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arXiv:1901.06912 · Physical Review Research

Maximal randomness from partially entangled states

Erik Woodhead, Jędrzej Kaniewski, Boris Bourdoncle, Alexia Salavrakos, Joseph Bowles, Antonio Acín, Remigiusz Augusiak

Published Nov 13, 2020• 37 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.2.042028

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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.
We report on an optical setup generating more than one bit of randomness from one entangled bit. The amount of randomness is certified through the observation of Bell nonlocal correlations. To attain this result we implemented a high-purity entanglement source and a nonprojective three-outcome measurement. Our implementation achieves a gain of 27% of randomness as compared with the standard methods using projective measurements. Additionally, we estimate the amount of randomness certified in a one-sided device-independent scenario, through the observation of Einstein-Podolsky-Rosen steering. Our results prove that nonprojective quantum measurements allow extending the limits for nonlocality-based certified randomness generation using current technology.
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.
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.
Previous theoretical works showed that all pure two-qubit entangled states can generate one bit of local randomness and can be self-tested through the violation of proper Bell inequalities. We report an experiment in which nearly pure partially entangled states of photonic qubits are produced to investigate these tasks in a practical scenario. We show that small deviations from the ideal situation make low entangled states impractical to self-testing and randomness generation using the available techniques. Our results show that in practice lower entanglement implies lower randomness generation, recovering the intuition that maximally entangled states are better candidates for device-independent quantum information processing.
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities -- entanglement, non-locality and randomness -- relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: For systems in a given entangled state, how many nonlocal correlations in a sequence can we obtain by measuring them repeatedly? And from this generated sequence of non-local correlations, how many random numbers is it possible to certify? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from an entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities – entanglement, non-locality and randomness – relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: Given a single entangled system, how many nonlocal correlations in a sequence can we obtain? And from this single entangled system, how many certified random numbers is it possible to generate? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from a single entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.