General bounds for sender-receiver capacities in multipoint quantum communications

We investigate the maximum rates for transmitting quantum information, distilling entanglement, and distributing secret keys between a sender and a receiver in a multipoint communication scenario, with the assistance of unlimited two-way classical communication involving all parties. First we consider the case where a sender communicates with an arbitrary number of receivers, a so-called quantum broadcast channel. Here we also provide a simple analysis in the bosonic setting where we consider quantum broadcasting through a sequence of beamsplitters. Then, we consider the opposite case where an arbitrary number of senders communicate with a single receiver, a so-called quantum multiple-access channel. Finally, we study the general case of all-in-all quantum communication where an arbitrary number of senders communicate with an arbitrary number of receivers. Since our bounds are formulated for quantum systems of arbitrary dimension, they can be applied to many different physical scenarios involving multipoint quantum communication.

Figure 1

Point-to-point adaptive protocol. Each transmission $a_i\to b_i$ through the quantum channel $\mathcal{E}$ is interleaved by two adaptive LOCCs, ${\mathrm{\Lambda }}_{i-1}$ and ${\mathrm{\Lambda }}_i$, applied to Alice's and Bob's local registers $\mathbf{a}$ and $\mathbf{b}$. After $n$ transmissions, Alice and Bob share an output state ${\rho }_{\mathbf{ab}}^n$.

Figure 2

Teleportation stretching of an adaptive point-to-point protocol. Top panels: The generic $i\mathrm{th}$ transmission through the channel $\mathcal{E}$ is simulated by an LOCC $\mathcal{T}$ and a resource state $\sigma$. Bottom panels: The resource state $\sigma$ is stretched back in time out of the adaptive LOCCs. This is repeated for all the $n$ transmissions, so that we accumulate the tensor-product state ${\sigma }^{\otimes n}$. All the LOCCs (those of the original protocol and those introduced by the simulation) are collapsed into a single trace-preserving LOCC $\overline{\mathrm{\Lambda }}$ (which also includes the initial state of the registers ${\rho }_{\mathbf{ab}}^0$).

Figure 3

Simulation of a teleportation-covariant quantum broadcast channel. We may replace the broadcast channel $\mathcal{E}:a\to bc$ by teleportation over its Choi matrix ${\rho }_{\mathcal{E}}$, with CCs to Bob and Charlie, who will implement correction unitaries. The broadcast channel is therefore Choi stretchable and its LOCC simulation is based on teleportation.

Figure 4

Stretching an adaptive protocol over a teleportation-covariant quantum broadcast channel. Top panels: The generic $i\mathrm{th}$ transmission $a_i\to \{b_i,c_i\}$ over the broadcast channel $\mathcal{E}$ (red line) is replaced by a teleportation over its Choi matrix ${\rho }_{\mathcal{E}}$ (following the procedure shown in Fig. 3). The input system and the upper half of the Choi matrix are subject to a Bell detection which becomes part of Alice's LO (upper LO). The result of the Bell detection $k$ is classically communicated to Bob and Charlie so that they can apply two correction unitaries which are then included in their respective LOs (middle and lower LOs). Bottom panels: The Choi matrix is stretched in time out of the adaptive LOCCs which are then collapsed into a single trace-preserving LOCC. After $n$ uses, we can express the output in terms of $n$ copies of the Choi matrix ${\rho }_{\mathcal{E}}$ of the broadcast channel, plus a trace-preserving single final LOCC $\overline{\mathrm{\Lambda }}$ as in Eq. (29).

Figure 5

Thermal-loss quantum broadcast channel ${\mathcal{E}}_{A^{\prime }\to B_1...B_M}$ from Alice (mode $A^{\prime }$) to $M$ Bobs (modes $B_1,...,B_M$), realized by a multisplitter, i.e., a sequence of $M+1$ beamsplitters with transmissivities $({\eta }_0,{\eta }_1,...{\eta }_M)$. The environmental modes $E_0,E_1...,E_M$ are in thermal states. Modes $E$ and $E^{\prime }$ describe leakage to the environment.

Figure 6

Simulation of a teleportation-covariant quantum multiple-access channel. We can replace the multiple-access channel $\mathcal{E}:a^1{a}^2\to b$ (left) by double teleportation over its tripartite Choi matrix ${\rho }_{\mathcal{E}}$ (right). This Choi matrix is obtained by sending halves ($A^{\prime 1}$ and $A^{\prime 2}$) of two EPR states $\mathrm{\Phi }$ through $\mathcal{E}$, with output $B$. Then, systems $a^1$ and $A^1$ are subject to a Bell detection with outcome $k_1$. Similarly, systems $a^2$ and $A^2$ are subject to a Bell detection with outcome $k_2$. The outcomes are CC'ed to Bob who applies a correction unitary on system $B$. Since the channel is teleportation-covariant, i.e., it commutes with the teleportation unitaries according to Eq. (53), Bob's correction unitary $V_k^{-1}$ on $B$ regenerates the original channel $\mathcal{E}:a^1{a}^2\to b$.

Figure 7

Teleportation stretching of an adaptive protocol implemented over a teleportation-covariant multiple-access channel (generic $m\mathrm{th}$ transmission shown on the left). After $n$ uses, we can express the output in terms of $n$ copies of the Choi matrix ${\rho }_{\mathcal{E}}$ of the quantum multiple-access channel, subject to a trace-preserving LOCC $\overline{\mathrm{\Lambda }}$.

Figure 8

Simulation of a teleportation-covariant quantum interference channel. The channel $\mathcal{E}:a^1{a}^2\to b^1{b}^2$ (left) can be simulated by its Choi matrix ${\rho }_{\mathcal{E}}$ (right). Systems $a^1$ and $A^1$ are subject to a Bell detection with outcome $k_1$. Similarly, systems $a^2$ and $A^2$ are subject to a Bell detection with outcome $k_2$. Both outcomes $k_1$ and $k_2$ are then classically communicated to Bob 1 and Bob 2 who apply two correction unitaries on $B^1$ and $B^2$. Since the channel is teleportation covariant, i.e., it commutes with the teleportation unitaries according to Eq. (63), the two Bobs are able to recover the original channel $\mathcal{E}:a^1{a}^2\to b^1{b}^2$ by applying correction unitaries ${\left(V_{l_1}^1\right)}^{-1}$ and ${\left(V_{l_2}^2\right)}^{-1}$.

Figure 9

Teleportation stretching of an adaptive protocol over a quantum interference channel (generic $m\mathrm{th}$ transmission shown on the left). After $n$ uses, we can express the output in terms of $n$ copies of the Choi matrix ${\rho }_{\mathcal{E}}$ of the quantum interference channel, subject to a trace-preserving LOCC $\overline{\mathrm{\Lambda }}$.