Split the support of a Pauli operator of weight at most $2t$ into two sets of size at most $t$. It is, up to a scalar phase, $E_a^{†}{E}_b$ for two errors in the correctable family. The Knill--Laflamme condition makes its compression to the code scalar, which is quantum error detection. Expansion in the Pauli basis extends the assertion to every operator supported on at most $2t$ qubits. The number $t$ counts affected qubits, not the cardinality of an arbitrary selected family of errors.

Expanding the covariant kinetic term of scalar electrodynamics gives a one-photon two-scalar vertex and a two-photon two-scalar seagull vertex. Their tree amplitudes and Ward identities depend on a consistent choice of charge-flow and gauge covariant derivative convention.

Vacuum polarization is the modification of a gauge field's propagation by quantum fluctuations of charged fields. In quantum electrodynamics, charged-particle loop contributions change the photon propagator; their proper two-point insertion is photon vacuum polarization.

Lorentz invariance and the Ward identity constrain the tensor structure. For an invariant vacuum it is transverse and can be written ${\Pi }^{\mu \nu }(q)=(q^2{g}^{\mu \nu }-q^{\mu }{q}^{\nu })\Pi (q^2)$, up to the stated sign convention. This momentum-dependent response contributes to the scale dependence of the effective electromagnetic coupling. The calculation describes a field correlation, rather than a literal medium of separately detectable virtual pairs. A mass term for the photon is not generated merely by renormalizing a gauge-invariant vacuum polarization.

The leading scattering amplitude is the sum of two tree-level Feynman diagrams, with the outgoing photons attached in either order to the fermion line. The Dirac propagator carries momentum $p-k_1$ or $p-k_2$ for incoming electron momentum $p$. Their sum obeys the two-photon fermion Ward identity, although individual diagrams need not. Crossing symmetry relates this amplitude to Compton scattering.

A finite-dimensional closed-system control model is unitarily operator controllable when its reachable propagators include every unitary operator. For the usual real Hamiltonian controls and freely chosen duration, this requires the Dynamical Lie algebra to be $\mathfrak{u}(N)$. If global phase is ignored, $\mathfrak{su}(N)$ also suffices; exact access to all of $U(N)$ still requires the central phase direction.

A closed system has density operator controllability when every pair of density operators with the same spectrum can be connected by its controls. Under the standard finite-dimensional Hamiltonian assumptions, this holds exactly for Dynamical Lie algebra $\mathfrak{u}(N)$ or $\mathfrak{su}(N)$. This definition ranges over every spectral orbit; a particular degenerate spectrum can be controllable under a smaller group.

For freely chosen control duration in a finite-dimensional closed system, pure-state controllability means the reachable group acts transitively on normalized states or their rays. Up to unitary conjugation, the possible Dynamical Lie algebras are $\mathfrak{u}(N)$, $\mathfrak{su}(N)$, and for even $N=2n$ the standard compact symplectic Lie algebra $\mathfrak{sp}(n)$, with or without $\mathbb{R}iI$. For $N\ge 4$, the symplectic possibilities are pure-state controllable but generally lack density operator controllability.

The dynamical Lie algebra of a finite-dimensional Hamiltonian control system is the real span of its skew-Hermitian drift and control generators and all their nested commutators. The sign choice $iH$ or $-iH$ gives the same algebra. Its connected group determines reachable propagators under the usual freely timed, real-control assumptions. An operator commuting with every generator also commutes with this algebra, imposing invariant sectors.

A finite-dimensional closed quantum control system with fixed drift and control Hermitian matrices, real control inputs, and dynamics linear in the state at fixed controls and affine in the controls at fixed state. Its Dynamical Lie algebra is generated over the reals by $i{H}_0,i{H}_1,\dots$. Reachability depends on the admissible controls, allowed duration and phase convention.

For an infrared-safe observable, loop corrections and integrated unresolved real emissions have compensating infrared divergences and collinear divergences. Keep a common regulator until they are added. At the first QCD correction to photon-mediated quark production, virtual gluon exchange corrects the two-parton state and real emission supplies the three-parton state. Their inclusive sum is finite after ultraviolet renormalization; resolved three-jet rates additionally depend on an infrared-safe resolution prescription.

A collimated collection of final hadrons associated with an energetic parton and its radiation and hadronization. A precise perturbative prediction requires an infrared-safe observable that specifies how particles are grouped into jets.

