Codex Wiki
quantum-theory.bigb
= Quantum theory
{wiki=Quantum_mechanics}
= Gauge covariance of the Schrödinger equation
{parent=Quantum theory}
{c}
For a particle of charge $-e$, the changes $A\mapsto A+\nabla f$ and $\phi\mapsto\phi-\partial_t f$ leave the minimally coupled Schrödinger equation invariant when $\psi\mapsto e^{-ief/\hbar}\psi$.
= Quantum measurement
{parent=Quantum theory}
{wiki=Measurement_in_quantum_mechanics}
A quantum measurement assigns probabilities to classical outcomes through positive operators that sum to the identity. A projective measurement uses mutually orthogonal projections.
= Amplitude amplification
{parent=Quantum theory}
{wiki=Amplitude_amplification}
If a state has good amplitude $\sin\theta$, the product of the reflection in the bad axis and the reflection in the state line rotates its good-bad plane by $2\theta$. After $k$ iterations the good probability is
$$
\sin^2((2k+1)\theta),
$$
so $O(1/\sin\theta)$ iterations raise a small good amplitude close to one.
= Density operator
{parent=Quantum theory}
{wiki=Density_matrix}
A finite-dimensional density operator is Hermitian, positive semidefinite, and has trace one. In dimension $N$ it has $N^2-1$ real parameters. Schrödinger evolution is
$$
\rho_S(t)=U(t)\rho_HU(t)^\dagger
$$
when $\rho_H$ denotes the time-independent Heisenberg-picture state.
= Von Neumann entropy
{parent=Density operator}
{c}
{wiki=Von_Neumann_entropy}
The Von Neumann entropy of a density operator is
$$
S(\rho)=-\operatorname{Tr}(\rho\log\rho).
$$
It vanishes for a pure state and measures mixedness and quantum entanglement in reduced states.
= Entanglement entropy
{title2=$S_A=-\operatorname{Tr}(\rho_A\log\rho_A)$}
{parent=Von Neumann entropy}
{wiki}
For a bipartite pure state, the entanglement entropy is the Von Neumann entropy of either <reduced density matrix>. The two reduced states have the same nonzero eigenvalues, so their entropies agree.
= Purity of a density operator
{parent=Density operator}
The purity of a density operator is $\operatorname{Tr}(\rho^2)$. It equals one exactly for a pure state; in finite dimension $d$ it lies between $1/d$ and $1$.
= Reduced density operator of a weakly coupled oscillator pair
{parent=Density operator}
If a weak interaction changes $|0,0\rangle$ to a normalized state proportional to $|0,0\rangle-g|1,1\rangle$, tracing out either oscillator gives eigenvalues $1/(1+g^2)$ and $g^2/(1+g^2)$.
= Bloch vector
{c}
{parent=Density operator}
{wiki=Bloch_sphere}
Every qubit state has the unique form
$$
\rho=\frac12(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z),
\qquad |r|\leq1.
$$
It is pure exactly when $|r|=1$, and
$r_i=2\langle S_i\rangle/\hbar$.
= Informationally complete three-observable qubit tomography
{parent=Bloch vector}
Three Hermitian expectation values determine an arbitrary qubit state exactly when the traceless parts of the three observables span the three-dimensional space generated by the Pauli matrices. Mere linear independence as Hermitian matrices is insufficient: $I,\sigma_x,\sigma_z$ are independent but cannot detect the sign or magnitude of the Bloch $y$ component.
= Quantum no-signalling
{parent=Quantum theory}
{wiki=No-communication_theorem}
An uncommunicated local trace-preserving operation cannot change the other subsystem's reduced state.
= Quantum cloning
{parent=Quantum theory}
{wiki=No-cloning_theorem}
A cloning operation for a family of pure states would map
$$
|\psi\rangle|0\rangle\longmapsto|\psi\rangle|\psi\rangle
$$
for every state in that family. A single physical operation cannot clone every unknown quantum state.
= No-cloning theorem for two pure states
{parent=Quantum cloning}
No <unitary operator> $U$ can satisfy
$$
U|c_j\rangle|0\rangle=|c_j\rangle|c_j\rangle,
\qquad j=0,1,
$$
for two distinct nonorthogonal pure states. Taking the inner product of these two equations would require
$$
\langle c_0|c_1\rangle
=\langle c_0|c_1\rangle^2,
$$
which is impossible when $0<|\langle c_0|c_1\rangle|<1$.
= Clone-assisted asymptotic state discrimination
{parent=Quantum cloning}
If a device produces arbitrarily many copies of either of two distinct pure-state rays, their $N$-copy overlap is
$$
|\langle\phi|\psi\rangle|^N\longrightarrow0.
$$
The Helstrom measurement on the copies therefore distinguishes the alternatives with success probability tending to one.
= Perfect discrimination implies cloning for a known state family
{parent=Quantum cloning}
If a device perfectly identifies which member of a known finite state family was supplied, its classical output can control state-preparation unitaries that prepare any requested number of fresh copies. The discrimination may destroy the supplied system.
