Measurement as an Operation

A measurement is an operation performed on a quantum system. It has an experimental setting, a set of possible classical outcomes, probabilities for those outcomes, and usually a state transformation that matters for later predictions.

This operational view is broader and safer than the phrase “reading off a value.” Some measurements are sharp and nearly repeatable. Others are noisy, weak, destructive, inefficient, or deliberately coarse-grained. In all cases, a trustworthy model must say what probabilities are assigned and what state is left when an outcome is known or ignored.

Operational Data

A finite-outcome measurement model should specify:

• the input system and its Hilbert space;
• the apparatus setting or measurement context;
• the set of reported outcomes $m$;
• the probability rule for those outcomes;
• the state transformation associated with each outcome;
• the nonselective state transformation when the outcome is ignored;
• any apparatus, environment, or record degrees of freedom that are being modeled explicitly.

The compact way to encode this data is an instrument, a family of outcome-resolved operations $\{\mathcal I_m\}$. For an input state $\rho$,

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

is the probability of outcome $m$. If outcome $m$ is known and $p(m)$ is nonzero, the conditional output state is

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

If the measurement is performed but the outcome is ignored, the nonselective output is

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

This is the basic operational skeleton behind projective measurements, generalized measurements, detector backaction, and trajectory updates.

Probabilities and Effects

Every instrument has associated POVM effects $\{F_m\}$ such that

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

In a Kraus representation,

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

and the corresponding effect is

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

The effects satisfy

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

for a complete measurement with no discarded failure branch. The effects determine outcome probabilities, but they do not determine the state transformation. Different devices can have the same $\{F_m\}$ and different backaction.

For the full probability-only formalism, see POVMs. For the outcome-resolved state-update object, see Quantum Instruments.

Apparatus Picture

The operational maps often come from a larger unitary model. Let the system be $S$ and an apparatus or pointer be $A$. Suppose the apparatus begins in state $\eta_A$, interacts with the system through a unitary $U$, and is then read with pointer projectors $\{R_m\}$. An outcome operation has the schematic form

$$\begin{aligned} \mathcal I_m(\rho) = \operatorname{Tr}_A \Big[ &(I_S\otimes R_m) U(\rho\otimes\eta_A)U^\dagger \\ &\times (I_S\otimes R_m) \Big]. \end{aligned}$$

This formula says what is physically happening:

• the system and apparatus interact;
• the apparatus carries a record correlated with the system;
• reading the pointer selects one outcome branch;
• tracing over the apparatus leaves an operation on the system.

The details of the apparatus determine the instrument, not only the list of outcome labels. This is why two measurements with the same probabilities can leave different post-measurement states.

Projective Measurements as a Special Case

An ideal projective measurement with projectors $\{\Pi_m\}$ has

$$\mathcal I_m(\rho) = \Pi_m\rho\Pi_m.$$

The associated effect is

$$F_m = \Pi_m,$$

because $\Pi_m^\dagger\Pi_m=\Pi_m$. The selective state is

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

and the unread state is

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

This special case is powerful, but it is not the definition of measurement in general. A detector can be physically meaningful without being represented by orthogonal projectors on the measured system.

Nonselective Operation

The nonselective operation

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

is a quantum channel when the measurement is complete. It predicts the later state of the system for an observer who knows that the apparatus acted but does not condition on the outcome.

For projective measurement, $\Phi$ often removes coherences between measured subspaces. For a weak or inefficient measurement, it may only partially reduce coherence. For a destructive detector, the output Hilbert space may differ from the input Hilbert space, and the channel must be defined accordingly.

The key point is that unread measurements can still disturb the state. Ignoring the record is not the same as the measurement not happening.

Repeatability and Disturbance

An ideal repeatable measurement has the property that immediately repeating the same measurement gives the same outcome with probability one, assuming no intervening dynamics. Projective measurements of nondegenerate eigenstates have this property in the ideal model.

But repeatability is not automatic. A measurement can:

• reveal information while disturbing the system;
• be nondestructive for one observable and destructive for another;
• be weak enough that many repetitions are needed;
• be noisy, so the reported outcome is not a perfectly sharp property;
• measure an apparatus pointer rather than a pre-existing system value.

Backaction is therefore part of the model, not a nuisance to mention afterward. The detailed discussion is Measurement Backaction.

