Positive semidefinite operator

Positivity: ρ ≥ 0 \rho \ge 0 ρ ≥ 0 , meaning ⟨ ψ , ρ ψ ⟩ ≥ 0 \langle \psi,\rho\psi\rangle \ge 0 ⟨ ψ , ρ ψ ⟩ ≥ 0 for all ψ ∈ H \psi\in H ψ ∈ H .
Unit trace: Tr ⁡ ( ρ ) = 1 \operatorname{Tr}(\rho)=1 Tr ( ρ ) = 1 , where Tr ⁡ \operatorname{Tr} Tr is the operator trace (<a class="knowl" href="/quantum-foundations/trace-operator/" data-knowl="/fragments/quantum-foundations/trace-operator/core.html" aria-expanded="false">Trace Operator ).

Let $H$ be a complex Hilbert space. A bounded linear operator $A:H\to H$ is positive semidefinite, written $A\ge0$ , if

\langle x,Ax\rangle\ge0

for every $x\in H$ . This condition implies that $A$ is self-adjoint and is equivalent to the existence of a bounded operator $B$ with $A=B^*B$ . In finite dimension it is also equivalent to every eigenvalue of $A$ being nonnegative.

Quantum interpretation

Positive operators bounded above by the identity are the effects in a POVM. Positive trace-class operators of trace $1$ are density operators.