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Nielsen and Chuang mention in Quantum Computation and Information that projective measurements has a property called repeatability.

Projective measurements are repeatable in the sense that if we perform a projective measurement once, and obtain the outcome m, repeating the measurement gives the outcome m again and does not change the state.

The authors then explain briefly why:

To see this, suppose $|\psi\rangle$ was the initial state. After the first measurement the state is $|\psi_m \rangle= \left( P_m | \psi \rangle \right) / \sqrt{\langle ψ|P_m|ψ \rangle}$. Applying $P_m$ to $|ψ_m \rangle$ does not change it, so we have $\langle ψ_m|P_m|ψ_m \rangle = 1$, and therefore repeated measurement gives the result m each time, without changing the state.

Since $P_m$ is a projector onto an eigenspace, once the state $\psi$ is projected and become $\psi_m$, applying $P_m$ to the state again does not change things because we are projecting the vector onto the same vector subspace in which the vector already lives.

What I don't understand is that how can this fact implies $\langle ψ_m|P_m|ψ_m \rangle = 1$ as highlighted in bold in the Nielsen and Chuang's explanation above.

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2 Answers 2

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The state as defined

$|\psi_m \rangle= \left( P_m | \psi \rangle \right) / \sqrt{\langle ψ|P_m|ψ \rangle}$.

is normalized.

Further, $P_m|\psi_m \rangle = |\psi_m \rangle$.

Thus, $$ \langle\psi_m|P_m|\psi_m \rangle = \langle\psi_m|\psi_m \rangle=1\ . $$

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If the projector $P_m$ does not alter the state $\psi_m$, then we have the equality:

$<\psi_m|P_m|\psi_m>=<\psi_m|\psi_m>=1$,

due to the normalisation of $\psi_m$.

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  • $\begingroup$ Thanks. But I cannot accept your answer because it is the same as another answer. $\endgroup$
    – Prksa
    Jun 22, 2021 at 17:07

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