Basic Distinguishability Theory

This refines Basic Decoding Theory.

Probability of successfully distinguishing between two equally likely pure states ∣α0⟩,∣α1⟩\left\lvert \alpha_0 \right\rangle, \left\lvert \alpha_1 \right\rangle where ⟨α0∣α1⟩=cos⁡θ\left\langle \alpha_0|\alpha_1 \right\rangle=\cos\theta is

Ps≤12(1+sin⁡θ)P_s\leq \frac{1}{2}(1+\sin\theta)

Example I

We want to find the optimal measurement to distinguish between two equally likely pure states ∣α0⟩,∣α1⟩\left\lvert \alpha_0 \right\rangle, \left\lvert \alpha_1 \right\rangle where ⟨α0∣α1⟩=cos⁡θ\left\langle \alpha_0|\alpha_1 \right\rangle=\cos\theta. We can try to do

∣α0⟩=∣0⟩∣α1⟩=∣+⟩\left\lvert \alpha_0 \right\rangle=\left\lvert 0 \right\rangle\quad\left\lvert \alpha_1 \right\rangle=\left\lvert + \right\rangle cos⁡θ=sin⁡θ=12\cos\theta =\sin\theta=\frac{1}{\sqrt{2}}

and there are is a ∣1⟩\left\lvert 1 \right\rangle basis that is ortho to ∣0⟩\left\lvert 0 \right\rangle. If we measure and get ∣1⟩\left\lvert 1 \right\rangle it couldn’t have come from ∣α0⟩\left\lvert \alpha_0 \right\rangle

But it isn’t optimal

Example II

The optimal way is to get the bisection angle between ∣α1⟩,∣α0⟩\left\lvert \alpha_1 \right\rangle,\left\lvert \alpha_0 \right\rangle

Ps(τ)=P(α0)P(measure k0 given ∣α0⟩)+P(α1)P(measure k1 given ∣α1⟩)P_s(\tau)=P(\alpha_0)P(\text{measure $k_0$ given $\left\lvert \alpha_0 \right\rangle$})+P(\alpha_1)P(\text{measure $k_1$ given $\left\lvert \alpha_1 \right\rangle$})\\ =12cos⁡2(τ)+12cos⁡2(π2−τ−θ)=\frac{1}{2}\cos^2(\tau)+\frac{1}{2}\cos^2(\frac{\pi}{2}-\tau-\theta)

Optimize to get

τ∗=12(π2−θ)\tau^*=\frac{1}{2}\left(\frac{\pi}{2}-\theta\right)\\ ⇒max⁡Ps(τ)=12cos⁡2(π4−θ2)+12cos⁡2(π4−θ2)\Rightarrow\max P_s(\tau)=\frac{1}{2}\cos^2\left(\frac{\pi}{4}-\frac{\theta}{2}\right)+\frac{1}{2}\cos^2(\frac{\pi}{4}-\frac{\theta}{2})\\ =12(1+sin⁡θ)=\frac{1}{2}\left(1+\sin\theta\right)