Measurement and State Update

Measurement formalism connects a preparation to possible recorded outcomes, assigns probabilities to those outcomes, and specifies conditional state assignments for later predictions. In its simplest form, a sharp measurement is represented by orthogonal projectors; in the generalized theory, positive effects determine probabilities and quantum instruments determine state changes.

This chapter gives the rules needed for standard quantum-mechanical calculations. It does not turn those rules into a claim about ontology, derive the occurrence of one definite outcome from unitary dynamics alone, or replace a physical model of a detector.

Three Questions That Must Stay Separate

The chapter follows a strict editorial distinction:

1. Measurement formalism asks which outcomes are possible, how their probabilities are computed, and which state is assigned after an outcome or after an unread measurement.
2. The measurement problem asks how the formalism relates to definite macroscopic records and whether state update represents a physical process, information update, branching structure, or something else.
3. Detector physics models an apparatus, its coupling, amplification, noise, finite resolution, calibration, readout, and environment.

Core Formalism owns the first question. The historical measurement problem provides historical orientation; the dedicated foundations volume treats interpretations. Realistic measurement dynamics, decoherence, continuous monitoring, and instruments belong to Measurement and Open Quantum Systems.

What This Chapter Owns

This chapter develops

• the operational data of a quantum measurement;
• ideal projective measurements and the Born rule for their outcomes;
• selective and nonselective state updates;
• degenerate outcomes and the Lüders update;
• probabilities for sequential measurements and order effects;
• finite-dimensional measurements in chosen bases;
• measurement operators, effects, and generalized measurements;
• a first precise definition of POVMs;
• the boundary between calculational rules and foundational claims.

The Born rule itself remains canonical in Probability and the Born Rule. Detailed POVM, Kraus-operator, instrument, dilation, and continuous-measurement theory is developed in the open-systems volume.

Projective Measurement Data

A discrete projective measurement with outcome labels $a$ is specified by orthogonal projectors $\{P_a\}$ satisfying

$$P_aP_b = \delta_{ab}P_a, \qquad P_a^\dagger=P_a, \qquad \sum_aP_a=I.$$

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

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

For a pure state $\rho=\lvert\psi\rangle\langle\psi\rvert$, this becomes

$$p(a) = \langle\psi\rvert P_a\lvert\psi\rangle.$$

If the projectors are the spectral projectors of an observable,

$$A = \sum_a aP_a,$$

then the labels $a$ are values of that observable. A projective measurement can also be discussed directly through its projectors without first emphasizing a numerical observable.

The projectors specify ideal alternatives, not a detailed apparatus. The same projective measurement can be implemented by different physical couplings, and real devices can approximate the same ideal measurement with different efficiencies and disturbances.

Selective and Nonselective Updates

When outcome $a$ is recorded and $p(a)>0$, the ideal Lüders conditional state is

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

For a pure input state, a pure conditional state can be represented by

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

No normalized conditional state is defined for a zero-probability outcome. Dividing by $p(a)=0$ is not a meaningful limiting prescription without additional structure.

If the measurement occurs but its outcome is ignored, averaged over, or unavailable, the nonselective state is

$$\rho' = \sum_aP_a\rho P_a.$$

This map removes coherence between distinct measured eigenspaces while preserving coherence within each degenerate eigenspace. It is not the same state assignment as conditioning on a known outcome.

Ideal repeatability means that an immediate repetition of the same sharp measurement, with no intervening dynamics, returns the same recorded value with probability one. Repeatability is a property of the ideal update model, not a claim that every laboratory measurement is nondestructive.

Degeneracy and Lüders Rule

When $a$ is degenerate, $P_a$ projects onto a subspace rather than a single ray. The Lüders rule retains superpositions within that subspace. It is therefore the minimally refining ideal update associated with the coarse outcome $a$.

A more detailed apparatus may distinguish additional labels inside the eigenspace, even if the displayed readout is later coarse-grained to $a$. If $P_a$ is decomposed into finer projectors,

$$P_a = \sum_\lambda P_{a\lambda},$$

then a refinement can produce the conditional transformation

$$\rho_a^{\mathrm{ref}} = \frac{ \sum_\lambda P_{a\lambda}\rho P_{a\lambda} }{\operatorname{Tr}(\rho P_a)},$$

for $\operatorname{Tr}(\rho P_a)>0$. This can destroy coherence that the Lüders update preserves. The outcome probabilities alone do not determine which physical refinement occurred.

Sequential Measurements

Suppose a projective measurement $\{P_a\}$ is followed by $\{Q_b\}$. Under the Lüders model, the joint probability for outcomes $a$ then $b$ is

$$p(a,b) = \operatorname{Tr} \!\left( Q_bP_a\rho P_aQ_b \right).$$

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

$$p(b\mid a) = \operatorname{Tr}(\rho_aQ_b).$$

Reversing the order generally changes both the intermediate state and the joint distribution. For commuting spectral projectors, a common refinement gives compatible sharp outcomes and the ideal sequential probabilities become order independent. For noncommuting measurements, order effects are expected, but their exact form depends on the instruments, not solely on the POVM effects.

An unread intermediate measurement can change later statistics even when no outcome is selected, because the nonselective map can remove coherence. Omitting a record and omitting the physical interaction are different operations.

