2 decohere during transmission. The quantum capacity Q is the coherent information I$_{c}$ [7], maximized over all possible in- put states. In Eq. (5), | Φ 〉 ∈ H$_{S}$ ⊗ H$_{R}$ is a purification of ρ$_{S}$ obtained by appending a reference Hilbert space H$_{R}$ to the system Hilbert space H$_{S}$ . The limit n → ∞ is neces- sary as I$_{c}$ is super-additive [11], which makes the evaluation of Q difficult. However, for degradable channels the coherent information I$_{c}$ reduces to the conditional entropy, which is subadditive and concave, from which it follows that for these channels Q = Q$_{1}$ (single-channel use) [51]. This is an im- portant simplification, which enables the explicit calculation of Q in a variety of interesting cases. and its environment or bath, followed by a partial trace Tr$_{B}$ over the bath, if and only if system and bath start from a purely classically correlated initial state [49], such as a product state: Here ρ$_{B}$ is the initial state of the bath. The conjugate ˜ E of a quantum channel E is defined as [50], A quantum channel is called degradable if it can be degraded to its conjugate, that is, there exists a CPTP map T such that ˜ E = T ◦ E [51]. We shall make use of degradable channels later on in this work. ρ$_{S}$ ↦→ E ( ρ$_{S}$ ) = Tr$_{B}$ [ U ( ρ$_{S}$ ⊗ ρ$_{B}$ ) U $^{†}$] . (1) ˜ E ( ρ$_{B}$ ) = Tr$_{S}$ [ U ( ρ$_{S}$ ⊗ ρ$_{B}$ ) U $^{†}$] . (2) Entanglement is a useful resource in quantum information transmission. For example, it can be used to enhance the per- formance of quantum error correcting codes [55], to enhance quantum channel capacities by sharing entanglement between sender and receiver prior to communication [56], or by encod- ing information into entangled states when making successive uses of the same channel [23–27, 29, 30]. If the sender and receiver share unlimited prior entanglement, the maximum amount of classical information reliably transmitted over the quantum channel is given by its entanglement-assisted clas- sical capacity C$_{E}$ [12, 13, 15]. This quantity is obtained by maximization of the quantum mutual information for single channel use, which yields Unlike classical channels at least four capacities are asso- ciated with quantum channels depending on the type of infor- mation transmitted (classical or quantum), protocols allowed, and auxiliary resources used [5]. We are interested in the classical capacity C , quantum capacity Q , and entanglement- assisted capacities C$_{E}$ , Q$_{E}$ = C$_{E}$/ 2 and C lim E of a quantum dephasing channel. Let S ( ρ ) = − Tr[ ρ log$_{2}$ ρ ] denote the von Neumann en- tropy. The maximum amount of classical information reliably transmitted over a quantum channel is given by its classical capacity C [6, 7], Here | Φ 〉 ∈ H$_{S}$ ⊗ H$_{R}$ is the shared entangled state, which is also a purification of the input state ρ$_{S}$ ∈ H$_{S}$ . The amount of pure-state entanglement consumed by this communication protocol is S ( ρ$_{S}$ ) ebits per channel use, where ρ$_{S}$ maximizes Eq. (7). In contrast to the classical and quantum capacities, C$_{E}$ is additive [15]. The entanglement-assisted quantum ca- pacity is given by Q$_{E}$ = C$_{E}$/ 2 , and can be attained by super- dense coding [56], and quantum teleportation [57]. Shor has given a trade-off curve showing the classical ca- pacity as a function of the amount of entanglement shared by the sender and receiver [18]. The end points of this curve are given by the classical capacity C and the entanglement- assisted classical capacity C$_{E}$ . If the amount of entanglement available P is less than S ( ρ$_{S}$ ) , then the classical capacity as- sisted by limited entanglement is given by It depends on the largest set of orthogonal input states distin- guishable during the transmission and not on the ability of a channel E to preserve phases of different superpositions. C is the Holevo information χ [52] maximized over all possible input ensembles ρ$_{S}$ = $^{∑}$$_{i}$p$_{i}$ ρ$_{S,i}$ , where { p$_{i}$ } is a probabil- ity distribution and { ρ$_{S,i}$ } a set of quantum states (“quantum alphabet” belonging to the n -fold tensor product of system Hilbert spaces H$_{S}$ ), in the limit n → ∞ of parallel or succes- sive channel uses. The limit can be avoided when the Holevo information is additive over channel uses, in which case the optimal ensembles which achieve the maximum in Eq. (3) are separable with respect to the n uses and C coincides with C$_{n}$ n for all n , and in particular with C$_{1}$ . Hastings recently provided counterexamples to the additivity of the minimum output en- tropy [53], which implies by a result of Shor that the classical capacity is not always additive [54]. The quantum capacity Q is the maximum amount of quan- tum information transmitted by a quantum channel per chan- nel use [8, 14, 20], C$_{E}$ = max ρ$_{S}$ $_{∈H$_{S}$}${ S ( ρ$_{S}$ )+ S [ E ( ρ$_{S}$ )] − S [( E ⊗I )( | Φ 〉〈 Φ | )] } . (7) C = lim n →∞ C$_{n}$ n , C$_{n}$ = max p$_{i}$,ρ$_{S,i}$ ∈H ⊗ n S χ (3) χ = S [ E ⊗ $^{n}$( ρ$_{S}$ )] − ∑ i p$_{i}$S [ E ⊗ $^{n}$( ρ$_{S,i}$ )] . (4) C lim E = max { ρ$_{S,i}$,p$_{i}$ } ∑ i p$_{i}$S ( ρ$_{S,i}$ ) + S [ E ( ∑ i p$_{i}$ρ$_{S,i}$ )] − ∑ i p$_{i}$S [( E ⊗ I )( | Φ$_{i}$ 〉〈 Φ$_{i}$ | )] , (8) Q = lim n →∞ Q$_{n}$ n , Q$_{n}$ = max ρ$_{S}$ ∈H ⊗ n S I$_{c}$ (5) I$_{c}$ = S [ E ⊗ $^{n}$( ρ$_{S}$ )] − S [( E ⊗ n ⊗ I )( | Φ 〉〈 Φ | )] . (6) subject to $^{∑}$$_{i}$p$_{i}$ S ( ρ$_{S,i}$ ) ≤ P . Here the maximization is over the probabilistic ensemble { ρ$_{S,i}$, p$_{i}$ } where ρ$_{S,i}$ ∈ H$_{S}$ , $^{∑}$$_{i}$p$_{i}$ = 1 , p$_{i}$ ≥ 0 , and as above the shared entangled states | Φ$_{i}$ 〉 are purifications of ρ$_{S,i}$ . The capacity C lim E reduces to the classical capacity C given by Eq. (3) for P = 0 , as the con- straint $^{∑}$$_{i}$p$_{i}$ S ( ρ$_{S,i}$ ) ≤ P implies that ρ$_{S,i}$ must then all be pure states. For sufficiently large P it gives the entanglement- assisted classical capacity C$_{E}$ . The proof of additivity of C lim E is an open problem. For a given number of channel uses n , it depends on the di- mension of the largest Hilbert subspace of H ⊗ n S that does not