Computation of mutual entropy in quantum markov chains

Luigi Accardi, Masanori Ohya, Hiroki Suyari

Yong‐Moon Park

Luigi Accardi, Masanori Ohya, Noboru Watanabe

Hiroki Suyari

Mutual entropy was introduced in classical communication theories to measure how much information can be transmitted from an input system to an output system. It has been expanded to quantum systems and applied to various physical models. By contrast, quantum Markov chains, introduced as an extension of classical Markov chains, have been used to explain the irreversibility of observed systems. In [1], a physical model used frequently in quantum optics was applied to a quantum Markov chain. The Stinespring-Kraus expression was derived to discuss mutual entropy on quantum Markov chains. Mutual entropy was then computed and irreversibility was examined using numerical experiments. To further the ongoing research, mean mutual entropy is applied here to the same quantum Markov chain as the previous one; necessary discussion is included. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(1): 104–112, 1997

Andreas Blühm, Ángela Capel, Pablo Costa Rico +1 more

Abstract It is well known that the conditional mutual information of a quantum state is zero if, and only if, the quantum state is a quantum Markov chain. Replacing the Umegaki relative entropy in the definition of the conditional mutual information by the Belavkin–Staszewski (BS) relative entropy, we obtain the BS-conditional mutual information, and we call the states with zero BS-conditional mutual information Belavkin–Staszewski quantum Markov chains. In this article, we establish a correspondence which relates quantum Markov chains and BS-quantum Markov chains. This correspondence allows us to find a recovery map for the BS-entropy in the spirit of the Petz recovery map. Furthermore, we show that, over the set of BS-quantum Markov chains, this correspondence constitutes an entanglement-breaking map. Moreover, we prove a structural decomposition of the Belavkin–Staszewski quantum Markov chains and also study states for which the BS-conditional mutual information is only approximately zero. We subsequently extend the aforementioned correspondence, structural decomposition, and recovery map to arbitrary pairs of states and quantum channels. As an application of the correspondence, we find a family of states with non-vanishing conditional mutual information for which it decays superexponentially fast with the size of the middle system.

Masanori Ohya, Hiroki Suyari

There are several damping phenomena in quantum optics.Such phenomena have been usually explained by open systems.In statistical physics, open system dynamics have been used to study the irreversibility and the approach to equilibrium.In this paper the dynamical change of the mutual entropy is discussed for a model of quantum Markov chain.The concrete Stinespring's expression for this model is obtained and applied to the derivation of the mutual entropy, and some computational results are presented.