State Update Rules

There is no single state update formula that applies to every measurement device. There is a hierarchy of update rules, and the correct one depends on what the apparatus measures, what outcome information is retained, and how much of the detector dynamics has been modeled.

The compact projective postulate is useful, but measurement theory needs a broader rule:

$$\text{probabilities come from effects,} \qquad \text{state updates come from instruments.}$$

This page is a comparison guide. It explains when to use each update rule and what assumptions come with it.

Decision Table

Situation	Probability object	State-update object	Conditional update
nondegenerate ideal projective measurement	rank-one projectors $P_a$	$P_a\rho P_a$	$\rho_a=P_a\rho P_a/\operatorname{Tr}(P_a\rho)$
degenerate ideal Lüders measurement	eigenspace projectors $P_a$	$P_a\rho P_a$	preserves coherence inside the eigenspace
refined projective measurement	projectors $P_{a\lambda}$	$\sum_\lambda P_{a\lambda}\rho P_{a\lambda}$ after coarse graining	can erase coherence inside a degenerate eigenspace
generalized measurement	POVM effects $F_m$	instrument maps $\mathcal I_m$	$\rho_m=\mathcal I_m(\rho)/\operatorname{Tr}\mathcal I_m(\rho)$
measurement operators	effects $F_m=M_m^\dagger M_m$	$M_m\rho M_m^\dagger$	one-Kraus-operator instrument for each outcome
unread measurement	all outcomes	channel $\Phi=\sum_m\mathcal I_m$	no conditioning; use $\rho'=\Phi(\rho)$
continuous monitoring	time-dependent record	stochastic update rule	conditional state depends on the whole record

The main warning is in the middle row: a POVM effect $F_m$ gives the probability of outcome $m$, but it does not by itself determine the post-measurement state.

Universal Instrument Form

A measurement instrument is a family of completely positive, trace-nonincreasing maps $\{\mathcal I_m\}$ indexed by outcomes. For input state $\rho$,

$$p(m) = \operatorname{Tr}\mathcal I_m(\rho).$$

If outcome $m$ is known and $p(m)\ne0$, the conditional state is

$$\rho_m = \frac{\mathcal I_m(\rho)} {\operatorname{Tr}\mathcal I_m(\rho)}.$$

If the outcome is ignored, the output state is

$$\rho' = \sum_m\mathcal I_m(\rho).$$

The sum over all outcomes must be trace preserving for an unconditional measurement with no discarded failure branch:

$$\operatorname{Tr} \left[ \sum_m\mathcal I_m(\rho) \right] = \operatorname{Tr}\rho.$$

This instrument form contains projective measurements, Kraus updates, detector inefficiency, coarse graining, postselection, and continuous-measurement time steps as special cases or limits.

Projection Postulate

For a rank-one ideal measurement in an orthonormal basis $\{|a\rangle\}$,

$$P_a=|a\rangle\langle a|.$$

If outcome $a$ occurs, the pure-state update is

$$|\psi\rangle \longmapsto \frac{P_a|\psi\rangle} {\sqrt{\langle\psi|P_a|\psi\rangle}} = |a\rangle$$

up to a phase when the probability is nonzero. In density-operator form,

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

This is the familiar projection postulate for an ideal sharp, nondegenerate outcome. It is not a model of detector inefficiency, finite resolution, weak measurement, or destructive readout.

Lüders Update

For a degenerate observable,

$$A=\sum_a aP_a,$$

where $P_a$ projects onto the whole eigenspace for eigenvalue $a$, the Lüders update for outcome $a$ is

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

The key feature is minimal disturbance inside the eigenspace. If $\rho$ contains coherence between two vectors in the same eigenspace, the Lüders update for that coarse outcome preserves it.

The nonselective Lüders update is

$$\rho \longmapsto \sum_a P_a\rho P_a.$$

It removes coherences between different eigenspaces, not within a single degenerate eigenspace.

For the focused treatment of degeneracy, refinement, repeatability, and the limits of this idealization, see Lüders Rule.

