with $N_P\gg 1/\kappa^2$, the total photon number in the signal beam can be inferred from a homodyne-detection measurement of the appropriate probe quadrature. In particular, the state of $\hat{a}'_P$ will be $|\sqrt{N}_P\rangle$ when the signal beam’s total photon number is zero, and its state will be $|(1+i\kappa)\sqrt{N}_P\rangle$ when the signal beam’s total photon number is one, where $\kappa \ll 1$ has been employed. Homodyne detection of the $\hat{a}'_{P2} \equiv {\rm Im}(\hat{a}'_P)$ quadrature thus yields a classical random-variable outcome $\alpha'_{P2}$ that is Gaussian distributed with mean zero and variance 1/4, in the absence of a signal-beam photon, and Gaussian distributed with mean $\kappa\sqrt{N}_P$ and variance 1/4 in the presence of a signal-beam photon. Note that these conditional distributions are independent of the polarization state of the signal-beam photon when it is present. Using the decision rule, “declare signal-beam photon present if and only if $\alpha'_{P2} > \kappa\sqrt{N}_P/2$,” it is easily shown that the QND error probability is bounded above by $\exp(-\kappa^2 N_P/2)/2 \ll 1$.

The preceding polarization independent, low error probability QND detection of the signal beam’s total photon number does *not disturb the polarization state of that beam. This is so because the probe imposes the same nonlinear phase shift 