= Remote state invariance under a local trace-preserving operation
{title2=$\operatorname{Tr}_A[(\mathcal E_A\otimes\operatorname{id}_B)(\rho_{AB})]=\rho_B$}
For local <Kraus operators> $A_i$ with $\sum_iA_i^\dagger A_i=I$, let $\rho'_{AB}=\sum_i(A_i\otimes I)\rho_{AB}(A_i^\dagger\otimes I)$. For every remote operator $B$, cyclicity of the <trace> gives $\operatorname{Tr}[(I\otimes B)\rho'_{AB}]=\sum_i\operatorname{Tr}[(A_i^\dagger A_i\otimes B)\rho_{AB}]=\operatorname{Tr}[(I\otimes B)\rho_{AB}]$. Hence the remote <reduced density matrix> is unchanged. In particular every remote <POVM> has the same outcome probabilities, proving the <no-communication theorem> from the measurement rule rather than assuming it as a physical postulate.
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