Refined Projective Updates

A device may measure more than the coarse observable $A$. Suppose the eigenspace for $a$ has an orthonormal basis $\{|a,\lambda\rangle\}$ and rank-one projectors

$$P_{a\lambda} = |a,\lambda\rangle\langle a,\lambda|.$$

If the device distinguishes $\lambda$ but later reports only $a$, the operation for the coarse outcome is

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

The normalized state is

$$\rho_a = \frac{ \sum_\lambda P_{a\lambda}\rho P_{a\lambda} }{ \operatorname{Tr}(P_a\rho) }.$$

This differs from the Lüders state $P_a\rho P_a/\operatorname{Tr}(P_a\rho)$ whenever the input has coherence between different $\lambda$ states in the same eigenspace. Therefore “measure $A$” is not precise enough if degeneracy is present.

Kraus and Measurement-Operator Updates

A common generalized measurement model assigns one operator $M_m$ to each outcome. The effects are

$$F_m=M_m^\dagger M_m, \qquad \sum_m F_m=I.$$

The probability is

$$p(m) = \operatorname{Tr}(\rho F_m) = \operatorname{Tr}(M_m\rho M_m^\dagger),$$

and the conditional update is

$$\rho_m = \frac{M_m\rho M_m^\dagger} {\operatorname{Tr}(M_m\rho M_m^\dagger)}.$$

This formula is broader than projective measurement, but still not the most general instrument. An outcome may have several microscopic detector histories $\alpha$ grouped into one reported result:

$$\mathcal I_m(\rho) = \sum_\alpha M_{m\alpha}\rho M_{m\alpha}^\dagger.$$

Then

$$F_m = \sum_\alpha M_{m\alpha}^\dagger M_{m\alpha}, \qquad p(m)=\operatorname{Tr}(\rho F_m).$$

The effect $F_m$ gives the probability, while the full collection of $M_{m\alpha}$ gives the output state.

Same POVM, Different Updates

The reason effects do not determine updates is visible in one line. If $F_m=M_m^\dagger M_m$, then

$$N_m=U_mM_m$$

with a unitary $U_m$ gives the same effect:

$$N_m^\dagger N_m = M_m^\dagger U_m^\dagger U_mM_m = F_m.$$

The probabilities are unchanged. The conditional state can change:

$$\frac{N_m\rho N_m^\dagger} {\operatorname{Tr}(N_m\rho N_m^\dagger)} = U_m \left( \frac{M_m\rho M_m^\dagger} {\operatorname{Tr}(M_m\rho M_m^\dagger)} \right) U_m^\dagger.$$

Physically, this could mean the detector applies an outcome-dependent kick after registering the same outcome statistics. A POVM alone cannot tell whether that kick occurred.

Nonselective Channel

If all outcomes are ignored, the instrument becomes a channel:

$$\Phi(\rho) = \sum_m\mathcal I_m(\rho).$$

In a Kraus representation,

$$\Phi(\rho) = \sum_{m,\alpha} M_{m\alpha}\rho M_{m\alpha}^\dagger, \qquad \sum_{m,\alpha} M_{m\alpha}^\dagger M_{m\alpha}=I.$$

This is the update rule for an unread measurement, an unconditioned detector, or a measurement record that has been averaged over. It is also the bridge to open-system dynamics, where environmental degrees of freedom are ignored rather than read.

Do not normalize one term in the sum and call it nonselective. Normalizing a single trace-decreasing operation is postselection.

Bayesian Analogy and Its Limit

Classically, conditioning on evidence updates probabilities by Bayes’ rule:

$$p(h|m) = \frac{p(m|h)p(h)} {p(m)}.$$

Quantum selective update has a similar conditional structure: divide the unnormalized conditional state by the probability of the outcome. But the analogy is limited. Quantum measurements can disturb the state, remove coherence, or apply outcome-dependent transformations. The state update is not determined only by prior probabilities and likelihoods; it depends on the physical instrument.

The Bayesian analogy is safest when used as a guide to conditioning, not as a replacement for specifying the measurement operation. For the full comparison, see Bayesian Quantum Measurement.

Continuous-Time Preview

In continuous measurement, the update is applied over small time steps and conditioned on a measurement record. A schematic discrete step has the same structure:

$$\rho(t+dt) = \frac{\mathcal I_{dr}(\rho(t))} {\operatorname{Tr}\mathcal I_{dr}(\rho(t))},$$

where $dr$ denotes a small record increment. Taking a continuous-time limit can lead to stochastic master equations or quantum trajectories. If the record is averaged over, one recovers an unconditional master equation.

