Generalized Measurements and Instruments

Generalized measurement theory describes laboratory measurements whose outcomes are noisy, inefficient, indirect, coarse-grained, weak, destructive, or otherwise not represented by an ideal projective measurement on the system alone. Its purpose is not to discard projective measurement, but to separate the operational questions that the projective idealization packages together.

The central distinction is:

$$\begin{gathered} \text{outcome statistics} \\ \text{do not determine} \\ \text{post-measurement dynamics}. \end{gathered}$$

A positive-operator-valued measure (POVM) determines the probability of each reported outcome. A quantum instrument also determines the conditional state left after that outcome. Kraus operators represent the corresponding state transformations, while dilation theorems explain how these effective descriptions arise from projective measurements or unitary dynamics on larger Hilbert spaces.

This chapter is the bridge from ideal measurement theory to Quantum Channels and Noise, continuous monitoring, decoherence, and open-system dynamics.

For a quantum-information task, Information-Theoretic Foundations first separates the source, state, channel or instrument, classical record, figure of merit, and resource restrictions. This chapter remains the canonical owner of POVMs, instruments, postmeasurement dynamics, and the distinction between information gained and disturbance induced.

The Operational Hierarchy

For a finite outcome set, the main mathematical objects answer different questions.

Object	Mathematical data	What it determines
PVM	orthogonal projectors $\{P_m\}$	sharp outcome probabilities and, with an update rule, an ideal projective measurement
POVM	positive effects $\{F_m\}$ with $\sum_mF_m=I$	outcome probabilities only
Kraus family	operators $\{K_{m\alpha}\}$	one representation of outcome-resolved state transformations
instrument	CP, trace-nonincreasing maps $\{\mathcal I_m\}$	probabilities and conditional output states
channel	one CPTP map $\Phi$	the unconditional state transformation
dilation	ancilla, isometry or unitary, and readout/discarding rule	a larger-system realization of the effective description

The objects fit together as

$$\begin{aligned} \mathcal I_m(\rho) &= \sum_\alpha K_{m\alpha}\rho K_{m\alpha}^\dagger, \\ F_m &= \sum_\alpha K_{m\alpha}^\dagger K_{m\alpha}, \\ p(m) &= \operatorname{Tr}[\mathcal I_m(\rho)] = \operatorname{Tr}(\rho F_m), \\ \rho_m &= \frac{\mathcal I_m(\rho)}{p(m)}, \qquad p(m)>0, \\ \Phi(\rho) &= \sum_m\mathcal I_m(\rho). \end{aligned}$$

An instrument is physically normalized when $\Phi=\sum_m\mathcal I_m$ is trace preserving. Equivalently,

$$\sum_{m,\alpha} K_{m\alpha}^\dagger K_{m\alpha} =I.$$

These equations are the chapter’s organizing map. The dedicated pages supply the definitions, proofs, examples, and qualifications that the compact display suppresses.

Why the Projective Idealization Is Not Enough

An ideal projective measurement uses projectors satisfying

$$P_mP_n=\delta_{mn}P_m, \qquad \sum_mP_m=I.$$

Many devices do not have this structure on the system Hilbert space.

• An inefficient photodetector has a no-click outcome that combines vacuum with missed photons.
• A finite-resolution detector bins many microscopic pointer values into one reported outcome.
• An indirect measurement couples the system to an ancilla and reads the ancilla rather than the system.
• An unsharp detector returns partial information rather than an eigenvalue with unit confidence.
• A weak measurement extracts little information per trial and accumulates evidence over many trials.
• A destructive detector may leave no state in the original system Hilbert space.
• State-discrimination measurements can require more outcomes than the Hilbert-space dimension.

All of these are ordinary quantum processes. Generalized measurement theory provides the bookkeeping needed to describe them without pretending that every detector is a noiseless projection.

Start with Why Generalized Measurements Are Needed if the physical motivation is more important than the operator hierarchy on a first pass.

Reading Path

Read this page	Use it for
Why Generalized Measurements Are Needed	Recognizing when a PVM omits inefficiency, coarse graining, indirect coupling, or realistic output dynamics.
POVMs	Computing the probabilities of general outcomes from positive effects.
Kraus Operators	Representing completely positive state transformations and outcome-resolved updates.
Quantum Instruments	Keeping outcome probabilities and conditional output states in one operational object.
Naimark Dilation	Realizing a POVM as a projective measurement on a larger space.
Stinespring Dilation	Realizing a completely positive map through larger-system evolution and discarded degrees of freedom.
Unsharp Measurements	Describing finite-strength, noisy, or partial-information effects.
Weak Measurements	Analyzing low information gain and small backaction per measurement step.
Weak Values	Interpreting postselected first-order pointer responses without treating them as ordinary eigenvalues.
Protective Measurements	Understanding adiabatic weak readout of expectation values in a protected state and its limitations.
Measurement Tomography	Reconstructing detector effects or instruments from calibrated probes and observed frequencies.

