Linear algebra, operators, Hilbert spaces
The mathematical grammar of quantum theory: Hermitian and unitary operators, Pauli matrices, projectors and tensor products.
Pauli commutant theorem
A $2\times2$ matrix commuting with all three Pauli matrices must be a multiple of the identity, because the Paulis act irreducibly on $\mathbb C^2$.
Linear algebra ↗Schur's lemma
For an irreducible complex representation, any operator commuting with every group element is scalar.
Linear algebra ↗Spectral theorem
Every finite-dimensional Hermitian operator has real eigenvalues and an orthonormal eigenbasis.
Linear algebra ↗Unitary eigenvalue theorem
If $U|v\rangle=\lambda|v\rangle$, then $\langle v|v\rangle=\langle Uv|Uv\rangle=|\lambda|^2\langle v|v\rangle$.
Linear algebra ↗Simultaneous diagonalization
Commuting Hermitian operators can be measured together.
Linear algebra ↗Trace of a commutator
From cyclicity, $\mathrm{Tr}(AB)=\mathrm{Tr}(BA)$.
Linear algebra ↗No finite-dimensional CCR
Taking the trace of $[X,P]=i\hbar I_d$ gives $0=i\hbar d$, a contradiction.
Linear algebra ↗Completeness of the Pauli basis
Every complex $2\times2$ matrix has a unique expansion, with $a_\mu=\tfrac12\mathrm{Tr}(\sigma_\mu A)$.
Linear algebra ↗Pauli multiplication
The full product rule for Pauli matrices.
Linear algebra ↗Pauli anticommutation
Distinct Paulis anticommute and each squares to $I$.
Linear algebra ↗Unitary invariance of norms
Unitary conjugation preserves singular values, hence the operator, trace, and Frobenius norms.
Linear algebra ↗Orthogonal projectors
A projector is idempotent and self-adjoint; it projects onto its range orthogonally.
Linear algebra ↗Projector rank and trace
Since the eigenvalues are 0 and 1, the trace counts the 1s, i.e. the dimension of the subspace.
Linear algebra ↗Positivity test
For Hermitian $A$ this is equivalent to all eigenvalues being nonnegative.
Linear algebra ↗Tensor-product dimensions
Composite systems multiply dimensions, so $n$ qubits need $2^n$ amplitudes.
Linear algebra ↗