Quantum theory

For a particle of charge $-e$, the changes $A\mapsto A+\nabla f$ and $\varphi \mapsto \varphi -{\partial }_{t}f$ leave the minimally coupled Schrödinger equation invariant when $\psi \mapsto e^{-ief/\hslash }\psi$.

A quantum measurement assigns probabilities to classical outcomes through positive operators that sum to the identity. A projective measurement uses mutually orthogonal projections.

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 isso $O(1/\sin \theta )$ iterations raise a small good amplitude close to one.

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 iswhen ${\rho }_H$ denotes the time-independent Heisenberg-picture state.

The Von Neumann entropy of a density operator isIt vanishes for a pure state and measures mixedness and quantum entanglement in reduced states.

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.

The purity of a density operator is $\mathrm{Tr}({\rho }^2)$. It equals one exactly for a pure state; in finite dimension $d$ it lies between $1/d$ and $1$.

If a weak interaction changes $|0,0⟩$ to a normalized state proportional to $|0,0⟩-g|1,1⟩$, tracing out either oscillator gives eigenvalues $1/(1+g^2)$ and $g^2/(1+g^2)$.

Every qubit state has the unique formIt is pure exactly when $|r|=1$, and
$r_i=2⟨S_i⟩/\hslash$.

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.

An uncommunicated local trace-preserving operation cannot change the other subsystem's reduced state.

A cloning operation for a family of pure states would mapfor every state in that family. A single physical operation cannot clone every unknown quantum state.

No unitary operator $U$ can satisfyfor two distinct nonorthogonal pure states. Taking the inner product of these two equations would requirewhich is impossible when $0<|⟨c_0|c_1⟩|<1$.

If a device produces arbitrarily many copies of either of two distinct pure-state rays, their $N$-copy overlap isThe Helstrom measurement on the copies therefore distinguishes the alternatives with success probability tending to one.

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.

There are a unitary $U$ and normalized environment states $|e_i⟩$ satisfyingexactly whenNecessity follows from preservation of inner products. For sufficiency, choose $⟨e_0|e_1⟩$ to make the input and output inner products equal; equal two-vector Gram matrices define an isometry that extends to a unitary.

Equiprobable pure states of overlap magnitude $c$ have optimal discrimination probability $\frac{1}{2}(1+\sqrt{1-c^2})$.

For ${\rho }_j=|{\alpha }_j⟩⟨{\alpha }_j|$, projecting onto the positive and negative eigenspaces of ${\rho }_0-{\rho }_1$ attains

For equiprobable states $|0⟩$ andthe orthonormal measurement basis at angle $\beta =-\pi /8$ to the computational basis attains

Two pure states can be distinguished with certainty in one measurement exactly when their inner product is zero.

To distinguish two gates in one use without an ancilla, choose an input $|\psi ⟩$ and discriminate the output states $U_1|\psi ⟩$ and $U_2|\psi ⟩$.

Perfect discrimination is possible exactly when some unit vector satisfies

The numerical range of an operator $A$ is

For a finite-dimensional normal matrix, the numerical range is the convex hull of its eigenvalues.

If $Uv=\lambda v$ and $U$ is unitary, norm preservation gives $|\lambda |=1$.

The spectral arc length $\theta (U)$ is the length of the shortest closed unit-circle arc containing every eigenvalue of $U$.

For $U_{\gamma }=\mathrm{diag}(1,e^{i\gamma })$ with $0\le \gamma <2\pi$,

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 $\pi$.

Two unitary gates are perfectly distinguishable in one use exactly when

$\Delta A\Delta B\ge \frac{1}{2}|⟨[A,B]⟩|$.

For a state with zero position and momentum means, positivity of $\Vert (p-isx)\psi {\Vert }^2$ for every real $s$ givesThe discriminant is nonpositive, hence $\Delta x\Delta p\ge \hslash /2$.

