All-optical entanglement recovery in a correlated noisy channel

Abstract

The task of protecting quantum entanglement is crucial for quantum communication. Conventionally, correlated noise correction requires an electro-optic modulation process, which limits the application of such correction schemes in constructing broadband quantum communication. Here, we experimentally demonstrate a scheme of all-optical entanglement recovery in a correlated noisy channel without electro-optic modulation. In our scheme, the correlated noisy channel is enabled by an Einstein−Podolsky−Rosen entangled state, a low-noise parametric amplifier, and a beam splitter, thereby avoiding electro-optic conversion. Quantum entanglement is destroyed after one half of the quantum entangled state passes through the noisy channel. Such quantum entanglement can be recovered in the auxiliary noisy channel which shares correlated noise with the noisy channel. Our results pave the way for deterministically implementing all-optical quantum secure communication.

Graphical abstract

Keywords

quantum entanglement / quantum communication / four-wave mixing

Quantum entanglement [1-3] plays an important role in quantum communication. Due to the introduction of quantum entanglement, the fidelity and security of communication are greatly improved. Therefore, quantum entanglement has been widely utilized in quantum communication [4-7]. In the continuous variable (CV) regime, a large number of quantum communication protocols have been developed based on quantum entanglement, including quantum key distribution [8-11], quantum teleportation [12-16], entanglement swapping [17-21], quantum state sharing [22-24], and so on. In CV quantum communication protocols, noise in the transmission channel could significantly affect the quality of the transmitted quantum entanglement between separated nodes [25]. Therefore, it is important to correct the noise.

With the development of optical communication, some transmission channels share correlated noise in today’s communication systems [26, 27]. Therefore, transmission channels with correlated noise should be considered. Due to the correlated noise between the noisy channel and auxiliary channel, quantum entanglement damaged in the noisy channel can be recovered in the auxiliary channel. Such entanglement recovery schemes have been experimentally implemented, such as Gaussian error correction of bipartite quantum entanglement [27] and entanglement recovery of tripartite entanglement [28]. However, the correlated noisy channel in these schemes involves electro-optic conversion, which limits the bandwidth of the experiment. To avoid electro-optic conversion, an all-optical scheme to protect quantum entanglement in a correlated noisy channel remains to be experimentally explored.

Recently, all-optical protocols have been experimentally implemented [29-36]. In these protocols, the electro-optic conversion can be replaced by a low-noise parametric amplifier (PA) [37] based on a four-wave mixing (FWM) process [38-43]. Such an all-optical scheme makes it promising to implement all-optical entanglement recovery (AOER) in a correlated noisy channel. Here, we experimentally demonstrate a scheme of AOER in a correlated noisy channel. In our scheme, quantum entanglement, which is destroyed after one half of the quantum entangled state passes through a noisy channel, can be recovered by AOER with the help of the correlated auxiliary noisy channel. Electro-optic conversion is replaced by an Einstein−Podolsky−Rosen (EPR) entangled state, a low-noise parametric amplifier, and a beam splitter.

Our AOER scheme is depicted in Fig. 1. Two EPR entangled states, generated from the double-\mathrm{\Lambda } configuration FWM process in an ⁸⁵Rb vapor cell, are utilized for implementing AOER. Specifically, EPR₁ serves as the initial entanglement source, and EPR₂ is used to construct the correlated noisy channel which consists of a noisy channel [formed by the conjugate beam of EPR₂{\hat{a}}_2 and a beam splitter (BS)] and an auxiliary noisy channel (formed by the probe beam of EPR₂{\hat{b}}_2 and a PA). In these FWM processes, pump beams are so strong that they can be considered as classical fields under the “undepleted pump” approximation [44]. This approximation implies that pump noises are not introduced into the system. In this way, the interaction Hamiltonian of these two FWM processes can be expressed as

