Lüders Rule

Lüders rule is the standard ideal update rule for a sharp projective measurement when the outcome may be degenerate. Its defining feature is minimal disturbance inside the reported eigenspace: the measurement removes coherence between different reported outcomes, but it does not refine the state any further inside a degenerate outcome subspace.

Helpful background. Projective measurements defines the PVM and Born probabilities. State update rules compares Lüders operations with other apparatus models.

Lüders operations for sharp outcomes

In finite dimension, let a self-adjoint observable have spectral decomposition

$$A=\sum_a aP_a,$$

where the sum is over distinct reported eigenvalues and the projectors satisfy

$$P_aP_b=\delta_{ab}P_a, \qquad \sum_a P_a=I.$$

For input density operator $\rho$, the probability of outcome $a$ is

$$p(a)=\operatorname{Tr}(P_a\rho).$$

If outcome $a$ is recorded and $p(a)\ne0$, Lüders rule gives the conditional state

$$\rho_a = \frac{P_a\rho P_a} {\operatorname{Tr}(P_a\rho)}.$$

The corresponding unnormalized operation is

$$\mathcal I_a^{L}(\rho) = P_a\rho P_a.$$

If the measurement is performed but the outcome is ignored, the nonselective Lüders channel is

$$\mathcal M_L(\rho) = \sum_a P_a\rho P_a.$$

This channel is trace preserving, completely positive, and idempotent:

$$\mathcal M_L(\mathcal M_L(\rho)) = \mathcal M_L(\rho).$$

What Minimal Disturbance Means

Decompose the density operator into blocks relative to the measurement projectors:

$$\rho = \sum_{a,b}P_a\rho P_b.$$

The nonselective Lüders channel keeps only the diagonal blocks:

$$\mathcal M_L(\rho) = \sum_a P_a\rho P_a.$$

Thus it destroys coherence between different reported outcomes $a\ne b$, but it preserves the full operator $P_a\rho P_a$ inside each reported eigenspace.

Here “minimal disturbance” refers to this precise preservation property, not to minimization of every possible distance between input and output states. If $P_a$ has rank greater than one, then a Lüders measurement does not choose a preferred basis inside $\operatorname{Ran}P_a$. It only says: the result was $a$, and the state has been compressed into the $a$ eigenspace.

Nondegenerate Measurements

If each projector is rank one,

$$P_a=\lvert a\rangle\langle a\rvert,$$

then Lüders rule reduces to the familiar projection postulate. For a pure input

$$\lvert\psi\rangle = \sum_a c_a\lvert a\rangle,$$

the probability is

$$p(a)=|c_a|^2,$$

and the conditional state is

$$\lvert\psi_a\rangle = \frac{P_a\lvert\psi\rangle}{\sqrt{p(a)}} = e^{i\theta_a}\lvert a\rangle$$

when $p(a)\ne0$. In the nondegenerate case there is no internal eigenspace coherence left to discuss.

Degenerate Measurements

Suppose $P_a$ has rank greater than one. Choose an orthonormal basis $\{\lvert a,\lambda\rangle\}$ for the $a$ eigenspace:

$$P_a = \sum_\lambda \lvert a,\lambda\rangle\langle a,\lambda\rvert.$$

For a pure input, Lüders rule gives

$$\lvert\psi_a\rangle = \frac{P_a\lvert\psi\rangle} {\sqrt{\langle\psi\rvert P_a\lvert\psi\rangle}}.$$

If the component inside the eigenspace is

$$P_a\lvert\psi\rangle = \sum_\lambda c_{a\lambda} \lvert a,\lambda\rangle,$$

then the relative amplitudes $c_{a\lambda}$ are not dephased by the Lüders update. The state is projected into the eigenspace but not refined to a particular $\lambda$.

The coarse probability is $p(a)=\sum_\lambda|c_{a\lambda}|^2$. Orthogonality makes this a sum of squared magnitudes, but it does not imply that the coherences inside the eigenspace have been destroyed. The label $\lambda$ is not part of the reported outcome.