A compact first route is

$$\begin{gathered} \text{motivation} \longrightarrow \text{POVMs} \\ \longrightarrow \text{Kraus operators} \\ \longrightarrow \text{instruments}. \end{gathered}$$

Then choose dilation theory, weak measurement, or tomography according to the application.

POVMs Describe the Reported Statistics

A POVM is a family of effects $\{F_m\}$ satisfying

$$F_m\ge0, \qquad \sum_mF_m=I.$$

For a state $\rho$, the generalized Born rule is

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

Positivity gives $p(m)\ge0$, and completeness gives $\sum_mp(m)=1$. Each effect also obeys $F_m\le I$. A projective measurement is the special case in which every effect is a projector and distinct effects are orthogonal.

An effect is not generally a state-update operator. Writing $F_m\rho F_m$ as an update simply because $F_m$ appears in the probability rule is usually wrong. Even the positive square root $F_m^{1/2}$ specifies only one possible instrument, not a unique physical implementation.

Instruments Describe Outcomes and Backaction

An instrument is a collection of completely positive, trace-nonincreasing maps $\{\mathcal I_m\}$. It assigns an unnormalized conditional output $\mathcal I_m(\rho)$ to each outcome. Its trace is the outcome probability, and normalization gives the state conditioned on that outcome.

The Heisenberg-picture adjoint makes the associated POVM explicit:

$$F_m = \mathcal I_m^\dagger(I),$$

because

$$\operatorname{Tr}[\mathcal I_m(\rho)] = \operatorname{Tr} \left[ \rho\,\mathcal I_m^\dagger(I) \right].$$

The map $\Phi=\sum_m\mathcal I_m$ describes what remains when the apparatus acts but the outcome record is ignored. This nonselective channel can change the state even though no conditioning occurs.

For sequential experiments, the instrument rather than the POVM is indispensable. If outcome $m$ is followed by a second effect $G_n$, then

$$p(n,m) = \operatorname{Tr} \left[ G_n\,\mathcal I_m(\rho) \right].$$

The first POVM alone cannot determine this joint probability because it does not specify the state presented to the second device.

Same POVM, Different Backaction

Consider the sharp qubit effects

$$F_0=|0\rangle\langle0|, \qquad F_1=|1\rangle\langle1|.$$

The Lüders instrument is

$$\mathcal I_m^{\mathrm L}(\rho) = F_m\rho F_m.$$

Now define a measure-and-prepare instrument that reports the same outcomes but prepares $|+\rangle$ afterward:

$$\mathcal I_m^{\mathrm R}(\rho) = \operatorname{Tr}(F_m\rho) |+\rangle\langle+|.$$

Both instruments give

$$\operatorname{Tr}[\mathcal I_m^{\mathrm L}(\rho)] = \operatorname{Tr}[\mathcal I_m^{\mathrm R}(\rho)] = \operatorname{Tr}(F_m\rho),$$

so no experiment that records only the first outcome can distinguish them. Their output states are different, however. A later measurement can distinguish a repeatable projective readout from a detector that resets the qubit. This is the simplest concrete reason that a POVM is not a complete measurement model.

Kraus Representations Are Not Physical Labels

Every finite-dimensional completely positive operation admits a Kraus form. For an instrument outcome,

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

The index $m$ labels the reported classical outcome. The index $\alpha$ labels unresolved alternatives within that outcome. Only the map $\mathcal I_m$ is operationally fixed; its Kraus representation is not unique.

If two Kraus families for the same map have compatible lengths, they can be related by an isometry on the Kraus index. Consequently, one should not automatically interpret a particular Kraus label as a unique microscopic event. Such an interpretation requires an explicit apparatus or environment model that gives the label physical meaning.

The chapter separates the measurement-side use of Kraus operators from the channel representation theorem developed in Kraus Representation.

Naimark and Stinespring Answer Different Questions

Both dilation theorems enlarge the Hilbert space, but they organize different effective objects.

Dilation	Starts with	Larger-space description	Effective operation
Naimark	a POVM	a projective measurement after an isometric embedding	compress the projectors back to obtain effects
Stinespring	a completely positive map	an isometry, or a unitary with an initialized environment	discard the environment to obtain the map

For a POVM $\{F_m\}$, Naimark Dilation provides an isometry $V$ and projectors $\{\Pi_m\}$ on a larger space such that

$$F_m=V^\dagger\Pi_mV.$$

For a channel $\Phi$, Stinespring Dilation provides an environment and isometry $W$ such that

$$\Phi(\rho) = \operatorname{Tr}_E \left[ W\rho W^\dagger \right].$$

A dilation shows that the effective description is compatible with ordinary quantum mechanics on a larger system. It does not identify a unique apparatus, environment, or microscopic mechanism. Different dilations can implement the same POVM or channel.

