Relative entropy convergence for depolarizing channels (Q2795547)

When a (quantum) signal \(\rho\) passes through a stochastic channel, the with time \(t\), the relative entropy \(f(t)=D(T_t(\rho)||\sigma)\) of the signal relative to the noise \(\sigma\) decreases. This decrease is usually exponential: \(\displaystyle\frac{df}{dt}\leq -2\alpha\cdot f\) for some \(\alpha>0\), hence \(f(t)\leq\exp(-2\alpha t)\cdot f(0)\). The main objective of this paper is to find the actual rate of relative entropy convergence, i.e., to find the largest \(\alpha\) for which the above inequalities hold. The authors provide an (almost explicit) formula describing this largest \(\alpha\) in terms of the smallest eigenvalue of the noise-describing matrix \(\sigma\): almost explicit, since it has the form \(\min(g(x):0\leq x\leq 1)\) for an explicit expression \(g(x)\).NEWLINENEWLINENEWLINENEWLINEThe corresponding technique is also used to come up with an improved formula for the concavity of von-Neumann entropy \(S(\rho)\) of quantum states. Concavity means that for every two states \(\rho\) and \(\rho'\) and for every \(\alpha\in [0,1]\), we have \(S(\alpha\cdot \rho+(1-\alpha)\cdot \rho')-\alpha\cdot S(\rho)-(1-\alpha')\cdot S(\rho')\geq 0\); the authors present a new inequality, in which the original zero right-hand side is replaced by a non-negative expression that contains \(\alpha\), the relative entropies \(D(\rho \|\rho')\) and \(D(\rho'\|\rho)\), and the smallest eigenvalues of the states \(\rho\) and \(\rho'\). The authors compare their formula with the previously proposed improvement for the concavity inequality. It turns out that the new inequality is better when \(\alpha\) is close to 0 or 1.