Bounds of the Accessible Information under the Influence of Thermal Noise

Upper bound of the accessible information obtained in quantum detection processes for Gaussian state signals under the influence of thermal noise is derived by means of the superoperator representation of quantum states. It is shown that the upper bound is obtained by replacing the parameters of the Gaussian quantum state with the renormalized parameters including the thermal noise effect in the accessible information in the absence of thermal noise. Let S and N be sets of the parameters characterizing the signal and thermal noise, and let I(S, N) [I 0(S) = I(S, 0)] be the accessible information in the presence [absence] of thermal noise. Then the inequality I(S,N) ≤ I o(S N ) is derived, where S N stands for a set of the renormalized parameters including the thermal noise effect. Of course, the inequality I(S, N) ≤ H(S,N) is satisfied, where H(S,N) is the Holevo bound of the accessible information. Furthermore, let I e(S, N) be the mutual information obtained in the detection process that minimizes the average probability of error. Then the inequality I e(S, N) ≤ I(S,N) is satisfied. Therefore the following inequality is obtained $${I_e}\left( {S,N} \right) \leqslant I\left( {S,N} \right) \leqslant \min \left[ {{I_0}\left( {{S_N}} \right),H\left( {S,N} \right)} \right]$$ . Furthermore the method used to derive the upper bound of the accessible information can be applied for obtaining the lower bound of the Bayes cost.