Randomness from causally independent processes

TL;DR

This work shows that uniform randomness can be distilled from outputs of two causally independent quantum channels acting on a potentially correlated input, leveraging two-process extractors Ext. It provides explicit constructions, including the inner product extractor for single-bit extraction and the DEOR scheme for multi-bit extraction, with precise entropy-based guarantees under min-entropy conditions. The results extend classical two-source extractor theory into the quantum domain, address robustness via smoothing, and connect to prior models such as Markov and GE adversaries. A key application is device-independent randomness amplification with quantum sources, where the framework supports randomness generation with quantum side information and nonlocal correlations, under realistic assumptions about entropy production. Overall, the paper establishes a flexible, physically justified approach to randomness extraction that broadens the settings in which quantum-secure randomness can be obtained.

Abstract

We consider a pair of causally independent processes, modelled as the tensor product of two channels, acting on a possibly correlated input to produce random outputs X and Y. We show that, assuming the processes produce a sufficient amount of randomness, one can extract uniform randomness from X and Y. This generalizes prior results, which assumed that X and Y are (conditionally) independent. Note that in contrast to the independence of quantum states, the independence of channels can be enforced through spacelike separation. As a consequence, our results allow for the generation of randomness under more practical and physically justifiable assumptions than previously possible. We illustrate this with the example of device-independent randomness amplification, where we can remove the constraint that the adversary only has access to classical side information about the source.

Figures (4)

• Figure 1: Spacetime diagram illustrating the generation of $X$ and $Y$. Two randomness generating processes begin at time $t_0$ and finish producing randomness by time $t_1$. Due to the spatial distance between the two experimentalists, the two processes $\mathcal{M}$ and $\mathcal{N}$ act independently on $A$ and $B$, which are spacelike separated regions of the Cauchy surface at time $t_0$.
• Figure 2: Circuit diagram of the setup. Two independent channels $\mathcal{M}$ and $\mathcal{N}$ are applied to an initial quantum state $\rho_{AB}$ to produce classical values $X$ and $Y$, respectively. Additionally, we allow the channels to produce quantum side information $S$ and $T$. The state $\rho_{AB}$ should be understood to capture all degrees of freedom that $\mathcal{M}$ and $\mathcal{N}$ may depend on (see also \ref{['fig:setup_spacetime']}). An extractor function $\text{Ext}$ produces a random bitstring $Z$, which should be uniformly distributed and independent from $S$ and $T$. The length of the generated bitstring $Z$ depends on the amount of randomness---measured in terms of entropy---produced by the channels $\mathcal{M}$ and $\mathcal{N}$. Note that one may also consider an extra purifying system $E$ for $\rho_{AB}$. This could be passed through $\mathcal{M}$ or $\mathcal{N}$, i.e., there is no need to explicitly model the identity channel on $E$.
• Figure 3: Diagram of a DIRA setup with a quantum source. We model the quantum SV source as a sequence of channels $\mathcal{S}_1, \ldots, \mathcal{S}_{2n}$, producing classical random variables. The first $n$ pairs of bits are used as the input to a Bell test (green boxes) which produces the measurement results $X^n$. An additional $n$ pairs of bits $Y^n$ are produced using the same quantum SV source which, together with $X^n$, are used to extract the final random bitstring $Z^m$.
• Figure 4: Diagramm of the alternative model studied in \ref{['lem:alt_model_equivalence']}. An instrument $\mathcal{N}_{YT|B}$ is applied to part of a cq state $\rho_{XB}$. The function $\mathrm{Ext}$ is applied to $\rho_{XYT}^{\mathrm{out}} \coloneqq \mathcal{N}_{YT|B}[\rho_{XB}]$ to extract the random bitstring $Z$.

Theorems & Definitions (70)

• Remark 2.1
• Definition 2.2: Instruments
• Definition 2.3: Adjoint channel
• Lemma 2.4: Stinespring dilation Stinespring_1955
• Definition 2.5: Trace norm
• Remark 2.6: Relation to $1$-norm
• Definition 2.7: Purified distance
• Remark 2.8
• Lemma 2.9: Tomamichel_2010
• Definition 2.10: Rényi entropies Muller_Lennert_2013Wilde_2014