arXiv subjects
Publications and source records attributed to Farruh Mukhamedov.
The program relative to the investigation of quantum Markov states for spin chains based on Canonical Anticommutation Relations algebra is carried on. This analysis provides a further step for a satisfactory theory of quantum Markov processes.
In this paper we consider nearest neighbour models where the spin takes values in the set $Φ=\{\z_1,\z_2,...,\z_q\}$ and is assigned to the vertices of the Cayley tree ${\G}^k$. The Hamiltonian is defined by some given $λ$-function. We find a condition for the function $λ$ to determine the type of the von Neumann algebra generated by the GNS - construction associated with the quantum Markov state corresponding to the unordered phase of the $λ$-model. Also we give some physical applications of the obtained result.
We clarify the meaning of diagonalizability of quantum Markov states. Then, we prove that each non homogeneous quantum Markov state is diagonalizable. Namely, for each Markov state $ϕ$ on the spin algebra $A:={\bar{\otimes_{j\in Z}M_{d_{j}}}^{C^{*}}}$ there exists a suitable maximal Abelian subalgebra $D\subset A$, a Umegaki conditional expectation $E:A\mapsto D$ and a Markov measure $μ$ on $spec(D)$ such that $ϕ=ϕ_μ\circ E$, the Markov state $ϕ_μ$, being the state on $D$ arising from the measure $μ$. An analogous result is true for non homogeneous quantum processes based on the forward or the backward chain. Besides, we determine the type of the von Neumann factors generated by GNS representation associated with translation invariant or periodic quantum Markov states.