Coherence-gated quantum devices via real-time weak measurement

Single-photon routers in cavity and circuit QED direct photons by the qubit's energy eigenstate -- a projective decision that destroys coherence. We propose a different primitive: coherence-gated routing, where the decision depends on the magnitude of the qubit's quantum coherence, estimated in real time from simultaneous weak measurements of $\sigma_x$ and $\sigma_z$. A photon is accepted if the coherence score $S(T) = \sqrt{\langle\sigma_x\rangle_c^2 + \langle\sigma_y\rangle_c^2}$, extracted from the conditional density matrix via the stochastic master equation, exceeds a tunable threshold $S_{\mathrm{th}}$. Certifying coherence at emission enables two applications conventional heralded sources cannot: (i) a quantum random number generator with min-entropy bounded by Bloch-sphere geometry, $H_\infty \geq -\log_2\!\bigl(\frac{1+\sqrt{1-S_{\mathrm{th}}^2}}{2}\bigr)$, and (ii) a phase-tracked photon source whose two-node coherence certification bounds the matter--matter entanglement fidelity after Bell-state measurement. The estimator is itself a security primitive. Benchmarking seven configurations, we find that underestimating detector efficiency ($\eta_{\mathrm{a}} < \eta_{\mathrm{true}}$) both stabilizes the numerics and suppresses overcertification. We trace this via a purity-monotonicity result, identify a geometric loophole amplifying purity undercertification into coherence overcertification by an order of magnitude ($\sim$12$\times$), and prove two complementary tail bounds: an Ornstein--Uhlenbeck comparison giving $9.0\%$ overcertification (empirical $6.3\%$ from $10^6$ trajectories) and an exponential supermartingale establishing structural exponential decay.