Free randomness can be amplified

Journal name:
Nature Physics
Year published:
Published online

Are there fundamentally random processes in nature? Theoretical predictions, confirmed experimentally, such as the violation of Bell inequalities1, point to an affirmative answer. However, these results are based on the assumption that measurement settings can be chosen freely at random2, so assume the existence of perfectly free random processes from the outset. Here we consider a scenario in which this assumption is weakened and show that partially free random bits can be amplified to make arbitrarily free ones. More precisely, given a source of random bits whose correlation with other variables is below a certain threshold, we propose a procedure for generating fresh random bits that are virtually uncorrelated with all other variables. We also conjecture that such procedures exist for any non-trivial threshold. Our result is based solely on the no-signalling principle, which is necessary for the existence of free randomness.

At a glance


  1. Illustration of the bipartite set-up.
    Figure 1: Illustration of the bipartite set-up.

    Spacelike-separated measurements are carried out using devices denoted D. The choices of measurement, A and B, are derived from bits generated by two sources of weak randomness, denoted S. These bits are only partially free; that is, they may be correlated (represented by the dashed line) with each other and with some other variables W (to be interpreted as parameters provided by a possible higher theory), which may also influence the supply of states being measured. By exploiting correlations between the outcomes, X and Y, we show that, in spite of the lack of perfectly free randomness to choose settings, the outcome X is arbitrarily close to being uniform and uncorrelated with W.

  2. Typical causal structure of a protocol.
    Figure 2: Typical causal structure of a protocol.

    A randomness amplification protocol for generating a random bit R may be initiated at a particular location and time represented by a spacetime point E. Depending on the protocol, information correlated to R may be generated at various locations within the causal future of E, depicted by the blue region. Any point in this region may therefore potentially be in the causal future of R. The bit R satisfies our definition of being free if it is uncorrelated with anything outside this region, indicated by the variable W. The protocol may invoke sources of random bits S1 and S2 at many locations. Technically, the only requirement on the causal structure is that the causal futures of each of these source bits (pink regions) lie in the blue region.


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  1. Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

    • Roger Colbeck &
    • Renato Renner
  2. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada

    • Roger Colbeck


Both authors contributed equally to this work.

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