Multipartite continuous-variable optical quantum entanglement: Generation and application

Quantum entanglement is a fundamental resource for various quantum applications and generation of large-scale entanglement is a key quantum technology. In recent years, continuous-variable optical systems have shown promising results in this direction, thanks to deterministic generation and multiplexing via rich degrees of freedom naturally occurring in the optical system. In this paper, we review the generation and applications of multipartite optical quantum entanglement. We begin with a theoretical overview of how to represent and verify multimode continuous-variable quantum states and entanglement. Then, we discuss the multiplexing technique in time, frequency, and spatial domains, and their applications in generation of large-scale entangled states. Afterward, we review the current status and development of the basic technology used in the generation of multipartite quantum entanglement. Finally, we discuss the applications of large-scale quantum entanglement and possible future perspectives.

Figure 1

Schematic diagram of the homodyne detector. The phase $\theta$ of the local oscillator determines the linear combination $\widehat{x}cos\theta +\widehat{p}sin\theta$ to be measured. The homodyne detector measures the difference of the current on the two photodiodes (${\widehat{I}}_1-{\widehat{I}}_2$).

Figure 2

Generation of arbitrary Gaussian states from vacuum states. The example here shows the case of five-mode Gaussian state. (a) Generation circuit based on the Bloch-Messiah decomposition. (b) Equivalent circuit based on offline squeezed states and linear optics. $\widehat{O}$ and ${\widehat{O}}^{\prime }$ are linear optics operation which are not unique as Bloch-Messiah decomposition is not a unique decomposition. $\widehat{Sq}\left(r\right)$ is a squeezing operation.

Figure 3

Bipartitions of four-partite state. Negating all of these possible configurations establishes genuine multipartite inseparability.

Figure 4

Cluster states generated from controlled-$Z$ gate. (a) A particular circuit for state generation. The black line corresponds to controlled-$Z$ gate and the number is the gain $g$. (b) Corresponding graphical representation determined by the adjacency matrix $\mathcal{A}$.

Figure 5

Graphical representation of the GHZ state. As an example, we show the case with five modes. (a) $\mathcal{H}$-graph representation as all-connected graph. (b) Representation using $\mathbf{Z}$ after the Fourier transform of a single mode as a star graph.

Figure 6

Conceptual diagram of time-domain multiplexing. Here we assume that the quantum state of light is generated with continuous wave, which is usually the case in the actual experiments. The shape of the wave packet $f\left(t\right)$ is assumed to be the same in all time bins of temporal width $\mathrm{\Delta }t$ that are labeled by index $k$. This assumption is made so that the temporal mode can interact with each other. The circle nodes correspond to each wave-packet mode that is usually shown in the graphical representation.

Figure 7

Various setups for generation of time-domain-multiplexed cluster states. Sqz. light: squeezed light. (a), (b) Setups for quad-rail lattice [36] and bilayer square lattice [37], while (c) and (d) are the experimental setups in Refs. [53] and [54], respectively.

Figure 8

Graphical representation of time-domain-multiplexed cluster state generated in [53] (a) and [54] (b) (reproduced from [53] and [54] with permission).

Figure 9

Conceptual diagram of frequency multiplexing. (a) Multiple resonant frequency mode from an optical parametric oscillator (OPO). Each peak is separated in the frequency by a free-spectral range (FSR). If the separation is large compared to the linewidth, each cavity peak can be considered as an independent bosonic mode. (b) Supermode, which is a linear combination of binned frequency mode within the phase-matching region of the nonlinear medium.

Figure 10

Experimental setup for generation of quantum entanglement multiplexed in frequency domain. (a) Generation using resonant peaks of cavity [57]. (b) Generation of frequency-multiplexed entanglement in supermode (reproduced from [60] with permission).

Figure 11

An example of the setup for the spatial-mode-multiplexed quantum entanglement generation. (a) Setup for generation of quantum entanglement in Laguerre-Gaussian modes. (b) The resulting entanglement in spatial-mode comb. Adapted with permission from [47].

Figure 12

Conceptual diagram of quantum teleportation.

Figure 13

Demonstration of measurement-based Gaussian operations using time-domain-multiplexed cluster state. (a), (b) From Refs. [55] and [56], respectively [(b) is reproduced from [56] with permission]. In both experiments, Gaussian operations are implemented via sequential CV teleportation in the time domain where the phase of the local oscillators is programmed according to the desired operations.