Measurement Theory

Measurement theory describes how a physical interaction between a system, an apparatus, and a record is represented in quantum mechanics. It is not a theory of human observation. It is the operational layer that connects states and observables to outcome probabilities, conditional state assignments, unread measurements, and measurement-induced disturbance.

This chapter focuses on ideal and near-ideal measurement structure: projective measurements, selective versus nonselective updates, Lüders rules, explicit von Neumann measurement models, and backaction. Generalized measurement theory continues in Generalized Measurements and Instruments, with Quantum Instruments as the canonical outcome-and-update formalism.

Central Distinction

The core distinction is:

$$\text{outcome probabilities} \quad \text{are not the same data as} \quad \text{post-measurement states}.$$

For an outcome-resolved operation $\mathcal I_m$, the probability of outcome $m$ is

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

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

$$\rho_m = \frac{\mathcal I_m(\rho)}{p(m)}.$$

If the apparatus acts but the outcome is ignored, the nonselective output is

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

Projective measurement is the sharp ideal case of this structure. POVMs, Kraus updates, and continuous monitoring generalize it.

Reading Path

Read This Page	Use It For
Measurement as an Operation	Separating outcome sets, probabilities, instruments, records, and state assignment.
Projective Measurements	Sharp measurements represented by orthogonal projectors, including selective and nonselective forms.
Degenerate Measurements	Understanding eigenspace-valued outcomes, Lüders updates, and refined apparatuses.
Selective and Nonselective Measurements	Distinguishing “we know the outcome” from “the measurement happened but the outcome was ignored.”
State Update Rules	Comparing Lüders updates, refined projective updates, Kraus updates, instruments, and continuous-record updates.
Lüders Rule	The standard ideal update for degenerate projective measurements.
Von Neumann Measurement Model	How system-apparatus entanglement produces pointer correlations and motivates state update.
Measurement Backaction	Why information gain and disturbance are linked, and when backaction is basis dependent.
Repeatability and QND Measurement	When repeated outcomes are stable and when monitoring can avoid demolishing a selected observable.
Compatible, Incompatible, and Sequential Measurements	How ordered records, compatibility, and measurement disturbance determine later probabilities.

The compact postulate-level treatment of ordered projective measurements remains in Sequential Measurements in Core Formalism. This chapter owns the instrument-level and disturbance-focused treatment.

Projective Starting Point

An ideal discrete projective measurement is specified by projectors $\{\Pi_m\}$ satisfying

$$\Pi_m\Pi_n = \delta_{mn}\Pi_m, \qquad \sum_m\Pi_m=I.$$

For state $\rho$, the outcome probability is

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

The ideal selective Lüders update is

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

The nonselective, unread measurement is

$$\rho' = \sum_m \Pi_m\rho\Pi_m.$$

This unread measurement is a channel: it can destroy coherences between different measured subspaces even when no outcome is retained.

Selective, Nonselective, and Conditioned

The words are easy to blur, so this chapter uses them carefully.

Term	Meaning
selective measurement	The outcome is known and the state is conditioned on that outcome.
nonselective measurement	The apparatus acts, but the outcome is ignored or averaged over.
conditioned state	A state assigned using a particular outcome or measurement record.
unconditional state	A state after averaging over outcomes or discarding the record.
postselection	Keeping only runs with a specified outcome, often changing normalization and sampling bias.

For a projective measurement, the nonselective map is often a dephasing operation in the measurement basis. It is not the same thing as learning a definite value and then forgetting it as a classical story. The mathematical object is the averaged quantum operation.

Backaction and Models

Measurement backaction is the change in the system state caused by the measurement interaction and the subsequent conditioning or averaging. In ideal projective measurements, the backaction is represented directly by projection. In a more physical model, the system first becomes correlated with an apparatus pointer.

