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In a covariant gauge, the photon propagator is the inverse of the gauge-fixed Maxwell kinetic operator.

For an on-shell amplitude $\mathcal{M}={ϵ}_{\mu }(k){\mathcal{M}}^{\mu }$ with an external photon, gauge invariance implies $k_{\mu }{\mathcal{M}}^{\mu }=0$.

Photon vacuum polarization is the one-particle-irreducible photon two-point function generated by charged-particle loops.

A Markovian continuous-time quantum channel obeys

A Stinespring dilation represents a quantum channel by an isometry $V={\sum }_i{K}_i\otimes |i⟩$ followed by tracing out the environment.

A finite-dimensional quantum channel is primitive when some power maps every nonzero positive operator to a positive-definite operator. Equivalently, eigenvalue one is simple and no other eigenvalue lies on the unit circle.

A quantum channel is strictly contractive in trace distance when it reduces the distance between every pair of distinct density operators by a uniform factor smaller than one.

A quantum channel is unital when $T(I)=I$. Every random unitary channel is unital, while the converse fails in dimension at least three.

A random unitary channel has $T(\rho )={\sum }_j{p}_j{U}_j\rho U_j^{†}$. Its Choi matrix is a convex combination of maximally entangled pure states.

For normalized maximally entangled $|\varphi ⟩=d^{-1/2}{\sum }_j|jj⟩$, the Choi matrix of $T$ is $C_T=(T\otimes \mathrm{id})(|\varphi ⟩⟨\varphi |)$. Choi's theorem says $T$ is completely positive exactly when $C_T\ge 0$.

Every finite-dimensional completely positive map has $T(\rho )={\sum }_j{K}_j\rho K_j^{†}$. It is trace preserving exactly when ${\sum }_j{K}_j^{†}{K}_j=I$.

For hypotheses ${\rho }_0,{\rho }_1$ with priors $p,1-p$, the optimal success probability isAn optimal measurement projects onto the positive and negative spectral subspaces of the weighted difference.

This is the minimum-error optimum in binary quantum state discrimination.

A dense linear order without endpoints is a linear order in which a third point lies strictly between every two distinct points and every point has points both below and above it.

The back-and-forth method constructs an isomorphism by alternately extending finite partial isomorphisms to include elements from each structure.

A quadrature rule is exact through degree $d$ precisely when its node weights reproduce the first $d+1$ moments of the integration measure.

If each $(n+1)$-node quadrature rule is exact through degree $n$ and has positive weights, then it converges on every continuous function. The Weierstrass approximation theorem reduces the error to the uniform polynomial-approximation error, while exactness on constants controls the sum of the weights.

If an $(n+1)$-node quadrature rule for a positive interval weight is exact through degree $2n$, every weight is positive. Indeed, applying the rule to the square of the corresponding degree-$n$ Lagrange cardinal polynomial isolates that weight.

No $(n+1)$-node quadrature rule for a positive interval weight can be exact through degree $2n+2$, because it evaluates the nonnegative nodal square $Q_{n+1}^2$ as zero although its integral is positive.

An $(n+1)$-node quadrature rule already exact through degree $n$ is exact through degree $n+1+k$ exactly when its nodal polynomialis orthogonal to every polynomial of degree at most $k$.

If $Q$ and $N$ are the nonzero square and nonsquare classes modulo eleven and $\omega =e^{2\pi i/11}$, thenTheir sum is $-1$, and their difference is the Quadratic Gauss sum $i\sqrt{11}$.