Unknown measurement statistics cannot be redundantly copied using finite resources

Measurements can be viewed as interactions between a measured system and a pointer system that imprint information about the system on the pointer. For so-called unbiased interactions, the measurement statistics--the information corresponding to the diagonal of the system's initial density operator--with respect to a chosen measurement basis are transferred accurately, even if the pointer is initially in a mixed state. However, establishing measurement outcomes as objective facts also requires redundancy. We therefore consider the problem of unitarily distributing the outcome statistics to several pointers or quantum memories. We show that the accuracy of this process is limited by thermodynamic restrictions on preparing the memories in pure states: exact duplication of unknown outcome statistics is impossible using finite resources. For finite-temperature memories, we put forward a lower bound on the entropy production of the duplication process. This \emph{Holevo-Landauer} bound demonstrates that the mixedness of the initial memory limits the ability to accurately transfer the same information to more than one memory component, thus fundamentally restricting the creation of redundancies while maintaining the integrity of the original information. Finally, we show how the outcome statistics can be recovered exactly in the classical limit--via coarse-graining or asymptotically as the number of subsystems of each memory component increases--thus elucidating how objective properties can emerge despite inherent imperfections.