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Suppose $|\psi \rangle = |1\rangle$ and $| \phi \rangle = | + \rangle = \frac{1}{\sqrt{2}} (|0 \rangle + |1 \rangle )$. Write a POVM that allows for imperfect distinguishability between two states.

POVM stands for Positive Operator-Values Measures, it's a set of operators $\{E_m\}$ where the probability of measuring result $m$ is $Pr(m)=⟨ψ|E_m|ψ⟩$ and $∑_mE_m=I$. These operators allow for more general forms of measurement, such as distinguishing non-orthogonal states. One can do this if $⟨ψ|E_i|ψ⟩=0$ but $⟨ϕ|Ei|ϕ⟩>0$ but $<1$ for some $E_i$ and vice-versa for $E_j$. This allows one to imperfectly identify if the system is in state $| \psi \rangle$ or $| \phi \rangle$ since there some measurements are impossible (probability of 0).

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    $\begingroup$ What is POVM and how is "imperfect distinguishability" defined? $\endgroup$
    – Korf
    Dec 14, 2020 at 23:40

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FYI this is probably better suited on the quantum computing stackexchange. But anyway, you can try taking $E_1 = c_1 |0\rangle \langle 0 |$, $E_2 = c_2 |-\rangle \langle - |$ and $E_3 = I - E_1 - E_2$. For some suitably chosen constants $c_1, c_2 > 0$ this should give you a POVM which satisfies $$ \langle \psi | E_1 |\psi\rangle = 0 \qquad \langle \phi | E_1 |\phi \rangle >0 $$ and $$ \langle \psi | E_2 |\psi\rangle > 0 \qquad \langle \phi | E_2 |\phi \rangle =0. $$

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