Lüders rule (${\rho }_i=P_i\rho P_i/\mathrm{Tr}(P_i\rho )$)

The Lüders rule specifies the ideal quantum instrument for a projective measurement. Conditional on an outcome with nonzero probability, it givesfor 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 ${\sum }_i{P}_i\rho P_i$.

The crucial feature for a degenerate eigenvalue is preservation of quantum coherence within its eigenspace. For a qutrit with orthonormal basis $|0⟩,|1⟩,|2⟩$, considerOn $|\chi ⟩=(|0⟩+|1⟩)/\sqrt{2}$, outcome zero has probability one and the Lüders rule leaves $|\chi ⟩$ unchanged. Measuring all three rank-one projectors and then forgetting whether the label was zero or one instead givesBoth procedures always report the coarse value zero, but a subsequent test with projector $|\chi ⟩⟨\chi |$ 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. ^[ref].

Repeatability follows briefly from $P_j{P}_i={\delta }_{ij}{P}_i$: after outcome $i$, $\mathrm{Tr}(P_j{\rho }_i)={\delta }_{ij}$. 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.

The post-measurement state is the conditional quantum state after a specified measurement outcome. For a projective measurement, it is given by the Lüders rule.

If the outcome of a projective measurement $\{P_i\}$ is discarded, its nonselective state update is the pinching channelIt removes coherence between distinct measurement subspaces.

For mutually orthogonal projectors summing to the identity, a pinching map discards coherences between their ranges while preserving each diagonal block. It is a quantum channel with Kraus operators $P_i$, and a unital quantum channel. A state is fixed by pinching exactly when it commutes with every projector.

In an orthonormal basis, rank-one dephasing is the quantum channel $\Delta (\rho )={\sum }_i|{\psi }_i⟩⟨{\psi }_i|\rho |{\psi }_i⟩⟨{\psi }_i|$. It removes off-diagonal entries while preserving the basis probabilities. Its output Von Neumann entropy is their Shannon entropy, and the relative-entropy identity for rank-one dephasing quantifies the entropy increase.

Because $\log \Delta (\rho )$ is diagonal, $\mathrm{Tr}(\rho \log \Delta (\rho ))=\mathrm{Tr}(\Delta (\rho )\log \Delta (\rho ))$. Hence $D_{\mathrm{rel}}(\rho \Vert \Delta (\rho ))=S(\Delta (\rho ))-S(\rho )$. Support inclusion under rank-one dephasing handles zero probabilities, and Klein's inequality makes the difference nonnegative. Equality holds exactly when the state was already diagonal.

If $p_i=⟨{\psi }_i|\rho |{\psi }_i⟩=0$, positivity gives $\sqrt{\rho }|{\psi }_i⟩=0$, so the corresponding basis vector is in $\ker \rho$. Therefore $\ker \Delta (\rho )\subseteq \ker \rho$, and $\mathrm{supp}\rho \subseteq \mathrm{supp}\Delta (\rho )$. This makes the quantum relative entropy of $\rho$ relative to its rank-one dephasing finite.