APPROXIMATION STATES AND FIXED POINTS OF QUANTUM CHANNELS*

In this note, we extend some results on positive and completely positive trace-preserving maps (called quantum channels in the CPTP case) from finite-dimensional to infinite-dimensional Hilbert space. Specifically, we mainly consider whether the fixed state of a quantum channel Φ on $T\left(\mathscr{H}\right)$ exists, where $T\left(\mathscr{H}\right)$ is the Banach algebra of all trace-class operators on the Hilbert space $\mathscr{H}$. We show that there exist the approximation states ρ_n for every quantum channel Φ. In particular, there is a quantum channel on $T\left(\mathscr{H}\right)$, which has not a fixed state. Also, we get the relationship between the fixed points of $\Phi \left(\left|A\right|\right)=\left|A\right|$ and Φ(A) = ωA, where ω is the complex number with |ω| = 1 and $A\in T\left(\mathscr{H}\right)$.