This paper presents some examples of quantum reliability function for the quantum communication system in which classical information is transmitted by quantum states. In addition, the quantum Cut off rate is defined. They will be compared with Gallager's reliability function for the same system.
We study the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Rényi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Rényi information.
The cut-off rate performance of quantum communication channels is considered when the signal quantum states are pure and symmetric. The properties of the quantum cut-off rate which is obtained by the entangled measurement of the received signal are investigated for communication channels in which the symmetric quantum states are both linearly independent and linearly dependent. It is found that the quantum cut-off rate becomes independent of the number of the signal quantum states when they are linearly dependent. Furthermore the properties of the semiclassical cut-off rate which is obtained by the disentangled measurement of the received signal are considered. The necessary and sufficient condition for the positive operator-valued measure of the disentangled measurement to maximize the semiclassical cut-off rate is derived. For binary quantum communication channels with pure signal states, the several parameters, such as the channel capacity and the cut-off rate, which characterize the quantum communication channel are compared.
Quantum communication with binary nonorthogonal pure states and single-shot measurements are discussed in terms of reliability function and cut-off rate. It is shown that some of different measurement strategies simultaneously attain the maximum single-shot cut-off rate. We also apply these measurements to the BPSK optical coherent states and compare them with the homodyne detection. Numerical results show that even including realistic imperfections, it is possible to exceed the homodyne limited cut-off rate.