Linear Operators and Matrix Elements

A linear operator $A$ on a Hilbert space $\mathcal{H}$ is a map $A: \mathcal{H} \to \mathcal{H}$ that respects addition and scalar multiplication: $A(|\psi\rangle + |\phi\rangle) = A|\psi\rangle + A|\phi\rangle$ and $A(c|\psi\rangle) = c A|\psi\rangle$. In quantum mechanics, operators represent physical observables, time evolution, and symmetries.

The set of linear operators on $\mathcal{H}$ is itself a vector space, denoted $L(\mathcal{H})$. Multiplication of operators is defined by composition: $(AB)|\psi\rangle = A(B|\psi\rangle)$. In general, operator multiplication is noncommutative: $AB \neq BA$. This noncommutativity is the root of uncertainty relations.

The identity operator $I$ satisfies $I|\psi\rangle = |\psi\rangle$ for all $|\psi\rangle$. The zero operator sends every ket $|\psi\rangle$ to the zero ket. Every operator has a matrix representation once a basis is chosen, but the operator itself is basis-independent.

Important examples include the Pauli matrices $\sigma_x, \sigma_y, \sigma_z$, which act on the two-dimensional Hilbert space of a spin-1/2 particle. These matrices are both Hermitian and unitary, and they form a basis for the space of 2x2 Hermitian matrices.

Once an orthonormal basis $\{|i\rangle\}$ is chosen, the matrix element of $A$ in row i and column j is $A_{ij} = \langle i|A|j\rangle$. The ket $|j\rangle$ selects the input basis state, the operator $A$ acts, and the bra $\langle i|$ extracts the i-th component of the output.

The full matrix representation is obtained by collecting all $A_{ij}$. A ket $|\psi\rangle$ is represented by the column vector $\psi_j = \langle j|\psi\rangle$, and the action $A|\psi\rangle$ is recovered by matrix multiplication: $(A\psi)_i = \sum_j A_{ij} \psi_j$.

Matrix elements depend on the basis. In the eigenbasis of $A$, the matrix is diagonal with the eigenvalues on the diagonal. Off-diagonal matrix elements measure the coupling between different basis states induced by $A$.

Dirac notation makes matrix elements intuitive. The expression $\langle i|A|j\rangle$ is read as the amplitude to go from $|j\rangle$ to $|i\rangle$ under $A$. This language is especially useful for scattering, transition amplitudes, and quantum circuits.

The product $AB$ of two operators acts on a ket by first applying $B$ and then $A$: $(AB)|\psi\rangle = A(B|\psi\rangle)$. The matrix element of a product is $\langle i|AB|j\rangle = \sum_k \langle i|A|k\rangle \langle k|B|j\rangle$, which is the familiar rule of matrix multiplication.

The identity $\sum_k |k\rangle\langle k| = I$ is inserted between $A$ and $B$ to obtain this sum. This is the operator version of the completeness relation. It shows that Dirac notation naturally encodes matrix multiplication without explicit indices.

Products of operators do not generally commute. The failure of $AB$ to equal $BA$ is measured by the commutator $[A,B] = AB - BA$. When $[A,B] = 0$, the operators are simultaneous diagonalizable in finite dimensions and are called compatible.

Operator powers are defined by repeated composition: $A^n = A A \cdots A$. A function of an operator can be defined through its Taylor series, through the spectral decomposition, or through the exponential series when the operator is normal.

The adjoint $A^\dagger$ of an operator $A$ is defined by the relation $\langle\phi|A^\dagger|\psi\rangle = \langle\psi|A|\phi\rangle^*$ for all kets $|\phi\rangle$, $|\psi\rangle$. In matrix language, $A^\dagger$ is the conjugate transpose of $A$.

The adjoint reverses products: $(AB)^\dagger = B^\dagger A^\dagger$. This mirrors the rule for transposes of matrix products and is essential for manipulating operator expressions. The adjoint of a scalar is its complex conjugate, and $(A^\dagger)^\dagger = A$.

A bra corresponding to $A|\psi\rangle$ is $\langle\psi|A^\dagger$. Thus the action of an operator on a ket corresponds to the action of its adjoint on the associated bra. This symmetry is built into Dirac notation.

The adjoint also governs preservation of inner products. An operator $U$ satisfies $\langle U\phi|U\psi\rangle = \langle\phi|\psi\rangle$ for all kets $|\phi\rangle$, $|\psi\rangle$ if and only if $U^\dagger U = I$, which is the defining property of a unitary operator.

An operator $A$ is Hermitian, or self-adjoint, if $A = A^\dagger$. Hermitian operators have real eigenvalues and orthogonal eigenvectors. In quantum mechanics, every observable is represented by a Hermitian operator because measurement outcomes must be real numbers.

An operator $U$ is unitary if $U^\dagger U = U U^\dagger = I$. Unitary operators preserve inner products and norms; they are the isometries of Hilbert space. Quantum time evolution is described by a unitary operator, so probabilities are conserved.

An operator $N$ is normal if $N N^\dagger = N^\dagger N$. Both Hermitian and unitary operators are normal. The spectral theorem guarantees that normal operators can be diagonalized by a unitary transformation, with orthogonal eigenvectors and possibly complex eigenvalues.

The Pauli matrices are Hermitian and unitary. Their eigenvalues are $\pm 1$, and their eigenvectors form orthonormal bases. The Hadamard gate $H = (\sigma_x + \sigma_z)/\sqrt{2}$ is unitary but not Hermitian; $H^2 = I$.

An operator $A$ is invertible if there exists an operator $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$. The inverse is unique when it exists. A finite-dimensional operator is invertible if and only if its determinant is nonzero, or equivalently if zero is not an eigenvalue.

Unitary operators are invertible, and their inverse equals their adjoint: $U^{-1} = U^\dagger$. This makes them especially easy to work with. Applying a unitary gate and then its inverse returns the system to its original state.

The identity operator $I$ has matrix representation equal to the identity matrix in every orthonormal basis. It is both Hermitian and unitary, and it satisfies $I^2 = I$. It plays the role of the number 1 in the algebra of operators.

Projectors are not invertible unless they equal $I$, because they collapse part of the space to zero. Their failure to be invertible reflects the irreversibility of quantum measurement in the standard Copenhagen formulation.

Consider the quantum gate $U$ defined by $U|0\rangle = |0\rangle$ and $U|1\rangle = i|1\rangle$ in the computational basis $\{|0\rangle, |1\rangle\}$. Find its matrix, verify that it is unitary, and compute $U|+\rangle$ where $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$.

Step 1 -- Matrix elements. $U_{00} = \langle 0|U|0\rangle = 1$, $U_{01} = \langle 0|U|1\rangle = 0$, $U_{10} = \langle 1|U|0\rangle = 0$, and $U_{11} = \langle 1|U|1\rangle = i$. Thus $U = \operatorname{diag}(1,i)$.

Step 2 -- Unitarity. $U^\dagger = \operatorname{diag}(1,-i)$. Then $U^\dagger U = \operatorname{diag}(1,(-i)(i)) = \operatorname{diag}(1,1) = I$. So $U$ is unitary. It is the phase gate $S$.

Step 3 -- Action on $|+\rangle$. $U|+\rangle = (U|0\rangle + U|1\rangle)/\sqrt{2} = (|0\rangle + i|1\rangle)/\sqrt{2}$. This state differs from $|+\rangle$ by a relative phase and is an eigenstate of the Hadamard basis measurement with modified probabilities.