= Lüders rule
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{title2=$\rho_i=P_i\rho P_i/\operatorname{Tr}(P_i\rho)$}
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The <Lüders rule> specifies the ideal <quantum instrument> for a <projective measurement>. Conditional on an outcome with nonzero <probability>, it gives
$$
\rho_i=\frac{P_i\rho P_i}{\operatorname{Tr}(P_i\rho)},\qquad
|\psi_i\rangle=\frac{P_i|\psi\rangle}{\sqrt{\langle\psi|P_i|\psi\rangle}}
$$
for a <density operator> and a <pure state>, respectively. The second formula applies only to a <pure state> input. If the result is discarded, the output is the <nonselective projective measurement> $\sum_iP_i\rho P_i$.
The crucial feature for a degenerate <eigenvalue> is preservation of <quantum coherence> within its <eigenspace>. For a <qutrit> with <orthonormal basis> $|0\rangle,|1\rangle,|2\rangle$, consider
$$
A=0P_0+1P_1,\qquad P_0=|0\rangle\langle0|+|1\rangle\langle1|,\quad P_1=|2\rangle\langle2|.
$$
On $|\chi\rangle=(|0\rangle+|1\rangle)/\sqrt2$, outcome zero has <probability> one and the <Lüders rule> leaves $|\chi\rangle$ unchanged. Measuring all three rank-one projectors and then forgetting whether the label was zero or one instead gives
$$
\widetilde\rho=\tfrac12|0\rangle\langle0|+\tfrac12|1\rangle\langle1|.
$$
Both procedures always report the coarse value zero, but a subsequent test with projector $|\chi\rangle\langle\chi|$ succeeds with <probability> one for the first and one half for the second. Resolving extra information destroyed coherence. This explains why a <quantum nondemolition measurement> of a degenerate quantity must specify what information the apparatus actually records.
Gerhart Lüders proposed this state-change rule in 1951, explicitly discussing degenerate quantities and distinguishing selected subensembles from the aggregate measured ensemble. Kirkpatrick's translation was first posted in 2004; equations (7) and (8) give the unnormalized selective update and the nonselective update. https://arxiv.org/abs/quant-ph/0403007[Lüders, 1951, translated by Kirkpatrick, 2004]{ref}.
Repeatability follows briefly from $P_jP_i=\delta_{ij}P_i$: after outcome $i$, $\operatorname{Tr}(P_j\rho_i)=\delta_{ij}$. This assumes a second application of the same ideal instrument without intervening dynamics. An instrument with the same effects can rotate or reset the output and need not satisfy this conclusion; see <POVM does not determine the post-measurement state>.
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