A exercise with Dirac notation

Question

I am dealing with the above exercise and not sure about my solution is correct or not,

It is possible for me to find another state that the payoff would be larger than 50? Or it is possible for me to prove that 50 is indeed the largest payoff?

Answer 1

In order to maximize the sum of two things, it is not enough to simply maximize the terms in the sum individually (which is what you attempted to do). You must consider the entire function.

To solve the problem correctly, first note that the payoff expression can be greatly simplified. In particular, note that $$ \cos^2\frac{\theta}{2} - \sin^2\frac{\theta}{2} = \cos\theta $$ (which can be seen by the double angle formula). Moreover, note that $$ \left\lvert \frac{\cos\frac{\theta}{2} \pm e^{i\varphi}\sin\frac{\theta}{2}}{\sqrt{2}}\right\rvert^2 = \frac{1\pm\cos\varphi\sin\theta}{2}. $$ Thus your expected payoff function $E(\theta,\varphi) = 100\cdot\frac{1}{2}(\lvert\langle \psi|0\rangle\rvert^2-\lvert\langle \psi|1\rangle\rvert^2+\lvert\langle \psi|+\rangle\rvert^2-\lvert\langle \psi|-\rangle\rvert^2)$ can be simplified to $$ E(\theta,\varphi) = 50(\cos\theta + \cos\varphi\sin\theta). $$ It is now not too difficult to see that the maximal value of $E$ is $50\sqrt{2}\approx 70.71$, but I'll leave it to you to figure out where the maximum is achieved.