content_image/353847.jpg content_image/353846.jpg content_image/353845.jpg content_image/353844.jpg content_image/353843.jpg
self-explanatory shorthand will be used for the state vectors of the bath 

<img src='content_image/353847.jpg'>

The Hilbert space H$_{B}$ of the bath will be defined as the linear span of the above states where the sequence { i$_{k}$ } contains only finitely many nonzero elements . When the Hamiltonian is defined below, it will be seen that the states which are not included into H$_{B}$ have infinite energy and zero equilibrium-state probabilities. Consequently such states will never enter into the discussion. 

In addition to this, defining H$_{B}$ in such a way makes it a separable Hilbert space having a countable orthonormal basis. To see this, note that the sequence { i$_{k}$ } can be considered as the digits of the base- d representation of an integer n , written in short as n = ( · · · i$_{3}$i$_{2}$i$_{1}$ )$_{d}$. The integer n is given in terms of the sequence as 

<img src='content_image/353846.jpg'>

Consequently, there is a one-to-one correspondence between nonnegative integers n and sequences { i$_{k}$ } that has finitely many nonzero elements. For this reason, the standard orthonormal basis of H$_{B}$ can be enumerated by a nonnegative integer n and therefore the alternative label 

<img src='content_image/353845.jpg'>

can be used. It is thus clear that the Hilbert space H$_{B}$ has a countable basis and hence it is separable. 

b. The isometry. Let q$_{i}$ ( i = 0 , 1 , . . . , d − 1) be the eigenvalues of Q arranged in nonde- creasing order and let {| i 〉} be the orthonormal basis of H$_{A}$ formed from the corresponding eigenvectors. 

The isometry U$_{AB}$ on the composite system AB is defined as follows 

<img src='content_image/353844.jpg'>

<img src='content_image/353843.jpg'>

for any allowed values of input labels. In words, the isometry sets the state of A to the standard state | ψ$_{0}$ 〉 and copies its contents onto the subsystem X$_{1}$. It also forms an infinite chain of information flow inside the bath as the contents of X$_{k}$ are copied onto X$_{k}$$_{+1}$ ( k ≥ 1) 

