Genuine certifiable randomness from a black-box

TL;DR

This work demonstrates genuine randomness certification in the black-box setting -- one in which no deterministic adversary, even with unlimited computational power, will succeed in getting their data certified.

Abstract

Randomness is intrinsic to quantum mechanics; the outcome of a measurement on a quantum state is a random variable. This feature has been applied to randomness certification, where one party must decide whether the data they receive is truly random. However, existing demonstrations are not black-box, to avoid falsely certifying deterministic data, assumptions must be made on how the data was generated. Here we demonstrate genuine randomness certification in the black-box setting -- one in which no deterministic adversary, even with unlimited computational power, will succeed in getting their data certified. We use it to provably generate random numbers using only measurements on single particle states and without a random seed.

Figures (3)

• Figure 1: (Quantum) estimation certified randomness (ECR). (a) One round of ECR involves: i) the verifier choosing a value of $\theta$ and encoding it into a quantum state $\left| \theta \right>$, ii) sending $\left| \theta \right>$ to the prover, iii) the prover returning an estimate $\hat{\theta}$ of $\theta$ which may or may not derive from a measurement of $\left| \theta \right>$, iv) the verifier testing the received estimate. (b) A mapping of the parameter $\theta$ to the quantum phase of a qubit, this is a quantum state $\left| \theta \right> = \frac{1}{\sqrt{2}}\left(\left| \uparrow \right> + \operatorname{exp}\left[i\mkern1mu \pi \theta\right] \left| \downarrow \right>\right)$. (c) Antipodal metric. The antipodal distance $\operatorname{d^{\circ}}\!\left[\theta_1,\theta_2\right]$ between two points $\theta_1,\theta_2$ is given by the (shortest) arc length on a circle connecting $\theta_1$ and $\theta_2$. We parametrize the circle by $\theta \in [0, 2)$, the positive angle (with the vertical) in $\pi$ radians. An antipodal pair of points, such as the purple circles, are two points where $\operatorname{d^{\circ}}\!\left[\theta_1,\theta_2\right]=\pi r$, and an antipodal probability distribution assigns the same probability to both points of an antipodal pair.
• Figure 2: Demonstration of non-remote estimation certified randomness with a single NV centre in diamond. (a) After selecting a value of $\theta$, the verifier prepares the NV spin state: $\left| \theta \right> = \frac{1}{\sqrt{2}}\left(\left| \uparrow \right> + \operatorname{exp}\left[-i\mkern1mu \pi \theta\right] \left| \downarrow \right>\right)$ by applying a resonant microwave $\pi/2$-pulse with phase $\pi\times\theta$. The prover performs an $X$-basis measurement of $\left| \theta \right>$ by applying a second $\pi/2$-pulse with phase $\pi\times\varphi = 0$ and projectively reading out the spin-state along $z$, denoting the outcomes '1' and '0'. The best estimate of $\theta$ is the measurement outcome. (b) Plots of the individual estimate squared error for 20 rounds of ECR, when the verifier selected $\theta = 0$ (top) and $\theta = 2/3$ (bottom). Four different estimate strategies are shown. A deterministic estimate without a measurement (blue, the sequence of estimates is a permutation of the binary representation of $e^{\pi}$), estimates using the experimental measurement outcome distributed according to \ref{['eq:meas_prob']} with $a\sim0.01, b\sim0.04$ (dark green) and $a\sim0.5, b\sim0.1$ (light green) and a simulation of the ideal estimate according to the Born rule (purple).
• Figure 3: Experimentally testing the randomness of data. The estimate mean squared error as a function of the number of ECR rounds when the verifier uniformly selected $\theta$ from $\Theta_6 := \{0, 1, 2, 3, 4, 5\}/3$ and the prover returned a one-bit estimate $\hat{\theta}$ based on four different strategies. (a) The prover's estimate is the outcome of a measurement $\left|\bm{X}_x \left| \theta \right>\right|^2$ with $a\sim0.5$, $b\sim0.1$ (light green) or a simulation of an ideal measurement with $a = 1$, $b = 0$ (purple). (b) The prover's estimate is the outcome of a measurement $\left|\bm{X}_x \left| \theta \right>\right|^2$ with $a\sim0.01$, $b\sim0.04$ (dark green) or a deterministic permutation of the binary representation of $e^\pi$ without a measurement (blue). Shaded regions show one standard deviation region of the data and grey is the five-sigma confidence region for a deterministic estimate to differ from $1/2$.

Theorems & Definitions (24)

• Remark 1: A single quantum state
• Example 1: No-measurement estimation bound for uniform prior and Euclidean error
• Definition 1: Mean squared error of an estimate
• Definition 2: Expected mean squared error of an estimate
• Definition 3: Antipodal (phase) estimation problem
• proof : Proof of \ref{['th:anti']}
• proof
• proof
• proof : Proof of \ref{['th:CRLB']}--(1)
• proof : Proof of \ref{['th:CRLB']}--(2)