Realistic prospects for testing a relativistic local quantum measurement inequality

TL;DR

The paper develops a practically usable bound for testing a relativistic local quantum measurement inequality with finite-size detectors and coherent-state inputs. By leveraging a Reeh--Schlieder-based approximation, it derives an explicit bound $P_\text{click}(f) \leq \left[ \mathcal{E}_{\zeta}(f) + \exp\left( \pi^2 /(2\zeta) \right) \sqrt{P_\text{dark}} \right]^2$ minimized over $\zeta$, and reframes it in terms of the coherent-state amplitude $\alpha$ for experimental relevance. The work specializes to a square-prism detector in a finite time window with a normally incident single-mode coherent state, introducing dimensionless variables and a key photon-number parameter $N = |\alpha_0|^2 V_\text{det}/V_\text{coh}$, and performing numerical analyses that reveal the expected trade-off: lowering dark counts tightens the allowed excitation response. The results retain the qualitative features of prior toy-model studies while clarifying how experimental parameters such as phase, detector geometry, and coherence scales influence the bound, guiding feasible experimental tests of relativistic locality in quantum measurements.

Abstract

We investigate the experimental prospects for testing a relativistic local quantum measurement inequality that quantifies the trade-off between vacuum insensitivity and responsiveness to excitations for finite-size detectors. Building on the Reeh--Schlieder approximation for coherent states, we derive an explicit and practically applicable bound for arbitrary coherent states. To connect with realistic photodetection scenarios, we model the detection region as a square prism operating over a finite time window and consider a normally incident single-mode coherent state. Numerical results exhibit the expected qualitative behavior: suppressing dark counts necessarily tightens the achievable click probability.

Figures (2)

• Figure 1: In the proposed experiment, the same detector is used to measure the click probability $P_\text{click}$ of a chosen non-vacuum state $| \psi \rangle$ (panel a) and the dark-count probability $P_\text{dark}$ in the vacuum (panel b). The two outcomes are then compared with the theoretical bound relating these quantities [Eq. \ref{['upper_bound_f_O_d_2']}]. The detector region is modeled as a square prism of dimensions $l \times L \times L$ operating over a finite time window $\tau$, and the input state is taken to be a normally incident single-mode coherent state characterized by amplitude $\alpha_0$, momentum $\mathbf{k}_0$ and coherence volume $V_\text{coh}$.
• Figure 2: Upper bound $P_\text{click}^\text{(max)}(f)$ on the click probability $P_\text{click}(f)$ as a function of the dark count $P_\text{dark}$. Results are shown for different combinations of parameters $(N, \Delta \varphi, a, \arg(\alpha_0))$, where $N = |\alpha_0|^2 (l + \tau) (L + \tau)^2 / V_\text{coh}$ estimates the number of photons effectively "seen" by the detector, $\Delta \varphi = k_0 (l+\tau)$ is the total optical phase accumulated across the thickness of the extended detection volume $\mathcal{V}_\text{det}$ of size $(l + \tau) \times (L + \tau) \times (L + \tau)$, $a = (l + \tau)/(L + \tau)$ is the aspect ratio of $\mathcal{V}_\text{det}$, $\arg(\alpha_0)$ is the phase of the coherent amplitude $\alpha_0$. The plots are displayed on a log--linear scale.