind of assumption has been considered by Cavalcanti et al. [30]. In their case, the assumption is concerned about the joint conditional probabilities for measurements performed on different systems, whereas in our case it is about the joint conditional probabilities for different measurements performed at different times but on a single and the same system.

  3. One may of course choose \(A = B\) and measure the same observable twice.

  4. Once the factorizability is achieved, the postulate of induction is further used in calculating the correlation between measurement outcomes at two other different times. It allows one to freely choose the measurement times, independent of the properties of the initially prepared state.

  5. For the Leggett–Garg function \(f^{LG}_4\), the associated randomness can also be obtained in a closed form as \(H_{\infty }(Q_{{\mathcal {T}}_{\alpha }},Q_{{\mathcal {T}}_{\beta }})\ge -\log _2(\frac{3}{2}-\frac{2+\epsilon }{4})\) (cf. Fig 1). The calculation is similar to Ref. [35], but the context is different here. While in [35], correlations between measurement results from two distantly located physical systems are considered; here the focus is on one and the same physical system to obtain the correlations between measurement outcomes at two different times.

  6. It would be worth mentioning here that like Bell’s scenario, in the case of temporal correlations too, we need some amount of seed randomness at the input. This is needed for freely choosing the measurement times.

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Acknowledgments

S.K.C. acknowledges fruitful discussions with Pankaj Agrawal. We also thank Guruprasad Kar for stimulating discussions. MB thankfully acknowledges comments from C. Brukner, C. Emary and A. J. Leggett. S.K.C. acknowledges support from the Council of Scientific and Industrial Research, Government of India (Scientists’ Pool Scheme). SM acknowledges support from the DST project SR/S2/PU-16/2007.

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Authors and Affiliations

  1. S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata, 700 098, India

    Shiladitya Mal

  2. Optics and Quantum Information Group, The Institute of Mathematical Sciences, C.I.T Campus, Tharamani, Chennai, 600 113, India

    Manik Banik

  3. Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700108, India

    Manik Banik

  4. Institute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha, 751005, India

    Sujit K. Choudhary

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  1. Shiladitya Mal
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Mal, S., Banik, M. & Choudhary, S.K. Temporal correlations and device-independent randomness. Quantum Inf Process 15, 2993–3004 (2016). https://doi.org/10.1007/s11128-016-1321-0

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