Does non-unitarity necessarily imply the probability leakage? I know that in quantum computing and also in the studies of the information loss inside black holes people often consider the following construction. The composite system, which consists of subparts $A$ and $B$, is evolving unitarily. However, from the point of view of the system $A$ the evolution is non-unitary. This can explain, for example, how the pure states of system $A$ can evolve into the mixed states of $A$.
I don't have much knowledge about, but to me it has always seemed that 'non-unitarity' is more or less having some real exponents ascribing time evolution. But this implies that the total probability is not conserved. Basically, that the matter goes somewhere from the system under consideration.
Can someone explain in simple words how this construction works? How do we explain the thermalization without the loss of matter?
 A: I am not going to go into the black hole stuff, but thermalization is never loss of matter and/or probability, but - so to say - only a "rearrangement" of density matrix eigenstates and corresponding probabilities. 
In other words, it is an evolution from an arbitrary initial state $\rho$, which may be pure ($\;\rho = \rho^2 = |\Psi\rangle\langle \Psi |\;$) or not ($\;\rho \neq \rho^2\;$), to some asymptotic equilibrium state $\rho_{eq}$. Since such an evolution does not preserve the number of density matrix eigenstates with non-zero probability, it cannot be unitary. But this doesn't mean that it does not conserve probability. 
In fact, any such evolution is necessarily required to 


*

*Preserve Hermiticity and be positive definite, so as to always take a density matrix into a density matrix with positive eigenvalues that represent well-defined probabilities.

*Remain positive definite in the presence of entanglement with another non-interacting system; this is called complete positivity, and is stronger then mere positivity (there are otherwise positive evolutions that are not completely positive).

*Preserve the trace of the density matrix, that is, conserve total probability.
As far as quantum computing is concerned, linear evolutions satisfying these conditions are very well-known and much used. They are described by Lindblad generators, 
$$
\dot{\rho} = \frac{i}{\hbar} [\rho(t), H] + \frac{1}{2}\sum_k{\left[ 2 A_k \rho(t) A_k^\dagger - A_k^\dagger A_k\rho(t) - \rho(t) A_k^\dagger A_k \right]}
$$
or equivalently, may be given in a Krauss representation, reading
$$
\rho(t) = \sum_j{M_j \rho(t) M_j^\dagger}\;\;\;\text{for}\;\;\;\sum_j{M_j^\dagger M_j} = I
$$
On this latter form it may be checked that requirements (1)-(3) are all satisfied.
As for thermalization, it has been found eventually that a true thermalizing evolution requires a specific form of the $A_k$, and the corresponding generators are known as thermalizing Davies generators. Say the system is described by a Hamiltonian H and is coupled to a thermal bath B by an interaction 
$$
H_{SB} = \sum_\alpha{S_\alpha \otimes B_\alpha}
$$
If in the interaction picture of the unperturbed Hamiltonian the system couplings $S_\alpha$ evolve according to 
$$
e^{iHt} S_\alpha e^{-iHt} = \sum_\omega{S_\alpha(\omega)e^{i\omega t}}
$$
then a thermalizing evolution is of the form
$$
\dot{\rho} = \frac{i}{\hbar} [\rho(t), H] +  \frac{1}{2} \sum_\omega{G_\alpha(\omega)\left[ 2 S_\alpha(\omega) \rho(t) S_\alpha(\omega)^\dagger - S_\alpha(\omega)^\dagger S_\alpha(\omega)\rho(t) - \rho(t) S_\alpha(\omega)^\dagger S_\alpha(\omega) \right]}
$$
with coefficients $G_\alpha(\omega)$ satisfying the so-called KMS relations
$$
G_\alpha(-\omega) = e^{-\beta\omega} G_\alpha(\omega) 
$$
The corresponding asymptotic state is of thermal type, $\rho_{eq} \sim \exp(-\beta H)$, and such that
$$
\rho_{eq} S_\alpha(\omega) = e^{\beta\omega} S_\alpha(\omega) \rho_{eq}
$$
See for instance the Intro to http://lanl.arxiv.org/pdf/1305.5591 for some details and refs. 