(1)\begin{array}{rl}& {\hat{H}}_1=\mathrm{i}\hslash {\gamma }_1{\hat{a}}_1^{†}{\hat{b}}_1^{†}+\mathrm{H}.\mathrm{c}.,\\ & {\hat{H}}_2=\mathrm{i}\hslash {\gamma }_2{\hat{a}}_2^{†}{\hat{b}}_2^{†}+\mathrm{H}.\mathrm{c}.,\end{array}

where {\hat{a}}_1^{†} ({\hat{a}}_2^{†}) and {\hat{b}}_1^{†} ({\hat{b}}_2^{†}) are the creation operators associated with conjugate and probe beams for EPR₁ (EPR₂), respectively. H.c. is the Hermitian conjugate. {\gamma }_1 ({\gamma }_2) is the interaction strength of FWM process for generating EPR₁ (EPR₂), and the intensity gain is G_1={\cosh }^2({\gamma }_1{\tau }_1)[G_2={\cosh }^2({\gamma }_2{\tau }_2)] for FWM₁ (FWM₂) [44]. {\tau }_1({\tau }_2) is the interaction timescale. Then, the probe beam of EPR₁ {\hat{b}}_1 is directly sent to Alice and measured by balanced homodyne detection (BHD). The conjugate beam of EPR₁ {\hat{a}}_1 is sent to Bob after passing through a noisy channel. In this noisy channel, the transmission of {\hat{a}}_1 is changed by a BS with the transmittance T. When , {\hat{a}}_1 will be attenuated, and additional noise from {\hat{a}}_2 will be introduced into {\hat{a}}_1, which gives {\hat{a}}_1^{\mathrm{\prime }}. In this way, the entanglement between {\hat{a}}_1 and {\hat{b}}_1 can be destroyed. In order to correct the additional noise in {\hat{a}}_1^{\mathrm{\prime }}, Bob couples {\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_2 by a low-noise PA based on the FWM process with a gain of G_3 in the auxiliary noisy channel. In this way, the output state {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} retrieved by Bob can be expressed as

(2)\begin{array}{rl}{\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}=& \sqrt{G_{3}T}{\hat{a}}_1-\sqrt{G_3(1-T)}{\hat{a}}_2\\ & +\sqrt{G_3-1}{\hat{b}}_2^{†}{\mathrm{e}}^{-\mathrm{i}\theta }.\end{array}

In order to retrieve {\hat{a}}_1, the gain of the low-noise PA is set to G_3=1/T (G_{3}T=1). Therefore, the output state {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} can be calculated as

(3)\begin{array}{rl}{\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}=& {\hat{a}}_1+\sqrt{G_3-1}({\hat{b}}_2^{†}{\mathrm{e}}^{-\mathrm{i}\theta }-{\hat{a}}_2)\\ =& {\hat{a}}_1+\sqrt{G_3-1}[{\hat{v}}^{†}(\sqrt{G_2}{\mathrm{e}}^{-\mathrm{i}\theta }-\sqrt{G_2-1})\\ & -\hat{u}(\sqrt{G_2}-\sqrt{G_2-1}{\mathrm{e}}^{-\mathrm{i}\theta })],\end{array}

where \hat{u} and \hat{v} are vacuum states involved in FWM₂. \theta is the relative phase between {\hat{a}}_2 and {\hat{b}}_2. Under the condition of the relative phase \theta =0, the last term in Eq. (3) vanishes in the limit of G_2\gg 1, leaving {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}\approx {\hat{a}}_1, which means that the noise introduced into {\hat{a}}_1 can be corrected. In other words, quantum entanglement destroyed in the noisy channel can be recovered. While it does not satisfy the condition of G_2\gg 1, the additional noise introduced by the last term decreases with the increase of T.