Simple Qubit Example

Consider a noisy two-outcome readout of a qubit in the $\sigma_z$ basis. One probability model is

$$F_0 = (1-\epsilon) \lvert0\rangle\langle0\rvert + \epsilon \lvert1\rangle\langle1\rvert,$$

and

$$F_1 = \epsilon \lvert0\rangle\langle0\rvert + (1-\epsilon) \lvert1\rangle\langle1\rvert, \qquad 0\le\epsilon\le\frac12.$$

These effects describe a detector that sometimes reports the wrong label. They determine the probabilities

$$p(0)=\operatorname{Tr}(F_0\rho), \qquad p(1)=\operatorname{Tr}(F_1\rho).$$

They do not, by themselves, say whether the readout strongly dephases the qubit, weakly dephases it, resets it, or destroys it. Those alternatives require an instrument. The probability model and the state-update model are separate pieces of operational data.

Measurement Is Not Interpretation

The operational formulas do not settle interpretive questions by themselves. They tell an agent how to assign probabilities and post-measurement states given a specified experimental arrangement and record.

For example, a von Neumann measurement model can show how a system becomes entangled with an apparatus pointer. A nonselective channel can show how local coherences disappear after the pointer or environment is ignored. These are physical statements about correlations and reduced states. Whether one describes the selective update as collapse, conditioning, branching, or information update depends on interpretive commitments outside the minimal formalism.

The site keeps those issues separate: measurement theory first specifies the operational object; interpretation pages discuss what, if anything, is inferred beyond that object.

Common Mistakes

• Treating a measurement as a passive look at a pre-existing classical value.
• Giving POVM effects but no instrument when later state predictions are needed.
• Assuming the unread state is the same as the input state.
• Forgetting that the apparatus setting is part of the measurement definition.
• Treating repeatability as automatic for every measurement.
• Using projective formulas for a noisy, weak, or inefficient detector.
• Confusing postselection with ordinary conditioning in an unbiased data set.

Exercises

Instrument Bookkeeping

Let an instrument have two operations

$$\mathcal I_0(\rho) = K_0\rho K_0^\dagger, \qquad \mathcal I_1(\rho) = K_1\rho K_1^\dagger,$$

with

$$K_0^\dagger K_0+K_1^\dagger K_1=I.$$

Write the outcome probabilities, selective states, and nonselective output.

Solution

The probabilities are

$$p(0) = \operatorname{Tr}(K_0\rho K_0^\dagger), \qquad p(1) = \operatorname{Tr}(K_1\rho K_1^\dagger).$$

When $p(m)$ is nonzero, the selective states are

$$\rho_0 = \frac{K_0\rho K_0^\dagger}{p(0)}, \qquad \rho_1 = \frac{K_1\rho K_1^\dagger}{p(1)}.$$

The nonselective output is

$$\rho' = K_0\rho K_0^\dagger + K_1\rho K_1^\dagger.$$

Same Probabilities, Different Backaction

Why can two devices with the same POVM effects produce different later measurement statistics?

Solution

The POVM effects determine only the probabilities of the current outcomes. Later statistics depend on the post-measurement state, which is determined by the instrument maps $\mathcal I_m$. Two instruments can have the same effects $F_m$ but different Kraus operators and therefore different output states.

Unread Does Not Mean Unperformed

An ideal $\sigma_z$ measurement is performed on a qubit, but the outcome is discarded. If the input state is $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$, what is the output state?

Solution

The input density operator is

$$\rho = \frac12 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}.$$

The unread projective measurement removes the off-diagonal terms in the $\sigma_z$ basis:

$$\rho' = \frac12 \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}.$$

Discarding the outcome does not undo the measurement interaction. The state used for later predictions is mixed.

Cross-Links

• Measurement Theory
• Projective Measurements
• Selective and Nonselective Measurements
• State Update Rules
• Von Neumann Measurement Model
• Measurement Backaction
• Repeatability and QND Measurement
• Compatible, Incompatible, and Sequential Measurements
• POVMs
• Kraus Operators
• Quantum Instruments
• Completely Positive Maps

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press (1955).
• K. Kraus, States, Effects, and Operations, Springer (1983).
• P. Busch, P. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, 2nd ed., 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).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).