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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 $|1⟩$.

This gate acts by the corresponding Pauli matrix, viewed as a unitary operator on one qubit.

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

This gate acts by the corresponding Pauli matrix, viewed as a unitary operator on one qubit.

The Hadamard gate is the unitary operatorwhere $I$ is the identity operator. It maps the computational basis to the Hadamard basis:

These outputs have equal computational-basis measurement probabilities, but their relative quantum phase differs. Applying another Hadamard gate sends them to different definite outputs. Calling this gate a classical coin toss would discard precisely the information that makes this possible. ^[ref].

Starting from $|0⟩$, compare two quantum circuits, reading matrix products from right to left:Here $Z$ is the Pauli Z gate. After the first Hadamard gate, the probability amplitudes are $(1/\sqrt{2},1/\sqrt{2})$. Inserting $Z$ changes them to $(1/\sqrt{2},-1/\sqrt{2})$ without changing their squared magnitudes. At the final gate, the contributions to output zero cancel and those to output one add. Without $Z$, cancellation occurs at output one instead. This is controlled interference, with the quantum phase determining which answer emerges.

The same calculation works for any phase gate $P_{\varphi }=\mathrm{diag}(1,e^{i\varphi })$:For example, $\varphi =\pi /3$ gives $P(1)=1/4$, while $\varphi =\pi /2$ gives $P(1)=1/2$. The output-one amplitude at $\varphi =\pi /2$ is $(1-i)/2$: its two complex contributions add before the Born rule is applied. A phase gate alone leaves the populations unchanged; recombination converts its relative quantum phase into observable probabilities.

If quantum decoherence completely removes the computational basis coherence between the gates, the intermediate density matrix becomes $I/2$. Both the phase gate and final Hadamard gate leave $I/2$ unchanged, so $P(1)=1/2$ for every $\varphi$. The disappearing fringe explains why preserving quantum coherence matters: it enables the gate sequence to carry phase information into the final measurement.

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

The variational method estimates the lowest energy of a Quantum Hamiltonian by choosing a trial wavefunction, evaluating its energy, and optimizing the trial parameters. It uses the quantum variational principle, the Rayleigh-Ritz variational principle expressed for quantum states. Let $H$ be a self-adjoint operator bounded below, and define $E_0=\inf \sigma (H)$, the bottom of its spectrum. Every normalized admissible trial quantum state satisfiesWhen $E_0$ is an energy eigenvalue, it is the ground-state energy; equality holds precisely for a quantum state in the ground-state subspace. Otherwise the bottom of the spectrum need not be attained by any normalizable wavefunction.

Admissible means finite form energy. Choose a real $b\le E_0$ and use the shifted quadratic form of a positive quantum Hamiltonian:The form domain $Q(H)$ is independent of this choice of $b$ and can be larger than the operator domain $D(H)$. For $\psi \in D(H)$, $E_{\psi }=⟨\psi ,H\psi ⟩$ is the usual expectation value; for nonzero unnormalized trials, divide $q_H[\psi ]$ by $\Vert \psi {\Vert }^2$ to get the form version of the Rayleigh quotient. See the quantum variational principle for the general domain justification.

Thus each trial energy is an upper bound on $E_0$, and minimizing within a trial family improves that upper bound. It is not a lower bound on $E_0$. The Gaussian variational energy for the harmonic oscillator reaches $E_0=1/2$ because its family contains an exact ground state. The exact variational solution in a two-level system makes the equality condition visible as vanishing excited-state probability amplitude.

The method is useful when solving the Schrödinger equation exactly is difficult but integrals of a trial wavefunction are tractable. Enlarging a trial family cannot increase its optimal energy, although an optimum need not be attained. The beauty is that even an imperfect trial gives a rigorous bound, provided its form energy is computed exactly: numerical integration errors require separate control. The Gaussian variational estimate for a solvable sextic potential provides an existing example beyond a quantum harmonic oscillator.

A short proof explains the direction. With a complete orthonormal eigenbasis of energy eigenstates, write $\psi ={\sum }_n{c}_n{\varphi }_n$, where $H{\varphi }_n=E_n{\varphi }_n$ and ${\sum }_n|c_n{|}^2=1$. ThenThis is a weighted mean of nonnegative energy differences; equality requires every occupied energy eigenstate to have energy $E_0$. ^[ref]. For continuous spectrum, the spectral theorem replaces the sum by an integral against the scalar spectral measure of $\psi$, with the same nonnegative integrand. This proves the bound without assuming an orthonormal eigenbasis exists.

The historical setting predates quantum mechanics: Rayleigh's The Theory of Sound, volume I, appeared in 1877, and Ritz's paper on solving variational problems of mathematical physics appeared in volume 135 of Crelle's journal, catalogued by its publisher as 1909, pp. 1–61. These are original works associated with the Rayleigh-Ritz method; the dates here identify the publications rather than assert priority. ^[ref]; ^[ref].

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

This intermediate basis occurs in attacks and discrimination calculations involving BB84.

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 $|x⟩$.

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.

For a Boolean function $f$, its standard oracle acts by

The Toffoli, or controlled-controlled-NOT, gate maps $|x_1{x}_{2}y⟩$ to $|x_1{x}_2,y\oplus x_1{x}_2⟩$.

The Born approximation replaces the unknown total field inside a scattering integral by the known incident field, thereby retaining only single scattering. In nonrelativistic quantum mechanics, first-order Lippmann-Schwinger iteration gives