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arXiv:1607.00471 · Physical Review A

Maximizing device-independent randomness from a Bell experiment by optimizing the measurement settings

Syed M. Assad, Oliver Thearle, Ping Koy Lam

Published Jul 5, 2016• 10 citations
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Abstract

The rates at which a user can generate device-independent quantum random numbers from a Bell-type experiment depend on the measurements that the user performs. By numerically optimizing over these measurements, we present lower bounds on the randomness generation rates for a family of two-qubit states composed from a mixture of partially entangled states and the completely mixed state. We also report on the randomness generation rates from a tomographic measurement. Interestingly in this case, the randomness generation rates are not monotonic functions of entanglement.

DOI
10.1103/physreva.94.012304

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The rates at which a user can generate device-independent quantum random numbers from a Bell-type experiment depend on the measurements that the user performs. By numerically optimizing over these measurements, we present lower bounds on the randomness generation rates for a family of two-qubit states composed from a mixture of partially entangled states and the completely mixed state. We also report on the randomness generation rates from a tomographic measurement. Interestingly in this case, the randomness generation rates are not monotonic functions of entanglement.
We investigate how much randomness can be extracted from a generic partially entangled pure state of two qubits in a device-independent setting, where a Bell test is used to certify the correct functioning of the apparatus.For any such state, we first show that two bits of randomness are always attainable both if projective measurements are used to generate the randomness globally or if a nonprojective measurement is used to generate the randomness locally.We then prove that the maximum amount of randomness that can be generated using nonprojective measurements globally is restricted to between approximately 3.58 and 3.96 bits.The upper limit rules out that a bound of four bits potentially obtainable with extremal qubit measurements can be attained.We point out this is a consequence of the fact that nonprojective qubit measurements with four outcomes can only be self-tested to a limited degree in a Bell experiment.
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Bell experiments can be used to generate private random numbers. An ideal Bell experiment would involve measuring a state of two maximally entangled qubits, but in practice any state produced is subject to noise. Here we consider how the techniques presented in arXiv:1309.3894 and arXiv:1309.3930, i.e. using an optimized Bell inequality, and taking advantage of the fact that the device provider is not our adversary, can be used to improve the rate of randomness generation in Bell-like tests performed on singlet states subject to either white or dephasing noise.
In this paper we investigate properties of several randomness generation protocols in the device-independent framework. Using Bell-type inequalities it is possible to certify that the numbers generated by an untrusted device are indeed random. We present a selection of certificates which guarantee two bits of randomness for each run of the experiment in the noiseless case and require the parties to share a maximally entangled state. To compare them we study their efficiency in the presence of white noise. We find that for different amounts of noise different operators are optimal for certifying most randomness. Therefore, the vendor of the device should use different protocols depending on the amount of noise expected to occur. Another of our results that we find particularly interesting is that using a single Bell operator as a figure of merit is rarely optimal.
The unpredictability of quantum physics gives rise to intrinsic randomness. In an adversarial scenario, any additional degrees of freedom must be attributed to an eavesdropper with correlations to the measurement set-up. The true randomness is then quantified by the probability that she correctly guesses the measurement outcomes, optimised over all possible strategies. Extremal measurements are appealing here, since they do not allow information to leak to such an eavesdropper. Beyond projective measurements, however, a simple question remains open: how much intrinsic randomness can be generated by a given extremal measurement? In a step towards solving it, we characterise the randomness generated by any unbiased extremal rank-one measurement acting on any state, solving the problem explicitly in dimension two. Four-outcome qubit measurements of this type are tomographic, so these results hold for fully source-device-dependent randomness too. The tetrahedral symmetric informationally complete (SIC) measurement, we find, has the least intrinsic randomness within this class. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that 2log⁡d2 \log d2logd bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC measurement.
The violation of the Bell inequality not only provides the most radical departure of quantum theory from classical concepts, but also paves the way for applications such as device-independent randomness certification. Here, we derive the tight upper bound of the maximum quantum value for the chained Bell inequality with an arbitrary number of measurements on each party. The constraints where the upper bound saturates are also presented. This method provides us the necessary and sufficient conditions for some quantum states to violate the chained Bell inequality with an arbitrary number of measurements. Based on the tight upper bound we present the lower bounds on the device-independent randomness with respect to the Werner states. In particular, we present lower bounds on the randomness generation rates of the chained Bell inequality for a different number of measurements, which are compared with the family of Bell inequalities proposed by Wooltorton et al. [Phys. Rev. Lett. 129, 150403 (2022)]. Our results show that the chained Bell inequality with three measurements has certain advantages at a low level of noise and could be used to improve randomness generation rates in practice.