Theory-independent randomness generation from spatial symmetries

TL;DR

This work establishes a theory-independent link between spacetime symmetries and quantum probabilistic structure by showing that rotational covariance combined with a spin-$J$ bound on the transmitted system fixes the set of achievable correlations in a simple prepare-and-measure scenario to the quantum set $\mathcal{Q}_{J,\alpha}$. The authors prove the central equivalence $\mathcal{Q}_{J,\alpha}=\mathcal{R}_{J,\alpha}$ for all $J$ and $\alpha$, including a detailed treatment of the $J=\tfrac{1}{2}$ case, and provide a robust framework for certifying private randomness $H^\star$ that remains valid under certain post-quantum extensions. They develop a comprehensive representation-theoretic analysis of ${\rm SO}(2)$, bounding overlaps via $\gamma=\cos(J\alpha)$ when $|J\alpha|<\tfrac{\pi}{2}$ and showing how rotation boxes capture all admissible probabilities under the symmetry constraint. The results imply that spacetime symmetries can determine at least part of quantum probabilistic structure and offer a theory-independent protocol for randomness generation with post-quantum security guarantees.

Abstract

We demonstrate a fundamental relation between the structures of physical space and of quantum theory: the set of quantum correlations in a rotational prepare-and-measure scenario can be derived from covariance alone, without assuming quantum physics. To show this, we consider a semi-device-independent randomness generation scheme where one of two spatial rotations is performed on an otherwise uncharacterized preparation device, and one of two possible measurement outcomes is subsequently obtained. An upper bound on a theory-independent notion of spin is assumed for the transmitted physical system. It turns out that this determines the set of quantum correlations and the amount of certifiable randomness in this setup exactly. Interestingly, this yields the basis of a theory-independent protocol for the secure generation of random numbers. Our results support the conjecture that the symmetries of space and time determine at least part of the probabilistic structure of quantum theory.

Figures (5)

• Figure 1: Setup. A preparation device $P$ is rotated by an angle $\alpha_x\in\{0,\alpha\}$ relative to the measurement device $M$, and then a fixed but arbitrary state is generated. The state is then sent to $M$, where a measurement yields one of two outcomes $b\in\{\pm 1\}$. Some additional classical random variable $\lambda$, unknown to the experimenters, may have been preshared by the devices.
• Figure 2: Illustration of a possible way to implement our prepare-and-measure scenario. The preparation consists of a single-photon source (SPS) which is rotated at angles $0$ and $\alpha$ for inputs $x=1$ and $x=2$ respectively. This leaves the photon unchanged or rotates its polarization by $\alpha$. The photon is then measured by a polarizing beam splitter (PBS) with two single-photon detectors corresponding to the binary outcomes $b\in\{+1,-1\}$. This setup realizes the $J=1$ case of our framework, where the mechanical rotation implements the unitary $U_\alpha={\rm diag}(e^{i\alpha},e^{-i\alpha})$ to rotate the photon polarization.
• Figure 3: The quantum sets $\mathcal{Q}_{J,\alpha}$ (dark blue) and the classical sets $\mathcal{C}_{J,\alpha}$ (dark red; line $E_1=E_2$), and the quantum and classical relaxed sets $\mathcal{Q}^\varepsilon_{J,\alpha}$ and $\mathcal{C}^\varepsilon_{J,\alpha}$ for $\varepsilon\in\{0.15,0.3\}$. We set $J=1$ and $\alpha=0.66$ in this figure.
• Figure 4: The quantum set $\mathcal{Q}_{J,a}$ (blue), the boundary of the set with error $\kappa$ (orange) and the boundary of the smallest relaxed quantum set $\mathcal{Q}_{J,a}^{\delta}$ (green) such that it includes the previous error set given by $\kappa$. The plot illustrates that $\mathcal{Q}_{J,a}^{\delta}$ always contains the set given by the error $\kappa$.
• Figure 5: $\mathcal{R}_{1/2}=\mathcal{Q}_{1/2}$ as a convex set.

Theorems & Definitions (11)

• Theorem 1
• proof
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 1
• Lemma 4
• proof