Measurement Theory

Measurement theory connects an apparatus interaction to outcome probabilities and to the state used for later predictions. This chapter begins with sharp measurements, then asks what changes when an outcome is ignored, a degenerate eigenspace is resolved, or a second measurement follows the first. The essential habit is to specify both what the detector reports and what it does to the system.

Required background. The volume introduction and One Interaction, Different Measurement Records distinguish probabilities, conditional states, and channels. Conventions fixes the notation used below.

A sharp outcome is not a complete apparatus model

For projectors $P_r$ with $P_rP_s=\delta_{rs}P_r$ and $\sum_rP_r=I$, the Born rule gives $p_r=\operatorname{Tr}(P_r\rho)$. An ideal Lüders instrument further specifies

$$\begin{gathered} \mathcal I_r(\rho)=P_r\rho P_r,\\ \rho_r=\frac{P_r\rho P_r}{p_r} \quad(p_r>0). \end{gathered}$$

It preserves coherence inside the selected eigenspace while removing components outside it. A device that resolves more detail and later hides part of its readout can have the same reported probabilities but a different output state.

For a concrete example, take $A=\operatorname{diag}(0,0,1)$ and prepare $(|0\rangle+|1\rangle)/\sqrt2$. An ideal Lüders measurement of $A$ certainly reports zero and leaves this state unchanged. A device that first distinguishes $|0\rangle$ from $|1\rangle$, then reports only their common eigenvalue, also certainly reports zero. Its output is instead the equal mixture of those states. The first readout cannot distinguish the devices; a later interference measurement can.

Build the measurement description in stages

Pages	What to learn
Measurement as an Operation and Projective Measurements	Specify outcomes and operations, then calculate sharp-measurement probabilities.
Lüders Rule	Determine which coherences survive within an eigenspace.
Selective and Nonselective Measurements and State Update Rules	Normalize a selected result or average the outcome-resolved operations.
Von Neumann Measurement Model and Measurement Backaction	Relate pointer separation and record resolution to the state change.
Repeatability and QND Measurement and Sequential Measurements	Predict repeated and ordered outcomes, including intervening dynamics.

For a first reading, follow operations → projective measurements → selective versus nonselective updates → pointer model. Return to degeneracy and sequential measurements when the distinction between a recorded outcome and its associated operation is comfortable.

A second measurement reveals backaction

Suppose the first apparatus implements $\mathcal I_r$ and the second has effect $F_s$. The joint probability is

$$p(r,s)=\operatorname{Tr}\!\left[F_s\mathcal I_r(\rho)\right].$$

There is no extra division by $p_r$ in this joint probability: the unnormalized operation already contains it. Divide by $p_r$ only to find the conditional probability $p(s\mid r)$. This formula also explains why the first POVM alone is insufficient for a sequential experiment.

An unread measurement averages these operations, $\rho'=\sum_r\mathcal I_r(\rho)$. It can change subsequent statistics even when its outcome is unavailable. For example, an unread $Z$ measurement turns a qubit $|+\rangle$ into $I/2$, making a later $X$ measurement random.

Repeatability has a physical timescale

Immediate repeatability concerns whether a repeated measurement returns the same outcome. A nondemolition measurement must also preserve the relevant observable through the measurement coupling and the intervening dynamics in the stated model. A projective readout alone does not guarantee that its result stays fixed while a Hamiltonian rotates the state.

Likewise, small pointer resolution and weak coupling are physical properties of a model, not consequences of merely naming an observable. When noisy, inefficient, weak, or destructive readout matters, continue to Generalized Measurements and Instruments.

Exercises

Resolve a degeneracy. In the three-level example, measure the projector onto $(|0\rangle+|1\rangle)/\sqrt2$ after the first apparatus. What is the probability for each implementation?

Solution

The Lüders implementation leaves the initial state unchanged, so the probability is one. The refined implementation leaves $(|0\rangle\langle0|+|1\rangle\langle1|)/2$, giving probability $1/2$. Reporting the same eigenvalue did not specify the same state change.

Joint versus conditional probabilities. A qubit starts in $|+\rangle$. Perform an ideal $Z$ measurement and then an ideal $X$ measurement. Find $p(Z=+1,X=+1)$ and $p(X=+1\mid Z=+1)$.

Solution

The first event has probability $1/2$ and leaves $|0\rangle$. The second then has conditional probability $1/2$. Thus the joint probability is $1/4$, while the conditional probability is $1/2$. The other three joint outcomes also have probability $1/4$.

References

• Busch, P., P. J. Lahti, and P. Mittelstaedt. The Quantum Theory of Measurement. 2nd edition, Springer, 1996. Sharp measurements, repeatability, and apparatus models.
• Kraus, K. States, Effects, and Operations: Fundamental Notions of Quantum Theory. Springer, 1983. Operations and sequential probabilities.
• Nielsen, M. A., and I. L. Chuang. Quantum Computation and Quantum Information. 10th anniversary edition, Cambridge University Press, 2010. Projective and generalized measurement examples.