How to calculate $Tr_{E} \underbrace{[\rho^{bath} \otimes \rho^{bath}......\otimes \rho^{bath}]}_{N \ times}$ if I know $Tr_{E} [\rho^{bath}]$?

Question

One can directly jump to the question and skip the first two paragraphs (motivation for the question)

Consider a system in a thermal gas of $N$ particles. If I want to study the reduced dynamics of the system alone, I need to find the partial trace (over the environment) of the total density matrix of the system and the $N$ environmental particles $\hat \rho_{tot}$.

If however the change in reduced density matrix for a system in a thermal gas by a single collision is $\Delta \hat \rho^{red}$ then for a very dilute gas where collisions can be looked at as "particle by particle" the total change in reduced density matrix should simply be $N \times \Delta \hat \rho^{red}$ and therefore, if one calculates the change in reduced density matrix for a system from system + just a "single colliding particle" density matrix (as opposed to the total density matrix), it should be enough.

I however want to see if I can reach $N \times \Delta \rho^{red}$ starting from writing the total density matrix for all $N$ particles.

So, I am asking the following question so that I can try to do that.

My question is:

If $$\hat \rho_{tot} = (\hat \rho)^{\otimes N}$$

then if I know $$\hat \rho^{red} = Tr_{E}(\hat \rho)$$

Is there a straight-forward way to find: $$\hat \rho_{tot}^{red} = Tr_{E}(\underbrace{\hat \rho \otimes \hat \rho ..... \otimes \hat \rho }_{N \ times} )$$

from the knowledge of $\hat \rho^{red}$?

Is simply $$\hat \rho_{tot}^{red} = (\hat \rho^{red})^{\otimes N}$$?

Answer 1

Is the first trace in the question title a trace over $N$ copies of the system? If so, the trace $\text{tr}^{\,}_{(N)} \left( A^{\otimes N} \right)$ of $N$ copies of an operator $A$, where the trace is over all $N$ copies, is simply equal to $\text{tr} (A)^N$. So then yeah, knowledge of one copy would tell you everything.

However, I suspect you mean to say that you have $N$ copies of the system coupled to the same bath. In this case, the reduced density matrix $\rho_{\text{red}}$ for one copy of the system comes from tracing out not only the environment $E$, but the other $N-1$ copies as well. Generically, this means that the reduced density matrix for one copy tells you nothing about the others. The only exception would be if the $N$ systems are prepared in a product state initially, never interact with one another, and the environment (bath) interacts with the $N$ systems in a permutation-symmetric manner so that each of the $N$ systems have the same reduced density matrix, with no entanglement between the $N$ systems, so that the reduced density matrix of the $N$ systems is $\rho_{\text{red}}^{\otimes N}$ upon tracing out the environment. But this is quite fine tuned.

Please let me know if I've misunderstood any aspect of your question or if anything above was unclear.