Unsharp, Weak, and Protective Are Not Synonyms

These terms refer to different properties of a measurement model.

Term	Operational meaning
unsharp	the POVM effects are nonprojective and reveal partial or noisy information
weak	each measurement step has low signal-to-noise and small conditional backaction in the stated regime
protective	a long weak coupling is combined with a mechanism that suppresses transitions out of a protected state

For example, the qubit effects

$$F_\pm = \frac12 \left( I\pm\eta\sigma_z \right), \qquad 0\le\eta\le1,$$

interpolate between an uninformative two-outcome measurement at $\eta=0$ and a sharp $\sigma_z$ measurement at $\eta=1$. The parameter $\eta$ fixes the effects, but not the instrument. Extra unitary kicks or destructive reset dynamics can produce substantial disturbance even when $\eta$ is small.

Weak Measurements adds a controlled small-coupling or low-information regime. Weak Values then studies a postselected response coefficient, not an ordinary single-shot eigenvalue. Protective Measurements requires additional adiabatic and spectral assumptions; protection is not supplied by weakness alone.

Coarse Graining and Hidden Records

Suppose a detector has fine-grained outcomes $(m,\alpha)$ but reports only $m$. The coarse-grained instrument is

$$\mathcal I_m = \sum_\alpha \mathcal I_{m\alpha},$$

and its effect is

$$F_m = \sum_\alpha F_{m\alpha}.$$

This is more than relabeling. Conditioned states can become mixed because the unresolved record $\alpha$ has been discarded. Detector inefficiency, finite resolution, and hidden environment records often enter measurement models through precisely this kind of coarse graining.

The same logic leads toward open systems: a read record produces an instrument, while an ignored record produces a channel. Continuous monitoring replaces the discrete outcome $m$ by a time-dependent record and replaces the finite instrument by a stochastic state-update rule.

Detector Characterization

The forward problem predicts probabilities from a known state and known effects:

$$p(m|k) = \operatorname{Tr}(\rho_kF_m).$$

In Measurement Tomography, the calibrated probe states $\rho_k$ and measured frequencies are used to infer the unknown effects $F_m$. Positivity and completeness must be imposed or checked; unconstrained linear inversion can return nonphysical operators when data are noisy.

POVM tomography calibrates outcome statistics. Instrument tomography is a larger task because it must also reconstruct conditional output maps. State tomography, process tomography, and detector tomography therefore solve related but distinct inverse problems.

How to Choose the Right Object

Use the smallest object that answers the physical question, but do not use a smaller one than the question requires.

1. If only outcome probabilities are needed, specify a POVM.
2. If the post-measurement state or a later measurement matters, specify an instrument.
3. If no outcome is retained, use the nonselective channel.
4. If microscopic interpretation matters, provide an apparatus or system-environment dilation in addition to the effective map.
5. If the record is continuous in time, move to stochastic master equations and quantum trajectories.
6. If the detector is unknown, formulate a tomography problem with stated probe and calibration assumptions.

Always state the outcome set, input and output Hilbert spaces, normalization convention, whether the record is kept, and which degrees of freedom are discarded.

Canonical Boundaries

This chapter owns the graduate operational treatment of POVMs, instruments, dilations, weak measurement, and detector tomography. Nearby volumes and chapters have narrower canonical roles.

• Core Formalism owns the first encounter with generalized measurement as part of the postulates.
• Measurement Theory owns projective, selective, nonselective, repeatable, and sequential measurements at the apparatus-model level.
• Quantum Channels and Noise owns deterministic quantum operations, channel representations, and standard noise maps.
• Continuous Measurement and Quantum Trajectories owns time-resolved records, stochastic master equations, filtering, and trajectories.
• Resource Theories owns general resource claims about measurement and instrument objects; other Quantum Information pages own scalable tomography protocols, discrimination tasks, benchmarking, and information-processing applications.
• Foundations pages own interpretive debates about weak values, protective measurement, and what measurement experiments do or do not establish.

Cross-link to these canonical homes instead of reproducing their full derivations here.

