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.

This page is the focused canonical page for Lüders rule in measurement theory. The broader comparison among update rules is State Update Rules, and the general projective-measurement framework is Projective Measurements.

Statement of the Rule

Let a discrete 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.

That last clause is the key point. 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 $\lambda$.

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

$$P_{a\lambda} = \lvert a,\lambda\rangle\langle a,\lambda\rvert$$

be rank-one projectors resolving the degeneracy of $P_a$. A device that measures the complete basis and later discards $\lambda$ 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 $\lambda\ne\mu$:

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

Lüders rule keeps them. The refined measurement erases them.

Three-Level Example

Consider a three-dimensional Hilbert space with basis

$$\{\lvert 0\rangle,\lvert 1\rangle,\lvert 2\rangle\}.$$

Let a coarse projective measurement have two outcomes:

$$P_+ = \lvert 0\rangle\langle 0\rvert + \lvert 1\rangle\langle 1\rvert, \qquad P_- = \lvert 2\rangle\langle 2\rvert.$$

For the input state

$$\lvert\psi\rangle = \frac{\lvert 0\rangle+\lvert 1\rangle}{\sqrt2},$$

the outcome $+$ occurs with probability one. The Lüders update leaves the state unchanged:

$$\rho_+^{L} = \lvert\psi\rangle\langle\psi\rvert.$$

A refined device that secretly distinguishes $\lvert0\rangle$ from $\lvert1\rangle$ but reports only $+$ gives

$$\rho_+^{\text{ref}} = \frac{1}{2} \lvert0\rangle\langle0\rvert + \frac{1}{2} \lvert1\rangle\langle1\rvert.$$

Both devices report the same outcome with the same probability. They differ only in the post-measurement state. This is the simplest reason why outcome probabilities alone do not determine a measurement update.

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 observable $B$ 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.

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 minimally disturbing ideal choice among these projective updates.

When to Use It

Use Lüders rule when all of the following are intended idealizations:

• The measurement is sharp and projective.
• The reported outcome corresponds to the projector $P_a$.
• The apparatus does not resolve hidden labels inside a degenerate eigenspace.
• The post-measurement system remains in the same Hilbert space.
• The measurement is modeled as minimally disturbing subject to the reported outcome.

Use a more general instrument when:

• the detector has finite efficiency, dark counts, or finite resolution,
• the measurement is weak or unsharp,
• the apparatus distinguishes additional degrees of freedom and then coarse grains them,
• the system is absorbed, lost, or reset,
• the backaction is known from a physical measurement model,
• a continuous measurement record is retained.

In the general case, the state update is not determined by the POVM effect alone. It is determined by the instrument map for the outcome.

Continuous Spectra

For an observable with continuous spectrum, projectors are replaced by a projection-valued measure $P(\Delta)$ over measurable outcome sets $\Delta$. The ideal conditional update for an outcome region $\Delta$ is formally

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

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.

Common Mistakes

Treating degeneracy as harmless

Degenerate measurements are exactly where Lüders rule matters. A Lüders measurement of a degenerate observable and a complete-basis measurement followed by coarse graining can have identical probabilities and different final states.

Assuming the POVM determines the update

The effect $E_a=P_a$ gives $p(a)=\operatorname{Tr}(P_a\rho)$, but many instruments can share the same effect. Lüders rule is one instrument choice, not a consequence of the probability formula alone.

Applying the rule to destructive detection

If the measured system is absorbed or replaced, the output state may live in a different Hilbert space. The simple formula $P_a\rho P_a/\operatorname{Tr}(P_a\rho)$ then need not describe the physical output.

Forgetting the nonselective case

If the outcome is not known, do not use a single conditional state $\rho_a$. Use the averaged channel

$$\rho \mapsto \sum_aP_a\rho P_a.$$

Exercises

Coherence inside a degenerate subspace

Let

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

and

$$\lvert\psi\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}.$$

Compute the Lüders update for outcome $P$ and compare it with the refined measurement that distinguishes $\lvert0\rangle$ and $\lvert1\rangle$.

Solution

Since $P\lvert\psi\rangle=\lvert\psi\rangle$, the Lüders update leaves the state pure:

$$\rho^L = \lvert\psi\rangle\langle\psi\rvert = \frac{1}{2} \left( \lvert0\rangle\langle0\rvert + \lvert0\rangle\langle1\rvert + \lvert1\rangle\langle0\rvert + \lvert1\rangle\langle1\rvert \right).$$

The refined measurement gives

$$\rho^{\text{ref}} = \frac{1}{2} \lvert0\rangle\langle0\rvert + \frac{1}{2} \lvert1\rangle\langle1\rvert.$$

The probabilities agree, but the refined measurement erases the coherence terms $\lvert0\rangle\langle1\rvert$ and $\lvert1\rangle\langle0\rvert$.

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

• G. Lüders, “Über die Zustandsänderung durch den Meßprozeß,” Annalen der Physik 443, 322–328 (1951).
• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, Springer (1996).
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic (1995).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010).