Expectation Functionals

A state can be specified by how it assigns expectations to observables. For a unit vector $\psi$, the assignment $A\mapsto\langle\psi,A\psi\rangle$ is a positive normalized linear functional on the bounded operators. This reformulation separates the state from a chosen coordinate representation and prepares the algebraic language. Unbounded observables require additional integrability conditions: their expectations cannot be appended to this bounded-operator functional without checking domains or spectral moments.

Required background. Physical States as Rays supplies phase-independent state representatives. Projection-Valued Measures supplies the scalar spectral probability measure of an observable.

Vector states on bounded operators

Let $\mathcal B(\mathcal H)$ be the algebra of bounded linear operators on a complex Hilbert space. For $\|\psi\|=1$, define

$$\omega_\psi(A)=\langle\psi,A\psi\rangle, \qquad A\in\mathcal B(\mathcal H).$$

The functional is complex-linear in $A$, satisfies $\omega_\psi(I)=1$, and is positive:

$$\omega_\psi(B^*B)=\|B\psi\|^2\geq0.$$

It is also bounded, since $|\omega_\psi(A)|\leq\|A\|$. Equality at $A=I$ shows that its functional norm is one. The defining properties therefore encode normalization, positivity of event probabilities, and stability under bounded observable perturbations.

Linearity here concerns the observable argument. The dependence on $\psi$ is quadratic, so in general

$$\omega_{a\psi+b\phi}(A) \ne |a|^2\omega_\psi(A)+|b|^2\omega_\phi(A).$$

The left side additionally needs normalization if the displayed sum is not a unit vector. A convex mixture of functionals describes a mixture of preparations, not their coherent vector sum.

The Cauchy–Schwarz inequality for vector states follows directly from the Hilbert-space inequality:

$$|\omega_\psi(A^*B)|^2 \leq\omega_\psi(A^*A)\,\omega_\psi(B^*B).$$

For bounded self-adjoint $A$, choosing $B=I$ yields the nonnegative variance $\omega_\psi(A^2)-\omega_\psi(A)^2$. The same inequality extends to abstract positive functionals, where positivity of $(A+zB)^*(A+zB)$ replaces the vector proof.

A functional retains quantum fluctuations

A state functional is generally not multiplicative. In the qubit state $|0\rangle$, for example,

$$\omega(\sigma_x)=0, \qquad \omega(\sigma_x^2)=\omega(I)=1.$$

Thus $\omega(A^2)$ is not the square of $\omega(A)$. Treating a quantum state as an assignment of sharp numbers to all observables would discard this distinction. A functional gives expectations; the observable’s spectral measure supplies its full outcome distribution.

For an effect $E$, positivity and normalization imply $0\leq\omega(E)\leq1$. For a vector state and a strongly countably additive PVM $F$, the map $\Delta\mapsto\omega_\psi(F(\Delta))$ is a countably additive probability measure. Vector states are normal: they respect bounded increasing limits of positive operators. Density operators produce the other normal states on $\mathcal B(\mathcal H)$ through $\omega_\rho(A)=\operatorname{Tr}(\rho A)$.

This characterization is specific to the full bounded-operator algebra. Restriction to a smaller algebra can erase distinctions between preparations, and a vector state can become mixed after restriction. General positive functionals, their representations, and possible nonnormal states belong to the operator-algebraic chapter.

Unbounded expectations are spectral integrals

Let $A$ be self-adjoint with spectral measure $E_A$, and write

$$\mu_\psi^A(\Delta)=\langle\psi,E_A(\Delta)\psi\rangle.$$

The finite expectation exists when

$$\int_{\mathbb R}|\lambda|\,d\mu_\psi^A(\lambda)<\infty, \qquad \mathbb E_\psi[A]=\int_{\mathbb R}\lambda\,d\mu_\psi^A(\lambda).$$

Absolute integrability prevents an undefined cancellation between positive and negative infinities. Its domain is $D(|A|^{1/2})$. The operator domain $D(A)$ instead requires the second moment $\int\lambda^2\,d\mu_\psi^A<\infty$. On $D(A)$, the spectral expectation equals the literal inner product $\langle\psi,A\psi\rangle$.

A finite first moment can exist outside $D(A)$. In that case $A\psi$ is not a Hilbert-space vector, so the literal inner product is unavailable. One can write the expectation as the quadratic form

$$\bigl\langle |A|^{1/2}\psi, \operatorname{sgn}(A)|A|^{1/2}\psi\bigr\rangle,$$

which has exactly the required first-moment domain. The sign operator is bounded and defined by spectral calculus.

For a positive observable, an extended expectation of $+\infty$ can also be meaningful. It is not a finite linear-functional value. Products, variances, and differences of extended expectations require separate care.

A principal value is not a mean

On $L^2(\mathbb R,dx)$ consider the normalized state

$$\psi(x)=\frac{1}{\sqrt{\pi(1+x^2)}}.$$

Its position distribution is the Cauchy density. Every symmetric truncation of the first-moment integral vanishes, but

$$\int_{\mathbb R}|x|\,|\psi(x)|^2dx=\infty.$$

The position expectation therefore does not exist as a finite Lebesgue integral. Reporting zero would silently substitute a principal-value prescription. Normalization of a state guarantees probabilities, not every moment of every observable.

Exercise: what bounded observables determine

If unit vectors $\psi,\phi$ satisfy $\omega_\psi(A)=\omega_\phi(A)$ for every bounded $A$, prove that they determine the same ray. Does equality for one chosen observable suffice?

Solution

Choose the bounded projection $A=P_\psi$. The assumed equality gives $1=|\langle\psi,\phi\rangle|^2$. Equality in Cauchy–Schwarz makes $\phi$ a phase multiple of $\psi$. One observable is generally insufficient: both $|0\rangle$ and $|1\rangle$ have zero expectation of $\sigma_x$, although their rays are orthogonal. The complete functional, not a single mean, identifies the state on the specified algebra.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — expectation values and spectral descriptions of observables.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Chapters 2–3 and Section 6.3 — forms, spectral integrals, and trace-class operators.