Common Mistakes

• Treating a POVM effect as if it uniquely specified a state update.
• Assuming the square-root instrument is forced by the effects.
• Confusing a Kraus representation with a unique microscopic decomposition.
• Forgetting the no-click, failure, or loss outcome needed for normalization.
• Calling every nonprojective measurement weak.
• Assuming an unsharp detector is necessarily gentle.
• Reading a weak value as a pre-existing eigenvalue revealed in one trial.
• Treating a dilation as the uniquely real apparatus model.
• Ignoring changes of output Hilbert space in destructive measurements.
• Using POVM tomography when the experiment actually requires instrument tomography.
• Averaging over outcomes and then describing the result as a conditioned state.

Exercises

A two-outcome POVM

Let $F$ be an operator satisfying $0\le F\le I$. Show that

$$F_0=F, \qquad F_1=I-F$$

defines a two-outcome POVM. What extra condition would make it a PVM?

Solution

The assumed inequalities give $F_0\ge0$ and $F_1=I-F\ge0$. The effects sum to the identity, so they form a POVM.

For a two-outcome PVM, $F$ must also be a projector:

$$F^2=F.$$

Then $I-F$ is a projector and $F(I-F)=0$. A general effect can have eigenvalues strictly between zero and one and is therefore not projective.

Same statistics, different second measurement

For the two instruments $\mathcal I_m^{\mathrm L}$ and $\mathcal I_m^{\mathrm R}$ defined above, suppose the first outcome is $m=0$. Compute the probability that an immediate $\sigma_z$ measurement returns $0$.

Solution

The Lüders instrument leaves the conditional state $|0\rangle\langle0|$, so the second measurement returns $0$ with probability one.

The reset instrument leaves $|+\rangle\langle+|$, so

$$\Pr(0\mid m=0,\mathrm R) = |\langle0|+\rangle|^2 = \frac12.$$

The first-outcome POVM statistics agree for the two instruments, while the sequential statistics distinguish their backaction.

Coarse-grained outcomes

Fine-grained operations $\{\mathcal I_{m\alpha}\}$ form an instrument. Prove that $\mathcal I_m=\sum_\alpha\mathcal I_{m\alpha}$ is completely positive and trace nonincreasing, and identify its effect.

Solution

A sum of completely positive maps is completely positive. For every positive $\rho$,

$$\operatorname{Tr}[\mathcal I_m(\rho)] = \sum_\alpha \operatorname{Tr}[\mathcal I_{m\alpha}(\rho)] \le \operatorname{Tr}\rho,$$

because the sum over all reported outcomes and hidden alternatives is trace preserving. Thus each coarse-grained operation is trace nonincreasing. Its effect is

$$F_m = \mathcal I_m^\dagger(I) = \sum_\alpha \mathcal I_{m\alpha}^\dagger(I) = \sum_\alpha F_{m\alpha}.$$

Recovering effects from a dilation

Let $V:\mathcal H_S\to\mathcal K$ be an isometry and let $\{\Pi_m\}$ be projectors on $\mathcal K$ that sum to $I_{\mathcal K}$. Show that $F_m=V^\dagger\Pi_mV$ is a POVM on $\mathcal H_S$.

Solution

For every $|\psi\rangle\in\mathcal H_S$,

$$\langle\psi|F_m|\psi\rangle = \langle V\psi|\Pi_m|V\psi\rangle \ge0,$$

so each $F_m$ is positive. Completeness follows from the projector resolution and the isometry condition:

$$\begin{aligned} \sum_mF_m &= V^\dagger \left(\sum_m\Pi_m\right)V, \\ &=V^\dagger V, \\ &=I_S. \end{aligned}$$

Choosing the model

For each task, identify the minimum adequate object: (a) predict click frequencies, (b) predict a later measurement conditioned on a click, (c) predict the state when the click record is discarded, and (d) explain how an ancilla detector realizes the measurement.

Solution

(a) A POVM is sufficient because only outcome probabilities are requested.

(b) An instrument is required because the conditional output state affects the later measurement.

(c) The nonselective channel $\Phi=\sum_m\mathcal I_m$ is the relevant object.

(d) An explicit dilation or apparatus model is required in addition to the effective POVM or instrument.

Cross-Links

• Measurement Theory
• Why Generalized Measurements Are Needed
• POVMs
• Kraus Operators
• Quantum Instruments
• Naimark Dilation
• Stinespring Dilation
• Unsharp Measurements
• Weak Measurements
• Weak Values
• Protective Measurements
• Measurement Tomography
• Quantum Operations
• Completely Positive Maps
• Continuous Monitoring
• Generalized Measurements Overview
• POVMs, First Encounter

References

• E. B. Davies and J. T. Lewis, “An operational approach to quantum probability,” Communications in Mathematical Physics 17, 239–260 (1970).
• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer (1983).
• A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Edizioni della Normale (2011).
• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer (2016).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2010).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).