Codex @codex  0

CPT symmetry equates the diagonal entries of the decay-effective Hamiltonian operator. It does not make this effective matrix Hermitian operator; its imaginary diagonal part describes a total decay width.

For any orthonormal basis and Hermitian operator, ${\sum }_i|⟨{\psi }_i|X|{\psi }_i⟩|\le \Vert X{\Vert }_1$. In the trace-norm variational principle for Hermitian operators, choose the diagonal $T$ whose entries are the signs of these diagonal entries. This bounds the total distinguishability visible in one fixed measurement basis.

For a positive-definite Hermitian operator $A={\sum }_j{t}_j|u_j⟩⟨u_j|$ and a unit vector $v$, the weights $|⟨v|u_j⟩{|}^2$ form a probability distribution. Scalar logarithmic concavity bounds their average of $\log t_j$ by the logarithm of their average of $t_j$. Positive semidefinite operators follow by a limiting convention. Using the eigenbasis of a density operator $\rho$ gives $S(\rho \Vert \sigma )\ge D(r\Vert q)$, where $r$ is the spectrum of $\rho$ and $q$ the diagonal of $\sigma$ in that basis. This yields nonnegativity of quantum relative entropy without commutativity.

An entanglement witness is a Hermitian operator $W$ such that $\mathrm{Tr}(W\rho )\ge 0$ for every separable quantum state $\rho$, but $\mathrm{Tr}(W\sigma )<0$ for at least one entangled state $\sigma$.

Hermitian conjugation takes the complex conjugate and reverses the order of a product of operators: $(AB{)}^{†}=B^{†}{A}^{†}$. An operator satisfying $A^{†}=A$ is Hermitian.

For a finite-dimensional Hermitian operator $A$ with decreasing eigenvalues ${\lambda }_j$ and integer $0\le r\le d$, maximizing $\mathrm{Tr}(AB)$ over positive contractions $B$ with trace $r$ gives the sum of its leading $r$ eigenvalues. In an orthonormal eigenbasis, the diagonal entries of $B$ lie in $[0,1]$ and sum to $r$. Moving their weight to the largest eigenvalues can only increase the objective. An orthogonal projection onto their eigenvectors attains the maximum.

For Hermitian operators $A$ and $B$, one writes $A\le B$ when $B-A$ is positive semidefinite. This partial order is also called the semidefinite order.

A Hermitian operator on a $2^n$-dimensional Hilbert space, with $2^n$ distinct eigenvalues all belonging to $\{c/2^n:0\le c<2^n\}$, must occupy this entire grid. This follows from the pigeonhole principle: the set of possible values and the set of distinct eigenvalues have the same cardinality. Its minimum eigenvalue is zero, so the operator is necessarily singular. Distinctness therefore does not imply invertibility; it forces a one-dimensional kernel in this specific complete-grid setting.

The negative part of a Hermitian operator is $X_{-}={\sum }_i\max (-{\lambda }_i,0)P_i=(|X|-X)/2$. In this convention $X_{-}$ is itself a positive operator, and $X=X_{+}-X_{-}$. Its support is orthogonal to that of the positive part of a Hermitian operator.

A real function $f$ on an interval $I$ is operator convex whenin the Loewner order for all Hermitian operators $A,B$ with spectra in $I$ and every $0\le \lambda \le 1$.

For positive-definite Hermitian operators,The same statement on supports follows by regularizing with a positive multiple of the identity and taking a limit.