This is why degeneracy makes the phrase “measure $A$” incomplete. An apparatus may report only $a$, but it may still physically distinguish additional labels inside the degenerate eigenspace.

Refined Measurement Versus Lüders Measurement

Let mutually orthogonal projectors refine each reported eigenspace:

$$P_a=\sum_\lambda P_{a\lambda}, \qquad P_{a\lambda}P_{a\mu} =\delta_{\lambda\mu}P_{a\lambda}.$$

These finer projectors need not have rank one. An ideal Lüders measurement of this finer PVM, followed by discarding $\lambda$ while retaining $a$, implements

$$\mathcal I_a^{\text{ref}}(\rho) = \sum_\lambda P_{a\lambda}\rho P_{a\lambda}.$$

The Lüders operation for the same reported outcome is

$$\mathcal I_a^{L}(\rho) = P_a\rho P_a = \sum_{\lambda,\mu} P_{a\lambda}\rho P_{a\mu}.$$

The difference is the off-diagonal terms with $\lambda\ne\mu$:

$$P_{a\lambda}\rho P_{a\mu}.$$

The coarse Lüders rule keeps them. The refined measurement erases coherence between distinct $\lambda$ subspaces while preserving coherence within each finer subspace. Complete refinement is the special case $P_{a\lambda}=\lvert a,\lambda\rangle\langle a,\lambda\rvert$ in which all these projectors have rank one.

Both operations give the same coarse probability:

$$\operatorname{Tr}\mathcal I_a^{\text{ref}}(\rho) =\sum_\lambda\operatorname{Tr}(P_{a\lambda}\rho) =\operatorname{Tr}(P_a\rho)=p(a).$$

For $p(a)>0$, the refined conditional state is

$$\rho_a^{\text{ref}} =\frac{\sum_\lambda P_{a\lambda}\rho P_{a\lambda}}{p(a)},$$

which generally differs from $\rho_a$. Discarding a label averages over the already measured branches; it does not itself create the disturbance. A finer PVM alone still does not specify a general apparatus’s backaction. The assumed Lüders implementation of that refinement matters, especially when a finer subspace remains degenerate; see quantum instruments.

A complex three-level example

For $P_0=|0\rangle\langle0|+|1\rangle\langle1|$, $P_2=|2\rangle\langle2|$, and $|\psi\rangle=(|0\rangle+i|1\rangle+|2\rangle)/\sqrt3$, the density matrix changes as

$$\rho=\frac13\begin{pmatrix}1&-i&1\\i&1&i\\1&-i&1\end{pmatrix} \quad\longmapsto\quad \mathcal M_L(\rho)=\frac13\begin{pmatrix}1&-i&0\\i&1&0\\0&0&1\end{pmatrix}.$$

The $0$–$1$ block retains its relative phase. The outcome probabilities are $2/3$ and $1/3$; conditional on the first, the pure state is

$$|\chi\rangle=\frac{|0\rangle+i|1\rangle}{\sqrt2}.$$

A refined basis measurement followed by forgetting which of 0 and 1 occurred would instead remove the two imaginary entries. Its unread output would be $I/3$, and its conditional output for the coarse outcome $0$ would be $(|0\rangle\langle0|+|1\rangle\langle1|)/2$.

To distinguish these predictions, immediately measure the two-outcome PVM $\{Q_\chi,I-Q_\chi\}$, with $Q_\chi=|\chi\rangle\langle\chi|$ and no intervening evolution. Among the runs conditioned on the first outcome $0$,

$$\Pr_L(Q_\chi\mid0)=1, \qquad \Pr_{\text{ref}}(Q_\chi\mid0)=\frac12.$$

If the first outcome is unread, the later probabilities instead include its weight $p(0)=2/3$:

$$\Pr_L(Q_\chi)=\frac23, \qquad \Pr_{\text{ref}}(Q_\chi)=\frac13.$$

The other first outcome leaves $|2\rangle$, which is orthogonal to $|\chi\rangle$ and contributes zero to this later probability. Repeated preparations can distinguish the two apparatus models statistically. A single $Q_\chi$ result does not identify the model, because both can produce it.

