What does density operator being same for two sytems tells us?

Yesterday I asked a question. I got it that if a density operator is given as $$\rho=\sum_{i=1}^{i=k}p_i|\psi_i\rangle \langle\psi_i| \tag{1}$$ then it means that the system is one of the states $|\psi_i\rangle$, but we don't know which.

But say I have two systems $S_1$ and $S_2$ and both are defined by same density operator given by $\rho$, then what I believe is that it is not necessary that both systems are clone of each other. Am I correct, because I have seen in books that density operators are equated $\rho_1=\rho_2$ , which is not making sense to me right now. What am I missing?

• what do you mean exactly by "system"? usually each system comes with its Hilbert space (more generally with their algebra of observables), so its hard to compare states, not to mention density matrices... Jan 9 '15 at 11:41

The outcome of measuring an observable $O$ on a system described by a density operator $\rho$ is given by $\mathrm{tr}[O\rho]$. Thus, if two systems are described by the same density operator, they cannot be distinguished by any measurement, i.e., they describe the same state. So if those density operators encode everything you know about the two states, then these two state are indeed identical for you.

Of course, if someone prepared the state $\rho$ in a specific way (e.g., by randomly preparing a specific state from an ensemble), then two states which from your perspective are described by the same density operator might be different for that person.

As @Phoenix87 said, you should claim what is the meaning of system in your question. If I have not missed something.

the system is one of the states $|ψ_i⟩$ but we don't know which.

Here the system means the system of a certain particle and a certain particle will be in a pure state $|ψ_i⟩$ but we don't know which.

two systems $S_1$ and $S_2$ and both are defined by same density operator given by $\rho$.

Here the system means the system of many (maybe infinite) particles which may be in a pure state $|ψ_i⟩$ but we don't know which. What we know is that the statistical law these particles obey. And two systems $S_1$ and $S_2$ are identical means they have the same statistical law.