The mother of all protocols: Restructuring quantum information's family tree

TL;DR

The paper certifies the mother protocol as a unifying hub in quantum information theory by presenting a direct, decoupling-based proof of fully quantum Slepian-Wolf (FQSW) that achieves simultaneous entanglement distillation and state merging. From this foundation, it derives one-shot and i.i.d. versions of the mother, and shows how the father and fully quantum reverse Shannon protocols emerge as natural children, as does distributed compression for correlated quantum sources. The results provide explicit rate expressions in terms of mutual informations and reveal a close operational link between entanglement distribution, state transfer, and channel simulation, all under a single operational framework. The work also proves that encoding can be done efficiently via Clifford operations and typical subspace techniques, while identifying open questions about the exact rate region and the role of squashed entanglement in distributed scenarios.

Abstract

We give a simple, direct proof of the "mother" protocol of quantum information theory. In this new formulation, it is easy to see that the mother, or rather her generalization to the fully quantum Slepian-Wolf protocol, simultaneously accomplishes two goals: quantum communication-assisted entanglement distillation, and state transfer from the sender to the receiver. As a result, in addition to her other "children," the mother protocol generates the state merging primitive of Horodecki, Oppenheim and Winter, a fully quantum reverse Shannon theorem, and a new class of distributed compression protocols for correlated quantum sources which are optimal for sources described by separable density operators. Moreover, the mother protocol described here is easily transformed into the so-called "father" protocol whose children provide the quantum capacity and the entanglement-assisted capacity of a quantum channel, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary: all protocols in the family are children of the mother.

Figures (6)

• Figure 1: a) The starting point for the FQSW protocol, a pure tripartite entangled state $|\psi\rangle = (|\varphi\rangle^{ABR})^{\otimes n}$. b) After execution of the protocol, Alice's portion of the original tripartite state has been transferred to Bob, so that Bob holds a purification of the unchanged reference system in his register $\widehat{B}$. He also shares pure state entanglement with Alice in the form of the state $|\Phi\rangle$.
• Figure 2: Partial quantum information theory family tree. The symbols  and  represent the "old" mother and father protocols from DHW04 and arrows indicate that a protocol accomplishing the task at the start of the arrow can be transformed into a protocol accomplishing the task at the end. The relationships between  ,  and their children are discussed in detail in DHW04DHW05. "QMAC" refers to the task of sending quantum data through a quantum multiple access channel HOW05YDH05, "broadcast" the task of sending quantum data through a quantum broadcast channel YHD06 and the environment-assisted quantum capacity is discussed in SVW05.
• Figure 3: Partial reduction from the father to the mother. Dotted lines are used to demarcate domains controlled by the different partipicants and solid lines represent quantum information. Note that Alice starts the protocol sharing one maximally entangled state with the reference, $|\Phi_0\rangle^{A_0 R}$, and another with Bob, $|\Phi_3\rangle^{A_3 B_3}$. The unitary transformation $U^{B_3 R}$ comes from an application of the FQSW theorem with $B_3 R$ replacing $A_1 A_2$. After the application of the unitary, the registers $R$ and $E$ are nearly decoupled, as desired, but unfortunately, because it requires acting on the reference system $R$, $U^{B_3 R}$ cannot be used in this way.
• Figure 4: Final version of the father protocol generated from FQSW. As Figure \ref{['fig:father1']} makes clear, $U^{B_3 B}$ was required to act on one half of a maximally entangled state, the other half of which found in $A_3 A_0$, register held by Alice. Thus, Alice can instead implement the encoding operation $W_2 = W_1 \circ U^T$. Bob performs the decoding operation $V$ mandated by FQSW, resulting in the one-shot father.
• Figure 5: a) The starting point for FQRS, a pair of pure entangled states. The system $A$ is a purification of Alice's input system $A'$ while $\tilde{A}\tilde{B}$ holds the entanglement that Alice-Bob will consume to execute the protocol. b) After execution of the protocol, the reference system $A$ is unchanged while Alice receives the environment feedback system $E$ and Bob receives his share $B$ of the state $|\psi\rangle^{ABE} = U_{{\cal N}}^{A'\rightarrow BE} |\psi\rangle^{AA'}$.

Theorems & Definitions (7)

• Theorem 4.1: One-shot, fully quantum Slepian-Wolf bound
• Theorem 4.2: Decoupling
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• Theorem 10.1
• Theorem 10.2