Analytic properties of cross-click operators in passive multi-basis photodetection: monotonicity, exact convergence rates, and dimension reduction for quantum key distribution
Cross-click operators, the POVM elements for simultaneous clicks in detectors assigned to different measurement bases, are used in QKD and entanglement-verification analyses with realistic threshold detectors to bound multiphoton contributions. Earlier applications verified the needed growth of the minimum eigenvalue $f^{(n)}$ on the $n$-photon subspace only numerically over finite sectors. This work gives an analytic characterization for passive linear-optical analyzers with arbitrary efficiency mismatch and dark counts. The key observation is that every silence operator is the second quantization $Γ(A)$ of an explicit single-photon contraction $A$, whose $n$-photon restriction is $A^{\otimes n}$ on $\mathrm{Sym}^n$. This yields: (i) monotonicity $f^{(n+1)}\ge f^{(n)}$; (ii) two-sided exponential bounds $\max_b γ_b|A_b|^n \le 1-f^{(n)} \le \sum_b γ_b|A_b|^n$, which determine the exact asymptotic convergence rate from single-photon spectral data; (iii) for ideal detectors and $n\ge1$, the exact formula $f^{(n)}=1-\sum_b p_b^n$; and (iv) an exact factorization $1-f^{(n_A,n_B)}=(1-f_A^{(n_A)})(1-f_B^{(n_B)})$ for the two-party cross-click operator. The results apply to polarization, time-bin, and spatial-mode analyzers within the stated threshold-detector model. As an application, we obtain closed-form photon-number weight bounds used in detection-efficiency-mismatch analyses, replacing finite-sector Fock-space numerics by formulas valid for all photon numbers.