= Environment-assisted two-state pure-state transformation
{parent=Quantum theory}
There are a unitary $U$ and normalized environment states $|e_i\rangle$ satisfying
$$
U|\phi_i\rangle|0\rangle=|\psi_i\rangle|e_i\rangle,
\qquad i=0,1,
$$
exactly when
$$
|\langle\phi_0|\phi_1\rangle|
\leq|\langle\psi_0|\psi_1\rangle|.
$$
Necessity follows from preservation of inner products. For sufficiency, choose $\langle e_0|e_1\rangle$ to make the input and output inner products equal; equal two-vector Gram matrices define an isometry that extends to a unitary.
= Helstrom-Holevo bound
{c}
{parent=Quantum theory}
{wiki=Helstrom_measurement}
Equiprobable pure states of overlap magnitude $c$ have optimal discrimination probability $\frac12(1+\sqrt{1-c^2})$.
= Helstrom measurement for two pure states
{parent=Helstrom-Holevo bound}
{c}
For $\rho_j=|\alpha_j\rangle\langle\alpha_j|$, projecting onto the positive and negative eigenspaces of $\rho_0-\rho_1$ attains
$$
P_s^{\rm opt}=\frac12+\frac14\|\rho_0-\rho_1\|_1
=\frac12\left(1+\sqrt{1-|\langle\alpha_0|\alpha_1\rangle|^2}\right).
$$
= Helstrom measurement for the zero and plus states
{parent=Helstrom measurement for two pure states}
{c}
For equiprobable states $|0\rangle$ and
$$
|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2},
$$
the orthonormal measurement basis at angle $\beta=-\pi/8$ to the computational basis attains
$$
P_s=\frac12\left(1+\frac1{\sqrt2}\right).
$$
= Perfect distinguishability of pure states
{parent=Helstrom-Holevo bound}
Two pure states can be distinguished with certainty in one measurement exactly when their inner product is zero.
= Unitary gate discrimination
{parent=Quantum theory}
To distinguish two gates in one use without an ancilla, choose an input $|\psi\rangle$ and discriminate the output states $U_1|\psi\rangle$ and $U_2|\psi\rangle$.
= Single-use perfect discrimination of two unitary gates
{parent=Unitary gate discrimination}
Perfect discrimination is possible exactly when some unit vector satisfies
$$\langle\psi|U_2^\dagger U_1|\psi\rangle=0.$$
= Numerical range
{parent=Unitary gate discrimination}
{wiki=Numerical_range}
The numerical range of an operator $A$ is
$$N(A)=\{\langle\psi|A|\psi\rangle:\|\psi\|=1\}.$$
= Numerical range of a normal matrix
{parent=Numerical range}
For a finite-dimensional normal matrix, the numerical range is the convex hull of its eigenvalues.
= Unit-circle spectrum of a unitary operator
{parent=Unitary gate discrimination}
If $Uv=\lambda v$ and $U$ is unitary, norm preservation gives $|\lambda|=1$.
= Spectral arc length of a unitary operator
{parent=Unitary gate discrimination}
The spectral arc length $\theta(U)$ is the length of the shortest closed unit-circle arc containing every eigenvalue of $U$.
= Spectral arc length of a phase gate
{parent=Spectral arc length of a unitary operator}
For $U_\gamma=\operatorname{diag}(1,e^{i\gamma})$ with $0\leq\gamma<2\pi$,
$$\theta(U_\gamma)=\min\{\gamma,2\pi-\gamma\}.$$
= Origin in the convex hull of unit-circle points
{parent=Spectral arc length of a unitary operator}
The convex hull of finitely many unit-circle points contains zero exactly when the points do not lie in any open semicircle, equivalently when their shortest containing arc has length at least $\pi$.
= Spectral-arc criterion for perfect unitary discrimination
{parent=Origin in the convex hull of unit-circle points}
Two unitary gates are perfectly distinguishable in one use exactly when
$$\theta(U_2^\dagger U_1)\geq\pi.$$
= Robertson uncertainty principle
{c}
{parent=Quantum theory}
{wiki=Robertson_uncertainty_relation}
$\Delta A\,\Delta B\ge\frac12|\langle[A,B]\rangle|$.
= Quadratic-norm proof of the Heisenberg uncertainty relation
{parent=Robertson uncertainty principle}
For a state with zero position and momentum means, positivity of $\lVert(p-isx)\psi\rVert^2$ for every real $s$ gives
$$
(\Delta p)^2+s^2(\Delta x)^2-s\hbar\geq0.
$$
The discriminant is nonpositive, hence $\Delta x\,\Delta p\geq\hbar/2$.
= Equality case of the Heisenberg uncertainty relation
{parent=Robertson uncertainty principle}
Equality requires the centred vectors $x'\psi$ and $p'\psi$ to be imaginary scalar multiples. Thus, for some $s>0$,
$$
(p-p_0)\psi=is(x-x_0)\psi.
$$
In the position representation this is a first-order differential equation, whose normalizable solutions are
$$
\psi(x)=C\exp\left(\frac{ip_0x}{\hbar}-\frac{s(x-x_0)^2}{2\hbar}\right).
$$
Hence the states saturating the position-momentum uncertainty relation are Gaussian wave packets, up to translation, modulation, and phase.
= Gaussian eigenstate for a quadratic potential
{parent=Quantum theory}
For $U(x)=kx^2$ and $k>0$, the ansatz $\psi=Ce^{-\alpha x^2}$ solves the stationary Schrödinger equation when
$$
\alpha=\frac1\hbar\sqrt{\frac{mk}{2}},
\qquad
E=\hbar\sqrt{\frac{k}{2m}},
\qquad
C=\left(\frac{2\alpha}{\pi}\right)^{1/4}.
$$
It has $\Delta x=1/(2\sqrt\alpha)$ and $\Delta p=\hbar\sqrt\alpha$, so it saturates the Heisenberg bound.