Equality requires the centred vectors $x^{\prime }\psi$ and $p^{\prime }\psi$ to be imaginary scalar multiples. Thus, for some $s>0$,In the position representation this is a first-order differential equation, whose normalizable solutions areHence the states saturating the position-momentum uncertainty relation are Gaussian wave packets, up to translation, modulation, and phase.

For $U(x)=k{x}^2$ and $k>0$, the ansatz $\psi =C{e}^{-\alpha x^2}$ solves the stationary Schrödinger equation whenIt has $\Delta x=1/(2\sqrt{\alpha })$ and $\Delta p=\hslash \sqrt{\alpha }$, so it saturates the Heisenberg bound.

$|\alpha ⟩=D(\alpha )|0⟩$ is normalized, satisfies $A|\alpha ⟩=\alpha |\alpha ⟩$, and evolves to a phase times $|\alpha e^{-i\omega t}⟩$.

First-order Lippmann-Schwinger iteration gives

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

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

Because $X|-⟩=-|-⟩$, a Boolean oracle with answer qubit $|-⟩$ acts as

For $f(x)=a\cdot x\oplus b$, phase kickback followed by a Walsh-Hadamard transform maps the uniform phase state toThe hidden linear string $a$ is therefore recovered with certainty.

If $f$ marks only $x_0$, phase kickback realizes $I_{x_0}=I-2|x_0⟩⟨x_0|$ on the search register.

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 $|-⟩$ kicks back the phase $-1$ exactly on $|x_0⟩$, using one query to $U_f$.

Starting the answer qubit in $|0⟩$, the sequencemaps $|x⟩|0⟩$ to $(-1{)}^{f(x)}|x⟩|0⟩$. It realizes a phase oracle with two standard-oracle queries while returning the answer register to its initial state.

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

For $|s⟩=H^{\otimes n}|0^n⟩$, the diffusion operator is $2|s⟩⟨s|-I=-H^{\otimes n}{I}_0{H}^{\otimes n}$, where $I_0=I-2|0^n⟩⟨0^n|$.

The diffusion operator sends every computational-basis amplitude $a_x$ to $2\overline{a}-a_x$, where $\overline{a}$ is their arithmetic mean.

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 $\sin ((2r+1)\theta )$.

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.

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

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

If $F|0⟩=N^{-1/2}{\sum }_i|i⟩$, then applying a phase oracle followed by $F^{-1}$ gives amplitudeon $|0⟩$. It has modulus one for a constant Boolean function and vanishes for a balanced one.

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

The root-of-unity filter givesThus ${\mathrm{QFT}}_N^4=I$, while the square is an involution whose eigenvalues lie in $\{1,-1\}$.

Let $r$ divide $N$ andThe geometric sum in the quantum Fourier transform vanishes unless $c$ is a multiple of $N/r$. Thus measurement after the transform returns the $r$ valueswith equal probabilities $1/r$.

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

For the exact sample $c=sN/r$, reducingrecovers 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.

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 $\phi (r)/r$, and an inverse-polylogarithmic lower bound for this ratio permits amplification to constant success probability with polynomially many repetitions.

For the cyclic shift $S|j⟩=|j+1\mathrm{mod}N⟩$ and the conventionthe Fourier states obeyConsequently $S={\mathrm{QFT}}_{N}D{\mathrm{QFT}}_N^{-1}$, where $D|k⟩=e^{-2\pi ik/N}|k⟩$.

If $U|\psi ⟩=e^{2\pi ij/N}|\psi ⟩$, controlled powers of $U$ applied to a uniform $N$-state control register produceThese states are the Fourier basis. Applying the inverse quantum Fourier transform and measuring returns $j$ with probability one.

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

For a bipartite density operator ${\rho }_{AB}$, the reduced state of subsystem $A$ is the partial trace ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$. Every measurement performed only on $A$ has outcome probabilities determined entirely by ${\rho }_A$.

A bipartite pure state is a product state when it factors as $|{\psi }_A⟩\otimes |{\psi }_B⟩$. Its reduced density matrices have rank one and zero entanglement entropy.

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.

