Phase Covariant Qubit Dynamics and Divisibility

: S.N. Filippov, A.N. Glinov, L. Leppäjärvi

Publisher: MAIK NAUKA/INTERPERIODICA/SPRINGER

: 2020

: Lobachevskii Journal of Mathematics

: LOBACHEVSKII JOURNAL OF MATHEMATICS

: LOBACHEVSKII J MATH

: 41

: 4

: 617

: 630

: 14

: 1995-0802

: 1818-9962

DOI: https://doi.org/10.1134/S1995080220040095

Phase covariant qubit dynamics describes an evolution of a two-level system under
simultaneous action of pure dephasing, energy dissipation, and energy gain with time-dependent
rates γz(t), γ−(t), and γ+(t), respectively. Non-negative rates correspond to completely positive
divisible dynamics, which can still exhibit such peculiarities as non-monotonicity of populations for
any initial state. We find a set of quantum channels attainable in the completely positive divisible
phase covariant dynamics and show that this set coincides with the set of channels attainable
in semigroup phase covariant dynamics. We also construct new examples of eternally indivisible
dynamics with γz(t) < 0 for all t > 0 that is neither unital nor commutative. Using the quantum
Sinkhorn theorem, we for the first time derive a restriction on the decoherence rates under which the
dynamics is positive divisible, namely, γ±(t) ≥ 0,
γ+(t)γ−(t)+2γz(t) > 0. Finally, we consider
phase covariant convolution master equations and find a class of admissible memory kernels that
guarantee complete positivity of the dynamical map.