= Coherent state
{parent=Quantum theory}
{wiki}
$|\alpha\rangle=D(\alpha)|0\rangle$ is normalized, satisfies $A|\alpha\rangle=\alpha|\alpha\rangle$, and evolves to a phase times $|\alpha e^{-i\omega t}\rangle$.
= Born approximation
{c}
{parent=Quantum theory}
{wiki}
First-order Lippmann-Schwinger iteration gives
$$
f(\mathbf k',\mathbf k)=-\frac{m}{2\pi\hbar^2}\widetilde V(\mathbf k'-\mathbf k).
$$
= Toffoli gate
{parent=Quantum theory}
{c}
{wiki}
The Toffoli, or controlled-controlled-NOT, gate maps $|x_1x_2y\rangle$ to $|x_1x_2,y\mathbin\oplus x_1x_2\rangle$.
= Boolean quantum oracle
{parent=Quantum theory}
{c}
For a Boolean function $f$, its standard oracle acts by
$$U_f|x\rangle|y\rangle=|x\rangle|y\mathbin\oplus f(x)\rangle.$$
= Quantum phase kickback
{parent=Boolean quantum oracle}
{wiki=Phase_kickback}
Because $X| -\rangle=-| -\rangle$, a Boolean oracle with answer qubit $| -\rangle$ acts as
$$U_f|x\rangle| -\rangle=(-1)^{f(x)}|x\rangle| -\rangle.$$
= Bernstein-Vazirani phase kickback
{parent=Quantum phase kickback}
{c}
For $f(x)=a\mathbin\cdot x\mathbin\oplus b$, phase kickback followed by a Walsh-Hadamard transform maps the uniform phase state to
$$
(-1)^b|a\rangle.
$$
The hidden linear string $a$ is therefore recovered with certainty.
= Marked-state phase oracle
{parent=Quantum phase kickback}
If $f$ marks only $x_0$, phase kickback realizes $I_{x_0}=I-2|x_0\rangle\langle x_0|$ on the search register.
= Marked-state phase oracle from a single faulty identity-oracle query
{parent=Marked-state phase oracle}
Suppose $U_f$ agrees with the reversible identity-function oracle except at $x_0$, where its answer differs by a fixed one-bit string $a$. Composing $U_f$ with the known identity oracle and preparing the affected answer qubit in $|-\rangle$ kicks back the phase $-1$ exactly on $|x_0\rangle$, using one query to $U_f$.
= Compute-phase-uncompute construction
{parent=Boolean quantum oracle}
Starting the answer qubit in $|0\rangle$, the sequence
$$
U_f,\qquad Z,\qquad U_f
$$
maps $|x\rangle|0\rangle$ to $(-1)^{f(x)}|x\rangle|0\rangle$. It realizes a phase oracle with two standard-oracle queries while returning the answer register to its initial state.
= Quantum circuit
{parent=Quantum theory}
{wiki}
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.
= Grover search algorithm
{c}
{parent=Quantum theory}
{wiki=Grover%27s_algorithm}
Grover search rotates a uniform state toward a marked subspace and finds one of $M$ marked entries using order $\sqrt{N/M}$ oracle calls.
= Grover diffusion operator
{parent=Grover search algorithm}
{c}
For $|s\rangle=H^{\otimes n}|0^n\rangle$, the diffusion operator is $2|s\rangle\langle s|-I=-H^{\otimes n}I_0H^{\otimes n}$, where $I_0=I-2|0^n\rangle\langle0^n|$.
= Inversion about the mean
{parent=Grover diffusion operator}
The diffusion operator sends every computational-basis amplitude $a_x$ to $2\overline a-a_x$, where $\overline a$ is their arithmetic mean.
= Grover rotation angle
{parent=Grover search algorithm}
{c}
If $M$ of $N$ items are marked and $\sin\theta=\sqrt{M/N}$, each Grover iteration rotates the state by $2\theta$ in the marked--unmarked plane, so after $r$ iterations the marked amplitude is $\sin((2r+1)\theta)$.
= Exact Grover search on four entries
{parent=Grover rotation angle}
For $N=4$ and one marked item, $\theta=\pi/6$. One Grover iteration gives marked amplitude $\sin(3\theta)=1$, so measurement returns the target with certainty.
= Quantum measurement in the computational basis
{parent=Quantum theory}
Measuring $\sum_xa_x|x\rangle$ in the computational basis returns $x$ with probability $|a_x|^2$ and leaves the measured register in $|x\rangle$.
= Deutsch-Jozsa algorithm
{c}
{parent=Quantum theory}
{wiki=Deutsch–Jozsa_algorithm}
The Deutsch-Jozsa algorithm uses phase kickback and interference to distinguish constant from perfectly balanced Boolean functions with one oracle call.
= Deutsch-Jozsa test with an arbitrary uniform-state unitary
{c}
{parent=Deutsch-Jozsa algorithm}
If $F|0\rangle=N^{-1/2}\sum_i|i\rangle$, then applying a phase oracle followed by $F^{-1}$ gives amplitude
$$
\frac1N\sum_i(-1)^{f(i)}
$$
on $|0\rangle$. It has modulus one for a constant Boolean function and vanishes for a balanced one.