The state $a|00⟩+b|11⟩$ is entangled exactly when $ab\ne 0$. Its reduced density matrix has nonzero eigenvalues $|a{|}^2$ and $|b{|}^2$.

Entanglement concentration probabilistically converts partially entangled pure states into fewer maximally entangled states by local operations and classical communication.

The Bell states $|{\Phi }^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$ and $|{\Phi }^{-}⟩=(|00⟩-|11⟩)/\sqrt{2}$ 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 allow separated parties to measure or transform their own subsystems and exchange ordinary classical messages.

Two parties can distinguish $|{\Phi }^{+}⟩$ from $|{\Phi }^{-}⟩$ by both measuring in the Hadamard basis and comparing outcomes: equal outcomes identify $|{\Phi }^{+}⟩$, while unequal outcomes identify $|{\Phi }^{-}⟩$.

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.

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.

When the shared resource is $|{\Psi }^{+}⟩$, Bell outcomes ${\Phi }^{+},{\Phi }^{-},{\Psi }^{+},{\Psi }^{-}$ leave the receiver with $X|\alpha ⟩,XZ|\alpha ⟩,|\alpha ⟩,Z|\alpha ⟩$, respectively, up to global phases.

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.

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

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.

Forthe nodeless wavefunction $e^{-x^4/4}$ is an exact eigenstate with energy ${\hslash }^2/m$, and hence is the ground state.

For the trial state $e^{-\alpha x^2/2}$, the dimensionless Rayleigh quotient isIts unique minimizer isand at the minimum $\epsilon ({\alpha }_{\ast })=2+2{\alpha }_{\ast }/3-1/{\alpha }_{\ast }$.

If $V_1\ge V_2$ pointwise for Hamiltonians with the same kinetic term, using the ground state of $H_1$ as a trial state for $H_2$ gives $E_1\ge E_2$.

For a normalized dilation ${\psi }_{\lambda }(x)={\lambda }^{d/2}\psi (\lambda x)$ and a homogeneous potential of degree $-n$, the energy is ${\lambda }^{2}T+{\lambda }^{n}V$. Stationarity at an eigenstate gives $2T+nV=0$.

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.

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.

The eigenvalues of $\left(\begin{array}{cc}{E}_1& h\\ h& E_2\end{array}\right)$ are their mean plus or minus $\sqrt{(E_2-E_1{)}^2/4+h^2}$, so nonzero coupling increases their separation.

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

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

A particle of charge $q$ and mass $m$ in a uniform magnetic field of magnitude $B$ has cyclotron angular frequency ${\omega }_c=|qB|/m$.

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.

In a planar region of area $A$, each spinless Landau level has guiding-centre degeneracyMagnetic periodic boundary conditions require this flux count to be an integer.

For a spin-one-half particle with the Pauli magnetic term and gyromagnetic factor two, the orbital energies $\hslash {\omega }_c(n+1/2)$ split into $\hslash {\omega }_{c}n$ and $\hslash {\omega }_c(n+1)$. The lowest level has one spin branch; every positive level has two.

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.

For a constant magnetic field,is the symmetric gauge.

For charge $e$, defineIn the symmetric gauge,

For a lattice vector $r$,acts in the symmetric gauge asIt commutes with $(p-eA{)}^2/(2m)+V(x)$ when $V(x+r)=V(x)$.

For constant $B$,

Magnetic translations along two lattice vectors commute when their cell flux obeys

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

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.

An energy band is a continuous branch $E_j(k)$ of the dispersion relation of a periodic quantum system.

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 $2|V_n|$.

Forwrite $k=n\pi /a+\kappa$. Near the $n$th Bragg crossing,At $\kappa =0$, the two branches differ by $2|V_n|$.

For a potential of period $a$, the Floquet matrix $M(E)$ maps Cauchy data through one cell:For a real Schrodinger equation, $\det M=1$.

The Floquet multipliers solveBounded Bloch waves occur when $|\mathrm{tr}M|\le 2$, with $\mu =e^{\pm iKa}$. Band edges satisfy $\mathrm{tr}M=\pm 2$; outside the bands the multipliers are real reciprocal numbers and one solution grows exponentially.

