Nonlocal continuous-variable quantum nondemolition gates by optical connections

Abstract

Nonlocal quantum gates, coupling quantum systems located at distance, are crucial for distributed quantum computing. High-capacity optical noiseless connections between these quantum systems are essential for transmitting large amounts of information per mode. We propose a library of feasible protocols to implement a necessary nonlocal continuous-variable (CV) quantum nondemolition (QND) gate between two distant users sharing a quantum channel with a newly available element - single-pass phase-sensitive optical parametric amplifiers (OPAs), allowing for both online squeezing and channel-loss compensation, and classical communication between them. The use of OPAs enhances quality of the resulting entangling gate in terms of both excess noise and logarithmic negativity. The proposed schemes are also applicable to CV cluster state fusion, providing a first step towards development of distributed CV measurement-based quantum computation.

Figures (6)

• Figure 1: Schemes for nonlocal QND gate implementation. The squeezing-based (${\rm SB}$) protocol (a) uses an offline-squeezed vacuum mediator with single quantum channel and one-way classical communication (CC). The entanglement-based (${\rm EB}$) scheme (b) exploits entanglement pre-sharing, by designing an optimized two-mode state that fits into QNDs $g_{A(B)}$, at the cost of two-way CC, while the Bell measurement (${\rm BM}$) one (c) replaces the prior resource distribution with online entanglement generation by QND-based Bell measurement, reducing the classical resources to one-way CC and feed-forward. Unlike the previous setups, the geometric phase (${\rm GP}$) protocol enables nonlocal gates independent of the mediator state, at the costs of double use of quantum channel, large offline squeezing $G_1\to 0$ and one more gate at Alice's side. In all setups, it is also possible to design local Gaussian gates that integrate QNDs together with the OPAs. Displacements might also be done by QND interaction with classical coherent light controlled by the measured results. We assume ideal homodyne detection, for if detectors have low quantum efficiency, one can use a further OPA as pre-amplifier Leonhardt1994.
• Figure 2: Logaritmic negativity $E_N^{(\rm s)}$, ${\rm s}={\rm SB},{\rm GP}$, (solid lines) for target gain $g=1$ as a function of the channel losses, equal to $-10\log_{10} (T)$ dB. ${\rm GP}$ scheme outperforms ${\rm SB}$, with a factor $2$ advantage for $1-T\ll 1$, where $\partial_T E_N^{({\rm GP})}=\partial_T E_N^{({\rm SB})}/2$. Light-colored lines are logarithmic negativities $E_{N,{\rm off(on)}}^{(\rm s)}$ with realistic OPAs ($\eta=0.7$) for the two cases of only offline squeezing and both online and offline OPAs, that both induce a maximum transmission loss. For $1-T\ll 1$, online OPA is useless for the ${\rm SB}$ and useful for the ${\rm GP}$, provided that $\eta >\eta_{\rm GP}(T)$. The ${\rm EB}$ and ${\rm BM}$ protocols are equivalent to the ${\rm SB}$.
• Figure 3: (a) Excess noise $\xi^{(\rm s)}$, ${\rm s}={\rm SB},{\rm GP}$, for target gain $g=1$ as a function of the channel losses. For $1-T\ll 1$, ${\rm GP}$ gives a factor $2$ advantage, as $\xi^{({\rm GP})}=2g(1-T)/(1+T) \approx g (1-T) =\xi^{({\rm SB})}/2$. Light-colored lines refer to the cases of lossy OPAs with $\eta=0.7$. Entanglement is lost when $\xi^{(\rm s)}$ beats the maximum tolerable noise $\xi_{\rm max}=2g$. (b) Entanglement ratio $R^{(\rm s)}_{\rm k}=E_{N,{\rm k}}^{(\rm s)}(\eta)/E_{N}^{(\rm s)}$, k=off,on, between the realistic case of lossy squeezers and the ideal OPA scheme for $g=1$ and different efficiencies $\eta$. When $1-T\ll 1$, online OPA is useless for the ${\rm SB}$ scheme as $\xi_{\rm on}^{({\rm SB})}(\eta) > \xi_{\rm off}^{({\rm SB})}(\eta)$, whereas, in the ${\rm GP}$ one, lossy online squeezing helps provided that the OPA efficiency is $\eta>\eta_{{\rm GP}}(T)$. The ${\rm EB}$ and ${\rm BM}$ protocols are equivalent to the ${\rm SB}$. (c) Threshold OPA efficiency $\eta_{{\rm GP}}(T)$ as a function of the channel losses.
• Figure 4: (a) QND-type cluster state of $2n$ modes. Ideally, this structure may be generated by the setup in Yokoyama2013Larsen2019, based on multiplexing of optical pulses in time-domain, where (passive) beam splitters are replaced with the (active) QND coupling (\ref{['eq:idealQND']}). Within the cluster, we distinguish modes that might describe different bosonic systems, e.g mechanical modes or collective spins, (bold line) from those ones considered optical (normal line). (b) Cluster fusion procedure. We consider two distant cluster states of sizes $2n+2$ between modes $\{A_1,B_1, \ldots, A_n,B_n,M_1,M_2\}$ and $2m$ between $\{ A_{n+1}, B_{n+1}, \ldots,A_{n+m},B_{n+m}\}$, respectively. We then send the edge mode $M_2$ of the first one to the second cluster through an OPA-assisted quantum channel and implement the ${\rm EB}$ scheme of Fig. \ref{['fig:01-Protocols']}(b). Eventually, we merge the two local clusters into a single bigger distributed cluster of size $2(n+m)$. (c) Nullifier variance ${\rm Var}[\mathbb{N}_{x(p)}^{(n)}]=S+\xi^{({\rm EB})}$ at the edge node $n$ for different levels of input squeezing $S$ and gain $g=1$. The entangled region is defined by ${\rm Var}[\mathbb{N}_{x(p)}^{(n)}]<2g$. Given the equivalence between ${\rm EB}$ and ${\rm SB}$ protocols, for ideal OPAs with $\eta=1$, ${\rm Var}[\mathbb{N}_{x(p)}^{(n)}] = S+2g(1-T)$, whereas if $\eta<1$ online OPA 3 in Fig. \ref{['fig:01-Protocols']}(b) is useless for $1-T\ll 1$.
• Figure 5: Log plot of the optimized parameters for the ${\rm SB}$ (a) and ${\rm GP}$ (b) protocols for target gain $g=1$ as a function of the channel losses in cases (off) and (on), respectively, and $\eta=0.7$. In the ${\rm GP}$ scheme, the optimal configuration of OPA 1 is obtained in the limit $G_{1,{\rm k}}^{({\rm GP})} \to 0$, $\rm k= off,on$, corresponding to large position-like squeezing of the mediator $M$.