What Counts as a Measurement

A measurement is a physical procedure that couples a system to a distinguishable record and assigns that record a calibrated meaning. Quantum theory represents its statistics and state change through an instrument. Whether measurement is also a fundamental kind of event, rather than an effective description of ordinary interactions, depends on the physical account being proposed.

Correlation needs calibration and a readout

An interaction $U$ between a system and a ready apparatus can correlate system alternatives with apparatus states. To use it as a measurement, specify which apparatus observables are read, which labels they produce, and how those labels relate to the system preparation. Record stability and error rates determine whether the procedure is useful.

Entanglement is neither a universal definition nor a universal requirement for every measurement implementation. Some inputs produce product outputs even in an excellent detector, and a classical randomization can implement an uninformative POVM without acquiring information about the input. The instrument, not a slogan about entanglement, provides the reproducible description.

A continuous family from weak to sharp information

For a qubit, consider effects

$$E_\pm=\frac{I\pm\eta Z}{2},\qquad 0\le\eta\le1.$$

At $\eta=1$, the effects are sharp $Z$ projectors. At $\eta=0$, they both equal $I/2$ and the outcome is an input-independent fair bit. Intermediate values provide partial information about $Z$. If $r_z=\operatorname{Tr}(\rho Z)$, the outcome probabilities are $(1\pm\eta r_z)/2$.

For equally likely inputs $|0\rangle,|1\rangle$, guessing the input from the corresponding sign succeeds with probability $(1+\eta)/2$. This makes “measurement strength” operational in this example. It does not fix disturbance: choosing Kraus operators $\sqrt{E_\pm}$ is one instrument, and adding an outcome-dependent unitary gives another with the same information statistics.

Record access and conditioning

A record can exist in an unattended memory. If its label is unavailable, predictions for the remaining system use the nonselective channel. If it is received, predictions can be conditioned on that label. These are different uses of one physical procedure, not competing claims about whether a person has magically activated a detector.

Measurement Without Observers analyzes stored records. Collapse as Postulate distinguishes conditional update from an objective collapse law. The technical instrument page supplies the full formalism.

Exercise: information without a sharp observable

For $\eta=0.4$, find the sign probabilities for inputs $|0\rangle$ and $|+\rangle$. Can a single result identify an arbitrary input state?

Solution

For $|0\rangle$, $r_z=1$ and the probabilities are $0.7,0.3$. For $|+\rangle$, $r_z=0$ and both are $0.5$. One binary result cannot identify an arbitrary state; even repeated use only estimates the component to which this POVM is sensitive. Informational completeness requires a sufficiently rich measurement family.

References

• Busch, Paul, Pekka J. Lahti, and Peter Mittelstaedt, The Quantum Theory of Measurement, 2nd edition, Springer (1996), doi:10.1007/978-3-540-37205-9.
• 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.