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arXiv:1707.00564 · Physical Review A

Device-independent certification of two bits of randomness from one entangled bit and Gisin's elegant Bell inequality

Ole Andersson, Piotr Badziąg, Irina Dumitru, Adán Cabello

Published Jan 16, 2018• 28 citations
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Abstract

We prove that as conjectured by Ac\'{\i}n et al. [Phys. Rev. A 93, 040102(R) (2016)], two bits of randomness can be certified in a device-independent way from one bit of entanglement using the maximal quantum violation of Gisin's elegant Bell inequality. This suggests a surprising connection between maximal entanglement, complete sets of mutually unbiased bases, and elements of symmetric informationally complete positive operator-valued measures, on one side, and the optimal way of certifying maximal randomness, on the other.

DOI
10.1103/physreva.97.012314

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We prove that as conjectured by Ac'{\i}n et al. [Phys. Rev. A 93, 040102(R) (2016)], two bits of randomness can be certified in a device-independent way from one bit of entanglement using the maximal quantum violation of Gisin's elegant Bell inequality. This suggests a surprising connection between maximal entanglement, complete sets of mutually unbiased bases, and elements of symmetric informationally complete positive operator-valued measures, on one side, and the optimal way of certifying maximal randomness, on the other.
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When the Clauser-Horne-Shimony-Holt (CHSH) inequality is maximally violated by quantum mechanics, at most 1.23 bits of randomness can be generated. In this paper, based on the quantum value of Gisin's elegant Bell inequality (EBI), we present lower bounds on global randomness. We compare our results with other Bell inequalities. Our results show that EBI has an advantage over the CHSH inequality and the chained inequality when the state is close to a maximally entangled 2-qubit state. They imply that it is feasible to improve the generation rate of device-independent randomness.
We present two novel families of bipartite Bell inequalities designed to achieve optimal global randomness certification for an arbitrary number of outputs ddd. We first use symmetry arguments to argue that their maximal quantum violations certify 2log⁡d2\log d2logd random bits. For the first family, we construct a quantum realization using d×dd\times dd×d maximally entangled states which provides a quantum violation that we conjecture to be optimal for any ddd. It is then numerically shown that the obtained quantum violation certifies optimal global randomness, up to numerical precision, for d=3,4d=3,4d=3,4. For the second family, we provide the optimal quantum violation and its quantum realization for any ddd, again using d×dd\times dd×d maximally entangled states and projective measurements over at least two unbiased bases on one of the parties. We self-test this realization for d=3d=3d=3, which implies the optimal certification of two fully random trits.
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While it has recently been demonstrated how to certify the maximal amount of randomness from any pure two-qubit entangled state in a device-independent way, the problem of optimal randomness certification from entangled states of higher local dimension remains open. Here we introduce a method for device-independent certification of the maximal possible amount of 2log23 random bits using pure bipartite entangled two-qutrit states and extremal nine-outcome general non-projective measurements. To this aim, we exploit a device-independent method for certification of the full Weyl-Heisenberg basis in three-dimensional Hilbert spaces together with a one-sided device-independent method for certification of two-qutrit partially entangled states.