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A bipartite pure state is maximally entangled when either reduced density matrix is maximally mixed.

For a bipartite state $|\psi ⟩$, measurement outcome $(x_j,y_k)$ has probability $|⟨x_j{y}_k|\psi ⟩{|}^2$.

The expectation of a Hermitian Pauli string can be obtained by basis rotation and measurement, or by a Hadamard test with the Pauli string as its controlled unitary.

Independent gates $A$ and $B$ on separate registers act jointly as $A\otimes B$.

A controlled-$U$ gate applies $U$ to its target exactly when its control qubit is one.

The inverse of a quantum circuit applies the adjoints of its gates in reverse temporal order. The Hadamard gate, Pauli Z gate, and controlled-NOT gate are each self-inverse.

The Pauli $X$ gate interchanges $|0⟩$ and $|1⟩$.

The Pauli $Z$ gate fixes $|0⟩$ and maps $|1⟩$ to $-|1⟩$.

The Hadamard gate maps

In a periodic potential, energy eigenstates have the form $e^{ik\cdot x}{u}_k(x)$ with lattice-periodic $u_k$.

A charged particle in a uniform magnetic field has equally spaced orbital energy levels with guiding-centre degeneracy.

The Rayleigh quotient of any normalized trial state is at least the ground-state energy.

The four Bell states form an orthonormal maximally entangled basis for two qubits.

The Breidbart basis bisects the computational and diagonal qubit bases. For equiprobable $|0⟩$ and $|-⟩$, whose overlap magnitude is $1/\sqrt{2}$, it realizes the Helstrom measurement and succeeds with probability

The BB84 protocol encodes random classical bits in independently chosen computational and Hadamard bases. Sender and receiver publicly compare bases and retain only positions where their bases agree; disturbance of a test sample detects eavesdropping.

The quantum Fourier transform maps $|k⟩$ to $N^{-1/2}{\sum }_j{e}^{2\pi ijk/N}|j⟩$.

The Deutsch-Jozsa algorithm uses phase kickback and interference to distinguish constant from perfectly balanced Boolean functions with one oracle call.

Measuring ${\sum }_x{a}_x|x⟩$ in the computational basis returns $x$ with probability $|a_x{|}^2$ and leaves the measured register in $|x⟩$.

Grover search rotates a uniform state toward a marked subspace and finds one of $M$ marked entries using order $\sqrt{N/M}$ oracle calls.

A quantum circuit is a finite composition of quantum gates acting on quantum registers. Reading the diagram from input to output gives the order in which its unitary operations are applied.