dly emphasized [@yuen04; @pla05; @yuen05qph; @nair06], we first consider this classical situation. Let $\theta_s$ be the signal point on the circle of Fig. 1, $x$ the data bit, $k'$ the running key segment that determines the basis. Before deliberate or noise randomization, $\theta_s(x,k')$ is uniquely determined by $x$ and $k'$. From $\theta_s$ one randomizes it to $\theta_r$ according to a probability density $p(\theta_r|\theta_s)$. We use continuous $\theta$’s here but the argument is identical for discrete $\theta$’s. More generally, let $\theta$ be Eve’s observed signal point, so that $\theta=\theta_r$ in a classical noiseless system with deliberate randomization. Then, $$\label{pdf}
p(\theta|x,k')=\int
p(\theta|\theta_r)p(\theta_r|\theta_s(x,k'))\mathrm{d} \theta_r.$$

In the classical noiseless case with just signal randomization, $p(\theta|\theta_r)=\delta(\theta-\theta_r)$, the BPSK signal may be correctly discriminated when the observed $\theta$ falls within the half-circle centred around $\theta_s$. Thus we pick $p(\theta_r|\theta_s)$ to be the uniform distribution on the half-circle with midpoint $\theta_s$. If $x$ is uniformly random, then from (1) $$\label{pdf2}
p(\theta|k')=\frac{1}{2} \sum_{x=0,1} p(\theta|x,k')$$ is the uniform distribution on the full circle independent of $k'$. This proves the observation of $\theta$ to Eve yields no information at all on $k'$. In other words, Eve’s channel on $k'$ has zero capacity from DSR and uniformly random data which acts as added noise unknown to her, similar to a nondegenerate nonrandom stream cipher.

For coherent-state noise described in the wedge approximation [@pla05; @nair06], whereupon a heterodyne or phase measurement the observed $\theta$ is taken to be uniformly distributed within a standard deviation around $\theta_r$ and zero outside, the same $k'-$independence for $p(\theta|k')$ obtains when $\theta_r$ is chosen in a discrete number of positions for given $\theta_s$ so that $p(\theta_r|\theta_s)$ fills out a uniform half-circle agai