Strong Commutation and Compatible Observables

Two sharp observables are compatible when their outcome statistics can be marginals of one joint measurement for every state. For self-adjoint operators, the precise criterion is commutation of all spectral projections, called strong commutation. A formal equation $[A,B]=0$ on a convenient dense test domain is not sufficient.

Helpful background. Projection-Valued Measures defines sharp observables. Adjoints of Unbounded Operators explains why the full domain matters.

Strong commutation compares bounded events

Self-adjoint $A$ and $B$ strongly commute if

$$E_A(\Delta)E_B(\Gamma)=E_B(\Gamma)E_A(\Delta)$$

for every pair of Borel sets. These bounded projections act on all vectors, so the condition avoids undefined products of unbounded operators.

Equivalent formulations include commutation of all bounded Borel functions of $A$ and $B$, and

$$e^{itA}e^{isB}=e^{isB}e^{itA} \qquad(s,t\in\mathbb R).$$

One direction follows immediately from bounded functional calculus. For the converse, spectral measures are determined by their Fourier transforms; commutation of every unitary in the two groups therefore recovers commutation of the spectral projections. Resolvents give another bounded criterion when the appropriate conjugate parameters are included.

For bounded self-adjoint operators, $AB=BA$ implies strong commutation by polynomial approximation followed by spectral limits. For unbounded operators, the products and their domains introduce additional requirements.

Commuting sharp events construct a joint measurement

When the spectral projections commute, set

$$G(\Delta\times\Gamma)=E_A(\Delta)E_B(\Gamma).$$

The product is a projection and has the desired marginals. It extends uniquely to a joint PVM on $\mathbb R^2$, developed in Joint Spectral Measures.

Conversely, existence of a joint POVM with these sharp marginals forces all their spectral projections to commute. The sharp-marginal theorem and proof establishes the stronger result that even one sharp marginal forces commutation with the other marginal’s effects. Its key input is positivity: an effect bounded above by a projection is supported inside that projection.

For two PVMs, the parent is therefore uniquely the product measure displayed above and is itself projection-valued. Permitting an unsharp parent cannot circumvent the incompatibility of two sharp observables. For two unsharp POVMs, by contrast, compatible noncommuting examples exist; this is a different criterion.

The quantifier “for every state” matters here. A single preparation may produce outcome laws that admit many classical joint distributions regardless of operator commutation. Such a state-specific coupling does not supply one apparatus whose marginals equal the proposed observables for all preparations. Strong commutation characterizes the latter, operator-level statement.

A dense-domain commutator can miss boundary physics

On $L^2(0,L)$, let $P_0$ and $P_\theta$ be self-adjoint momenta with periodic and quasiperiodic domains:

$$f(L)=f(0), \qquad f(L)=e^{i\theta}f(0),$$

respectively, where $\theta\notin2\pi\mathbb Z$. Both act as $-i\,d/dx$. On $C_c^\infty(0,L)$ their actions agree, the domain is invariant under both derivatives, and $P_0P_\theta f=P_\theta P_0f$.

Nevertheless they do not strongly commute. The normalized constant function is an eigenvector of $P_0$, whereas $L^{-1/2}e^{i\theta x/L}$ is an eigenvector of $P_\theta$. These vectors are neither proportional nor orthogonal: their overlap is nonzero for the stated $\theta$. Their rank-one spectral projections therefore do not commute.

The compactly supported domain is dense but not a core for either chosen periodic realization. It suppresses the boundary data responsible for the incompatibility. A formal commutator calculation on that space cannot determine the relation between the full observables.

Compatibility differs from uncertainty and disturbance

Commuting observables can have broad distributions and correlated outcomes in a state. Compatibility does not require either variance to vanish.

For noncommuting observables, some particular states can still have a vanishing commutator expectation. That is a state-specific scalar condition, far weaker than commutation of every spectral event. The Robertson lower bound can therefore vanish without implying compatibility.

Nor does compatibility prescribe every apparatus interaction. A joint PVM specifies sharp joint statistics; a chosen instrument may still disturb other observables. For discrete commuting PVMs, their Lüders instruments leave each other’s probabilities unchanged, but this describes a particular implementation.

Exercises

Show that the sharp spin observables $\sigma_x$ and $\sigma_z$ have no joint POVM with exactly those marginals.

Solution

Their positive-outcome projections are $P_x=(I+\sigma_x)/2$ and $P_z=(I+\sigma_z)/2$. Since $[P_x,P_z]=[\sigma_x,\sigma_z]/4\neq0$, the sharp-marginal theorem above rules out a joint POVM. Adding noise can change the problem by replacing these projections with nonprojective effects.

If $B$ is bounded and $A$ self-adjoint, explain why the assumptions $BD(A)\subseteq D(A)$ and $AB\psi=BA\psi$ on all of $D(A)$ are stronger than dense-test commutation.

Solution

For $z\notin\mathbb R$ and $\phi=(A-z)^{-1}\eta$, the assumptions give $(A-z)B\phi=B\eta$. Applying the resolvent yields $B(A-z)^{-1}=(A-z)^{-1}B$. This identity holds on the whole Hilbert space and controls the full realization. If $B$ is also self-adjoint, bounded functional calculus and spectral limits yield strong commutation.

References

• P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016, doi:10.1007/978-3-319-43389-9 — joint observables and compatibility of sharp marginals.
• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — unbounded commutation and domain caveats.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, doi:10.1007/978-94-007-4753-1 — strong commutation and spectral calculus.