Positive State Functionals on Bounded Operators

A state can be specified by its expectation-value rule rather than by a particular vector or matrix. On the algebra of bounded operators, this rule is a positive normalized linear functional. The formulation unifies vectors and density operators and identifies exactly where additional continuity assumptions enter; the general abstract-algebra construction belongs to operator-algebraic quantum mechanics.

Helpful background. Density operators supplies the trace-rule interpretation; positive operators supplies the positive cone.

Expectation values as a linear functional

Let $\mathcal H$ be a complex Hilbert space. A state on $\mathcal B(\mathcal H)$ is a complex-linear map $\omega:\mathcal B(\mathcal H)\to\mathbb C$ satisfying

$$\omega(A^*A)\geq0 \quad(A\in\mathcal B(\mathcal H)), \qquad \omega(I)=1.$$

Every positive bounded operator has a square root, so the first condition is equivalent to $\omega(B)\geq0$ for every $B\geq0$. For self-adjoint $A$, write $A=A_+-A_-$ using its positive and negative spectral parts. Both expectations are real, so $\omega(A)$ is real. Decomposing a general operator into real and imaginary self-adjoint parts then gives

$$\omega(A^*)=\overline{\omega(A)}.$$

Complex linearity packages expectations of all self-adjoint observables into a single algebraic object. It does not assert that every complex operator is itself one measured observable.

Cauchy–Schwarz from positivity

Define a semidefinite form on operators by

$$(A,B)_\omega=\omega(A^*B).$$

Expanding $\omega((A+zB)^*(A+zB))\geq0$ for arbitrary complex $z$ and minimizing the resulting quadratic expression gives

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

If $\omega(B^*B)=0$, varying the magnitude and phase of $z$ forces the cross term to vanish, so the inequality still holds in the degenerate case. This is the positive-functional Cauchy–Schwarz inequality.

Taking one argument to be $I$ and using $A^*A\leq\|A\|^2I$ yields

$$|\omega(A)|^2\leq\omega(A^*A)\leq\|A\|^2.$$

Thus $\omega$ is automatically continuous in operator norm with norm one. This continuity is weaker than normality: norm continuity alone does not permit passing through strongly convergent projection sums.

The same form is the starting point of the GNS construction. One quotients out its null vectors and completes to obtain a Hilbert space. The full representation theorem and its abstract assumptions belong to the GNS construction.

Vectors, density operators, and indistinguishable preparations

For a normalized vector $\psi$,

$$\omega_\psi(A)=\langle\psi,A\psi\rangle$$

is a state. For a density operator $\rho$, $\omega_\rho(A)=\operatorname{Tr}(\rho A)$ is a state because trace positivity gives $\omega_\rho(A^*A)\geq0$ and the trace is one.

Different ensembles can define the same functional. For example, the equally weighted mixtures of the two computational qubit states and the two $x$-spin eigenstates both give $\omega(A)=\operatorname{Tr}(A)/2$. No bounded observable on that qubit distinguishes the ensembles once their preparation label is discarded.

Conversely, if two density operators agree on all bounded observables, they are equal. Their difference has zero trace pairing with every rank-one matrix unit, so all its matrix elements vanish. Agreement on a restricted observable algebra is weaker: distinct density operators can agree on that algebra.

For example, restricting to matrices diagonal in a basis retains only the diagonal entries of $\rho$. A superposition and a suitable mixture can then define the same restricted functional. Purity and distinguishability must therefore be stated relative to the algebra of allowed observables.

What a state functional does not guarantee

The bounded-operator algebra contains neither an unbounded Hamiltonian nor an unbounded position operator. Their spectral projections are bounded and belong to the algebra, but finite moments are additional conditions. In a normal state these projections define a countably additive probability distribution; the mean requires the integrability condition in Density Operators.

Moreover, not every positive normalized functional on an infinite-dimensional $\mathcal B(\mathcal H)$ has the trace form. A nonnormal state can assign zero to every finite-rank projection and one to the identity. The normal-state representation theorem identifies the missing monotone continuity hypothesis.

Bratteli and Robinson develop the general theory of state functionals and their representations. The calculations here isolate the bounded-operator version needed to understand why ordinary Hilbert-space states fit into that wider theory.

Exercises

1. Show that the variance of a bounded self-adjoint observable $A$ is nonnegative using only positivity of $\omega$.

Solution

Let $a=\omega(A)\in\mathbb R$. Positivity of $(A-aI)^2$ gives $0\leq\omega(A^2)-\omega(A)^2$. This recovers the expectation-value variance without choosing a vector or a density-matrix representation.

2. If $\omega(P)=0$ for a projection $P$, prove $\omega(PA)=\omega(AP)=0$ for every bounded $A$.

Solution

Cauchy–Schwarz gives $|\omega(PA)|^2\leq\omega(P)\omega(A^*A)=0$. Apply adjoint preservation to obtain the other equality. Zero probability for the projection annihilates these cross expectations; no division by $\omega(P)$ is allowed.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1: C- and W*-Algebras, Symmetry Groups, Decomposition of States*, 2nd ed., Springer, 1987 — positive functionals, their Cauchy–Schwarz inequality, and representations. Publisher.
• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd revised and expanded ed., De Gruyter, 2019 — states and statistical predictions in Hilbert space. Publisher.