The delta-comb Kronig-Penney potential is

For $E={\hslash }^2{k}^2/(2m)$ and $\gamma =m{V}_0/({\hslash }^{2}k)$,Thus the allowed bands satisfy $|\cos (ka)+\gamma \sin (ka)|\le 1$.

If one symmetric cell has reflection and transmission amplitudes $r,t$, thenAt a periodic or antiperiodic band edge, with $\mathrm{tr}M=2\sigma$ and $\sigma =\pm 1$, setting $z=e^{-ika}$ gives

A tight-binding model expands a particle's state in localized orbitals and represents tunnelling between sites by off-diagonal Hamiltonian matrix elements.

For $N$ identical sites of spacing $a$ with periodic boundary conditions, translation eigenstates have coefficientsThus $k=2\pi j/(Na)$ modulo the reciprocal-lattice period $2\pi /a$.

The first Brillouin zone is a fundamental cell in reciprocal space. For a one-dimensional lattice of spacing $a$, one conventional choice is

The Hadamard gate maps

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

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

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.

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

The controlled-NOT gate maps $|a,b⟩$ to $|a,b\oplus a⟩$.

The SWAP gate maps $|x⟩|y⟩$ to $|y⟩|x⟩$.

Applying controlled-NOT gates with directions $1\to 2$, $2\to 1$, and $1\to 2$ mapsand therefore implements the SWAP gate.

The three-qubit GHZ state is

For $|\alpha ⟩=a|0⟩+b|1⟩$, a Bell measurement on $|\alpha ⟩$ and the first leg of a GHZ state leaves the other two legs in $a|00⟩\pm b|11⟩$, after applying $X$ to both legs for the two $\Psi$ outcomes.

Starting from $|000⟩$, apply a Hadamard gate to the first qubit and then controlled-NOT gates from the first qubit to each of the other two qubits.

If two senders share a GHZ state with a receiver, one sender can encode two bits with $X^x{Z}^z$ 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.

The Hadamard test estimates the real part of $⟨\psi |U|\psi ⟩$ from one ancilla measurement.

If $U|\psi ⟩=e^{2\pi i\theta }|\psi ⟩$, the ordinary Hadamard test returns ancilla zero with probability

Adding a phase gate to the Hadamard test rotates the interference signal and measures the imaginary part of $⟨\psi |U|\psi ⟩$.

The swap test on pure states $|a⟩,|b⟩$ returns ancilla zero with probability

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

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.

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

For two labelled orthonormal bases, the event of equal labels is represented by ${\sum }_j|x_j{y}_j⟩⟨x_j{y}_j|$.

For a product state, probabilities of local measurement outcomes multiply.

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

For real orthogonal $U$, the Bell state $2^{-1/2}{\sum }_j|jj⟩$ is invariant under $U\otimes U$ because $U{U}^T=I$.

For the real Bell state, the joint amplitude in real basis vectors $x,y$ equals $⟨x|y⟩/\sqrt{2}$.

Shor's factoring algorithm uses quantum period finding to determine a modular order and then extracts factors classically with greatest common divisors.

If $r$ is even, $a^r\equiv 1(\mathrm{mod}N)$, and $a^{r/2}\not\equiv -1(\mathrm{mod}N)$, then $\gcd (a^{r/2}-1,N)$ is a nontrivial factor unless the order was not minimal.

Quantum order finding Fourier-transforms a superposition of arguments having the same modular-exponentiation value, producing peaks near multiples of the reciprocal period.

The Fourier amplitude of equally spaced basis states is a finite geometric sum with peaks where the Fourier phase increment is nearly one.

For $A$ terms with phase ratio $q$, the amplitude factor is $(1-q^A)/(1-q)$, with limiting value $A$ when $q=1$.

If a measured ratio approximates a reduced $k/r$ within $1/(2r^2)$, continued-fraction convergents recover the candidate denominator $r$.

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.