Reversibility of entanglement assisted coding

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For any two quantum channels [math]S[/math] and [math]T[/math], define the entanglement assisted capacity [math]C_\text{E}(T,S)[/math] of [math]T[/math] for [math]S[/math]-messages as the supremum of all rates [math]r[/math] such that, for large [math]n[/math], [math]rn[/math] parallel copies of [math]T[/math] may be simulated by [math]n[/math] copies of [math]S[/math], where the simulation involves arbitrary coding and decoding operations using (if necessary) arbitrarily many entangled pairs between sender and receiver, and where the errors go to zero as [math]n\to\infty[/math].

Show that [math]C_\text{E}(T,S)=C_\text{E}(S,T)^{-1}[/math].


As for other capacities, the two-step coding inequality [math]C_\text{E}(T,S) \,C_\text{E}(S,R)\leq C_\text{E}(T,R)[/math] is easy to show. Hence [math]C_\text{E}(T,S)C_\text{E}(S,T)\leq 1[/math]. Equality means here, that the two channels are essentially equivalent as a resource for simulating other channels [math]R[/math] (apart from a constant factor): [math]C_\text{E}(R,S)=\text{const} C_\text{E}(R,T)[/math] (with [math]\text{const}=C_\text{E} (T,S)[/math]). In this case we call [math]S[/math] and [math]T[/math] reversible for entanglement assisted coding.

For ordinary capacity [math]C(T,S)[/math] (without entanglement assistance) reversibility fails in general: When [math]S[/math] is an ideal classical 1 bit channel, and [math]T[/math] is an ideal 1 qubit quantum channel, we have [math]C(S,T)=1[/math], but [math]C(T,S)=0[/math], because quantum information cannot be sent on classical channels. On the other hand, with entanglement assistance we have [math]C(S,T)=2[/math] by superdense coding and [math]C(T,S)=1/2[/math] by teleportation.

Because all ideal channels [math]S[/math] are equivalent as reference channels, we can define [math]C_\text{E}(T)=C_\text{E}(T,S_1)[/math], with [math]S_1[/math] the ideal classical 1 bit channel as the entanglement assisted capacity of [math]T[/math]. For this quantity there is an explicit formula (coding theorem) by [1]. The problem stated above appears in [2] as the "Reverse Shannon Theorem".

Partial Results

The problem is solved for the special case of a known "tensor power source", i.e. a source emitting the same, known, density matrix at each time step. Recent efforts by P. Shor focus on the unknown tensor power source and the known "tensor product source" where the density matrix of the source is a tensor product [3].


  1. C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Entanglement-assisted classical capacity of noisy quantum channels, Phys. Rev. Lett. 83, 3081 (1999), quant-ph/9904023.
  2. C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem, quant-ph/0106052
  3. P. W. Shor, private communication (2003).