Observables Properties or Measurement Procedures

A quantum observable specifies possible outcome statistics, while a measurement procedure also specifies a physical intervention and its state change. A result recorded in one procedure is not automatically a context-independent value that existed before every possible procedure. Keeping these statements separate connects operational measurement theory to foundational property claims.

An observable is not a complete apparatus model

For a sharp observable $A=\sum_a aP_a$, the spectral projectors give probabilities $p(a)=\operatorname{Tr}(\rho P_a)$. More general measurements use POVM effects. These objects describe outcome distributions for each input state. They do not, by themselves, specify the complete conditional transformation of the measured system.

An instrument provides completely positive maps $\mathcal I_a$ whose trace gives the outcome probability. Its output state conditioned on an outcome $a$ with $p(a)>0$ is $\mathcal I_a(\rho)/p(a)$. See Quantum Instruments for the general definition and examples. Two instruments can have the same effects and different postmeasurement states.

Same outcome, different later physics

Consider a qubit with projectors $P_0=|0\rangle\langle0|$ and $P_1=|1\rangle\langle1|$. One instrument uses $K_0=P_0,K_1=P_1$, retaining the measured eigenstate. Another uses

$$L_0=|+\rangle\langle0|,\qquad L_1=|+\rangle\langle1|.$$

Both have $K_a^\dagger K_a=L_a^\dagger L_a=P_a$, so every input gives the same zero/one probabilities. The second instrument resets the output to $|+\rangle$ regardless of the recorded label. A later $X$ measurement is therefore certain for it, whereas the first instrument produces equal $X$ probabilities conditional on either label.

The recorded label identifies which effect occurred. It need not mean that the output remains in the corresponding eigenspace. Repeatability is an additional instrument property, not a consequence of merely naming the measured observable.

When is a property attribution justified?

If a state is a sharp eigenstate and an ideal measurement is specified, the formalism assigns a certain outcome. Turning certainty into an ontological statement about a pre-existing property is a further interpretive step, highlighted by EPR. Assigning values simultaneously to all sharp observables while preserving their compatible functional relations meets the Kochen–Specker obstruction in dimension at least three.

This does not mean every use of property language is forbidden. A theory can assign selected properties, context-dependent values or additional variables. It must state which assignments are allowed and how they relate to actual procedures. The theorem excludes a particular global assignment rule, not meaningful physical description in general.

Exercise: repeatability test

Apply each instrument above to $|0\rangle$, then immediately measure $Z$ again using the first instrument. Compare the probability that the second label is zero.

Solution

The first instrument leaves $|0\rangle$, so the second zero probability is one. The reset instrument leaves $|+\rangle$, giving probability $1/2$. The first outcome probabilities were identical, but a sequential experiment distinguishes the implementations.

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.
• Spekkens, Robert W., “Contextuality for preparations, transformations, and unsharp measurements,” Physical Review A 71, 052108 (2005), doi:10.1103/PhysRevA.71.052108.