# Open Quantum Systems: Born-Approximation and the preservation of Trace, Hermicity and Positivity

This is related to a previous question of mine.

We consider a density matrix $$\sigma(t)$$ operating on a Hilbert space $$\mathscr{H}_{s}\otimes \mathscr{H}_b$$ with Hamiltonian $$H = H_s \otimes \mathbb{I}_B + \mathbb{I}_s \otimes H_b + g H_{int}$$. We are interested in the reduced density matrix $$\rho(t) := \mathrm{Tr}_{B}[ \sigma(t) ]$$.

In the Born Approximation, the evolution of the reduced density matrix is described (in the interaction-picture) by $$\frac{d\rho_I(t)}{dt} \simeq - g^2 \int_0^t ds\ \mathrm{Tr}_b\bigg( \big[ V(t), [V(s), \rho_I(s) \otimes \varrho_{B}] \big] \bigg)$$ where $$\rho_{I}(t) = e^{+ i H_s t} \rho(t) e^{- i H_s t}$$, and $$V(t)$$ is just $$H_{int}$$ in the interaction-picture, and $$\varrho_B$$ is the initial state of the reservoir (where $$\sigma(0) = \rho(0) \otimes \varrho_B$$).

Suppose for concreteness that $$\mathscr{H}_s$$ is finite-dimensional, and the interaction is of the form $$V(t) = \sum_{jk} A_{j} \otimes B_{k}$$ Then the above equation of motion takes the form $$\frac{d\rho_I(t)}{dt} \simeq g^2 \sum_{jk\ell m} \int_0^t ds\ \bigg( \big[ A_{\ell}(s) \rho_I(s), A_{j}(t) \big] \langle B_{k}(t) B_{m}(s) \rangle + \big[ A_{j}(t), \rho_I(s) A_{\ell}(s) \big] \langle B_{m}(s) B_{k}(t) \rangle \bigg)$$ where we define the correlator $$\langle B_{k}(t) B_{m}(s) \rangle := \mathrm{Tr}_B\left[ B_{k}(t) B_{m}(s) \varrho_B \right]$$.

My Question: Suppose $$\sigma(0)$$ is trace 1, Hermitian and positive. This means that $$\varrho(0)$$ is also trace 1, Hermitian and positive. Are these properties preserved for $$\varrho(t)$$ with $$t>0$$?

I am not sure. Here are my thoughts:

1. Preservation of Trace: taking the trace of the above evolution equation yields $$\mathrm{Tr}[\dot{\varrho}_I(t)]=0$$ because the equation contains commutators of finite-dimensional matrices. As far as I can tell this is only true because of my particular choice of $$H_{int}$$ which is a tensor product.

2. Preservation of Hermicity: I would like to show explicitly that $$\frac{d}{dt}\big[ \varrho(t)^\dagger - \varrho(t) \big]=0$$ for the state evolved by the above approximate equation.

3. Preservation of Positivity: I don't know if this is true at all, or how to show it.

In my previous post, it was pointed out that $$\sigma(t)$$ is trace 1, Hermitian and positive. Taking the partial trace of a positive operator yields a positive operator, which also implies that the reduced density matrix is Hermitian (since bounded positive operators are self-adjoint). I may be confused, but I think this is not an answer to my question: the evolved state $$\rho(t)$$ from the above approximate equation of motion does not contain the full information about $$\sigma(t)$$. I would like to know if these properties hold true for the $$\rho(t)$$ predicted by the above approximation.