3 B. Exact Solution III. QUANTUM DEPHASING CHANNEL A. The Model At any given time t , the state of the system qubit ρ$_{S}$ ( t ) can be written in the Kraus representation as [58], where the Kraus operators satisfy the completeness relation ∑ i,j K † $_{ij}$K$_{ij}$ = I$_{S}$ . After a transformation to the interaction picture defined by H$_{S}$ + H$_{B}$ , these operators can be expressed as where we have introduced the spectral decomposition ρ$_{B}$ = ∑ i λ$_{i}$ | i 〉〈 i | of the initial bath state. For the Gibbs thermal state chosen here the eigenbasis states {| i 〉} are N -fold tensor products of the σ z eigenstates, which gives We consider the case of an exactly solvable spin star sys- tem of N + 1 localized spin- 1 2 particles as shown in Fig. 1. The system input state ρ$_{S}$ (0) is carried by the central (system) spin. This spin interacts with N noncentral spins comprising the bath. The bath spins do not interact with each other di- rectly. The interaction between the system spin and bath is given by the Ising Hamiltonian ρ$_{S}$ ( t ) = E ( ρ$_{S}$ (0)) = ∑ i,j K$_{ij}$ρ$_{S}$ (0) K † $_{ij}$, (15) K$_{ij}$ = $^{√}$λ$_{i}$ 〈 j | exp( − iH$_{I}$t ) | i 〉 , (16) ρ$_{B}$ = ∑ i exp( − βE$_{i}$ ) Z | i 〉〈 i | , (17) FIG. 1: Communication protocol: a qubit passes through an Ising spin-star channel. This is repeated in parallel over many identical such channels. where E$_{i}$ = 1 2 ∑ N n $_{=1}$Ω$_{n}$ ( − 1) i$_{n}$ is the energy of each bath eigenstate | i 〉 ( i = i$_{1}$, i$_{2}$, . . . , i$_{N}$ is the binary expansion of the integer i , where i ∈ [0 , . . . , 2 $^{N}$− 1] ) and Z = $^{∑}$$_{i}$exp( − βE$_{i}$ ) is the partition function. Therefore, the Kraus operators are where we work in$_{ℏ}$ = 1 units, g$_{n}$ ∈ [ − 1 , 1] are dimensionless real-valued coupling constants, and α > 0 is the coupling strength having the dimension of frequency. The system and bath Hamiltonians are given by with λ$_{i}$ = exp( − βE$_{i}$ ) /Z , and ˜ E$_{i}$ given by K$_{ij}$ = $^{√}$λ$_{i}$ exp( − itα ˜ E$_{i}$σ $^{z}$) δ$_{ij}$, (18) H$_{I}$ = ασ$_{z}$ ⊗ N ∑ n =1 g$_{n}$σ z $_{n}$, (9) ˜ E$_{i}$ = N ∑ n =1 g$_{n}$ ( − 1) i$_{n}$ − Tr { ∑ n g$_{n}$σ z $_{n}$ρ$_{B}$ } , = N ∑ n =1 g$_{n}$ [( − 1) i$_{n}$ − β$_{n}$ ] , (19) H$_{S}$ = 1 2 ω$_{0}$σ $^{z}$, (10) H$_{B}$ = 1 2 N ∑ n =1 Ω$_{n}$ σ z $_{n}$. (11) ρ (0) = ρ$_{S}$ (0) ⊗ ρ$_{B}$, (12) ρ$_{B}$ = exp( − βH$_{B}$ ) / Tr [exp( − βH$_{B}$ )] . (13) ρ$_{S}$ ( t ) = Tr$_{B}$ { U ( t ) ρ (0) U $^{†}$( t )) } , (14) where β$_{n}$ = tanh( − 1 $_{2}$β Ω$_{n}$) . The CPTP map ρ$_{S}$ (0) E → ρ$_{S}$ ( t ) with the Kraus operators given by Eq. (18), represents a quan- tum dephasing channel, since the Kraus operators are diagonal in the reference basis {| 0 〉 , | 1 〉} ∈ H$_{S}$ (eigenstates of σ z with eigenvalues ± 1 ) of the system. Moreover, as it corresponds to a non-Markovian model [38], E represents a quantum dephas- ing channel with memory. We now determine the information transmission capacities of this channel. The frequencies ω$_{0}$ and Ω$_{n}$ are restricted to the interval [ − 1 , 1] , in frequency units. Initially, the total system is as- sumed to be in the product state with the bath in the Gibbs thermal state at inverse temperature β = 1 / ( kT ) given by Since ρ$_{B}$ commutes with H$_{I}$ the bath state is stationary throughout the dynamics: ρ$_{B}$ ( t ) = ρ$_{B}$ . The state of the sys- tem qubit is obtained by performing a partial trace over the bath Hilbert space where U ( t ) = exp[ − it ( H$_{S}$ + H$_{I}$ + H$_{B}$ )] . The analytical solution of this model was worked out in detail in Ref. [38], and we present a brief summary next. Dephasing channels have the characteristic property of transmitting states of a preferential orthonormal basis without introducing any error [2]. These basis states can be used to en- code classical information, which makes these channels noise- less for the transmission of classical information [28]. Super- positions of the basis states will decohere, however, therefore IV. CAPACITIES OF QUANTUM DEPHASING CHANNEL A. Classical Capacity