Measurement and State Update

A measurement calculation has two parts: the probabilities of the recorded outcomes, and the state used for later predictions. Knowing one does not always determine the other. This chapter begins with ideal projective measurements and then explains why more general measurements require both effects and state-changing maps.

An outcome and an unread measurement

Let $P_a$ be the projectors of a discrete sharp measurement. In the ideal Lüders model,

$$p(a)=\operatorname{Tr}(\rho P_a),\qquad \rho_a=\frac{P_a\rho P_a}{p(a)}\quad\text{if }p(a)>0.$$

$\rho_a$ predicts later observations in the subensemble with the specified outcome. When the measurement occurs but the outcome is ignored, use

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

These are different questions. Prepare $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ and measure in the $0,1$ basis. Selecting outcome $0$ gives $\lvert0\rangle$. Ignoring the outcome gives the equal mixture of $0$ and $1$. That mixture gives a later $+$ result with probability $1/2$, whereas omitting the intermediate measurement altogether gives $+$ with certainty.

The Lüders model preserves coherence within a degenerate measured eigenspace. A device that resolves additional information can destroy that coherence while displaying the same coarse outcome. More generally, a POVM fixes outcome probabilities; a quantum instrument also fixes the update.

Reading sequence

Question	Article	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

From ideal rules to physical measurements

Begin with projective measurement, state update, and degeneracy, then calculate a sequence of two measurements. Use the generalized-measurement articles when projectors do not describe the available outcomes.

For coupling to an apparatus, noise, and continuous monitoring, continue to measurement theory. For the relation between these predictive rules and definite macroscopic outcomes, read what the formalism does not settle.

Try it. In the example above, condition instead on outcome $1$. What is the probability of a later $+$ result?

Solution

It is $\lvert\langle+\rvert1\rangle\rvert^2=1/2$, just as for outcome $0$. Averaging these two conditional predictions gives $1/2$ for the unread measurement. The phase coherence needed for certainty has been removed.

References

• P. Busch, P. J. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016, doi:10.1007/978-3-319-43389-9.
• E. B. Davies, Quantum Theory of Open Systems, Academic Press, 1976.
• K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer, 1983.
• G. Lüders, “Über die Zustandsänderung durch den Meßprozeß,” Annalen der Physik 8, 322–328, 1951. Translation and discussion by K. A. Kirkpatrick.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.