Commuting Observables and Spectral Multiplicity

A complete set of commuting observables, or CSCO, is a compatible family whose joint spectral labels leave no unresolved multiplicity. In a discrete spectrum, each joint eigenspace is one-dimensional. In continuous spectral theory, completeness concerns almost-everywhere fiber dimension, not a basis of normalizable eigenvectors.

Helpful background. Joint Spectral Measures constructs the joint outcome measure. Compatible Observables states the strong-commutation requirement for unbounded observables.

Completeness in a discrete spectrum

Let $A_1,\ldots,A_n$ strongly commute and suppose their joint PVM is purely atomic. The joint spectral projector is

$$P_{\boldsymbol a} =E(\{(a_1,\ldots,a_n)\}),$$

with range the common eigenspace. The family is complete in the usual discrete sense if every nonzero $P_{\boldsymbol a}$ has rank one and the strong sum of these projections is the identity.

Then a joint eigenvalue tuple determines an eigenvector up to a phase. Adding another observable can remove degeneracy only if it strongly commutes with the existing family and acts nontrivially within its unresolved eigenspaces.

Completeness is relative to the declared Hilbert space. A family complete for spatial wavefunctions may cease to be complete after a spin or internal factor is included. The missing labels are encoded by the dimension of the joint eigenspaces.

Two spins give a finite worked example

For two spin-$1/2$ particles, define $\mathbf J=\mathbf S_1+\mathbf S_2$. The operator $J_z$ alone has a two-dimensional zero eigenspace spanned by $|\uparrow\downarrow\rangle$ and $|\downarrow\uparrow\rangle$.

The commuting operator $J^2$ resolves this space into

$$|1,0\rangle =\frac{|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle}{\sqrt2},$$

and

$$|0,0\rangle =\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2}.$$

Their $J^2$ eigenvalues are $2\hbar^2$ and zero. Together with $|1,\pm1\rangle$, every pair $(j(j+1)\hbar^2,m\hbar)$ now labels a one-dimensional space. The pair $(J^2,J_z)$ is complete on this four-dimensional Hilbert space.

The pair $(S_{1z},S_{2z})$ is another complete commuting family with a different joint eigenbasis. Completeness does not select one preferred basis independently of the physical question.

Continuous labels carry spectral multiplicity

In a separable Hilbert space, a joint spectral representation can be written as

$$\mathcal H\simeq\int_X^\oplus\mathcal K_x\,d\nu(x),$$

where the observables multiply by coordinate functions and $\mathcal K_x$ is a multiplicity space. A multiplicity-one representation has $\dim\mathcal K_x=1$ almost everywhere.

If the fibers have dimension greater than one, the same joint spectral label still permits several independent state components. Additional commuting observables may refine the labels and split those fibers. Dimensions and identities are understood almost everywhere; values on measure-zero sets do not determine extra physical channels.

For the one-dimensional free particle, $H=P^2/(2m)$ has two momentum branches at every positive energy. Energy alone therefore has multiplicity two. Adding $\operatorname{sgn}(P)$ distinguishes the branches, since

$$p=s\sqrt{2mE}, \qquad s\in\{-1,+1\}.$$

The exceptional point $p=0$ has zero projection in the usual $L^2(\mathbb R,dp)$ representation. Jointly, energy and direction recover the momentum label almost everywhere. No normalizable exact-energy vector is needed for this completeness statement.

A precise algebraic formulation

Take the von Neumann algebra $\mathcal M$ generated by the joint spectral projections. It is abelian because the observables strongly commute. A useful rigorous version of completeness is

$$\mathcal M'=\mathcal M,$$

meaning that $\mathcal M$ is maximal abelian in $\mathcal B(\mathcal H)$.

In the standard separable spectral representation, this is equivalent to multiplicity one: otherwise bounded operators acting within a nontrivial multiplicity fiber commute with every spectral multiplication operator without being scalar functions of the spectral label.

This formulation avoids ambiguous requests for a “maximal number” of observables. One self-adjoint operator with simple spectrum can already generate a maximal abelian algebra. Conversely, many functions of one degenerate operator can leave its degeneracy untouched. The number of written symbols is not a test of completeness.

Complete labels do not imply complete state tomography

A CSCO uniquely identifies its joint eigenvectors up to phase, but its outcome probabilities do not determine an arbitrary quantum state. In a discrete joint basis, a pure state $\psi=\sum_\alpha c_\alpha e_\alpha$ produces probabilities $|c_\alpha|^2$. Relative phases are absent.

Likewise, the density operator’s off-diagonal entries are not fixed by a single commuting measurement algebra. Reconstructing a general state requires additional incompatible measurement settings or an informationally complete POVM. The word “complete” in CSCO refers to spectral multiplicity, not informational completeness.

Exercises

Show that adding $A^2,A^3,\ldots$ to a degenerate discrete observable $A$ cannot resolve its eigenspace degeneracy.

Solution

On the eigenspace with eigenvalue $a$, every function $f(A)$ acts as the scalar $f(a)I$. All vectors in that eigenspace retain the same joint labels. Resolving the multiplicity requires an operator that acts nontrivially within the eigenspace while commuting with its spectral projection.

Give two distinct pure states with identical probabilities for a complete qubit observable $\sigma_z$.

Solution

The states $(|0\rangle+|1\rangle)/\sqrt2$ and $(|0\rangle-|1\rangle)/\sqrt2$ each give probabilities $1/2,1/2$ for $\sigma_z$. They are orthogonal and are distinguished by $\sigma_x$. The $\sigma_z$ eigenspaces are one-dimensional, but its statistics do not reveal relative phase.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — spectral multiplicity and angular momentum.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, doi:10.1007/978-94-007-4753-1 — spectral representations and commuting operator families.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — cyclic representations and quantum spectral examples.