The conceptual distinction is already present here: conditioned states follow records; unconditioned states average over records.

Choosing an Update Rule

Before updating a state, ask:

1. What are the reported outcomes?
2. Is the outcome known, ignored, postselected, or coarse grained?
3. Is the measurement ideal projective, or does it require a generalized instrument?
4. If projective, is the measured outcome degenerate?
5. If degenerate, does the device preserve or resolve coherence inside the eigenspace?
6. Are no-click, loss, or failure outcomes included?
7. Is there outcome-dependent feedback or a detector kick?
8. Is the record continuous in time?

If the answer to any question is uncertain, the update rule is part of the model and should be stated explicitly.

Common Mistakes

• Treating the projection postulate as the update rule for every detector.
• Using a POVM effect as though it were a Kraus operator.
• Assuming a degenerate measurement always selects a basis vector.
• Erasing coherence inside a degenerate eigenspace when the Lüders update is intended.
• Using a nonselective channel for a postselected ensemble.
• Ignoring no-click outcomes and accidentally losing trace.
• Calling a Bayesian update complete without specifying measurement backaction.
• Reading microscopic meaning into a Kraus representation without checking representation nonuniqueness.

Cross-Links

• Projective Measurements
• Selective and Nonselective Measurements
• State Update Rule
• Degenerate Measurements and Lüders Rule
• POVMs: First Encounter
• Generalized Measurements Overview
• What the Postulates Do Not Say
• Bayesian Quantum Measurement
• Common Misconceptions

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• G. Lüders, “Über die Zustandsänderung durch den Messprozeß,” Annalen der Physik 8, 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, 1995.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.

Exercises

1. Let $P_0=|0\rangle\langle0|+|1\rangle\langle1|$ and $P_2=|2\rangle\langle2|$. For

$$|\psi\rangle = \frac{|0\rangle+|1\rangle+|2\rangle}{\sqrt3},$$

find the Lüders-updated state after outcome $0$.

Solution

The probability of outcome $0$ is

$$p(0) = \langle\psi|P_0|\psi\rangle = \frac{2}{3}.$$

The projected state is $(|0\rangle+|1\rangle)/\sqrt3$. Normalizing gives

$$|\psi_0\rangle = \frac{|0\rangle+|1\rangle}{\sqrt2}.$$

The coherence between $|0\rangle$ and $|1\rangle$ is preserved.

2. For the same input state, suppose the device distinguishes $|0\rangle$ and $|1\rangle$ but reports only the coarse outcome $0$. What conditional state results after outcome $0$?

Solution

The refined operation for coarse outcome $0$ is

$$\mathcal I_0(\rho) = |0\rangle\langle0|\rho|0\rangle\langle0| + |1\rangle\langle1|\rho|1\rangle\langle1|.$$

For the given state, after normalization by $p(0)=2/3$ this gives

$$\rho_0 = \frac{1}{2}|0\rangle\langle0| + \frac{1}{2}|1\rangle\langle1|.$$

This differs from the Lüders pure state because the refined measurement erased the coherence between $|0\rangle$ and $|1\rangle$.

3. Let $M_0=|0\rangle\langle0|$ and $N_0=X M_0$, where $X$ swaps $|0\rangle$ and $|1\rangle$. Do $M_0$ and $N_0$ define the same effect for outcome $0$? Do they give the same conditional output state?

Solution

They define the same effect:

$$N_0^\dagger N_0 = M_0^\dagger X^\dagger X M_0 = M_0^\dagger M_0 = |0\rangle\langle0|.$$

But the conditional output differs. The $M_0$ update outputs $|0\rangle\langle0|$ when outcome $0$ occurs, while the $N_0$ update outputs $|1\rangle\langle1|$. The same effect gives the same probability, not the same backaction.

4. A calculation uses $\rho_m=\mathcal I_m(\rho)/\operatorname{Tr}\mathcal I_m(\rho)$ and then averages over $m$ with equal weights. What is wrong unless all outcomes are equally probable?

Solution

The nonselective state is the probability-weighted average:

$$\rho' = \sum_m p(m)\rho_m, \qquad p(m)=\operatorname{Tr}\mathcal I_m(\rho).$$

Equal weights are correct only if the outcome probabilities are equal. Otherwise the calculation describes a different ensemble from the one produced by the measurement.