Coherent communication with linear optics

We show how to implement several continuous-variable coherent protocols with linear optics. Noise can accumulate when implementing each coherent protocol with realistic optical devices. Our analysis bounds the level of noise accumulation. We highlight the connection between a coherent channel and a nonlocal quantum nondemolition interaction and give two new protocols that implement a coherent channel. One protocol is superior to a previous method for a nonlocal quantum nondemolition interaction because it requires fewer communication resources. We then show how continuous-variable coherent superdense coding implements two nonlocal quantum nondemolition interactions with a quantum channel and bipartite entanglement. We finally show how to implement continuous-variable coherent teleportation experimentally and provide a way to verify the correctness of its operation.

Figure 1

(Color online) We outline the elements used in our optical circuits. (a) An amplitude modulator displaces the position quadrature of an optical mode. A classical control signal controls the amount of displacement. (b) A phase modulator kicks the momentum quadrature of an optical mode. A classical signal also controls it. (c) A photodetector. (d) A phase shifter. (e) A polarizing beam splitter. (f) A mirror. (g) A half-wave plate with variable transmittivity. (h) A beam splitter.

Figure 2

(Color online) FMA’s linear-optical scheme for a quantum nondemolition interaction [22]. LO is an abbreviation for “local oscillator.” The two input modes are orthogonally polarized. The circuit uses two offline squeezers, homodyne detection, and feedforward control to give a quantum nondemolition interaction with unity transfer gain.

Figure 3

(Color online) Coherent channel implemented with entanglement and classical communication. The protocol has similarities to previous protocols [17, 31, 32, 33], but has significant differences as well. Alice and Bob share a three-mode entangled state. Alice possesses the first two modes and Bob possesses the third. Alice has a mode $A$ that she wants to send through a coherent channel. She performs teleportationlike measurements on mode $A$ and her first mode in the entangled state. She performs feedforward control on her second mode of the entangled state. She sends one variable over a classical communications channel so that Bob can perform feedforward control. The resulting state is equivalent to one sent through a coherent channel.

Figure 4

(Color online) The above linear-optical circuit implements coherent superdense coding or, equivalently, two-nonlocal quantum nondemolition interactions. All phase shifters in the above circuit rotate by $\pi$. Alice possesses two modes that she wants to send through a coherent channel. Alice and Bob share two entangled modes. Alice performs some local operations on her two modes and her half of the entangled state. She sends one mode over a quantum channel to Bob. Bob performs some local operations. The resulting states are equivalent to those that would result from Alice sending her two modes through two coherent channels. The states are also equivalent to those that would result from performing two nonlocal quantum nondemolition interactions.

Figure 5

(Color online) The above circuit implements the coherent teleportation protocol with a linear-optical circuit. All phase shifters in the above circuit rotate by $\pi$. Alice possesses one mode that she wants to teleport and shares an entangled state with Bob. Alice performs some local operations and sends her two modes through the circuit for coherent superdense coding. Coherent superdense coding gives Alice two modes and Bob two modes. Bob performs some local operations. The result is that Alice’s original mode teleports to mode three. Alice and Bob additionally share two entangled states at the end of the protocol.