For our AOER in a correlated noisy channel, the entanglement of the two beams can be tested by the positivity under partial transposition (PPT) criterion. The PPT criterion, which is expressed in terms of the symplectic eigenvalue of the partially transposed (PT) covariance matrix (CM) [45-49], is a sufficient and necessary condition for bipartite entanglement in CV systems. In addition, the degree of entanglement can be quantified by the smallest symplectic eigenvalue \nu. The smaller the \nu, the stronger the entanglement is. Entanglement exists only when the \nu is less than 1 [45-49]. In our system, the CM is defined as \sigma =⟨{\xi }^T\xi ⟩, where \xi =({\hat{X}}_{\hat{a}},{\hat{Y}}_{\hat{a}},{\hat{X}}_{\hat{b}},{\hat{Y}}_{\hat{b}}). {\hat{X}}_{\hat{a}} = ({\hat{a}}^{†}+\hat{a}) [{\hat{X}}_{\hat{b}} = ({\hat{b}}^{†}+\hat{b})] and {\hat{Y}}_{\hat{a}} = \mathrm{i}({\hat{a}}^{†}-\hat{a}) [{\hat{Y}}_{\hat{b}} = \mathrm{i}({\hat{b}}^{†}-\hat{b})] represent the amplitude and phase quadratures of the optical fields, respectively.

The detailed experimental setup of AOER in a correlated noisy channel is shown in Fig. 2. A Gaussian laser beam is emitted by a cavity-stabilized Ti:sapphire laser with a frequency of about 1 GHz blue detuned from the ⁸⁵Rb D1 line (5S_{1/2},F=2\to 5P_{1/2}). Then, it is divided into four parts by three polarization beam splitters (PBSs) and three half-wave plates (HWPs). The first part with a power of about 200 mW passes through the acousto-optic modulator (AOM) to obtain the conjugate beam {\hat{a}}_0 which has a blueshift of about 3.04 GHz from the pump beam and a power of about 0.3 μW. The waist of {\hat{a}}_0 is about 270 μm at the center of the vapor cell. The second part, which has a power of about 122 mW, is used as the pump beam of FWM₁ for generating EPR entangled beams {\hat{a}}_1 and {\hat{b}}_1. The corresponding pump waist is about 650 μm at the center of a hot ⁸⁵Rb vapor cell which is 12 mm long and stabilized at about 117 °C. The pump beam and {\hat{a}}_0 cross in the center of the ⁸⁵Rb vapor cell at an angle of about 7 mrad. In this FWM process, two pump photons convert to one photon of probe beam which is red detuned from the pump beam, and one photon of conjugate beam which is blue detuned from the pump beam shown in the inset of Fig. 2. The third part with a power of about 300 mW is used as the pump beam of FWM₂ for generating EPR entangled beams {\hat{a}}_2 and {\hat{b}}_2. Then, {\hat{b}}_1 and {\hat{b}}_2 are directly sent to Alice and Bob, respectively. There is a piezoelectric transducer (PZT) placed in the path of {\hat{b}}_2 to change the relative phase \theta between {\hat{a}}_2 and {\hat{b}}_2. {\hat{a}}_1 is sent to Bob after passing through a noisy channel which consists of {\hat{a}}_2 and a BS with a transmittance of 0.65. The BS is constructed by two PBSs and one HWP. By coupling {\hat{a}}_1 and {\hat{a}}_2 with such a BS, the additional noise from {\hat{a}}_2 is introduced into {\hat{a}}_1. In this way, {\hat{a}}_1^{\mathrm{\prime }} is obtained. After such a noisy channel, the entanglement between {\hat{a}}_1 and {\hat{b}}_1 can be destroyed. To recover the entanglement, Bob couples {\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_2 by a low-noise PA with a gain of G_3 = 1/T\approx 1.54 in an auxiliary noisy channel. The fourth part of the laser beam serves as the pump beam of such a low-noise PA. The pump beam, {\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_2 cross in the center of the ⁸⁵Rb vapor cell on the same plane. The angle between {\hat{a}}_1^{\mathrm{\prime }} ({\hat{b}}_2) and the pump beam is about 7 mrad. In our scheme, two flip mirrors (FMs) are inserted in the beam path, which ensures that the CM of {\hat{a}}_{\mathrm{o}\mathrm{u}\mathrm{t}} ({\hat{a}}_1, {\hat{a}}_1^{\mathrm{\prime }}, {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}) and {\hat{b}}_1 can be measured by BHDs at Victor’s station. The LO beams of BHD with a power of about 430 μW are obtained by setting up a similar FWM process in the PA of the Bob station, which is located a few millimeters below the current beam. The balanced homodyne detector has a transimpedance gain of 10⁵ V/A and a quantum efficiency of 97%. The phase locking of BHD is enabled by microcontrol unit (MCU), and the phase stability is better than 0.1 rad.

