Joint Spectral Measures

A finite family of strongly commuting self-adjoint operators has one projection-valued measure on the Cartesian product of its outcome spaces. This joint measure makes several compatible observables into ordinary random variables within each state, while retaining the full operator structure needed to compare different states.

Helpful background. Compatible Observables establishes the strong-commutation criterion. Projection-Valued Measures develops countable additivity and scalar probabilities.

Joint spectral theorem on rectangles

Let $A_1,\ldots,A_n$ be self-adjoint and pairwise strongly commuting. There is a unique PVM $E$ on $\mathbb R^n$ satisfying

$$E(\Delta_1\times\cdots\times\Delta_n) =\prod_{j=1}^n E_{A_j}(\Delta_j).$$

The product is independent of order because all its factors commute. Each marginal recovers the original PVM, and coordinate integration recovers the observables:

$$A_j=\int_{\mathbb R^n}x_j\,dE(x).$$

This statement concerns a finite family. Infinite collections require additional care about the outcome sigma-algebra and the topology used for probability extension.

Products on rectangles are not by themselves the full proof: one must extend them consistently to all Borel sets and prove strong countable additivity. A standard construction uses the commuting bounded spectral algebras, decomposes them into cyclic representations, represents continuous functions by scalar measures, and assembles the resulting multiplication PVMs. Uniqueness follows because rectangles generate the Borel sigma-algebra and determine all scalar matrix-element measures.

For a finite discrete spectrum the construction is elementary. The joint eigenspace projector for values $(a,b)$ is $P_aQ_b$, possibly zero. Orthogonality and the marginal completeness sums directly establish the PVM.

Joint probabilities and correlation

For a normalized vector or a density operator, define

$$\mu_\psi(\Omega)=\langle\psi,E(\Omega)\psi\rangle, \qquad \mu_\rho(\Omega)=\operatorname{Tr}(\rho E(\Omega)).$$

These are ordinary probability measures on the joint outcome space. On rectangles,

$$\mathbb P_\rho(A\in\Delta,\ B\in\Gamma) =\operatorname{Tr}\bigl(\rho E_A(\Delta)E_B(\Gamma)\bigr).$$

Commutation does not imply statistical independence. Independence is the extra state-dependent condition that the joint probabilities factor into their marginals for every rectangle.

For example, on $L^2(\mathbb R)$ let $A=Q$ and $B=Q^2$. They strongly commute because both are functions of the same observable. Their joint measure acts as

$$(E(\Omega)\psi)(x)=\mathbf1_\Omega(x,x^2)\psi(x).$$

It is supported on a parabola, so the two outcomes are perfectly functionally related. Even though the marginal spectra are $\mathbb R$ and $[0,\infty)$, the joint spectral support is not their full Cartesian product.

Multivariable functions and domains

A finite-valued Borel function $f:\mathbb R^n\to\mathbb C$ defines a normal operator

$$f(A_1,\ldots,A_n)=\int f(x)\,dE(x)$$

on

$$D(f)=\left\{\psi:\int|f(x)|^2\,d\mu_\psi(x)<\infty\right\}.$$

Real-valued $f$ gives a self-adjoint operator. This is the correct setting for functions such as a sum, a product, or the Euclidean magnitude of compatible momentum components.

The integral for $x+y$ can have a larger domain than the natural sum $A+B$ on $D(A)\cap D(B)$ because cancellation can improve integrability. The maximal joint-calculus operator is the closure of that natural sum for strongly commuting $A,B$; joint spectral cutoffs provide a common core proving the closure statement.

A simple example is $B=-A$ for unbounded self-adjoint $A$. The algebraic sum on $D(A)$ is zero on a proper domain, whereas the function $x+y$ vanishes on the joint spectral support and defines zero on all of $\mathcal H$. The closure restores the full bounded zero operator.

Joint support and degeneracy

A point belongs to the joint spectral support if every open neighborhood has nonzero projection. A singleton projection detects a common eigenspace:

$$\operatorname{Ran}E(\{(a_1,\ldots,a_n)\}) =\bigcap_j\ker(A_j-a_j).$$

The intersection may have dimension zero, one, or greater. Thus a joint PVM can remove some degeneracies while leaving others.

For position components on $L^2(\mathbb R^d)$, the joint PVM multiplies by spatial-region indicators. Every singleton projection vanishes, but the support is all of $\mathbb R^d$. No basis of normalizable simultaneous position eigenvectors is needed.

For a tensor product with a spin factor, spatial position still leaves spin multiplicity unresolved. Including a compatible spin component refines the joint measure by adding the spin label. Complete Sets of Commuting Observables interprets this refinement.

Exercises

For two qubits, let $A=\sigma_z\otimes I$ and $B=I\otimes\sigma_z$. Compute the joint probabilities in $(|00\rangle+|11\rangle)/\sqrt2$.

Solution

The joint projectors are $|jk\rangle\langle jk|$. The outcomes $(+1,+1)$ and $(-1,-1)$ have probability $1/2$ each, while the other two have probability zero. Each marginal is uniform, so the outcomes are correlated rather than independent despite strong commutation.

For $A=Q$ and $B=Q^2$, compute the projection for the event $A<0$ and $B<1$.

Solution

The preimage under $x\mapsto(x,x^2)$ is $(-1,0)$. The projection is therefore multiplication by $\mathbf1_{(-1,0)}$. In a vector state the probability is $\int_{-1}^0|\psi(x)|^2\,dx$.

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 marginals.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, doi:10.1007/978-94-007-4753-1 — strongly commuting operators and joint spectral calculus.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — spectral representations and operator domains.