= Quantum Fourier transform
{title2=$\operatorname{QFT}_N$}
{parent=Quantum theory}
{wiki}
The quantum Fourier transform maps $|k\rangle$ to $N^{-1/2}\sum_j e^{2\pi ijk/N}|j\rangle$.
= Square of the quantum Fourier transform
{parent=Quantum Fourier transform}
The <root-of-unity filter> gives
$$
\operatorname{QFT}_N^2|x\rangle=|-x\bmod N\rangle.
$$
Thus $\operatorname{QFT}_N^4=I$, while the square is an involution whose eigenvalues lie in $\{1,-1\}$.
= Quantum Fourier transform of a periodic coset state
{parent=Quantum Fourier transform}
Let $r$ divide $N$ and
$$
|\alpha\rangle=\frac1{\sqrt{N/r}}
\sum_{j=0}^{N/r-1}|x_0+jr\rangle.
$$
The geometric sum in the quantum Fourier transform vanishes unless $c$ is a multiple of $N/r$. Thus measurement after the transform returns the $r$ values
$$
c=0,\frac Nr,\ldots,(r-1)\frac Nr
$$
with equal probabilities $1/r$.
= Quantum period finding
{parent=Quantum Fourier transform}
For a function of period $r$ dividing $N$, prepare a uniform superposition of inputs, evaluate the function into a second register, and measure that register. The first register becomes a periodic coset state. Applying the <quantum Fourier transform> and measuring gives
$$
c=s\frac Nr,
\qquad 0\leq s<r.
$$
= Exact period recovery from a Fourier sample
{parent=Quantum period finding}
For the exact sample $c=sN/r$, reducing
$$
\frac cN=\frac sr
$$
recovers denominator $r/\gcd(s,r)$. Thus one sample reveals the full period exactly when $s$ is coprime to $r$; otherwise it reveals only a proper divisor and the algorithm must obtain more information.
= Heralded exact quantum period finding when the period divides the register size
{parent=Exact period recovery from a Fourier sample}
Suppose a function on $\mathbb Z_N$ has least period $r\mid N$ and is injective on each period. A Fourier sample $c=sN/r$ gives the candidate denominator $q=r/\gcd(s,r)$. If equality of function values can be tested efficiently, then $q$ is the full period exactly when $f(q)=f(0)$, so each successful run is certified. A sample succeeds with probability $\varphi(r)/r$, and an inverse-polylogarithmic lower bound for this ratio permits amplification to constant success probability with polynomially many repetitions.
= Cyclic shift diagonalization by the quantum Fourier transform
{parent=Quantum Fourier transform}
For the cyclic shift $S|j\rangle=|j+1\bmod N\rangle$ and the convention
$$
\operatorname{QFT}_N|k\rangle=N^{-1/2}\sum_j e^{2\pi ijk/N}|j\rangle,
$$
the Fourier states obey
$$
S\operatorname{QFT}_N|k\rangle=e^{-2\pi ik/N}\operatorname{QFT}_N|k\rangle.
$$
Consequently $S=\operatorname{QFT}_N D\operatorname{QFT}_N^{-1}$, where $D|k\rangle=e^{-2\pi ik/N}|k\rangle$.
= Exact quantum phase estimation
{parent=Quantum Fourier transform}
If $U|\psi\rangle=e^{2\pi ij/N}|\psi\rangle$, controlled powers of $U$ applied to a uniform $N$-state control register produce
$$
|\phi_j\rangle=\frac1{\sqrt N}\sum_{k=0}^{N-1}e^{2\pi ijk/N}|k\rangle.
$$
These states are the Fourier basis. Applying the inverse quantum Fourier transform and measuring returns $j$ with probability one.
= BB84 protocol
{c}
{parent=Quantum theory}
{wiki=BB84}
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.
= Breidbart basis
{c}
{parent=Quantum theory}
{wiki=BB84}
The Breidbart basis bisects the computational and diagonal qubit bases. For equiprobable $|0\rangle$ and $|-\rangle$, whose overlap magnitude is $1/\sqrt2$, it realizes the Helstrom measurement and succeeds with probability
$$
\frac12\left(1+\frac1{\sqrt2}\right)=\cos^2\frac\pi8.
$$
= Bell state
{title2=$|\Phi^\pm\rangle,|\Psi^\pm\rangle$}
{c}
{parent=Quantum theory}
{wiki}
The four Bell states form an orthonormal maximally entangled basis for two qubits.
= Reduced density matrix
{title2=$\rho_A$}
{parent=Bell state}
{wiki}
For a bipartite density operator $\rho_{AB}$, the reduced state of subsystem $A$ is the partial trace $\rho_A=\operatorname{Tr}_B\rho_{AB}$. Every measurement performed only on $A$ has outcome probabilities determined entirely by $\rho_A$.
= Product state
{parent=Reduced density matrix}
{wiki}
A bipartite pure state is a product state when it factors as $|\psi_A\rangle\otimes|\psi_B\rangle$. Its reduced density matrices have rank one and zero <entanglement entropy>.
= Entangled state
{parent=Reduced density matrix}
{wiki=Quantum_entanglement}
A bipartite pure state is entangled when it cannot be written as a product of one state for each subsystem. Equivalently, either reduced density matrix has rank greater than one.
