On S-mixing entropy of quantum channels

Junfang Cheng, Yanjun Chu, Fang Huang

Inspired by a conjecture related to numerical investigations for a quantum channel in a system of large size, we shall study relations between two different entropies of a quantum channel. One of them is defined by Gour and Wilde as the generalization of quantum relative entropy and maximally mixed state. The second one is defined as the difference between the entropy of the Choi state of the quantum channel and a constant number. At first, we show that the latter entropy is an upper boundary of the former entropy. Moreover, we also give a sufficient and necessary condition that the two entropies are equal. That is, if and only if the relative entropy reaches its supremum on the maximal entangled pure state in the former entropy. At last, to study the difference between the two entropies, we expect to find some quantum channels in which their related entropies do not reach their supremum on the maximal entangled pure states for the former entropy. By calculating the two entropies, we find that amplitude-damping channels are desired examples.

R. S. Ingarden, Andrzej Kossakowski, M. Ohya

Alexey E. Rastegin

Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.

Yanjun Chu, Fang Huang, Mingxiao Li +1 more

Noboru Watanabe

In classical information theory, the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov–Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes–Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy were proposed [10,[Formula: see text]3,[Formula: see text]8,[Formula: see text]39,[Formula: see text]15,[Formula: see text]44,[Formula: see text]9,[Formula: see text]27,[Formula: see text]28,[Formula: see text]2,[Formula: see text]19,[Formula: see text]45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S-mixing entropy by Ohya [26,[Formula: see text]27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S-mixing entropy. In this paper, based on research into the dynamical entropy and mean mutual entropy of quantum systems, this study investigates the characteristics of quantum channels that exhibit entanglement, a property unique to quantum systems, and investigates the behaviour of quantum entropy for compound quantum channels by changing the perspective from information transmission between input and output to a new perspective of information transmission from the initial state to the final state.

Masanori Ohya, Dėnes Petz, Noboru Watanabe