Observables as PVMs

A projection-valued measure specifies sharp outcome probabilities, but it does not select a state-update instrument. Even repeatability generally leaves freedom inside degenerate eigenspaces. The canonical Projective Measurements article supplies the physical framework; this page characterizes that remaining freedom and explains how event additivity constrains a fixed instrument.

Helpful background. Projection-valued measures introduces the measure axioms, spectral domains, and continuous-outcome examples. Instruments supplies event operations, while Kraus representations supplies their countable operator sums.

A sharp observable still permits many instruments

Consider a discrete PVM $(P_i)$ on a separable Hilbert space, with $\sum_iP_i=I$ strongly. An instrument implementing it has operations $\mathcal I_i$ satisfying

$$\mathcal I_i^*(I)=P_i, \qquad p_\rho(i)=\operatorname{Tr}(\rho P_i).$$

The Lüders choice is $\mathcal L_i(\rho)=P_i\rho P_i$. But for any unitaries $U_i$, the operations

$$\mathcal I_i(\rho)=U_iP_i\rho P_iU_i^*$$

have exactly the same effects and probabilities. An outcome can therefore be statistically sharp while its apparatus rotates the selected subspace afterward. If the unitary carries that subspace elsewhere, immediate repetition need not return the same outcome.

An even stronger change is $\mathcal I_i(\rho)=\operatorname{Tr}(\rho P_i)\tau_i$, where each $\tau_i$ is a chosen output state. This device reads the sharp event and then prepares a state. The PVM cannot distinguish it from the Lüders instrument using first-measurement statistics alone.

Repeatability as a support condition

Assume the output Hilbert space is the same as the input, so the same PVM can be measured again. The first instrument is repeatable for its outcome labels when

$$\operatorname{Tr}\bigl(P_i\mathcal I_i(\rho)\bigr) =\operatorname{Tr}\mathcal I_i(\rho)$$

for every input state. The equality includes impossible outcomes through their zero unnormalized output. For positive outputs it is equivalent to

$$\mathcal I_i(\rho)=P_i\mathcal I_i(\rho)P_i.$$

Indeed a positive operator with zero trace against $I-P_i$ has support in $P_i\mathcal H$. This is a statement about the instrument’s range, not merely about its probability effect.

Choose Kraus operators $K_{i\alpha}$ for the branch, with $\sum_\alpha K_{i\alpha}^*K_{i\alpha}=P_i$. If $\phi\in\ker P_i$, then

$$\sum_\alpha\|K_{i\alpha}\phi\|^2=0,$$

so every Kraus operator satisfies $K_{i\alpha}=K_{i\alpha}P_i$. Repeatability additionally forces $(I-P_i)K_{i\alpha}=0$: apply the output-support condition to pure inputs and use nonnegativity of each squared norm. Thus the exact criterion is

$$K_{i\alpha}=P_iK_{i\alpha}P_i \quad\text{for every }\alpha.$$

Conversely this condition guarantees repeatability. A repeatable instrument acts internally within the selected eigenspace, and arbitrary trace-preserving dynamics on that eigenspace remains possible.

Nondegenerate and degenerate outcomes differ

If $P_i$ has rank one, the support criterion gives $K_{i\alpha}=c_{i\alpha}P_i$. Completeness forces $\sum_\alpha|c_{i\alpha}|^2=1$, so the branch operation is uniquely $P_i\rho P_i$. Thus a rank-one sharp observable has a unique repeatable instrument on the same system, although it has many nonrepeatable implementations.

If $P_i$ has higher rank, repeatability does not imply the Lüders rule. One may rotate within its range, dephase an internal basis, or replace the input by a fixed state supported there. All these choices return label $i$ on repetition but can disagree on every later measurement sensitive to internal coherence.

The distinction explains why an eigenvalue alone does not specify a pure post-measurement state when its eigenspace is degenerate.

Event additivity constrains a fixed instrument

An instrument must add operations for disjoint events. For a fixed discrete Lüders instrument, its event map is therefore

$$\mathcal I(\Delta)(\rho)=\sum_{i\in\Delta}P_i\rho P_i,$$

with trace-norm convergence. For positive $\rho$, the summands are positive and their traces sum to at most one; the trace norm of the positive tail is exactly its omitted probability. Extension to arbitrary trace-class inputs follows by their decomposition into four positive parts.

By contrast, assigning $\Delta\mapsto P(\Delta)\rho P(\Delta)$ on every event is generally not additive. The coarse-versus-refined measurement example demonstrates the surviving cross terms and computes the different conditional states. The coarse Lüders operation belongs to a different instrument, selected for that coarse observable; it cannot replace all event operations of the original fine instrument simultaneously.

In particular, the total operation $\mathcal I(\Omega)$ of a sharp Lüders measurement is generally dephasing, even though its total effect is $P(\Omega)=I$. Normalization requires trace preservation of the total operation, not identity action on every input state. Confusing these two assertions would erase the physical distinction between measuring and doing nothing.

Continuous outcomes do not supply singleton updates

A continuous PVM can have $P(\{x\})=0$ for every point. Dividing $P(\{x\})\rho P(\{x\})$ by its zero probability does not define a state. Nor does replacing this projector by a generalized eigenket solve the normalization or instrument-additivity problem.

A continuous measurement requires a specified instrument. Conditional state densities, where constructed, are defined almost everywhere relative to the input outcome law. Finite detector events remain ordinary events, but their conditional output still depends on the chosen instrument rather than on the PVM alone.

Exercises

1. Let a degenerate sharp outcome have projection $P$ and choose a density operator $\tau$ supported in its range. Is the branch $\rho\mapsto\operatorname{Tr}(\rho P)\tau$ repeatable?

Solution

Yes. Its effect is $P$, and every nonzero output is supported in $P\mathcal H$. It is generally different from the Lüders branch because it erases the input’s internal information. This supplies an explicit alternative within projective measurement models.

2. For the qubit $\sigma_z$ PVM, take branch Kraus operators $K_0=|1\rangle\langle0|$ and $K_1=|0\rangle\langle1|$. Is the measurement sharp? Is it repeatable?

Solution

Its effects are $K_i^*K_i=|i\rangle\langle i|$, so the first measurement has exactly the sharp $\sigma_z$ probabilities. Conditional on label $i$, the output is the opposite basis state. Immediate repetition gives $1-i$ with certainty, violating the range condition $K_i=P_iK_iP_i$. Sharp statistics alone do not imply repeatability.

References

• Busch, Paul, Pekka Lahti, Juha-Pekka Pellonpää, and Kari Ylinen, Quantum Measurement, Springer, 2016 — sharp observables, repeatability, and instrument structure. Publisher.
• Heinosaari, Teiko, and Mário Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement, Cambridge University Press, 2012, Chapter 5 — measurement models and conditional transformations. Publisher.