The typical noise power results at 1.5 MHz sideband of our AOER in a correlated noisy channel when T=0.65, G_3=1/T\approx 1.54 are shown in Fig. 3. The quadrature variances of the corresponding shot-noise limits (SNLs) measured by blocking {\hat{a}}_{\mathrm{o}\mathrm{u}\mathrm{t}} and {\hat{b}}_1 are represented by the black traces. The variances of amplitude quadrature difference and phase quadrature sum between {\hat{a}}_1 and {\hat{b}}_1 ({\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_1), which are measured by locking the phase of BHD via MCU, are represented by the purple (green) solid line. We can see the variances of amplitude quadrature difference ({\hat{X}}_{{\hat{a}}_1}-{\hat{X}}_{{\hat{b}}_1}) and phase quadrature sum ({\hat{Y}}_{{\hat{a}}_1}+{\hat{Y}}_{{\hat{b}}_1}) of EPR₁ are 1.70 \pm 0.06 dB and 1.72 \pm 0.07 dB below the corresponding SNL, respectively. The corresponding smallest symplectic eigenvalue of PT CM for {\hat{a}}_1 and {\hat{b}}_1 can be calculated as 0.55 \pm 0.01 < 1, which demonstrates the existence of the initial EPR entanglement between {\hat{a}}_1 and {\hat{b}}_1 [45-49]. The green solid line represents the variance of amplitude quadrature difference (phase quadrature sum) between {\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_1, which are 0.66 \pm 0.08 dB and 0.64 \pm 0.10 dB above the corresponding SNL, respectively. The corresponding smallest symplectic eigenvalue of PT CM for {\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_1 is 1.05 \pm 0.02 > 1. This means that the entanglement between {\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_1 is destroyed when the noise of {\hat{a}}_2 is introduced in the noisy channel. Finally, the quadrature variances of the recovered pair ({\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} and {\hat{b}}_1) measured by scanning the relative phase \theta between {\hat{a}}_2 and {\hat{b}}_2 are shown as the minima of the red trace in Fig. 3(a) [amplitude quadrature difference Var({\hat{X}}_{{\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}}-{\hat{X}}_{{\hat{b}}_1})] and Fig. 3(b) [phase quadrature sum Var({\hat{Y}}_{{\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}}+{\hat{Y}}_{{\hat{b}}_1})] [30]. The Var({\hat{X}}_{{\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}}-{\hat{X}}_{{\hat{b}}_1}) and Var({\hat{Y}}_{{\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }}}+{\hat{Y}}_{{\hat{b}}_1}) are 0.31 \pm 0.09 dB and 0.27 \pm 0.11 dB below the corresponding SNL, respectively. The corresponding smallest symplectic eigenvalue of PT CM for {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} and {\hat{b}}_1 is 0.81 \pm 0.02 < 1, which means that {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} and {\hat{b}}_1 are entangled. In other words, we successfully recover the entanglement in a correlated noisy channel.

