States, Density Operators, and Trace

The finite-dimensional formula $\langle A\rangle=\operatorname{Tr}(\rho A)$ survives in infinite-dimensional quantum mechanics, but the word “trace” acquires analytic content. This chapter explains which operators have traces, why normal states are represented by density operators, and what extra assumptions are needed for unbounded expectations, purification comparisons, and entropy.

For the preparation and probability rules, see Core Formalism; for the subsystem meaning of partial trace, see Composite Systems. Here we establish the convergence, representation, and domain statements that make these constructions reliable beyond finite matrices.

Enter this chapter

Readers should be comfortable with orthonormal bases, bounded operators, spectral decompositions of compact positive operators, and elementary convergence of nonnegative series. The operators and domains chapter supplies the general operator framework.

There are three useful routes through the material:

1. Establish the operator classes. Begin with positive operators, then compact operators, Hilbert–Schmidt operators, and trace-class operators. The question is which infinite sums converge and in which norm.
2. Characterize the states. Read density operators, states as positive functionals, and normal states. This route separates the trace-class formulation from the larger algebraic state space.
3. Control approximations and composites. Continue with trace versus operator norm, trace and partial trace, purification, entropy and trace, and fidelity and trace distance. This route answers which physical predictions survive a cutoff or a limit.

For the geometry of preparation, pure and mixed states gives the classification, convex state space explains its topology, and extreme points proves the rank-one characterization.

The operator hierarchy

On an infinite-dimensional Hilbert space, the relevant strict inclusions are

$$\mathcal S_1 \subsetneq\mathcal S_2 \subsetneq\mathcal K \subsetneq\mathcal B.$$

Trace-class operators have summable singular values; Hilbert–Schmidt operators have square-summable singular values; compact operators have singular values tending to zero; bounded operators need satisfy none of these summability conditions.

Density operators occupy the positive trace-one slice of $\mathcal S_1$. Their square roots lie in $\mathcal S_2$. Observables used as bounded tests lie in $\mathcal B$. The formula $\operatorname{Tr}(\rho A)$ is well-defined because trace class is an ideal: multiplying a trace-class operator by a bounded operator preserves trace class.

For an unbounded self-adjoint observable, start with its spectral projections, which are bounded. They define the outcome distribution in a normal state. A finite mean then requires a first absolute moment; a finite variance requires a second moment. These are separate statements from normalization of $\rho$.

Four distinctions that prevent false conclusions

Positive does not mean boundedly invertible. An infinite-dimensional positive operator can have a trivial kernel while its spectrum approaches zero. A faithful density operator therefore need not have a bounded inverse.

Small operator norm does not mean small total probability error. The states $P_N/N$ converge to zero in operator norm, although their traces stay one. Trace norm controls all bounded-observable predictions uniformly.

A state functional need not be normal. Positivity and normalization imply norm continuity, but not preservation of increasing strong limits. On $\mathcal B(\mathcal H)$, normality is exactly what recovers a density operator.

A valid state need not have finite entropy or energy. A summable probability distribution may have divergent entropy or moments. Likewise, trace-norm convergence can coexist with entropy divergence. Additional tail or energy constraints must be stated where they are used.

Each distinction is supported by an explicit counterexample in the corresponding article. The point is to identify a missing hypothesis before a calculation treats a formal infinite expression as a finite matrix identity.

A model calculation to connect the routes

For $0<q<1$, consider the diagonal state on $\ell^2(\mathbb N_0)$

$$\rho_q=(1-q)\sum_{n=0}^{\infty}q^n|n\rangle\langle n|.$$

The trace equals one, its purity is $(1-q)/(1+q)$, and its mean occupation is $q/(1-q)$. It has a norm-convergent purification $\sqrt{1-q}\sum_nq^{n/2}|n\rangle|n\rangle$. The missing weight after retaining levels $0,\ldots,N$ is $q^{N+1}$; normalized spectral truncation has exactly that trace distance from $\rho_q$.

As $q\to1$, probability spreads into high levels. The purity tends to zero, but there is no trace-one limit proportional to the identity. This one example links operator ideals, convex geometry, purification, and the distinction between a controlled finite cutoff and an uncontrolled infinite-temperature limit.

Check your understanding

Before moving on, verify that you can explain why the unilateral-shift commutator can have nonzero trace, why an infinite-rank state cannot be purified with a finite ancilla, and why a finite-energy constraint needs a confining Hamiltonian to make the state set compact. The relevant leaf pages give full arguments and solved exercises.

Continue toward measurement theory for effects, instruments, and positive maps, or toward operator-algebraic quantum mechanics for general observable algebras and representations. Simon is the main operator-ideal reference; Holevo connects the analysis to quantum states and information, while Bratteli and Robinson develop the algebraic setting.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987 — positive functionals, normality, and state-space geometry. Publisher.
• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd revised and expanded ed., De Gruyter, 2019 — density operators, composite states, and infinite systems. Publisher.
• Simon, Barry, Trace Ideals and Their Applications, 2nd ed., Mathematical Surveys and Monographs 120, American Mathematical Society, 2005 — compact operators and trace ideals. Publisher.