2 2 4 Fe b 20 06 Quantum capacity of channel with thermal noise

The quantum capacity of thermal noise channel (which is addressed as Gaussian quantum channel before) is studied, it turns out to be the maximum of the coherent information of a single use of the channel. The quantum capacity is achieved when the input state is a thermal noise state with infinitive energy. The capacity is achievable with quantum error-correction code. The quantum capacities of thermal noise channels with damping or amplification are also given. One of the most important issues of classical information theory is the Shannon formula, which is the capacity of an additive white Gaussian noise channel. It is achieved when the input is a Gaussian noise source with power constraint [1] [2]. C = log2(1 + S N ), (1) where S is the power of the source andN is the power of the noise, the bandwidthW should be multiplied when it is considered. The formula guide the design of the practical communication system for decades. Correspondingly, in quantum information theory, although a lot of works have been done [3], such a formula is remained to be discovered. The Shannon formula comes from the Shannon noisy coding theorem, the later gives the capacity of any noisy channel: C = sup X I(X ;Y ), (2) where the supremum is taken over all inputs X , I(X ;Y ) is the Shannon mutual information and Y is the output. The counterpart of mutual information in quantum information theory is the coherent information (CI) Ic(ρ, E) = S(E(ρ))− S(ρ ′ ) [4] [5]. Here S(̺) = −Tr̺ log2 ̺ is the von Neumann entropy, ρ is the input state, the application of the channel E resulting the output state E(ρ); ρRQ′ = (E ⊗ I)(|ψ〉 〈ψ|), |ψ〉 is the purification of the input state ρ. The quantum channel capacity is Q = lim n→∞ sup ρn Ic(ρn, E). (3) The righthand side of above formula was firstly proved to be the upper bound of quantum channel capacity [6]. The equality was proved at the postulation of hashing inequality [3]; the hashing inequality was lately established [7]. The quantum capacity of a noisy quantum channel is the maximum rate at which coherent information can be transmitted through the channel and recovered with arbitrarily good fidelity. For quantum information channel can be supplemented by oneor two-way classical channel, thus quantum capacities should be defined with these supplementary resources. We here deal with the quantum capacity without any supplementary classical channel[3]. Quantum capacity exhibits a kind of nonadditivity [8] that makes it extremely hard to deal with. Until now, quantum capacity has not been carried out except for quantum erasure channel[9]. As we will elucidate below, for thermal noise quantum channel (which is addressed as Gaussian quantum channel before [10]), quantum capacity can be carried out. The general description of the channel is to map the state ρ to another state E(ρ), where E is a trace preserving completely positive map. The map E has a Krauss operator sum representation. That is E(ρ) = ∑ iAiρA † i with ∑ iA † iAi = I. For additive quantum Gaussian channel, it is quite simple to choose Aα proportional to displacement operator [11] D (α) = exp[αa † − αa]. The output state will be E(ρ) = 1 πN ∫ dα exp(− |α| /N)D (α) ρD(α). (4)