The same letter can stand for different kinds of data in different calculations. Keep the following types fixed while reading a formula. In this appendix, vectors are columns, operators compose from right to left, and \(A^*\) denotes the conjugate transpose.
| Symbol | Object | What it records |
|---|---|---|
| \({\mathcal H},\psi\) | Hilbert space and state vector | They specify the space of addresses and its coherent amplitudes. |
| \({\mathcal A}\) | Declared operator algebra | It specifies the available linear operations. |
| \(P_a\) | Orthogonal packet projector | It selects an entire subspace. |
| \(\rho\) | Positive matrix of trace one | It records populations and coherences. |
| \(E_k\) | Positive measurement effect | It gives the outcome probability \({\operatorname{Tr}}(E_k\rho)\). |
| \(K_a\) | Kraus operator | It describes one channel branch and need not be a projector. |
| \(H\) | Self-adjoint Hamiltonian | It generates evolution in real time through \(e^{-{\mathrm i} tH}\). |
| \(C,\nabla\) | Packet transport generator and connection | They describe moving subspaces and differentiation within their ranges. |
| \(P,Q\) | Retained and hidden projectors | They define a decomposition with \(Q=I-P\). |
| \(\Sigma(z)\) | Self-energy | It records the spectral influence of the hidden sector. |
| \(L,T(\tau)\) | Nonnegative operator and Euclidean transfer | They determine decay factors and finite heat trace diagnostics. |
Normalization and indexing
The tensor factors are ordered from left to right, so \(X_1=X\otimes I\otimes I\). A normalized pure state gives \(\rho=\psi\psi^*\). A coefficient matrix used to count a seed orbit need not be normalized. Scaling a nonzero seed does not change its generated linear span.
A Hamiltonian and a Hadamard gate are both traditionally denoted \(H\). Where both appear, the gate is \(H_{\rm Had}\) and the Hamiltonian keeps the letter \(H\). Numerical labels on a drawing are coordinates or exact example values, not observations of a physical device.
Read this as
Check the type before interpreting the picture. An idempotent projector, a density matrix, and a channel matrix have different defining equations.