Main Mathematical Objects

A quantum prediction combines several objects with different roles: a state supplies probabilities, an observable specifies possible measurement events, and a dynamical law transports states or observables. A Hilbert space is the setting for one representation of those objects. Keeping their types distinct prevents a vector, an eigenvalue, a probability density, and a measurement operation from being treated as interchangeable symbols.

A Hilbert space hosts a representation of states and observables

A complex Hilbert space $\mathcal H$ supplies inner products, norm limits, and orthogonality. It does not by itself identify a physical system. The model also specifies which operators represent position, energy, spin, or other observables, together with dynamics and the interpretation of those quantities.

For a system whose accessible bounded observables form $B(\mathcal H)$, a unit vector defines a pure state up to phase. More general normal states are density operators. In an algebraic formulation, the initial object is instead an observable algebra $\mathcal A$, and a representation $\pi:\mathcal A\to B(\mathcal H)$ realizes it on a Hilbert space. The same abstract algebra can admit different representations.

Hilbert Space as State Space explains what the Hilbert description assumes. Operator Algebra Minimum separates the algebra from its representation.

States are rules for expectation values

A density operator satisfies $\rho\ge0$, $\rho$ trace class, and $\operatorname{Tr}\rho=1$. It produces the functional

$$\omega_\rho(B)=\operatorname{Tr}(\rho B), \qquad B\in B(\mathcal H).$$

For a vector state, $\rho=|\psi\rangle\langle\psi|$. The vector, projector, and functional are different mathematical objects that encode the same state under these assumptions. An ensemble decomposition of $\rho$ into pure states is generally not unique.

An abstract state on a unital C*-algebra is a positive normalized linear functional. It need not have a density-operator form in an arbitrarily chosen representation. Thus “state” identifies a probabilistic role, while “density operator” identifies one realization of that role. See Positive State Functionals and Normal States.

An observable assigns operators to outcome events

A sharp real-valued observable has a PVM $E$, mapping Borel sets $\Delta$ to orthogonal projections. A general measurement observable has a POVM $F$, mapping events to positive effects with $F(\mathbb R)=I$. The scalar probability is obtained only after a state is supplied:

$$\Pr_\rho(\Delta)=\operatorname{Tr}(\rho F(\Delta)).$$

An effect is therefore neither a state nor a probability. It is the operator to which a state assigns the probability of an event. For sharp observables, spectral integration recovers a self-adjoint operator $A=\int\lambda\,dE(\lambda)$ with its domain. This connects the familiar operator and event-based descriptions.

A POVM specifies outcome statistics but does not in general specify the post-measurement state. An instrument adds completely positive operations for the outcomes. This distinction is visible even in two dimensions, as the example below shows.

Dynamics transforms states or observables

A closed-system unitary $U$ acts on density operators by $\rho\mapsto U\rho U^*$. Equivalently, it acts on bounded observables by $B\mapsto U^*BU$. The identity

$$\operatorname{Tr}(U\rho U^*B) =\operatorname{Tr}(\rho U^*BU)$$

expresses agreement of the Schrödinger and Heisenberg descriptions. A quantum channel generalizes the state map to a completely positive trace-preserving linear map. Its dual is unital and completely positive on bounded observables, with normality qualifications in infinite dimension.

A Hamiltonian is a generator of a one-parameter unitary group, not an arbitrary name for a channel. Stone’s Theorem makes the closed-system generator statement precise.

Worked example: one qubit, several object types

Let $\mathcal H=\mathbb C^2$, $\rho=(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z)/2$, and $|\mathbf r|\le1$. A sharp $z$ measurement has projections $P_\pm=(I\pm\sigma_z)/2$, giving

$$p_\pm=\operatorname{Tr}(\rho P_\pm)=\frac{1\pm r_z}{2}.$$

Here $\rho$ is the state, $P_\pm$ are effects, and $p_\pm$ are numbers. Choosing a Hamiltonian $H=\hbar\Omega\sigma_x/2$ adds dynamics; it does not alter the definition of the measurement effects. Evolving the state before measuring can change $r_z$ and hence the probabilities.

For the same effects, the Lüders operation is $\mathcal I_\pm(\rho)=P_\pm\rho P_\pm$. A different valid instrument is $\mathcal J_\pm(\rho)=\operatorname{Tr}(\rho P_\pm)\tau_\pm$, where $\tau_\pm$ are fixed prepared states. Both have the same outcome probabilities. The second prepares chosen states after the measurement and need not preserve the detected eigenspace. Outcome statistics alone therefore do not identify the physical state update.

Exercise: distinguish a channel from an event

For the Lüders instrument, find the unconditioned state after the $z$ measurement and compare it with the probability of the positive outcome.

Solution

Summing the operations gives $\rho'=P_+\rho P_++P_-\rho P_-=(I+r_z\sigma_z)/2$. This is a density operator, the output of a trace-preserving channel. In contrast, $p_+=(1+r_z)/2$ is a scalar. The effect $P_+$ and operation $\rho\mapsto P_+\rho P_+$ are also distinct: one evaluates a probability, while the other returns an unnormalized conditional state.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987, doi:10.1007/978-3-662-02520-8 — algebras, states, and representations.
• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — Hilbert spaces, spectral observables, and unitary dynamics.
• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed., De Gruyter, 2019, doi:10.1515/9783110642490 — states, observables, and completely positive maps.