= Entanglement criterion for a two-term correlated state
{parent=Entangled state}
The state $a|00\rangle+b|11\rangle$ is entangled exactly when $ab\ne0$. Its reduced density matrix has nonzero eigenvalues $|a|^2$ and $|b|^2$.
= Entanglement concentration
{parent=Entangled state}
{wiki}
Entanglement concentration probabilistically converts partially entangled pure states into fewer maximally entangled states by local operations and classical communication.
= Local indistinguishability of Bell-state phase
{parent=Reduced density matrix}
The Bell states $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$ and $|\Phi^-\rangle=(|00\rangle-|11\rangle)/\sqrt2$ both have reduced state $I/2$ on either qubit. No measurement on one qubit alone can distinguish their relative sign.
= Local operations and classical communication
{title2=LOCC}
{parent=Bell state}
{wiki}
Local operations and classical communication allow separated parties to measure or transform their own subsystems and exchange ordinary classical messages.
= LOCC discrimination of two Bell states
{c}
{parent=Local operations and classical communication}
Two parties can distinguish $|\Phi^+\rangle$ from $|\Phi^-\rangle$ by both measuring in the Hadamard basis and comparing outcomes: equal outcomes identify $|\Phi^+\rangle$, while unequal outcomes identify $|\Phi^-\rangle$.
= Quantum teleportation
{parent=Local operations and classical communication}
{wiki}
Quantum teleportation transfers an unknown qubit using one shared Bell pair, a Bell-basis measurement by the sender, two classical bits, and a Pauli correction by the receiver.
= No-programming theorem
{parent=Quantum teleportation}
{wiki=No-programming_theorem}
A deterministic finite-dimensional universal programmable quantum gate cannot implement every unitary exactly: programs for physically distinct unitaries would have to be mutually orthogonal, but a finite-dimensional program register contains only finitely many mutually orthogonal states.
= Teleportation with the psi-plus Bell state
{parent=Quantum teleportation}
When the shared resource is $|\Psi^+\rangle$, Bell outcomes $\Phi^+,\Phi^-,\Psi^+,\Psi^-$ leave the receiver with $X|\alpha\rangle,XZ|\alpha\rangle,|\alpha\rangle,Z|\alpha\rangle$, respectively, up to global phases.
= Quantum dense coding
{parent=Bell state}
{wiki=Superdense_coding}
Quantum dense coding uses one shared Bell pair and one transmitted qubit to communicate two classical bits. Alice applies one of four Pauli operators to her qubit, producing four orthogonal Bell states that Bob can distinguish jointly. The transmitted qubit alone is maximally mixed for every message and reveals no information to an interceptor.
= Variational method
{parent=Quantum theory}
{wiki}
The Rayleigh quotient of any normalized trial state is at least the ground-state energy.
= Nodeless theorem for a one-dimensional ground state
{parent=Variational method}
For a regular one-dimensional confining potential, the bound-state eigenfunctions can be ordered by their number of nodes. The unique nodeless normalizable eigenfunction is the ground state.
= Exact ground state of a solvable sextic potential
{parent=Variational method}
For
$$
V(x)=\frac{\hbar^2}{2m}(x^6-3x^2+2),
$$
the nodeless wavefunction $e^{-x^4/4}$ is an exact eigenstate with energy $\hbar^2/m$, and hence is the ground state.
= Gaussian variational estimate for a solvable sextic potential
{parent=Variational method}
For the trial state $e^{-\alpha x^2/2}$, the dimensionless Rayleigh quotient is
$$
\varepsilon(\alpha)=2+\frac\alpha2-\frac3{2\alpha}+\frac{15}{8\alpha^3}.
$$
Its unique minimizer is
$$
\alpha_*^2=\frac{-3+\sqrt{54}}2,
$$
and at the minimum $\varepsilon(\alpha_*)=2+2\alpha_*/3-1/\alpha_*$.
= Ground-state energy monotonicity under potential ordering
{parent=Variational method}
If $V_1\geq V_2$ pointwise for Hamiltonians with the same kinetic term, using the ground state of $H_1$ as a trial state for $H_2$ gives $E_1\geq E_2$.
= Quantum virial identity from scaling
{parent=Variational method}
For a normalized dilation $\psi_\lambda(x)=\lambda^{d/2}\psi(\lambda x)$ and a homogeneous potential of degree $-n$, the energy is $\lambda^2T+\lambda^nV$. Stationarity at an eigenstate gives $2T+nV=0$.
= Fall to the centre for a supercritical inverse-power potential
{parent=Quantum virial identity from scaling}
For an attractive potential $-\alpha/|x|^n$ with $n>2$, dilations make the negative potential energy grow faster than kinetic energy, sending the Rayleigh quotient to minus infinity.
= Exact variational solution in a two-level system
{parent=Variational method}
If a normalized trial family covers the entire two-dimensional Hilbert space, minimizing its Rayleigh quotient gives the exact lower eigenvalue rather than merely an upper bound.
= Level repulsion in a two-level Hamiltonian
{parent=Exact variational solution in a two-level system}
The eigenvalues of $\left(\begin{smallmatrix}E_1&h\\h&E_2\end{smallmatrix}\right)$ are their mean plus or minus $\sqrt{(E_2-E_1)^2/4+h^2}$, so nonzero coupling increases their separation.
