Almost all Pauli Channels have non-zero quantum capacities

Quantum transmission rate may be improved by using input of entangled state instead of product state for Pauli channel. Using GHZ state as input for the depolarizing channel (a special Pauli channel), the channel fidelity of non-zero quantum capacity was improved from > 0.81071 to > 0.80944. We will introduce orthogonal and complete code basis to evaluate the coherent information per channel use when the input is the maximal mixture of stabilizer codewords. In the code basis, the output density matrix is diagonal, the joint output is block diagonal. The coherent information is worked out by counting the weights of error operators. Quantum capacity is analytically proved to be non-zero for all values of channel fidelity except 1 4 in depolarizing channel, for all cases except px + py = 1 2 in Pauli channel. Quantum information can be transmitted in almost all Pauli channels, even in zero fidelity channel. PACS: 03.67.Hk, 03.67.Pp, 03.65.Ud The basic issue in quantum information theory is quantum coding theorem. After ten year’s efforts, quantum coding theorem had at last been proven. The rate of faithfully transmitting quantum information per use of quantum channel is limited by quantum capacity, the capacity is asymptotically achievable [1] [2] [3] [4]. Quantum capacity is the maximization of coherent information [5] over all input states. Unfortunately, since coherent information is non-additive [6], quantum capacity in single letter form is not available except for degradable [7] or anti-degradable channels. Regulation is need, that is, block input with infinitive number of qubits should be used to calculate quantum capacity in general. We may only obtain the lower bound of quantum capacity . The most prominent example is the depolarizing channel which is a special case of Pauli channel. The single letter lower bound of quantum capacity for depolarizing channel is not zero only for very low noise level p < 0.06310 or high fidelity f = 1 − 3p > 0.81071. With the input of GHZ state, the allowed fidelity was improved to f > 0.80944 (p < 0.06352) [6]. Further but limited improvement by degenerate code can be found in [8]. The obstacle of further improving the domain of fidelity with other multipartite input state is obvious, the dimension of the state increases exponentially with the the number of the input qubits, making the calculation of the output entropy and the entropy exchange (thus the coherent information) an awful work. We will greatly reduce the complexity of diagonalizing the output density matrix by introducing quantum error-correcting code (QECC) as the input state. The theory of QECCs was established more than a decade ago as the tool for fighting decoherence in quantum computers and quantum communication systems [9]. Maybe the most impressive development in quantum errorcorrection theory is the use of the stabilizer formalism[10] [11] [12] [13]. The power of the stabilizer formalism comes from the clever use of group theory. The n-fold Pauli operators {I,X, Y, Z}⊗n together with the possible overall factors ±1,±i form a group Gn under multiplication, the n-fold Pauli group. Suppose S is an abelian subgroup of Gn. Stabilizer coding space T is the simultaneous +1 eigenspace of all elements of S, T = {|ψ〉 : M |ψ〉 = |ψ〉 , ∀M ∈ S}. For an [[n, k, d]] stabilizer code, which encodes k logical qubits into n physical qubits, T has dimension 2 and S has 2n−k elements. The generators of S are denoted as Mi (i = 1, . . . , n− k) which are Hermitian. There are many elements in Gn that commute with every elements of S but not actually in S. The set of elements in Gn that commute with all of S is defined as the centralizer C(S) of S in Gn. Clearly S ⊂ C(S). Denote Ω = ∏ i(I+Mi). Due to the properties that the elements of stabilizer group S commute andM 2 i = I, it follows Mj ∏ i(I +Mi) = ∏ i(I +Mi). Further, we have ΩMjΩ = Ω 2 = 2n−kΩ. For error operator Ea that anti-commutes with at least one of the generators Mi, we have ΩEaΩ = 0, (1) 1 this is due to (I+Mi)(I−Mi) = I−M 2 i = 0, the operator factor (I−Mi) comes from Ea(I+Mi) = (I−Mi)Ea if Ea anti-commute with Mi. In Krauss representation, Pauli channel map E acting on qubit state ρ can be written as E (ρ) = fρ + pxXρX+ pyY ρY + pzZρZ, where px(y,z) ∈ [0, 1] are the probabilities, f = 1− px − py − pz ∈ [0, 1] is the fidelity of the channel. For depolarizing channel, px = py = pz = p, f = 1 − 3p. For n use of depolarizing channels with n qubits input state ρ, we have the output state ρ′ = E⊗n (ρ) = ∑ a ηaEaρE † a, with ηa = f pxp j yp l z for Ea = X Y Z . The purification of ρ is |Ψ〉 , we have the joint output state ρe = (E ⊗n ⊗ I⊗n) (|Ψ〉 〈Ψ|) , whose entropy is the entropy exchange. The coherent information is Ic = S(ρ ′) − S(ρe), where S(·) is the von Neumann entropy. To illustrate non-zero capacity even for zero fidelity of Pauli channel, we first consider the example of [[5, 1, 3]] QECC as the input state to the depolarizing channel. The stabilizer code has four generatorsMi (i = 1, . . . , 4). The codewords are ∣