Just a friendly reminder as to why the density matrix formalism is so important and why it should be thought along side the state vector formalism.

Nothing that is done in one system can instantaneously affect the density matrix of another isolated system, although it can affect the state vector.

Nobel Laureate Steven Weinberg, [1]

[1] “Quantum mechanics without state vectors”, Steven Weinberg, Phys. Rev. A 90, 042102 (2014).