Definition Index

A definition specifies a type of object, a domain of quantification, and a condition. Omitting any one can turn a valid statement into a different claim. This index records the predicates most often needed in a rigorous quantum calculation and directs the reader to their complete definitions. For ordinary-language distinctions between similar terms, use the Glossary.

Definitions for operators and their domains

In this table, operators are linear and the Hilbert-space domains are part of their specification. Adjoint statements assume a densely defined operator.

Object	Defining condition	Full definition
Closed operator	Its graph $\{(\psi,A\psi):\psi\in D(A)\}$ is closed in $\mathcal H\oplus\mathcal H$	Closed and closable operators
Operator core $\mathcal D$	$\mathcal D\subset D(A)$ is dense for $\lVert\psi\rVert+\lVert A\psi\rVert$	Dense domains
Symmetric operator	$A\subseteq A^\dagger$	Symmetric operators
Self-adjoint operator	$A=A^\dagger$, with equal domains	Self-adjoint operators
Essentially self-adjoint operator	$A$ is symmetric and its graph closure is self-adjoint	Essential self-adjointness
Resolvent point $z$	$A-zI:D(A)\to\mathcal H$ is bijective with bounded inverse	Resolvent operators
Strongly commuting self-adjoint pair	Every spectral projection of one commutes with every spectral projection of the other	Compatible observables

For a densely defined semibounded quadratic form $q$, closedness is defined through completeness of its form domain in a norm obtained by adding a sufficiently large multiple of $\|\psi\|^2$ to $q[\psi]$. That definition belongs to Quadratic Forms. It is not the graph-closedness test in the first row.

Notice the quantifier in strong commutation: it concerns all Borel spectral projections. The statement $AB\psi=BA\psi$ for vectors in one selected subspace has different quantifiers and need not imply it.

Definitions for states, measurements, and dynamics

The state rows refer to a unital C*-algebra $\mathcal A$ or the indicated normal-state setting. State maps below are linear maps on trace-class operators; their Heisenberg adjoints act on bounded observables.

Object	Defining condition	Full definition
Algebraic state	$\omega(A^\dagger A)\ge0$ for all $A\in\mathcal A$, and $\omega(I)=1$; $\omega$ is linear	Algebraic states
Density operator	$\rho\ge0$, $\rho$ trace class, $\operatorname{Tr}\rho=1$	Density operators
Pure state	An extreme point of the state space of the specified algebra	Extreme points
Normal positive functional	Preserves suprema of increasing bounded nets of positive observables	Normal states
Effect	A bounded operator $0\le E\le I$	Effects and effect algebras
POVM	Positive operator measure, normalized by $F(\Omega)=I$ and countably additive weakly	Generalized observables
Instrument	A countably additive CP operation-valued measure with trace-preserving total operation	Instruments
Channel	A CP trace-preserving linear map on trace-class operators; its adjoint is normal	Infinite-dimensional channels
Conservative $C_0$ semigroup	Channels $\Phi_t$ with $\Phi_0=\mathrm{id}$, $\Phi_{t+s}=\Phi_t\Phi_s$, and trace-norm continuity of each orbit	Quantum dynamical semigroups

For instruments on normal states, countable additivity is in trace norm after application to a trace-class input. For complete positivity, every finite matrix amplification must preserve positivity. These qualifications are developed in the linked articles; “positive outputs on the examples tested” is weaker than either definition.

Test object type before testing inequalities

A function that takes density operators to density operators is not automatically a channel. Linearity on the surrounding trace-class space is part of the definition, because physical randomization must have a consistent action on mixtures.

For example, in finite dimension define

$$F(\rho)=\frac{\rho^2}{\operatorname{Tr}\rho^2}.$$

This always returns a density operator. It fixes each pure projector, but for $\rho=\operatorname{diag}(3/4,1/4)$,

$$F(\rho)=\operatorname{diag}(9/10,1/10).$$

A linear map that fixes both basis projectors would instead fix every mixture of them. Thus $F$ is not a channel, despite positivity and normalization of every output. Asking whether its Choi matrix is positive before checking linearity would already be the wrong test.

Conversely, a linear CP operation need not preserve trace. A selective measurement branch has its trace interpreted as the outcome probability. Normalizing that branch produces a conditional state rule, whose linearity must be reconsidered; see State Update Maps.

Exercises

An effect is not an instrument. Given a qubit effect $E=\operatorname{diag}(1/4,3/4)$, what does it determine, and what additional object is needed to predict a state after that outcome?

Solution

It determines the probability $\operatorname{Tr}(\rho E)$. It does not determine a unique CP branch. For any unitary $U$, the single-Kraus branch with $K=U\sqrt E$ has $K^\dagger K=E$ and hence the same probability, but generally a different output. An instrument supplies the branch map as well as the effect. This is the definition-level distinction behind measurement backaction.

A channel family is not yet a semigroup. Set $H=\operatorname{diag}(0,1)$ and $U_t=e^{-it^2H}$ in dimensionless units. Which defining condition fails for $\Phi_t(\rho)=U_t\rho U_t^\dagger$?

Solution

Each map is a channel, the family starts at the identity, and it is continuous. The homogeneous composition law fails: at $t=s=1$, $\Phi_2$ multiplies the matrix unit $|1\rangle\langle0|$ by $e^{-4i}$, whereas $\Phi_1\Phi_1$ multiplies it by $e^{-2i}$. This is an observable relative phase, not a removable global phase. The defining condition that fails is the homogeneous composition law. The Markovianity language page distinguishes that law from the existence of intermediate channels.

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. Algebraic states and normality.
• Hall, Brian C. Quantum Theory for Mathematicians. Graduate Texts in Mathematics 267, Springer, 2013. doi:10.1007/978-1-4614-7116-5. Operators and spectral definitions.
• Holevo, Alexander S. Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed. De Gruyter, 2019. doi:10.1515/9783110642490. Normal states, measurements, and channels.