= Landau level
{c}
{parent=Quantum theory}
{wiki}
A charged particle in a uniform magnetic field has equally spaced orbital energy levels with guiding-centre degeneracy.
= Magnetic minimal coupling
{parent=Landau level}
{wiki=Minimal_coupling}
For a particle of charge $q$ in a vector potential $\mathbf A$, the canonical momentum in the free Hamiltonian is replaced by $\mathbf p-q\mathbf A$, giving
$$
H=\frac1{2m}(-i\hbar\nabla-q\mathbf A)^2.
$$
= Cyclotron frequency
{title2=$\omega_c=|qB|/m$}
{parent=Landau level}
{wiki}
A particle of charge $q$ and mass $m$ in a uniform magnetic field of magnitude $B$ has cyclotron angular frequency $\omega_c=|qB|/m$.
= Landau gauge
{parent=Landau level}
{c}
{wiki}
For $\mathbf B=B\mathbf e_z$, the vector potential $\mathbf A=(0,Bx,0)$ is a Landau gauge. Translation invariance in $y$ turns the transverse Hamiltonian into a family of harmonic oscillators in $x$ whose centres are labelled by the conserved $y$-momentum.
= Degeneracy of a Landau level
{parent=Landau level}
In a planar region of area $A$, each spinless Landau level has guiding-centre degeneracy
$$
D=\frac{|qB|A}{2\pi\hbar}=\frac{|\Phi|}{h/|q|}.
$$
Magnetic periodic boundary conditions require this flux count to be an integer.
= Spin splitting of Landau levels
{parent=Landau level}
For a spin-one-half particle with the Pauli magnetic term and gyromagnetic factor two, the orbital energies $\hbar\omega_c(n+1/2)$ split into $\hbar\omega_c n$ and $\hbar\omega_c(n+1)$. The lowest level has one spin branch; every positive level has two.
= Landau levels in crossed electric and magnetic fields
{parent=Landau level}
{c}
A uniform electric field perpendicular to a uniform magnetic field shifts each cyclotron oscillator's guiding centre and makes its energy depend linearly on the conserved momentum. The formerly degenerate Landau levels therefore become tilted bands.
= Symmetric gauge
{parent=Landau level}
{wiki}
For a constant magnetic field,
$$
A(x)=\frac12B\times x
$$
is the symmetric gauge.
= Kinetic momentum and magnetic pseudomomentum
{parent=Symmetric gauge}
For charge $e$, define
$$
\pi=p-eA,\qquad K=p+eA.
$$
In the symmetric gauge,
$$
[K_i,\pi_j]=0,
\qquad
[K_i,K_j]=-ie\hbar\varepsilon_{ijk}B_k.
$$
= Magnetic translation operator
{parent=Kinetic momentum and magnetic pseudomomentum}
{wiki=Magnetic_translation}
For a lattice vector $r$,
$$
\mathcal T_r=\exp\left(\frac i\hbar r\cdot K\right)
$$
acts in the symmetric gauge as
$$
(\mathcal T_r\psi)(x)
=\exp\left(\frac{ie}{2\hbar}r\cdot(B\times x)\right)\psi(x+r).
$$
It commutes with $(p-eA)^2/(2m)+V(x)$ when $V(x+r)=V(x)$.
= Magnetic translation algebra
{parent=Magnetic translation operator}
For constant $B$,
$$
\mathcal T_r\mathcal T_{r'}
=\exp\left(\frac{ie}{\hbar}(r\times r')\cdot B\right)
\mathcal T_{r'}\mathcal T_r.
$$
= Magnetic flux quantum
{parent=Magnetic translation algebra}
{wiki}
Magnetic translations along two lattice vectors commute when their cell flux obeys
$$
\frac{e\Phi}{\hbar}\in2\pi\mathbb Z,
\qquad\text{equivalently}\qquad
\Phi\in\frac he\mathbb Z.
$$
= Bloch theorem
{c}
{parent=Quantum theory}
{wiki=Bloch%27s_theorem}
In a periodic potential, energy eigenstates have the form $e^{ik\cdot x}u_k(x)$ with lattice-periodic $u_k$.
= Periodic potential
{parent=Bloch theorem}
{wiki=Periodic_potential}
A periodic potential satisfies $V(x+a)=V(x)$ for some lattice period $a$. Its Fourier coefficients couple plane waves whose wavevectors differ by reciprocal-lattice vectors.
= Energy band
{parent=Bloch theorem}
{wiki=Electronic_band_structure}
An energy band is a continuous branch $E_j(k)$ of the <dispersion relation> of a periodic quantum system.
= Energy bands
{synonym}
= Nearly-free electron model
{parent=Bloch theorem}
{wiki}
The nearly-free electron model treats a weak <periodic potential> by <degenerate perturbation theory>. At a Bragg crossing, a Fourier coefficient $V_n$ couples the two degenerate plane waves and opens an energy gap of width $2|V_n|$.