Measurements in a Chosen Basis

For an orthonormal measurement basis $\{\lvert e_j\rangle\}$,

$$P_j = \lvert e_j\rangle\langle e_j\rvert, \qquad p_j = \bigl\lvert\langle e_j\rvert\psi\rangle\bigr\rvert^2.$$

A coordinate change transforms state components and projectors together and leaves probabilities invariant. Choosing a different measurement basis changes the projectors and therefore changes the experiment. This distinction is central in qubit, spin, and quantum-information calculations.

The practical workflow is to identify the measured basis or eigenspaces, express the state and projectors in one consistent representation, compute probabilities, and apply an update only if a post-measurement prediction is requested.

Generalized Measurements and Instruments

A generalized measurement can be represented by measurement operators $M_{a\alpha}$, where $a$ is the recorded outcome and $\alpha$ labels unresolved alternatives. They satisfy

$$\sum_{a,\alpha} M_{a\alpha}^\dagger M_{a\alpha} = I.$$

The effect associated with outcome $a$ is

$$E_a = \sum_\alpha M_{a\alpha}^\dagger M_{a\alpha},$$

so the POVM conditions and probability rule are

$$E_a\ge0, \qquad \sum_aE_a=I, \qquad p(a)=\operatorname{Tr}(\rho E_a).$$

The normalized conditional state is

$$\rho_a = \frac{ \sum_\alpha M_{a\alpha}\rho M_{a\alpha}^\dagger }{p(a)}.$$

The effects $\{E_a\}$ determine outcome probabilities for every input state, but they do not uniquely determine the operators $M_{a\alpha}$ or the state update. A quantum instrument supplies the completely positive outcome maps and therefore both probabilities and conditional transformations. This distinction prevents a common mistake: a POVM is not, by itself, a complete disturbance model.

Page Map

Question	Canonical page	Main distinction
What does measurement mean inside the formalism?	Measurement in the Formalism	calculational rule versus physical mechanism
What is an ideal sharp measurement?	Projective Measurement	orthogonal alternatives versus apparatus model
Which state is assigned after an outcome?	State Update Rule	selective versus nonselective update
What changes for degenerate outcomes?	Degenerate Measurements and Lüders Rule	eigenspace projection versus refined measurement
How are measurements performed in order?	Sequential Measurements	conditional order versus one-time probability
How is a specified basis used?	Measurement in a Chosen Basis	changed coordinates versus changed measurement
Why go beyond projectors?	Generalized Measurements Overview	measurement operators versus projectors
What information does a POVM contain?	POVMs: First Encounter	outcome effects versus state-changing instrument
Which foundational questions remain open?	What Measurement Formalism Does Not Settle	predictive rules versus interpretation

These nine articles form the planned chapter.

Suggested Routes

First systematic pass

Read measurement in the formalism, projective measurement, state update, and the Lüders-rule article. Add sequential measurements before applying the formalism to multi-step experiments.

Finite-dimensional route

Read measurement in a chosen basis after the projective pages, then practice computational, Hadamard, and spin-axis measurements. Continue to qubits and quantum circuits in the quantum-information volume.

Generalized-measurement route

Read the overview and first POVM encounter, then continue to Why Generalized Measurements?, Kraus Operators, and quantum instruments.

Foundations-aware route

Read What Measurement Formalism Does Not Settle immediately after the core rules, then pair it with What Decoherence Does Not Solve.

Measurement Sanity Checks

• Verify positivity and completeness of projectors or POVM effects.
• Check that outcome probabilities are real, nonnegative, and normalized.
• Normalize a selective state only when the conditioning outcome has nonzero probability.
• Confirm that nonselective maps preserve trace and positivity.
• For degenerate outcomes, state whether the apparatus implements a Lüders update or a refinement.
• In a sequence, apply each update before computing the next conditional probability.
• Distinguish a basis-coordinate change from changing the measured projectors.
• Specify an instrument, not only a POVM, when post-measurement states or disturbance matter.

Common Mistakes

• Calling the probability rule a complete apparatus model. Projectors and effects omit coupling, noise, amplification, and readout physics.
• Using state update before an outcome is conditioned on. Selective and nonselective descriptions answer different questions.
• Normalizing a zero-probability branch. The formal conditional state is undefined there.
• Replacing a degenerate eigenspace by an arbitrary eigenvector. The ideal coarse outcome projects onto the whole subspace.
• Assuming Lüders update is the only implementation with the same outcomes. Refinements and other instruments can share outcome statistics.
• Ignoring an unread measurement. Its nonselective channel can alter later interference.
• Treating POVM effects as Kraus operators. Effects determine probabilities; instruments determine transformations.
• Equating formal state update with one settled ontology. The rule alone does not choose an interpretation.
• Claiming decoherence by itself selects one actual outcome. Decoherence explains suppression of interference in reduced descriptions, not a universally agreed outcome mechanism.

Cross-Links

• Probability and the Born Rule
• Observables and Operators
• Compatibility, Commutators, and Uncertainty
• Density Operators
• Measurement Theory
• POVMs
• Environment-Induced Decoherence
• POVM glossary entry

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• G. Lüders, “Über die Zustandsänderung durch den Meßprozeß,” Annalen der Physik 443, 322–328, 1950.
• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer, 1983.
• E. B. Davies, Quantum Theory of Open Systems, Academic Press, 1976.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• P. Busch, P. J. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016, doi:10.1007/978-3-319-43389-9.