General symmetry in the reduced dynamics of two-level system

Abstract

We study general transformation on the density matrix of two-level system that keeps the expectation value of observable invariant. We introduce a set of generators that yields hermiticity and trace preserving general transformation which casts the transformation into simple form. The general transformation is in general not factorized and not completely positive. Consequently, either the parameter of transformation or the density matrix it acts on needs to be restricted. It can transform the system in the forward and backward direction with regard to its parameter, not as a semigroup in the time translation symmetry of dynamical maps. The general transformation can rotate the Bloch vector circularly or hyperbolically, dilate it or translate it. We apply the general transformation to study the general symmetry of amplitude damping and phase damping in two-level system. We generalize the generators to higher level systems.

Damping modes of harmonic oscillator in open quantum systems

“…( 13), we obtain the following results. For KL equation (24), we find that η0,KL (t) = −2bγ , η1,KL (t) = 0 = η2,KL (t) . We reproduce the commutation relations of the generators [19] in Table B.3 for the convenience of the readers.…”

“…(13), we obtain the following results. For KL equation (24), we find thatη 0,KL (t) = −2bγ ,η 1,KL (t) = 0 =η 2,KL (t) .…”

“…Examples of damping modes are provided by the relaxation of two-level systems or qubits. The damping in qubits [14,15,24] can be analyzed by following the change in the Bloch ball [8]. Unitary rotations generated by the Pauli matrices change the axis of the Bloch ball.…”