To show the entanglement sideband that AOER can successfully recover in a correlated noisy channel, we scan and analyze side frequency from 1.4 to 2.6 MHz with intervals of 0.3 MHz under the same experimental conditions. These results are shown in Fig. 4. The upper bound of the smallest symplectic eigenvalue in the presence of entanglement is denoted by the black solid line. The purple dots represent the smallest symplectic eigenvalues \nu of PT CMs for the initial EPR₁ ({\hat{a}}_1 and {\hat{b}}_1) from 1.4 to 2.6 MHz, which are all less than 1. These results prove that {\hat{a}}_1 and {\hat{b}}_1 are entangled from 1.4 to 2.6 MHz. The smallest symplectic eigenvalues \nu of PT CMs for {\hat{a}}_1^{\mathrm{\prime }} and {\hat{b}}_1 from 1.4 to 2.6 MHz are shown as green dots, which all exceed the upper bound of the smallest symplectic eigenvalue in the presence of entanglement. Therefore, the initial EPR entanglement between {\hat{a}}_1 and {\hat{b}}_1 is destroyed. The red dots represent the smallest symplectic eigenvalues \nu of PT CMs for {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} and {\hat{b}}_1 from 1.4 to 2.6 MHz after AOER. It can be seen that these values are all less than 1, which means that the entanglement between {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} and {\hat{b}}_1 is recovered after AOER. From these results, it can be clearly seen that we successfully recover the entanglement between {\hat{a}}_1^{\mathrm{\prime }\mathrm{\prime }} and {\hat{b}}_1 from 1.4 to 2.6 MHz by AOER in a correlated noisy channel. The bandwidth of our AOER is mainly limited by the bandwidth of EPR entanglement from the FWM process whose bandwidth is mainly limited by atomic linewidth and pump power. With the help of THz sideband CV squeezing based on the periodically poled LiNbO₃ waveguide [50], it is promising to demonstrate the ultra-broadband AOER in the future.

In conclusion, we have experimentally implemented a scheme of AOER in a correlated noisy channel. In our scheme, quantum entanglement is destroyed after one half of the quantum entangled state passes through a noisy channel. With the help of the correlated auxiliary noisy channel, such quantum entanglement can be recovered by AOER. In addition, we have shown that AOER can recover the entanglement for the bandwidth ranging from 1.4 to 2.6 MHz. Such an AOER scheme provides a promising paradigm for implementing all-optical fault-tolerant quantum communication in a correlated noisy channel.

References

[1] A. Einstein , B. Podolsky , and N. Rosen , Can quantum-mechanical description of physical reality be considered complete, Phys. Rev. 47(10), 777 (1935)

[2] L. Tian , S. Shi , Y. Tian , Y. Wang , Y. Zheng , and K. Peng , Resource reduction for simultaneous generation of two types of continuous variable nonclassical states, Front. Phys. (Beijing) 16(2), 21502 (2021)

[3] G. Zuo , Y. Zhang , G. Li , P. Zhang , P. Yang , Y. Guo , S. Zhu , and T. Zhang , 10-Hertz squeezed light source generation on the cesium D2 line using single photon modulation, Front. Phys. (Beijing) 18(3), 32301 (2023)

[4] S. L. Braunstein and P. van Loock , Quantum information with continuous variables, Rev. Mod. Phys. 77(2), 513 (2005)

[5] K. Zhang , S. Liu , Y. Chen , X. Wang , and J. Jing , Optical quantum states based on hot atomic ensembles and their applications, Photonics Insights 1(2), R06 (2022)

[6] Y. Yan , L. Zhou , W. Zhong , and Y. Sheng , Measurement-device-independent quantum key distribution of multiple degrees of freedom of a single photon, Front. Phys. (Beijing) 16(1), 11501 (2021)

[7] F. Li , X. Zhang , J. Li , J. Wang , S. Shi , L. Tian , Y. Wang , L. Chen , and Y. Zheng , Demonstration of fully-connected quantum communication network exploiting entangled sideband modes, Front. Phys. (Beijing) 18(4), 42303 (2023)

[8] Ch. Silberhorn , N. Korolkova , and G. Leuchs , Quantum key distribution with bright entangled beams, Phys. Rev. Lett. 88(16), 167902 (2002)

[9] L. S. Madsen , V. C. Usenko , M. Lassen , R. Filip , and U. L. Andersen , Continuous variable quantum key distribution with modulated entangled states, Nat. Commun. 3(1), 1083 (2012)

[10] N. Wang , S. Du , W. Liu , X. Wang , Y. Li , and K. Peng , Long-distance continuous-variable quantum key distribution with entangled states, Phys. Rev. Appl. 10(6), 064028 (2018)

[11] O. Kovalenko , Y. S. Ra , Y. Cai , V. C. Usenko , C. Fabre , N. Treps , and R. Filip , Frequency-multiplexed entanglement for continuous-variable quantum key distribution, Photon. Res. 9(12), 2351 (2021)

