Positive Semi-Definite (PSD) Operator

An operator P^\hat{P} is PSD if and only if

  • P^\hat{P} is Hermitian

  • this holds true for all states ∣ψ⟩\left\lvert \psi \right\rangle :

⟨ψ∣P^∣ψ⟩≥0\begin{gather*} \left\langle \psi \right\rvert\hat{P}\left\lvert \psi \right\rangle\geq 0 \end{gather*}

This means the angle between ∣ψ⟩\left\lvert \psi \right\rangle and P^∣ψ⟩\hat{P}\left\lvert \psi \right\rangle is less than 90 degrees.

Example