Equal reduced states give identical local statistics

Suppose two density operators on $R\otimes E$ have the same reduced density matrix ${\sigma }_R$. Every measurement confined to $R$ then has the same outcome probabilities for the two preparations: a POVM effect $F_R$ has probability $\mathrm{Tr}(F_R{\sigma }_R)$ in either case. This remains true when $R$ contains several laboratories brought together for a joint measurement.

For independent copies, the accessible states are both ${\sigma }_R^{\otimes n}$. No collective measurement on those copies can distinguish the original preparations. Adaptive local processing also cannot help, since applying the same quantum operation to equal inputs gives equal outputs. Access to an additional environment measurement record is a change of resources, and can reveal differences hidden by the partial trace.