Partial trace positivity from product vectors ($⟨u,{\mathrm{Tr}}_B(\rho )u⟩={\sum }_b⟨u\otimes e_b,\rho (u\otimes e_b)⟩$)

The partial trace of a positive operator is positive by this sum of nonnegative expectations in an orthonormal basis of the traced factor. Its trace equals the original trace, using the product orthonormal basis. Basis independence follows from $\mathrm{Tr}[({\mathrm{Tr}}_B\rho )A]=\mathrm{Tr}[\rho (A\otimes I)]$, which uniquely determines the reduced operator. Thus reductions of density matrices remain density matrices. For a positive trace-class operator the same argument uses convergent nonnegative basis sums.