[12] S. L. Braunstein and H. J. Kimble , Teleportation of continuous quantum variables, Phys. Rev. Lett. 80(4), 869 (1998)

[13] A. Furusawa , J. L. Sørensen , S. L. Braunstein , C. A. Fuchs , H. J. Kimble , and E. S. Polzik , Unconditional quantum teleportation, Science 282(5389), 706 (1998)

[14] M. Yukawa , H. Benichi , and A. Furusawa , High-fidelity continuous-variable quantum teleportation toward multistep quantum operations, Phys. Rev. A 77(2), 022314 (2008)

[15] N. Lee , H. Benichi , Y. Takeno , S. Takeda , J. Webb , E. Huntington , and A. Furusawa , Teleportation of nonclassical wave packets of light, Science 332(6027), 330 (2011)

[16] M. Huo , J. Qin , J. Cheng , Z. Yan , Z. Qin , X. Su , X. Jia , C. Xie , and K. Peng , Deterministic quantum teleportation through fiber channels, Sci. Adv. 4(10), eaas9401 (2018)

[17] R. E. S. Polkinghorne and T. C. Ralph , Continuous variable entanglement swapping, Phys. Rev. Lett. 83(11), 2095 (1999)

[18] S. M. Tan , Confirming entanglement in continuous variable quantum teleportation, Phys. Rev. A 60(4), 2752 (1999)

[19] P. van Loock and S. L. Braunstein , Unconditional teleportation of continuous-variable entanglement, Phys. Rev. A 61(1), 010302(R) (1999)

[20] X. Jia , X. Su , Q. Pan , J. Gao , C. Xie , and K. Peng , Experimental demonstration of unconditional entanglement swapping for continuous variables, Phys. Rev. Lett. 93(25), 250503 (2004)

[21] N. Takei , H. Yonezawa , T. Aoki , and A. Furusawa , High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables, Phys. Rev. Lett. 94(22), 220502 (2005)

[22] T. Tyc and B. C. Sanders , How to share a continuous-variable quantum secret by optical interferometry, Phys. Rev. A 65(4), 042310 (2002)

[23] A. M. Lance , T. Symul , W. P. Bowen , B. C. Sanders , T. Tyc , T. C. Ralph , and P. K. Lam , Continuous-variable quantum-state sharing via quantum disentanglement, Phys. Rev. A 71(3), 033814 (2005)

[24] A. M. Lance , T. Symul , W. P. Bowen , B. C. Sanders , and P. K. Lam , Tripartite quantum state sharing, Phys. Rev. Lett. 92(17), 177903 (2004)

[25] T. Fan , C. Wen , J. Jing , and J. Wang , Genuine tripartite entanglement and geometric quantum discord in entangled three-body Unruh–DeWitt detector system, Front. Phys. (Beijing) 19(5), 54201 (2024)

[26] D. Kretschmann and R. F. Werner , Quantum channels with memory, Phys. Rev. A 72(6), 062323 (2005)

[27] M. Lassen , A. Berni , L. S. Madsen , R. Filip , and U. L. Andersen , Gaussian error correction of quantum states in a correlated noisy channel, Phys. Rev. Lett. 111(18), 180502 (2013)

[28] X. Deng , C. Tian , X. Su , and C. Xie , Avoiding disentanglement of multipartite entangled optical beams with a correlated noisy channel, Sci. Rep. 7(1), 44475 (2017)

[29] T. C. Ralph , All-optical quantum teleportation, Opt. Lett. 24(5), 348 (1999)

[30] S. Liu , Y. Lou , and J. Jing , Orbital angular momentum multiplexed deterministic all-optical quantum teleportation, Nat. Commun. 11(1), 3875 (2020)

[31] S. Liu , Y. Lou , Y. Chen , and J. Jing , All-optical optimal N-to-M quantum cloning of coherent states, Phys. Rev. Lett. 126(6), 060503 (2021)

