Information Trade-Offs for Optical Quantum Communication

Recent work has precisely characterized the achievable trade-offs between three key information processing tasks—classical communication (generation or consumption), quantum communication (generation or consumption), and shared entanglement (distribution or consumption), measured in bits, qubits, and ebits per channel use, respectively. Slices and corner points of this three-dimensional region reduce to well-known protocols for quantum channels. A trade-off coding technique can attain any point in the region and can outperform time sharing between the best-known protocols for accomplishing each information processing task by itself. Previously, the benefits of trade-off coding that had been found were too small to be of practical value (viz., for the dephasing and the universal cloning machine channels). In this Letter, we demonstrate that the associated performance gains are in fact remarkably high for several physically relevant bosonic channels that model free-space or fiber-optic links, thermal-noise channels, and amplifiers. We show that significant performance gains from trade-off coding also apply when trading photon-number resources between transmitting public and private classical information simultaneously over secret-key-assisted bosonic channels.

Figure 1

A sketch of the trade-off coding protocol for communication of classical and quantum information without any entanglement assistance (this case is the Devetak-Shor protocol [6]). The sender begins by encoding qubits $|{\phi }_1⟩$, $|{\phi }_2⟩$, and $|{\phi }_3⟩$ into different quantum error-correcting codes, each constructed from a particular state $|{\varphi }_x⟩$ and the channel $\mathcal{N}$. To encode classical information, the sender permutes the quantum systems emerging from the outputs of the encoders according to some classical message $m$. The sender then transmits these systems through many independent uses of the noisy channel $\mathcal{N}$. The receiver obtains the outputs of channels and performs an HSW measurement to determine the classical message $m$. If the success probability of the measurement is asymptotically close to 1, then it causes an asymptotically negligible disturbance to the state on which it acts. The receiver then knows the permutation, unpermutes the quantum systems, and exploits the decoders of the quantum error-correcting codes to decode the qubits $|{\phi }_1⟩$, $|{\phi }_2⟩$, and $|{\phi }_3⟩$. The Wilde-Hsieh protocol [31] extends this idea by permuting entanglement-assisted quantum codes in a similar way.

Figure 2

(a) The $(C,Q)$ trade-off: A lossy bosonic channel with transmissivity $\eta =3/4$ can reliably transmit a maximum of ${log}_2(3/4)-{log}_2(1/4)\approx 1.58$ qubits per channel use [8], and $N_S=200$ photons per mode at the channel input is sufficient to nearly achieve this quantum capacity. A trade-off coding strategy that lowers the quantum data rate to about 1.4 qubits per use while retaining the same mean photon budget allows the transmission of an additional 4.5 classical bits per channel use, while time sharing would only allow for an additional 1 classical bit per channel use with this photon budget. (b) The $(C,E)$ trade-off: The sender and the receiver share entanglement, and the sender would like to transmit classical information while minimizing the consumption of entanglement. With a mean photon budget of $N_S=200$ photons per channel use, the sender can reliably transmit a maximum of about 10.7 classical bits per channel use while consuming entanglement at a rate of about 9.1 entangled bits per channel use [17, 36, 37]. With trade-off coding, the sender can significantly reduce the entanglement consumption rate to about 5 entangled bits per channel use while still transmitting about 10.5 classical bits per channel use, i.e., only a 0.08 dB decrease in the rate of classical communication for a 2.6 dB decrease in the entanglement consumption rate.