Nonselective Channel

The unread Lüders measurement is a dephasing channel in the decomposition defined by the projectors:

$$\rho = \sum_{a,b}P_a\rho P_b \quad\longmapsto\quad \sum_aP_a\rho P_a.$$

Its fixed points are precisely the states block diagonal in the measured decomposition:

$$\mathcal M_L(\rho)=\rho \quad\Longleftrightarrow\quad \rho=\sum_aP_a\rho P_a.$$

Equivalently,

$$[\rho,P_a]=0$$

for every $a$ when the projectors form the full measured decomposition.

For any self-adjoint matrix $B$ that commutes with all measurement projectors,

$$[B,P_a]=0 \qquad \text{for all }a,$$

the unread Lüders measurement does not change its expectation value:

$$\operatorname{Tr}\!\left[B\,\mathcal M_L(\rho)\right] = \operatorname{Tr}(B\rho).$$

For observables with off-block components, expectation values can change because the measurement has removed coherence between outcome sectors.

Degeneracy and compatible labels

Symmetry often supplies additional labels inside a degenerate eigenspace. For example, a rotationally invariant Hamiltonian can have energy levels degenerate in angular-momentum projection; measuring energy need not measure that projection. A complete set of commuting observables resolves the common eigenspaces to one dimension in the ideal finite-dimensional setting. Measuring only some of those observables can give a partial refinement with higher-rank projectors.

Listing a complete set or choosing a common eigenbasis does not establish which quantities an apparatus measured. Its coupling and record determine which compatible labels it distinguishes. The algebraic conditions belong to compatible observables; angular momentum and selection rules supply symmetry examples.

Repeatability

If outcome $a$ is obtained and the same projective measurement is immediately repeated, Lüders rule predicts outcome $a$ with probability one:

$$\operatorname{Tr}(P_a\rho_a) = \frac{\operatorname{Tr}(P_aP_a\rho P_a)} {\operatorname{Tr}(P_a\rho)} = 1.$$

This repeatability is part of the ideal projective-measurement model. It should not be confused with nondestructive measurement in every laboratory sense. A detector can destroy the physical carrier while still being represented by a projective effect on the pre-measurement Hilbert space, in which case a post-measurement system state may not be available in the same Hilbert space.

Repeatability also does not uniquely imply Lüders rule in degenerate subspaces. Other instruments may leave the state inside the $a$ eigenspace but rotate, dephase, or refine it. Lüders rule is the choice that preserves every state within each eigenspace, including all of its internal coherences.

Choosing the idealization

Use this instrument when a sharp readout preserves every state already supported in its reported eigenspace. The system must remain available in the same Hilbert space, with no additional conditional rotation or unresolved refinement. A coarse outcome label alone does not establish these properties.

Use quantum instruments when detector noise, absorption, reset, or additional backaction must be included. A finite-resolution measurement may still be a Lüders measurement of sharp bins; it need not be one if neighboring detector responses overlap.

Continuous Spectra

For continuous spectra, a spectral measure assigns a projector $P(\Delta)$ to each measurable region $\Delta$. An ideal two-outcome Lüders measurement of “inside $\Delta$ or outside?” has, for a positive-probability inside result, the well-defined update

$$\rho \mapsto \frac{P(\Delta)\rho P(\Delta)} {\operatorname{Tr}[P(\Delta)\rho]}.$$

This operation describes a coarse region measurement, not a finer position readout whose detailed result is later discarded. Infinitely sharp outcomes, such as an exact position value, require care: ideal eigenstates may not be normalizable vectors in the Hilbert space. Real measurements have finite resolution and are usually better described by POVMs or instruments with finite-width response functions.

