temperature and number of bath spins N is not too large. This is somewhat similar to the Zeno behavior pointed out in Ref. [38]. of N , in contrast to the results for random couplings and fre- quencies. Clearly, as N gets larger, these recurrences become sharper, until in the limit N → ∞ they become isolated peaks, as shown in the right-side panels of Figs. 4 and 5. In the low temperature limit β Ω → ∞ and | Π$_{N}$ | → exp( N 2 β Ω) , but so does the partition function Z = ∑ 2 $^{N}$− 1 i =0 e − 1 $_{2}$β Ω ∑ N n $_{=1}$( − 1) i n = (2 cosh[ f (0)]) N → exp( N 2 β Ω) , so that χ$_{3}$, χ$_{4}$ → 1 , and all the capacities are saturated at their maximum values. For small, but finite temperatures, the capacities exhibit oscilla- tions with an amplitude that grows with N , as can be seen in the left-side panels of Figs. 4 and 5. This is in contrast to the case of random couplings seen in Figs. 2 and 3; there destruc- tive interference caused a cancellation of these oscillations, while in the case of equal couplings the capacity oscillations survive and grow with the number of bath spins, reflecting the increased information transfer from the system to the bath as a function of bath size. We have studied an exactly solvable spin-star system for transmission of classical and quantum information. The in- formation is encoded into a system spin which interacts with a spin-bath via arbitrary Ising couplings. We considered the “parallel uses” setting of a memoryless quantum channel, where multiple copies of the system spin are transmitted si- multaneously via the same number of copies of the spin bath. As our model is described by the dephasing channel, classical information can be transmitted noiselessly, while the quan- tum capacity can be determined via the “single-letter” for- mula Q = Q$_{1}$ , i.e., it suffices to consider a single copy of the spin-star system. We analytically determined the quan- tum capacities of this communication system, which exhibit a strong dependence on the couplings of the bath spins with the system, and on the bath temperature. The Ising spin bath be- comes noisier as the temperature is increased, and the capac- ities rapidly deteriorate. For random couplings and frequen- cies, recurrences are of small amplitude and die out rapidly. However, for equal couplings and frequencies full periodic recurrences occur independently of the number of bath spins. These recurrences are a signature of the non-Markovian nature of the spin-bath. At low temperature the quantum capacities remain high when the number of bath spins is not too large. V. SUMMARY AND CONCLUSIONS 6 p$_{1}$ ( x$_{1}$, x$_{2}$ ) = cos $^{2}$x$_{1}$ cos $^{2}$x$_{2}$ , p$_{2}$ ( x$_{1}$, x$_{2}$ ) = sin $^{2}$x$_{1}$ cos $^{2}$x$_{2}$ , p$_{3}$ ( x$_{1}$, x$_{2}$ ) = cos $^{2}$x$_{1}$ sin $^{2}$x$_{2}$ , p$_{4}$ ( x$_{1}$, x$_{2}$ ) = sin $^{2}$x$_{1}$ sin $^{2}$x$_{2}$ , (A1) N.A. was supported by Higher Education Commission Pak- istan under grant no. 063-111368-Ps3-001 and IRSIP-4-Ps- 08. D.A.L. was supported by the National Science Foundation under grants no. PHY-803304 and CHE-924318. We prove Eq. (30). Without loss of generality the ensem- ble of orthogonal states given by Eq. (29) can be assumed to appear with probabilities parametrized as Without symmetries in the coupling constants or frequen- cies the capacities rapidly decrease to their minimum values and we find no recurrences for N ≫ 1 , high temperature and uniformly distributed random values of g$_{n}$ and Ω$_{n}$ . However, partial recurrences occur in this situation for small tempera- ture. Appendix A: Derivation of the result for the classical capacity assisted by limited entanglement Acknowledgments 2. Large N 3. Short Times FIG. 4: Capacities C$_{E}$ (solid, thick), Q$_{E}$ (solid, thin) and Q (dashed) of a qubit coupled to an Ising spin bath with N = 4 , for g$_{n}$ = 1 and Ω$_{n}$ = 1 ∀ n . Left: β = 10 , right: β = 1 . FIG. 5: Same as Fig. 4, with N = 100 . where 0 ≤ θ, x$_{1}$, x$_{2}$ ≤ π $_{4}$. We will show by explicit cal- culation that for our pure dephasing model only the second term in Eq. (8) for the classical capacity assisted by limited entanglement depends on the parameters x$_{1}$, x$_{2}$ . Hence the maximization can be carried out, for fixed θ , by maximizing only this second term. The capacities are flat initially and do not decay exponen- tially in the limit of short times αt ≪ 1 , provided that the