# Proof of repeatability property of projective measurements

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.

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

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 $$\psi_m$$.

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