Exercises

Coherence inside a degenerate subspace

Let

$$P=\lvert0\rangle\langle0\rvert+\lvert1\rangle\langle1\rvert$$

and

$$\lvert\psi\rangle = \alpha\lvert0\rangle+\beta\lvert1\rangle+\gamma\lvert2\rangle, \qquad |\alpha|^2+|\beta|^2+|\gamma|^2=1.$$

Find the probability $p$ of outcome $P$. For $p>0$, compute the Lüders conditional state and compare it with the ideal refined measurement that distinguishes $\lvert0\rangle$ and $\lvert1\rangle$ but reports only $P$. Find the probability of $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ in a later projective measurement with no intervening evolution. Recover the equal-superposition case $\alpha=\beta=1/\sqrt2$, $\gamma=0$.

Solution

The coarse outcome has probability

$$p=\langle\psi|P|\psi\rangle=|\alpha|^2+|\beta|^2.$$

Orthogonality removes cross terms from this probability, not from the conditional density operator. In the ordered basis $\{|0\rangle,|1\rangle\}$ of the reported subspace, the two conditional states are

$$\rho^L =\frac1p \begin{pmatrix} |\alpha|^2&\alpha\beta^*\\ \alpha^*\beta&|\beta|^2 \end{pmatrix}, \qquad \rho^{\text{ref}} =\frac1p \begin{pmatrix} |\alpha|^2&0\\ 0&|\beta|^2 \end{pmatrix}.$$

The Lüders state is the pure projector onto $(\alpha|0\rangle+\beta|1\rangle)/\sqrt p$. The refined state loses the off-diagonal coherence while retaining the same coarse outcome probability. Consequently,

$$\Pr_L(+\mid P)=\frac{|\alpha+\beta|^2}{2p}, \qquad \Pr_{\text{ref}}(+\mid P)=\frac12.$$

If the first outcome is unread, these become $|\alpha+\beta|^2/2$ and $p/2$, since the complementary outcome leaves $|2\rangle$. If $p=0$, conditioning on $P$ is undefined; that outcome never occurs.

For $\alpha=\beta=1/\sqrt2$ and $\gamma=0$, $p=1$: the Lüders output is the unchanged pure state $|+\rangle\langle+|$, while the refined output is $(|0\rangle\langle0|+|1\rangle\langle1|)/2$. Their later $+$ probabilities are $1$ and $1/2$, respectively.

Idempotence

Show that the nonselective Lüders channel

$$\mathcal M_L(\rho)=\sum_aP_a\rho P_a$$

satisfies $\mathcal M_L(\mathcal M_L(\rho))=\mathcal M_L(\rho)$.

Solution

Use $P_aP_b=\delta_{ab}P_a$:

$$\mathcal M_L(\mathcal M_L(\rho)) = \sum_{a,b} P_aP_b\rho P_bP_a = \sum_{a,b} \delta_{ab}P_a\rho P_a = \sum_aP_a\rho P_a.$$

Thus applying the same unread ideal measurement a second time changes nothing further.

Expectation of a commuting observable

Let $B$ commute with every $P_a$. Prove that the unread Lüders measurement does not change $\langle B\rangle$.

Solution

Using cyclicity of the trace and $[B,P_a]=0$,

$$\operatorname{Tr}\!\left[B\sum_aP_a\rho P_a\right] = \sum_a\operatorname{Tr}(P_aBP_a\rho) = \operatorname{Tr} \!\left[ B\left(\sum_aP_a\right)\rho \right] = \operatorname{Tr}(B\rho).$$

The measurement may still change expectations of observables that do not commute with the projectors.

References

• P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, 2nd ed., Springer (1996). Publisher.
• G. Lüders, “Concerning the state-change due to the measurement process,” Annalen der Physik 15, 663–670 (2006), English translation of the 1951 article. DOI.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018), chapter 2. Author’s text.