Triangle of Dualities between Quantum Communication Protocols

We show that three quantum communication protocols—the “feedback father,” “fully quantum reverse Shannon,” and “fully quantum Slepian-Wolf” protocols—are pairwise related by three different dualities—the resource-reversal, time-reversal, and source-channel dualities.

Figure 1

A hypothetical decomposition of (many copies of) the state ${\psi }^{RAB}$ into three entangled pure states. The values of the nodes are the local entropy rates. The values of the links are the corresponding amounts of entanglement, which are uniquely determined by the local entropy values via (10). As in the GHZ case, it is only the dynamic version of this decomposition that holds.

Figure 2

Duality between FQSW and FQRS. The source is sending states ${\phi }^{RS}$ initially purified by some reference system $R$. A protocol implementing FQSW (16) transforms $n$ copies of the state ${\psi }^{RAB}=({\mathcal{U}}^{S\to AB}){\phi }^{RS}$ into $n$ copies of ${\phi }^{R\widehat{B}}={\mathrm{id}}^{S\to \widehat{B}}({\phi }^{RS})$ plus $\approx nI(A;B{)}_{\psi }/2$ shared ebits. Alice encodes with an isometry $V_1\mathrm{\text{:}}A\to A_1{A}_2$ and sends $A_1$, of size $\approx nI(R;A{)}_{\psi }/2$ qubits, through a quantum channel ${\mathrm{id}}^{A_1\to B_1}$ to Bob, and Bob decodes with an isometry $V_2\mathrm{\text{:}}B{B}_1\to B_2\widehat{B}$. The time reversal of this protocol implements the FQRS resource inequality (18), which transforms $n$ copies of ${\phi }^{R\widehat{B}}$ plus $\approx nI(A;B{)}_{\psi }/2$ shared ebits into $n$ copies of ${\psi }^{RAB}$. Bob encodes with $V_2^{-1}$ and sends $B_1$ through the channel ${\mathrm{id}}^{B_1\to A_1}=({\mathrm{id}}^{A_1\to B_1}{)}^{-1}$, and Alice decodes with $V_1^{-1}$.

Figure 3

A triangle of dualities.