5 a maximally entangled state, an ensemble of orthogonal states is shared prior to the communication, where 0 ≤ θ ≤ π $_{4}$. As above the first and second qubits belong to the sender and receiver, respectively. We show in Appendix A that the clas- sical capacity assisted by limited entanglement [Eq. (8)] is at- tained when all states {| Φ$_{i}$ 〉} 4 i =1 are equiprobable, and that this yields: real, random values of g$_{n}$ and Ω$_{n}$ uniformly distributed in the interval [ − 1 , 1] and plot average capacities for 50 random en- sembles. In Fig. 2, we plot the capacities of the system spin coupled to a bath with N = 4 spins. We plot the capacities at low and high temperatures in order to study the effect of bath temperature. For low temperature ( β = 10 ), the bath is not too noisy and the system spin retains its coherence well. The capacities do not acquire their minimum values and partial re- currences occur, with an amplitude that diminishes over time. At high temperature ( β = 1 ) the capacities rapidly decrease to their minimum values and the recurrences are of smaller amplitude. As the system spin loses its coherence to the Ising bath, the entanglement shared between the sender and receiver is destroyed and C$_{E}$ is reduced to its minimum value of one. This corresponds to the qubit in possession of the receiver prior to the communication. The quantum capacity Q , which is a measure of the coherent information transmitted, is re- duced to zero as the system spin decoheres completely. As we increase the number of bath spins to N = 100 , we observe a similar dependence on bath temperature. The main difference compared to the case of a small number of bath spins is the drastically diminished amplitude of the recurrences. As noted in Ref. [38], this behavior is due to the averaging of the pos- itive and negative oscillations which arise for different values of the parameters g$_{n}$ and Ω$_{n}$ . | Φ$_{1}$ 〉 = cos θ | 00 〉 + sin θ | 11 〉 , | Φ$_{2}$ 〉 = sin θ | 00 〉 − cos θ | 11 〉 , | Φ$_{3}$ 〉 = cos θ | 01 〉 + sin θ | 10 〉 , | Φ$_{4}$ 〉 = sin θ | 01 〉 − cos θ | 10 〉 , (29) C lim E = − [cos 2 θ log$_{2}$ cos 2 θ + sin 2 θ log$_{2}$ sin 2 θ ] + 1 + 4 ∑ i =1 ω$_{i}$ log$_{2}$ ω$_{i}$, (30) ω$_{3}$ = 1 2 [1 + { ( 2 cos θ sin θ Z | Π$_{N}$ | ) 2 + cos $^{2}$2 θ } 1 2 ] , ω$_{4}$ = 1 2 [1 − { ( 2 cos θ sin θ Z | Π$_{N}$ | ) 2 + cos $^{2}$2 θ } 1 2 ] . (31) with ω$_{1}$ = ω$_{2}$ = 0 and For θ = 0 the states given by Eq. (29) are product states and we recover the classical capacity which is equal to one. The capacity C lim E increases as we increase the value of θ , attaining its maximum for θ = π 4 for which the states are maximally entangled and Eq. (30) reduces to Eq. (27). If the bath spins have equal frequencies Ω$_{n}$ ≡ Ω ∀ n , and couplings g$_{n}$ ≡ g ∀ n with the system spin then Eq. (26) re- duces to D. Limiting Cases 1. Equal Couplings and Frequencies Π$_{N}$ = 2 $^{N}$− 1 ∑ i =0 e − f ( t ) ∑ N n $_{=1}$( − 1) i n = N ∑ k =0 $^{(}$N k$^{)}$ e (2 k − N ) f ( t ) = (2 cosh[ f ( t )]) $^{N}$, (32) f ( t ) = 1 2 β Ω + 2 iαtg. (33) where FIG. 2: Capacities C$_{E}$ = 1 + Q (solid, thick), Q$_{E}$ = C$_{E}$/ 2 (solid, thin) and Q (dashed) of qubit coupled to an Ising spin bath with N = 4 , for random values of g$_{n}$ and Ω$_{n}$ . Left: β = 10 , right: β = 1 . See text for details. FIG. 3: Same as Fig. 2, with N = 100 . The second equality in Eq. (32) follows from the fact that the term ∑ N n $_{=1}$( − 1) i$_{n}$ = N − 2 k for i with Hamming weight k , of which there are ( N k ) cases for i ∈ [0 , . . . , 2 N − 1] . Therefore | Π$_{N}$ | = 2 N $^{√}$(cos $^{2}$(2 αtg ) cosh $^{2}$( β $_{2}$Ω) + sin$^{2}$(2 αtg ) sinh $^{2}$( β $_{2}$Ω)) N is periodic with period T$_{p}$ = π/ (2 αg ) , and the same is true of all the capacities computed above. At these times the bath spins destructively interfere and the dephasing channel becomes noiseless for information transmission. In the high temper- ature limit β Ω → 0 , and | Π$_{N}$ | → (2 cos(2 αtg )) $^{N}$, so that the capacities exhibit full periodic recurrences independently Plots of capacities for random values of couplings g$_{n}$ and bath frequencies Ω$_{n}$ , are given in Figs. 2 and 3. We generate