Distance between quantum states in the presence of initial qubit-environment correlations: A comparative study

Abstract

The time evolution of the trace distance between two states of an open quantum system may increase due to initial system-environment correlations, thus exhibiting a breakdown of distance contractivity of the reduced dynamics. We analyze how the time evolution of the distance depends on the chosen distance measure. Here we elucidate the behavior of the trace distance, the HilbertSchmidt distance, the Bures distance, the Hellinger distance and the quantum Jensen-Shannon divergence for two system-environment setups, namely a qubit bi-linearly coupled to an infinite and a finite size environment with the latter composed of harmonic oscillators.

Two-Member Markov Processes toward an Equilibrium from a Continuum of Initial States

“…Elongated ellipses on the frequency space are also seen for interference patterns of light polarization [12,13]. In a few physical examples observed in real space, we find a similar array of shape changes with varying parameters [5,10]. These phenomena on two-dimensional real space are of particular importance from the viewpoint of phase transitions and topological physics [2,3,13].…”

“…Therefore, non-Markovian processes involving fast optical pulses or switching are not under our investigation [6,8]. In addition, our Markov process corresponds to the steady increase in indistinguishability among the member states [9,10,12,13]. Along the same line of reasoning, Markov processes governed by random transition matrices normally erase the initial-state dependence as events progress [5].…”

“…From both left and right eigenvectors [1], its Markov process is described by the following transient dynamics [10].…”

“…Such two-level systems form the basis for the interferometers [9][10][11][12][13]. In addition, the Schwinger boson representation for even-numbered spin systems is admissible, thus revealing a bosonic feature of the collective ensemble of fermionic atoms [11].…”

Unified scheme for correlations using linear relative entropy

“…Clearly, Hilbert-Schmidt norm is not the unique distance which can be defined in the space of quantum states. Several distances are possible (trace distance, Bures distance, ...) with their own advantages and drawbacks and each one might be useful for some appropriate purpose [36,37,38,39].…”