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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 $i\sqrt{11}$.

The roots of $a{x}^2+bx+c=0$, where $a\ne 0$, are

For symmetric $A$, the quadratic form $x^{T}Ax$ has gradient $2Ax$.

A vector $v$ is isotropic for a quadratic form $q$ when $q(v)=0$.

A subspace $E$ is totally isotropic for $B$ when $B(u,v)=0$ for all $u,v\in E$. For a real nondegenerate bilinear form of inertia $(p,q)$, its dimension is at most $\min (p,q)$.

Every real quadratic form is congruent to a diagonal form with entries $+1,-1,0$, and the numbers of entries of each kind are invariant.

A real symmetric matrix $A$ is positive definite whenfor every nonzero real vector $x$. It therefore defines the inner product $⟨x,y{⟩}_A=x^{T}Ay$ and is invertible.

A real symmetric or complex Hermitian matrix $A$ is positive semidefinite when $x^{†}Ax\ge 0$ for every vector $x$, equivalently when all its eigenvalues are nonnegative.

A symmetric bilinear form is positive semidefinite when $B(v,v)\ge 0$ for every $v$, and positive definite when equality is possible only for $v=0$.

Over a field of characteristic other than two, a quadratic form recovers its symmetric bilinear form by

The signature of a real quadratic form is the number of positive squares minus the number of negative squares in its diagonal normal form.