# Distinguish two states with their priors probability

EDIT: This is a computer programming / coding exercise

The states $$\left|\psi\right>$$ and $$\left|\phi\right>$$ are defined as

$$∣ϕ⟩=\cos(θ_ϕ)\left|0\right>+\sin(θ_ϕ) \left|1\right>$$ with prior $$p(\phi)$$

$$∣\psi⟩=\cos(θ_\psi)\left|0\right>+\sin(θ_\psi) \left|1\right>$$ with prior $$p(\psi)$$

With one measurement, what is the best average case probability ​that one can distinguish these two states?

Here is how I am solving it, and getting stuck

• If the measurement is $$0$$, then we guess $$\phi, \psi$$ with probability $$e$$ and $$1-e$$, respectively. If the measurement is $$1$$, then we guess $$\phi, \psi$$ with probability $$e_1$$ and $$1-e_1$$, respectively
• Probability of success when guessing $$\phi$$ is $$p_s(\phi):=p(\phi)e\cos^2\theta_\phi+e_1\sin^2\theta_\phi)$$. Probability of success when guessing $$\psi$$ is $$p_s(\psi):=p(\psi)[(1-e)\cos^2\theta_\psi+(1-e_1)\sin^2\theta_\psi]$$

It is tempting to say the answer is $$\max(p_s(\phi),p_s(\psi))$$, but I think if we are talking about average case probability, then we need to weighted sum $$p_s(\cdot)$$.

The task seems to be finding maximum $$e, e_1$$ and plug that to $$P$$.

\begin{align*} P&=p(\phi)(e\cos^2\theta_\phi+e_1\sin^2\theta_\phi)+p(\psi)((1-e)\cos^2\theta_\psi+(1-e_1)\sin^2\theta_\psi)\\ &=p(\phi)e\cos^2\theta_\phi+p(\phi)e_1\sin^2\theta_\phi+\\&p(\psi)\cos^2\theta_\psi -p(\psi)e\cos^2\theta_\psi+p(\psi)\sin^2\theta_\psi-p(\psi)e_1\sin^2\theta_\psi\\ &=p(\psi)+e(p(\phi)\cos^2\theta_\phi-p(\psi)\sin^2\theta_\psi)+e_1(p(\phi)\sin^2\theta_\phi+p(\psi)\sin^2\theta_\psi) \end{align*}

But now coeff of $$e, e_1$$ are constants, and the maximum is achieved when $$e=e_1=1$$ which doesn't make any sense.

• If I understand correctly, we have $p(\psi)=1-p(\phi)$? In what context are you interested in such an exercise? Is it voluntary that you're only considering measurements in the computational basis? Commented Dec 21, 2023 at 15:47
• @TristanNemoz Yes $p(ψ)=1−p(ϕ)$. This is a coding exercise. I could consider the measurements in any basis. Commented Dec 21, 2023 at 19:02
• Oh it's a coding exercise! In which case you probably want to optimize over the different POVM and select the one leading to the largest probability. The reason why I was asking about the context was precisely in case it's an exercise: how much help do you want? Just get some clues, an actual solution, etc...? Also, you should definitely mention that the exercise is meant to be a coding one, it's a totally different way to approach the question. Commented Dec 21, 2023 at 19:27
• I edited the question. Think that you already gave me a hint so I would like to try my hand at this once more before asking for more. Thank you and Merry Christmas Commented Dec 24, 2023 at 21:48