[32] Y. Lou , S. Liu , and J. Jing , Experimental demonstration of a multifunctional all-optical quantum state transfer machine, Phys. Rev. Lett. 126(21), 210507 (2021)

[33] S. Liu , Y. Lou , Y. Chen , and J. Jing , All-optical entanglement swapping, Phys. Rev. Lett. 128(6), 060503 (2022)

[34] Y. Chen , Q. Zhu , X. Wang , Y. Lou , S. Liu , and J. Jing , Deterministic all-optical quantum state sharing, Adv. Photonics 5(2), 026006 (2023)

[35] Y. Lou , Y. Lv , J. Wang , S. Liu , and J. Jing , Orbital angular momentum multiplexed deterministic all-optical quantum erasure-correcting code, Phys. Rev. Lett. 132(4), 040601 (2024)

[36] S. Liu , Y. Lv , X. Wang , J. Wang , Y. Lou , and J. Jing , Deterministic all-optical quantum teleportation of four degrees of freedom, Phys. Rev. Lett. 132(10), 100801 (2024)

[37] Q. Kang , Z. Zhao , T. Zhao , C. Liu , and L. Hu , Improved phase sensitivity of an SU(1,1) interferometer based on the internal single-path local squeezing operation, Front. Phys. (Beijing) 20(5), 053200 (2025)

[38] C. F. McCormick , V. Boyer , E. Arimondo , and P. D. Lett , Strong relative intensity squeezing by four-wave mixing in rubidium vapor, Opt. Lett. 32(2), 178 (2007)

[39] V. Boyer , A. M. Marino , R. C. Pooser , and P. D. Lett , Entangled images from four-wave mixing, Science 321(5888), 544 (2008)

[40] A. M. Marino , R. C. Pooser , V. Boyer , and P. D. Lett , Tunable delay of Einstein–Podolsky–Rosen entanglement, Nature 457(7231), 859 (2009)

[41] R. C. Pooser , A. M. Marino , V. Boyer , K. M. Jones , and P. D. Lett , Low-noise amplification of a continuous-variable quantum state, Phys. Rev. Lett. 103(1), 010501 (2009)

[42] S. Liu , Y. Lou , and J. Jing , Interference-induced quantum squeezing enhancement in a two-beam phase-sensitive amplifier, Phys. Rev. Lett. 123(11), 113602 (2019)

[43] X. Hu , C. Wei , Y. Yu , and Z. Zhang , Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light, Front. Phys. (Beijing) 11(3), 114203 (2016)

[44] M. Jasperse , L. D. Turner , and R. E. Scholten , Relative intensity squeezing by four-wave mixing with loss: An analytic model and experimental diagnostic, Opt. Express 19(4), 3765 (2011)

[45] R. Simon , Peres–Horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84(12), 2726 (2000)

[46] F. A. S. Barbosa , A. S. Coelho , A. J. de Faria , K. N. Cassemiro , A. S. Villar , P. Nussenzveig , and M. Martinelli , Robustness of bipartite Gaussian entangled beams propagating in lossy channels, Nat. Photonics 4(12), 858 (2010)

[47] R. F. Werner and M. M. Wolf , Bound entangled Gaussian states, Phys. Rev. Lett. 86(16), 3658 (2001)

[48] A. S. Coelho , F. A. S. Barbosa , K. N. Cassemiro , A. S. Villar , M. Martinelli , and P. Nussenzveig , Three-color entanglement, Science 326(5954), 823 (2009)

[49] X. Pan , S. Yu , Y. Zhou , K. Zhang , K. Zhang , S. Lv , S. Li , W. Wang , and J. Jing , Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor, Phys. Rev. Lett. 123(7), 070506 (2019)

[50] T. Kashiwazaki , T. Yamashima , N. Takanashi , A. Inoue , T. Umeki , and A. Furusawa , Fabrication of low-loss quasi-single-mode PPLN waveguide and its application to a modularized broadband high-level squeezer, Appl. Phys. Lett. 119(25), 251104 (2021)

RIGHTS & PERMISSIONS

Higher Education Press