Realizing the Quantum Relative Entropy of Two Noisy States using the Hudson-Parthasarathy Equations

The idea of noisy states can be derived through a quantum relative entropy over a given time period and construct the average value of X at time based on the system variables. A random Hermitian matrix is used to represent the quantum system observables with BATH states. The Hudson-Parthasarathy (HP equation) context for stochastic processes allows us to simulate quantum relative entropy using quantum Brownian motion. The Sudarshan-Lindblad's density evolution matrix equation was already derivable in generalized form in my previous work. This paper's goal is to illustrate how the HP equation may be used to estimate the density matrix for noise in a perturbed quantum system of a stochastic process. The last stage involves using MATLAB to estimate and simulate a random density matrix and measure the quantum average T_r (ρ(t)X) at various times. These formulas would be helpful in determining how sensitive the evolving/evolved states are to changes in the Hamiltonian of the noise operators in a sensitivity/robustness study of quantum systems.