Reduced dynamics

 ( Quantum Information Science ) 

In quantum mechanics, especially in the study of open quantum systems, reduced dynamics refers to the time evolution of a density matrix for a system coupled to an environment. Consider a system and environment initially in the state $\backslash rho\_\{SE\}\; (0)\; \backslash ,$ (which in general may be entangled) and undergoing unitary evolution given by $U\_t\; \backslash ,$. Then the reduced dynamics of the system alone is simply

$\backslash rho\_S\; (t)\; =\; \backslash mathrm\{Tr\}\_E\; U\_t$

If we assume that the mapping $\backslash rho\_S(0)\; \backslash mapsto\; \backslash rho\_S(t)$ is linear map and completely positive, then the reduced dynamics can be represented by a quantum operation. This mean we can express it in the operator-sum form

$\backslash rho\_S\; =\; \backslash sum\_i\; F\_i\; \backslash rho\_S\; (0)\; F\_i^\backslash dagger$

where the $F\_i\; \backslash ,$ are operators on the Hilbert space of the system alone, and no reference is made to the environment. In particular, if the system and environment are initially in a product state $\backslash rho\_\{SE\}\; (0)\; =\; \backslash rho\_S\; (0)\; \backslash otimes\; \backslash rho\_E\; (0)$, it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.

---

Notes

• Nielsen, Michael A. and Isaac L. Chuang (2000). Quantum Computation and Quantum Information, Cambridge University Press,

Categories: Quantum Information Science

Post Comment