In Chapter 8 of Nielsen and Chuang's Quantum Computation and Quantum Information, a mathematical framework is developed to describe the dynamics of open quantum systems.
Suppose the initial state of principal system is $\rho$ and the initial state of the environment is $\rho_{env}$. Let the principal-system-environment system start out in the product state $\rho \otimes \rho_{env}$. Let's say after some unitary evolution $U$, we want to extract the "evolved" state of the principal system. We do this by tracing over the environment i.e. averaging over the states of environment as shown below
$$\mathcal{E}(\rho) = tr_{env}(U(\rho \otimes \rho_{env})U^{\dagger})$$
Let ${|e_k\rangle}$ be some orthonormal basis of the state space of the environment and $\rho_{env} = |e_0\rangle\langle e_0|$. Then we have
Now in the text, it is said that $tr(\mathcal{E}(p))$ can be less than 1 or $\sum_k E_kE^{\dagger}_k <I$. But how can this be true? Is the basis ${|e_k\rangle}$ not complete?
In the text, non-trace preserving that follow the above properties are said to "not provide a complete description of the process that may occur in the system". An example is used to illustrate this :
Granted $\mathcal{E}_0(\rho)$ and $\mathcal{E}_1(\rho)$ don't provide full description of the system but that is only because the basis for these operations are not complete. So are non trace preserving quantum operations those that don't account for a complete orthonormal basis of environment? If not, is there a better example or physical intuition of understanding this "non trace preserving" property?