Collapse as Postulate

A collapse postulate tells a user of the theory how to assign a state after learning a measurement outcome. Interpreting that update as an objective physical process is a further claim. Keeping the two uses separate makes textbook calculations precise and exposes what a dynamical-collapse proposal must add.

Helpful background. Born Rule defines outcome probabilities.

Conditional update for a sharp measurement

For an ideal Lüders measurement with projectors $P_a$, the probability and conditional state are

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

The rule predicts later statistics conditional on an obtained label. For a degenerate projector it preserves coherence inside the corresponding subspace. Measuring a finer observable and discarding part of its result can instead destroy that coherence, even if the coarse outcome probabilities agree. A measurement observable alone does not determine every possible state update.

The instrument treatment supplies the general completely positive maps $\mathcal I_a$, with $p(a)=\operatorname{Tr}\mathcal I_a(\rho)$ and normalized conditional state $\mathcal I_a(\rho)/p(a)$. The postulate above is one important special case.

Conditioning and ignoring an outcome

If the result is not retained, the ensemble output is

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

For an input $|+\rangle$ and a $Z$ measurement, the conditional states are $|0\rangle$ or $|1\rangle$, while the nonselective state is $I/2$. No observer who lacks the result may consistently substitute a chosen conditional branch for that ensemble when predicting accessible frequencies.

Classical conditioning also changes an information state when data arrive. Quantum measurement can additionally disturb later statistics: measuring $Z$ and discarding its result changes a subsequent $X$ measurement on $|+\rangle$ from certainty to equal probabilities. Calling collapse an information update does not erase this physical disturbance, which belongs to the measurement instrument.

What an objective collapse law must specify

If collapse is a physical law applying without measurement terminology, one must state when it occurs, which operators or spatial variables it acts on, its stochastic probabilities, and its behavior in composite systems. Parameters and conservation-law consequences become experimentally relevant. The objective-collapse overview and tests address those models.

An instruction to collapse “when observed” is adequate only if observation is already a defined operational boundary for the calculation. It is not a microscopic law until the boundary has a physical criterion. Conversely, a useful operational update rule need not pretend to settle ontology in order to predict conditional data correctly.

Exercise: coarse and fine measurements

In three dimensions let $P=|0\rangle\langle0|+|1\rangle\langle1|$ and prepare $(|0\rangle+|1\rangle)/\sqrt2$. Compare a Lüders measurement of $\{P,I-P\}$ with a computational-basis measurement whose zero and one labels are later grouped together.

Solution

Both produce the coarse result $P$ with certainty. The Lüders update leaves the original coherent vector unchanged. The fine measurement followed by discarded labels produces $(|0\rangle\langle0|+|1\rangle\langle1|)/2$. A later interference measurement inside the two-dimensional subspace distinguishes them. Identical effects do not imply identical instruments.

References

• Davies, E. B., and J. T. Lewis, “An operational approach to quantum probability,” Communications in Mathematical Physics 17, 239–260 (1970), doi:10.1007/BF01647093 — instruments and conditional predictions.
• Lüders, Gerhart, “Concerning the state-change due to the measurement process,” Annalen der Physik 15, 663–670 (2006), translation of the 1951 paper, doi:10.1002/andp.200610207.