= Nearly-free electron dispersion near a one-dimensional band gap
{parent=Nearly-free electron model}
For
$$
V(x)=\sum_{n\in\mathbb Z}V_ne^{2\pi inx/a},
$$
write $k=n\pi/a+\kappa$. Near the $n$th Bragg crossing,
$$
E_\pm(k)=V_0+\frac{\hbar^2}{2m}
\left[\left(\frac{n\pi}{a}\right)^2+\kappa^2\right]
\pm\sqrt{
\left(\frac{\hbar^2n\pi\kappa}{ma}\right)^2+|V_n|^2}.
$$
At $\kappa=0$, the two branches differ by $2|V_n|$.
= Floquet matrix for a one-dimensional periodic potential
{parent=Bloch theorem}
{c}
For a potential of period $a$, the Floquet matrix $M(E)$ maps Cauchy data through one cell:
$$
\begin{pmatrix}\psi(x+a)\\\psi'(x+a)\end{pmatrix}
=M(E)\begin{pmatrix}\psi(x)\\\psi'(x)\end{pmatrix}.
$$
For a real Schrodinger equation, $\det M=1$.
= Floquet discriminant and energy bands
{parent=Floquet matrix for a one-dimensional periodic potential}
{c}
The Floquet multipliers solve
$$
\mu^2-\operatorname{tr}(M)\mu+1=0.
$$
Bounded Bloch waves occur when $|\operatorname{tr}M|\leq2$, with $\mu=e^{\pm iKa}$. Band edges satisfy $\operatorname{tr}M=\pm2$; outside the bands the multipliers are real reciprocal numbers and one solution grows exponentially.
= Kronig-Penney model
{parent=Bloch theorem}
{c}
{wiki=Kronig%E2%80%93Penney_model}
The delta-comb Kronig-Penney potential is
$$
V(x)=V_0\sum_{n\in\mathbb Z}\delta(x-na).
$$
= Floquet discriminant of the delta-comb Kronig-Penney model
{parent=Kronig-Penney model}
{c}
For $E=\hbar^2k^2/(2m)$ and $\gamma=mV_0/(\hbar^2k)$,
$$
\frac12\operatorname{tr}M(E)
=\cos(ka)+\gamma\sin(ka).
$$
Thus the allowed bands satisfy $|\cos(ka)+\gamma\sin(ka)|\leq1$.
= Scattering-amplitude equations for one-dimensional band edges
{parent=Kronig-Penney model}
If one symmetric cell has reflection and transmission amplitudes $r,t$, then
$$
\operatorname{tr}M
=\frac{(t^2-r^2)e^{ika}+e^{-ika}}t.
$$
At a periodic or antiperiodic band edge, with $\operatorname{tr}M=2\sigma$ and $\sigma=\pm1$, setting $z=e^{-ika}$ gives
$$
z^2-2\sigma tz+t^2-r^2=0,
\qquad
z=\sigma t\pm r.
$$
= Tight-binding model
{parent=Bloch theorem}
{wiki=Tight_binding}
A tight-binding model expands a particle's state in localized orbitals and represents tunnelling between sites by off-diagonal Hamiltonian matrix elements.
= Finite periodic tight-binding ring
{parent=Tight-binding model}
For $N$ identical sites of spacing $a$ with periodic boundary conditions, translation eigenstates have coefficients
$$
c_n=\frac{1}{\sqrt N}e^{ikna},
\qquad
e^{ikNa}=1.
$$
Thus $k=2\pi j/(Na)$ modulo the reciprocal-lattice period $2\pi/a$.
= Brillouin zone
{parent=Bloch theorem}
{c}
{wiki=Brillouin_zone}
The first Brillouin zone is a fundamental cell in reciprocal space. For a one-dimensional lattice of spacing $a$, one conventional choice is
$$
-\frac{\pi}{a}\leq k<\frac{\pi}{a}.
$$
= Hadamard gate
{parent=Quantum theory}
{c}
{wiki=Hadamard_transform}
The Hadamard gate maps
$$
|0\rangle\mapsto\frac{|0\rangle+|1\rangle}{\sqrt2},
\qquad
|1\rangle\mapsto\frac{|0\rangle-|1\rangle}{\sqrt2}.
$$
= Pauli Z gate
{title2=$Z$}
{parent=Quantum theory}
{c}
{wiki=Pauli_matrices}
The Pauli $Z$ gate fixes $|0\rangle$ and maps $|1\rangle$ to $-|1\rangle$.
= Pauli X gate
{title2=$X$}
{c}
{parent=Quantum theory}
{wiki=Pauli_matrices}
The Pauli $X$ gate interchanges $|0\rangle$ and $|1\rangle$.
= Inverse quantum circuit
{parent=Quantum theory}
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.
= Controlled unitary gate
{parent=Quantum theory}
{wiki=Controlled_NOT_gate\#Controlled_U_gates}
A controlled-$U$ gate applies $U$ to its target exactly when its control qubit is one.
= Controlled-NOT gate
{parent=Controlled unitary gate}
{wiki}
The controlled-NOT gate maps $|a,b\rangle$ to $|a,b\mathbin\oplus a\rangle$.
= SWAP gate
{c}
{parent=Controlled-NOT gate}
{wiki=Swap_gate}
The SWAP gate maps $|x\rangle|y\rangle$ to $|y\rangle|x\rangle$.
= Three-CNOT decomposition of the SWAP gate
{parent=SWAP gate}
{c}
Applying controlled-NOT gates with directions $1\to2$, $2\to1$, and $1\to2$ maps
$$
(x,y)\mapsto(x,x\oplus y)\mapsto(y,x\oplus y)\mapsto(y,x),
$$
and therefore implements the <SWAP gate>.
