7 Since this state is diagonal (“classical”) it is invariant under the dephasing channel with Kraus operators given by Eq. (18). Therefore the eigenvalues of the output state (A5) are The states input to the quantum dephasing channel obtained from Eq. (29) are therefore, for all ρ$_{S,i}$ This results in the following expression for the first term in Eq. (8): Finally, we calculate the third term in Eq. (8): υ$_{1}$ = cos 2 θ cos 2 x$_{1}$ + sin 2 θ sin 2 x$_{1}$, υ$_{2}$ = cos 2 θ sin 2 x$_{1}$ + sin 2 θ cos 2 x$_{1}$, (A7) ρ$_{S,}$$_{1}$ = Tr$_{R}$ ( | Φ$_{1}$ 〉〈 Φ$_{1}$ | ) = cos 2 θ | 0 〉〈 0 | + sin 2 θ | 1 〉〈 1 | , ρ$_{S,}$$_{2}$ = Tr$_{R}$ ( | Φ$_{2}$ 〉〈 Φ$_{2}$ | ) = sin 2 θ | 0 〉〈 0 | + cos 2 θ | 1 〉〈 1 | , ρ$_{S,}$$_{3}$ = Tr$_{R}$ ( | Φ$_{3}$ 〉〈 Φ$_{3}$ | ) = cos 2 θ | 0 〉〈 0 | + sin 2 θ | 1 〉〈 1 | , ρ$_{S,}$$_{4}$ = Tr$_{R}$ ( | Φ$_{4}$ 〉〈 Φ$_{4}$ | ) = sin 2 θ | 0 〉〈 0 | + cos 2 θ | 1 〉〈 1 | , (A2) S ( ρ$_{S,i}$ ) = − [cos 2 θ log$_{2}$ cos 2 θ + sin 2 θ log$_{2}$ sin 2 θ ] . (A3) S [ E ( ∑ i p$_{i}$ ( x$_{1}$, x$_{2}$ ) ρ$_{S,i}$ )] = − 2 ∑ i =1 υ$_{i}$ log$_{2}$ υ$_{i}$. (A8) and which for all | Φ$_{i}$ 〉 has eigenvalues ω$_{1}$ = ω$_{2}$ = 0 and ω$_{3}$, ω$_{4}$ are given in Eq. (31). The third term in Eq. (8) is thus ( E ⊗ I )( | Φ$_{i}$ 〉〈 Φ$_{i}$ | ) = ∑ i,j ( K$_{ij}$ ⊗ I )( | Φ$_{i}$ 〉〈 Φ$_{i}$ | )( K † ij ⊗ I ) , (A9) ∑ i p$_{i}$ ( x$_{1}$, x$_{2}$ ) S ( ρ$_{S,i}$ ) = − (cos 2 x$_{1}$ cos 2 x$_{2}$ + sin 2 x$_{1}$ cos 2 x$_{2}$ + cos 2 x$_{1}$ sin 2 x$_{2}$ + sin 2 x$_{1}$ sin 2 x$_{2}$ ) × [cos 2 θ log$_{2}$ cos 2 θ + sin 2 θ log$_{2}$ sin 2 θ ] , = − [cos 2 θ log$_{2}$ cos 2 θ + sin 2 θ log$_{2}$ sin 2 θ ] , (A4) where we have used the normalization ∑ 4 i =1 p$_{i}$ ( x$_{1}$, x$_{2}$ ) = 1 , ∀ x$_{1}$, x$_{2}$ . This yields the first term in Eq. (30). Next we calculate the second term in Eq. (8). The output state is ∑ i p$_{i}$ ( x$_{1}$, x$_{2}$ ) S [( E ⊗ I )( | Φ$_{i}$ 〉〈 Φ$_{i}$ | )] = − 4 ∑ i =1 ω$_{i}$ log$_{2}$ ω$_{i}$, (A10) E ( ∑ i p$_{i}$ ( x$_{1}$, x$_{2}$ ) ρ$_{S,i}$ ) = ∑ i,j K$_{ij}$ ( ∑ i p$_{i}$ ( x$_{1}$, x$_{2}$ ) ρ$_{S,i}$ ) K † $_{ij}$, (A5) ∑ i p$_{i}$ ( x$_{1}$, x$_{2}$ ) ρ$_{S,i}$ = [cos 2 θ (cos 2 x$_{1}$ cos 2 x$_{2}$ + cos 2 x$_{1}$ sin 2 x$_{2}$ ) + sin 2 θ (sin 2 x$_{1}$ sin 2 x$_{2}$ + sin 2 x$_{1}$ cos 2 x$_{2}$ )] | 0 〉〈 0 | + [cos 2 θ (sin 2 x$_{1}$ sin 2 x$_{2}$ + sin 2 x$_{1}$ cos 2 x$_{2}$ ) + sin 2 θ (cos 2 x$_{1}$ cos 2 x$_{2}$ + cos 2 x$_{1}$ sin 2 x$_{2}$ )] | 1 〉〈 1 | = (cos 2 θ cos 2 x$_{1}$ + sin 2 θ sin 2 x$_{1}$ ) | 0 〉〈 0 | + (cos 2 θ sin 2 x$_{1}$ + sin 2 θ cos 2 x$_{1}$ ) | 1 〉〈 1 | . (A6) where, as for Eq. (A4), we have used the normalization ∑ 4 i =1 p$_{i}$ = 1 . This yields the third term in Eq. (30). Thus, indeed only the second term in Eq. (8) depends on x$_{1}$, x$_{2}$ , and for a given value of θ , the classical capacity as- sisted by limited entanglement is maximized by maximizing Eq. (A8). The maximum is attained when the output state (A5) is fully mixed, i.e., when its eigenvalues υ$_{1}$ = υ$_{2}$ = 1 / 2 . This occurs when x$_{1}$ = x$_{2}$ = π $_{4}$, i.e., when we have an equiprobable ensemble of the states. This gives rise to the 1 in Eq. (30). where [4] C. Shannon, Bell Sys. Tech. J. 27 , 379 (1948). [5] C. H. Bennett and P. W. Shor, Science 303 , 1784 (2004). [6] P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, and W. K. Wootters, Phys. Rev. A 54 , 1869 (1996). [7] B. Schumacher and M. D. Westmoreland, Phys. Rev. A 56 , 131 (1997). [8] S. Lloyd, Phys. Rev. A 55 , 1613 (1997). [1] C. Bennett and P. Shor, IEEE Trans. Inf. Theory 44 , 2724 (1998). [2] M. Nielsen and I. Chuang, Quantum Computation and Quan- tum Information (Cambridge University Press, Cambridge, England, 2000). [3] T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).