The gauge-covariant kinetic term for two massless quarks separates into left- and right-handed parts. Independent constant unitary quark flavor rotations preserve these terms, giving classical $U(2{)}_L\times U(2{)}_R$ internal symmetry in addition to local color $SU(3)$. The singlet axial $U(1{)}_A$ has a chiral anomaly, while the nonsinglet chiral symmetry and vector baryon number remain exact quantum symmetries, up to their shared discrete identifications. Equal nonzero quark masses reduce the continuous flavor symmetry to $SU(2{)}_V\times U(1{)}_V$.

An ancilla qubit is a two-dimensional quantum ancilla. It is the quantum version of an ancilla bit; an auxiliary quantum register can contain several such qubits.

For a binary relation with no symmetry or loop restrictions, an extension axiom demands a fresh element realizing any prescribed choices of incoming edges, outgoing edges and its own loop relative to a finite list of distinct parameters. All such axioms together eliminate quantifiers: the quantifier-free type of a parameter tuple determines which witness patterns can occur. These axioms differ from the undirected loop-free theory of the random graph.

For a positive nonsquare integer $D$ and positive integers $a,q$, the integer $D{q}^2-a^2$ is nonzero. Therefore $|\sqrt{D}-a/q|=|D{q}^2-a^2|/(q^2(\sqrt{D}+a/q))$. If $a/q\le \sqrt{D}+1$, the denominator factor is at most $2\sqrt{D}+1$; otherwise the approximation error already exceeds one. This proves that a quadratic irrational of this form is a badly approximable number. For $D=2$, splitting instead at $a/q=2$ gives the strict bound $|\sqrt{2}-a/q|>1/(4q^2)$.

// Target: analytic-number-theory.bigb

A real quadratic form is indefinite when it takes both positive and negative values. For a real symmetric matrix representing it, this means that its eigenvalues include both signs. For example, $q(x,y)=x^2-y^2$ has $q(1,0)=1$ and $q(0,1)=-1$; a zero value at the nonzero vector $(1,1)$ does not imply positive semidefiniteness.^[ref]

Let $Q:V\to {\mathbb{F}}_2$ be a quadratic form whose polar bilinear form of a quadratic form $B$ is nondegenerate, so $\dim V=2m$. For a symplectic basis $e_1,f_1,\dots ,e_m,f_m$, defineThis does not depend on the symplectic basis. The hyperbolic plane $Q(x,y)=xy$ has Arf invariant zero; $Q(x,y)=x^2+xy+y^2$ has Arf invariant one and vanishes only at zero. Every such quadratic form is an orthogonal direct sum of hyperbolic planes, with one copy of the latter plane replacing a hyperbolic plane when the Arf invariant is one.^[ref]

A useful basis-independent test isOn one two-dimensional summand with symplectic basis $e,f$, write $Q(xe+yf)=ax+by+xy$; the four terms sum to $2(-1{)}^{ab}$. Multiplication over the orthogonal direct sum proves the formula and the invariance of its sign. Consequently the number of zeros is $2^{2m-1}+(-1{)}^{\mathrm{Arf}(Q)}{2}^{m-1}$. For a fixed $B$ and $m\ge 1$, the numbers of binary quadratic refinements with Arf invariant zero and one are respectively $2^{m-1}(2^m+1)$ and $2^{m-1}(2^m-1)$.

Its symmetric matrix is $11^T-I$. The line spanned by $1$ has eigenvalue $n-1$ and its orthogonal complement has eigenvalue $-1$. Thus for $n>1$ it is nondegenerate with inertia $(1,n-1,0)$; for $n=1$ it vanishes. This gives a useful explicit example for Sylvester's law of inertia.

An integral quadratic lattice is a finite-rank free abelian group with a nondegenerate integral symmetric bilinear form. Its norm convention here is $q(x)=B(x,x)$. A basis gives an integral Gram matrix; basis changes preserve its determinant up to multiplication by the square of a unit.

The displayed bilinear form is the polar form of a quadratic form. In odd characteristic of a field, $Q(x)=B_Q(x,x)/2$ recovers the quadratic form; in characteristic of a field two, different forms can have the same alternating bilinear form. Over ${\mathbb{F}}_2$, two such forms differ by a linear map to the field, giving the affine parametrization of binary quadratic refinements.