= Partial trace positivity from product vectors
{title2=$\langle u,\operatorname{Tr}_B(\rho)u\rangle=\sum_b\langle u\otimes e_b,\rho(u\otimes e_b)\rangle$}
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 $\operatorname{Tr}[(\operatorname{Tr}_B\rho)A]=\operatorname{Tr}[\rho(A\otimes I)]$, 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.
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