Quantum capacity of additive Gaussian quantum channel

For a quantum channel with additive Gaussian quantum noise, at the large input energy side, we prove that the one shot capacity is achieved by the thermal noise state for all Gaussian state inputs, it is also true for non-Gaussian input in the sense of first order perturbation. For a general case of n copies input, we show that up to first order perturbation, any non-Gaussian perturbation to the product thermal state input has a less quantum information transmission rate when the input energy tend to infinitive. PACS number(s): 03.67.-a, 89.70.+c, 42.50.Dv Quantum capacity is one of the main issues in quantum information theory. It is concerned with the transmission ability of unknown quantum state on a given quantum channel. The critical quantity involved in the quantum capacity is the coherent information (CI) Ic(σ, E) = max{0, S(E(σ))−S(σQR ′ )} [1] [2]. Here S(̺) = −Tr̺ log2 ̺ is the von Neumann entropy, σ is the input state, the application of the channel E results the output state E(σ); σQR′ = (E ⊗ I)(|ψ〉 〈ψ|), with R referred to the ’reference’ system[1] (the system under process is Q system with annihilation and creation operators a and a, we denote σ as σ for simplicity), |ψ〉 is the purification of the input state σ. The quantum channel capacity is[3][4][5] Q = lim n→∞ sup σn 1 n Ic(σn, E). (1) Quantum capacity exhibits a kind of nonadditivity [6] that makes it extremely hard to deal with. The first example with calculable quantum capacity is quantum erasure channel[7]. Other examples are depolarizing qubit channel, and continuous variable lossy channel[8], where the channels are either degradable or anti-degradable. Gaussian quantum channel [9] (we may call it thermal noise channel, it differs from the terminology of quantum Gaussian channel which is a general mapping that maps Gaussian state to Gaussian state) is quite essential in quantum information theory. Unfortunately, this channel is neither degradable nor anti-degradable[10], makes the technics developed for calculating the quantum capacity unapplicable. The quantum capacity of the Gaussian quantum channel has been conjectured as[11] Q = max{0,− log2(eNn)}, (2) where Nn specifies the Gaussian quantum channel. It can be achieved by quantum error-correction codes[9]. For additive Gaussian quantum channel, we have [9] [12] E(σ) = 1 Nn ∫ dα π exp(− |α| /Nn)D (α)σD(α), (3) where D (α) = exp(αa − αa) is the displacement operator. Any quantum state σ can be equivalently specified by its characteristic function χσ(μ) = tr[σD(μ)], and inversely σ = ∫