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Guaranteed randomness

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You have received a device that is claimed to produce random numbers, but you don't trust it. Can you check it without opening it? In some cases, you can, thanks to the bizarre nature of quantum physics.

From the stock market to the weather, we are surrounded by processes that are best described by unpredictable, random elements. But randomness is a notoriously difficult property to test. Worse still, when it is used to protect personal details, the random elements must be private. On page 1021 of this issue, Pironio and co-workers1 describe a method for obtaining numbers that are guaranteed to be random and private from an unknown process, provided that the numbers are certified as being derived from measurements on quantum systems. Some may surmise that this claim is trivial, because quantum physics has long been known to produce randomness. Some may even surmise that this claim is wrong, on the basis of the idea illustrated by the Dilbert comic strip 'Tour of Accounting' reproduced here. But the result is both new and correct.

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Starting with the comic strip, Dilbert's guide is uttering a scientific truth: the list 9, 9, 9, 9, 9, 9 is as valid an output of a generator of random numbers as is 1, 2, 3, 4, 5, 6 or 4, 6, 7, 1, 3, 8. In fact, in a long enough sequence of lists, any list of numbers should appear with the same frequency. Classic, 'black-box', tests of randomness exploit this idea: they check for the relative frequencies of lists. But, in practice, no such test can distinguish a sequence generated by a truly random process from one generated by a suitable deterministic algorithm that repeats itself after, for example, 1023 numbers. Moreover, in this way, whether the numbers are private cannot be checked: even if the sequence had initially been generated by a truly random process, it could have been copied several times, and the random generator may just be reading from a record.

It seems therefore that only a careful characterization of the process — that is, an 'open-box' test — can guarantee randomness, especially of the private kind. But nature sometimes dares to go where abstract thinking alone cannot. Pironio and colleagues1 take an alternative route for generating guaranteed, private randomness in their study, using one of the most remarkable phenomena of quantum physics: the violation of the 'Bell inequalities'2, which has been observed in numerous different experiments in the past three decades3.

To understand what Bell inequalities are, first consider two quantum systems, for example two photons emitted by the same source and propagating away from one another. Consider further that a measurement is made of each of the photons, which are now spatially distant. For example, the polarization of each photon can be measured, assigning the value 0 if a photon is transmitted through a polarizer and the value 1 if it is reflected. For certain specific sources, the outcomes of the two measurements are not independent. For instance, when the two polarizers are set in the same direction, only the pairs of outcomes (0, 1) and (1, 0) are observed: the photons are never both transmitted or both reflected. Such correlations between distant events are striking, raising the question of where the connection is. There is surely no communication between the two photons, because the signal would need to propagate faster than light. The only plausible hypothesis therefore is that the photons leave the source with a common 'list of instructions', which dictates the outcomes of each possible measurement.

Bell inequalities are criteria that, when applied, allow this latter hypothesis to be proved false. If the statistics of the measurement outcomes violate the inequalities, then the observed correlations cannot arise from a pre-established common list. Quantum correlations violate Bell inequalities and thus cannot come from a pre-established list either, and there is no classical mechanism that explains those correlations.

So how can random numbers be obtained using Bell inequalities? Following on from the example above, many pairs of photons are taken, and the measurement procedure is repeated: two sequences of 0s and 1s are produced, one at each measurement location. If these sequences violate Bell inequalities, they are guaranteed to be private random numbers: random, because there was no information about them before their generation; and private, because, given that the information did not exist, nobody else could have had access to it.

This line of qualitative reasoning has been known for many years. However, only recently have physicists been realizing and using its full power by stressing that Bell inequalities depend solely on the statistics of the observed outcomes and not on the description of the physical system or on the measurements that are carried out on it. In other words, Bell inequalities define a black-box test. The only caveat is that the statistics that are computed are conditional on some choices by the users: the box must allow input. A box that produces numbers without any input from the external world, like the troll in the comic strip, cannot be checked by this method.

Before the study by Pironio and colleagues1, the idea of exploiting Bell inequalities had proved fruitful in quantum cryptography4 and in assessing the quality of a source for producing 'quantum entanglement'5; another investigation6 had also addressed the issue of randomness. In their study, Pironio et al. meet three previously unmet challenges.

First, they obtain a quantitative estimate of how many random numbers can be extracted in a real experiment, in which the measurement outcome correlations are not necessarily the ideal ones and the sequences are not infinitely long.

Second, the 'boxes' require that the user makes 'choices', so the users must supply some of the initial randomness. But, in the authors' study, the initial randomness is cheap (it can come from the user's brain), and the protocol can generate many more random numbers than were initially supplied.

Third, such random numbers are actually produced by an experiment1 involving two atoms. Experimental results that violate Bell inequalities are not new, but an additional constraint must be met for black-box assessments: the detection process needs to be very efficient. For two photons, as in the example, this is not possible at present. But it is possible for two atoms, hence the choice of these as quantum systems by Pironio and colleagues1. Admittedly, the experiment is not a fully black-box experiment: we need to trust that atoms are indeed being measured and that atoms in different metallic boxes do not 'talk' to each other. But these are the only features that need to be trusted: there is no need to know which properties are measured or how they are measured.

A final point to consider is that Bell inequalities are independent of quantum physics. Their violation falsifies the existence of a common list of instructions, and a falsification is a fact that will remain true regardless of the nature of the physics, whether quantum physics or an as-yet-unveiled form of physics. With their study, Pironio and colleagues1 demonstrate a method for generating guaranteed, private randomness that will be useful for the ages to come.

References

  1. 1

    Pironio, S. et al. Nature 464, 1021–1024 (2010).

  2. 2

    Bell, J. S. Physics 1, 195–200 (1964).

  3. 3

    Scarani, V. Quantum Physics: A First Encounter (Oxford Univ. Press, 2006).

  4. 4

    Acín, A. et al. Phys. Rev. Lett. 98, 230501 (2007).

  5. 5

    Bardyn, C.-E. et al. Phys. Rev. A 80, 062327 (2009).

  6. 6

    Colbeck, R. PhD thesis, Univ. Cambridge (2007); available at http://arxiv.org/abs/0911.3814 (2009).

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Scarani, V. Guaranteed randomness. Nature 464, 988–989 (2010). https://doi.org/10.1038/464988a

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