A schematic von Neumann measurement correlates system eigenstates with pointer states:

$$\lvert a\rangle\lvert A_0\rangle \longmapsto \lvert a\rangle\lvert A_a\rangle.$$

For a superposition, linearity gives entanglement:

$$\left( \sum_a c_a\lvert a\rangle \right) \lvert A_0\rangle \longmapsto \sum_a c_a \lvert a\rangle\lvert A_a\rangle.$$

The update rule depends on which pointer information is read, ignored, coarse-grained, or lost into an environment. That is why measurement theory naturally leads into instruments, channels, decoherence, and quantum trajectories.

Boundary with Generalized Measurements

Projective measurements are not the only physical measurements. A realistic detector can be inefficient, noisy, weak, coarse-grained, or destructive. Then outcome probabilities are usually described by POVM effects $\{F_m\}$:

$$F_m\ge0, \qquad \sum_mF_m=I, \qquad p(m)=\operatorname{Tr}(F_m\rho).$$

But the effects alone do not determine the post-measurement state. That is the role of an instrument. This chapter prepares the projective and operational ideas; the next chapter develops POVMs, Kraus Operators, and Quantum Instruments.

Common Decisions

When facing a measurement problem, ask:

1. What is the outcome set?
2. Is the measurement ideal projective, generalized, weak, continuous, or only phenomenological?
3. Are you keeping a particular outcome, discarding the outcome, or conditioning on a whole time record?
4. Does the problem ask only for probabilities, or also for the post-measurement state?
5. Is degeneracy present, and if so, does the apparatus resolve the degeneracy?
6. Are you modeling the apparatus explicitly or using an effective update rule?

These decisions determine whether projectors, POVM effects, measurement operators, instruments, channels, or stochastic master equations are the right object.

Common Mistakes

• Treating every measurement as projective.
• Computing outcome probabilities from a POVM and then inventing a state update not specified by an instrument.
• Forgetting that unread measurements can still disturb the state.
• Treating degeneracy as harmless when the apparatus may resolve more information than the reported eigenvalue.
• Confusing postselection with ordinary conditioning in an unbiased ensemble.
• Reading a measurement-update formula as an interpretation of what “really happened” rather than as an operational state-assignment rule.

Exercises

Unread Qubit Measurement

A qubit state has density matrix

$$\rho = \begin{pmatrix} a&c\\ c^*&b \end{pmatrix}$$

in the $\sigma_z$ basis. What is the nonselective output after an ideal $\sigma_z$ measurement?

Solution

The projectors are

$$\Pi_0 = \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix}, \qquad \Pi_1 = \begin{pmatrix} 0&0\\ 0&1 \end{pmatrix}.$$

The nonselective state is

$$\rho' = \Pi_0\rho\Pi_0+\Pi_1\rho\Pi_1 = \begin{pmatrix} a&0\\ 0&b \end{pmatrix}.$$

The populations are preserved, while the coherence in this basis is removed.

Probability Versus State Update

Why does a POVM effect $F_m$ not determine the post-measurement state?

Solution

The effect determines only the probability

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

Different instruments can have the same effect $F_m$ but produce different output states. The post-measurement state requires an outcome-resolved operation $\mathcal I_m$, not just the probability effect.

Degeneracy Question

An observable has a degenerate eigenspace. Why is it not enough to know the eigenvalue outcome when predicting the post-measurement state?

Solution

The same eigenvalue can correspond to several orthogonal states inside one eigenspace. An ideal Lüders measurement preserves coherence inside that eigenspace, while a refined apparatus may measure additional degrees of freedom and destroy those coherences. The reported eigenvalue alone does not specify which instrument the apparatus implemented.

Cross-Links

• Notation and Conventions
• Projective Measurements
• Selective and Nonselective Measurements
• State Update Rules
• Lüders Rule
• Von Neumann Measurement Model
• Measurement Backaction
• Repeatability and QND Measurement
• Compatible, Incompatible, and Sequential Measurements
• POVMs
• Quantum Instruments
• Common Misconceptions

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press (1955).
• G. Lüders, “Concerning the state-change due to the measurement process,” Annalen der Physik 15, 663-670 (2006 English translation of 1951 article).
• K. Kraus, States, Effects, and Operations, Springer (1983).
• P. Busch, P. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, 2nd ed., Springer (1996).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010).