= GHZ state
{parent=Controlled-NOT gate}
{c}
{wiki=Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state}
The three-qubit GHZ state is
$$
|\operatorname{GHZ}\rangle
=\frac{|000\rangle+|111\rangle}{\sqrt2}.
$$
= Bell measurement on one qubit and one leg of a GHZ state
{parent=GHZ state}
For $|\alpha\rangle=a|0\rangle+b|1\rangle$, a Bell measurement on $|\alpha\rangle$ and the first leg of a GHZ state leaves the other two legs in $a|00\rangle\pm b|11\rangle$, after applying $X$ to both legs for the two $\Psi$ outcomes.
= Quantum circuit preparation of a three-qubit GHZ state
{parent=GHZ state}
Starting from $|000\rangle$, apply a Hadamard gate to the first qubit and then controlled-NOT gates from the first qubit to each of the other two qubits.
= Three-party dense coding with a GHZ state
{parent=GHZ state}
If two senders share a GHZ state with a receiver, one sender can encode two bits with $X^xZ^z$ and the other one bit with $X^b$. Sending both qubits to the receiver produces one of eight orthogonal GHZ-basis states, so a joint measurement recovers all three bits.
= Hadamard test
{parent=Controlled unitary gate}
{c}
{wiki}
The Hadamard test estimates the real part of $\langle\psi|U|\psi\rangle$ from one ancilla measurement.
= Eigenphase Hadamard-test probability
{parent=Hadamard test}
If $U|\psi\rangle=e^{2\pi i\theta}|\psi\rangle$, the ordinary Hadamard test returns ancilla zero with probability
$$
p_0=\frac{1+\cos(2\pi\theta)}2=\cos^2(\pi\theta).
$$
= Imaginary-part Hadamard test
{parent=Hadamard test}
Adding a phase gate to the Hadamard test rotates the interference signal and measures the imaginary part of $\langle\psi|U|\psi\rangle$.
= Swap test
{parent=Controlled unitary gate}
{wiki=Swap_test}
The swap test on pure states $|a\rangle,|b\rangle$ returns ancilla zero with probability
$$
p_0=\frac12\left(1+|\langle a|b\rangle|^2\right).
$$
= Parallel quantum gates
{parent=Quantum theory}
Independent gates $A$ and $B$ on separate registers act jointly as $A\otimes B$.
= Measurement of a Pauli observable
{parent=Quantum theory}
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.
= Born rule for a product-basis measurement
{parent=Quantum theory}
{c}
For a bipartite state $|\psi\rangle$, measurement outcome $(x_j,y_k)$ has probability $|\langle x_jy_k|\psi\rangle|^2$.
= Matching-outcome projector
{parent=Born rule for a product-basis measurement}
For two labelled orthonormal bases, the event of equal labels is represented by $\sum_j|x_jy_j\rangle\langle x_jy_j|$.
= Product-state factorization of measurement probabilities
{parent=Born rule for a product-basis measurement}
For a product state, probabilities of local measurement outcomes multiply.
= Maximally entangled state
{parent=Quantum theory}
{wiki}
A bipartite pure state is maximally entangled when either reduced density matrix is maximally mixed.
= Orthogonal-basis invariance of a real Bell state
{parent=Maximally entangled state}
For real orthogonal $U$, the Bell state $2^{-1/2}\sum_j|jj\rangle$ is invariant under $U\otimes U$ because $UU^T=I$.
= Bell-state correlation in two real bases
{parent=Maximally entangled state}
{c}
For the real Bell state, the joint amplitude in real basis vectors $x,y$ equals $\langle x|y\rangle/\sqrt2$.
= Shor algorithm
{parent=Quantum theory}
{c}
{wiki=Shor%27s_algorithm}
Shor's factoring algorithm uses quantum period finding to determine a modular order and then extracts factors classically with greatest common divisors.
= Factor extraction from an even modular order
{parent=Shor algorithm}
If $r$ is even, $a^r\equiv1\pmod N$, and $a^{r/2}\not\equiv-1\pmod N$, then $\gcd(a^{r/2}-1,N)$ is a nontrivial factor unless the order was not minimal.
= Quantum order finding
{parent=Shor algorithm}
Quantum order finding Fourier-transforms a superposition of arguments having the same modular-exponentiation value, producing peaks near multiples of the reciprocal period.
= Fourier transform of a finite periodic comb
{parent=Quantum order finding}
{c}
The Fourier amplitude of equally spaced basis states is a finite geometric sum with peaks where the Fourier phase increment is nearly one.
= Finite geometric Fourier amplitude
{parent=Fourier transform of a finite periodic comb}
For $A$ terms with phase ratio $q$, the amplitude factor is $(1-q^A)/(1-q)$, with limiting value $A$ when $q=1$.
= Continued-fraction recovery in quantum order finding
{parent=Quantum order finding}
If a measured ratio approximates a reduced $k/r$ within $1/(2r^2)$, continued-fraction convergents recover the candidate denominator $r$.
= Uniqueness of a rational approximation with bounded denominator
{parent=Continued-fraction recovery in quantum order finding}
Two distinct reduced fractions with denominators below $N$ differ by more than $1/N^2$, so an interval of radius $1/(2N^2)$ contains at most one.