Abstract
1 Introduction
Quantum information,1 which utilizes the laws of quantum mechanics to process and communicate information, is one of the key directions in quantum physics. Due to the introduction of the quantum effect, quantum information processing greatly improves the capacity of information processing and the security of communication. Therefore, quantum information has attracted extensive attention all over the world. In quantum information, discrete variable (DV)2 and continuous variable (CV)3 systems are two important platforms. In the DV system, the physical quantity that has a discrete spectrum is used to describe the quantum state, such as polarization and orbital angular momentum. The DV system has the advantage of being insensitive to losses.2 In contrast, in the CV system, the quantum state is described by the physical quantity that has a continuous spectrum, such as amplitude quadrature and phase quadrature. The CV system has the advantage of deterministic implementation.3 With the development of quantum technology, a series of quantum information protocols based on DV and CV systems have been developed, including quantum key distribution,4,5 quantum teleportation,6
Among them, quantum state sharing (QSS), which enables secure state distribution and reconstruction, is an important quantum information protocol for constructing a quantum network. In this protocol, the dealer encodes a secret state into shares and distributes them to players. Any players () can cooperate to reconstruct the secret state, while the rest of the players get nothing. Due to this feature, QSS can be used in quantum error correction in which up to nodes malfunction.33 QSS can also be utilized in the construction of scalable quantum information networks, the transmission of entanglement over faulty channels, and the implementation of multipartite quantum cryptography.33 The QSS was first proposed in the DV regime.34,35 Then, such protocol was transplanted to the CV regime.36 In the CV regime, the (2, 3) threshold deterministic QSS has been studied both in theory37 and experiment.37,38 However, the feedforward technique is needed for implementing QSS in the CV regime.38 The feedforward technique involves the optic-electro and electro-optic conversions, which limits the bandwidth of QSS. Therefore, to broaden the bandwidth of QSS, optic-electro and electro-optic conversions should be avoided.
In CV regime, all-optical QSS (AOQSS) based on a phase-insensitive amplifier (PIA), which avoids the feedforward technique in QSS, has been theoretically proposed.37 However, it is difficult to directly control the inherent noise coupled into the amplified output state of PIA.37 Therefore, such AOQSS has never been experimentally implemented. Here, we experimentally demonstrate (2, 3) threshold deterministic AOQSS by utilizing a low-noise PIA based on a double- configuration four-wave mixing (FWM) process.39
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2 Results
2.1 Principle of AOQSS and Experimental Setup
The experimental setup of the deterministic AOQSS protocol is shown in Fig. 1. The Ti:sapphire laser, whose frequency is about 1 GHz blue detuned from the line (, 795 nm) of , is divided into two by a polarization beam splitter (). The vertically polarized one with a power of about 100 mW serves as the pump beam for generating Einstein–Podolsky–Rosen (EPR) entangled source45 based on the FWM process.39
Figure 1.The detailed experimental setup of the deterministic AOQSS protocol. (a) The detailed experimental scheme. (b)
The additional Gaussian noise is denoted by , which has a mean of and variance of . The () and () are the amplitude quadrature and phase quadrature of the state, respectively. Note that the additional Gaussian noise is introduced naturally in our EPR generation process. (See Sec. 1 of the Supplemental Material for the noise power spectra of , , , and .)
For the (2, 3) threshold deterministic AOQSS protocol, there are three different reconstruction protocols. In this sense, we send these shares into three reconstruction boxes, which are indicated by Figs. 1(b)–1(d) for , , and structures, respectively. For the reconstruction structure shown in Fig. 1(b), and are combined with . The piezoelectric transducer () placed in the path of is used to change the relative phase between and . After locking the relative phase between and with a microcontrol unit,46 the has one bright output and one vacuum output. The bright output is the reconstructed state of the structure. The quadratures of the output reconstructed state can be expressed as37
It can be seen that the structure can completely reconstruct the secret state. is the corresponding adversary structure, in which carries no information on the secret state.
For the structure, the secret state reconstruction is implemented by amplifying with the help of in a PIA based on the FWM process. As shown in Fig. 1(c), the strong one from serves as the pump for the PIA. It crosses with and symmetrically at the center of the second 12-mm long vapor cell, whose temperature is stabilized at 118°C. The angle between and the pump beam is about 7 mrad. The placed in the path of is used to change the relative phase between and . is the corresponding adversary structure. When the intensity gain of the PIA () is set to 2, the quadratures of the output reconstructed state can be expressed as
The structure is equivalent to the structure in the (2, 3) threshold QSS.38 The structure is shown in Fig. 1(d). is the corresponding adversary structure. It can be seen that the functions of PIA are coupling the quadratures of into the output state and reaching the unity gain point when the gain of PIA is set to 2. In other words, the physical essence of this PIA is reconstructing the secret state.
To quantify the quality of the reconstructed state of deterministic AOQSS protocol, we utilize its fidelity , which is defined as .47 Assuming that all states involved are Gaussian and the secret state is a coherent state, the fidelity can be expressed as37,38
To check the performance of the AOQSS, the reconstructed state is measured by a balanced homodyne detection (BHD). The local oscillator (LO) is obtained by setting up a similar bright-seed setup, which is a few millimeters above the current beams. The relative phase between and LO is changed by . The transimpedance gain of the balanced detector is , and the quantum efficiency of balanced detector is about 97%. The variances of the amplitude (locking the phase of BHD to 0) and phase (locking the phase of BHD to ) quadratures of are analyzed by a spectrum analyzer (SA).
2.2 Experimental Results and Discussion
Figure 2 shows the typical noise power results for reconstruction structure. As shown in Figs. 2(a) and 2(b), the blue dashed traces (orange solid traces) are the measured variances of amplitude quadrature and phase quadrature of () with modulation signals at 1.5 MHz. The overlap between the modulation signal peaks of and shows that the input state and output state have the same amplitude ( and are almost 1). To quantify the fidelity of reconstruction structure, we turn off the modulation signals of the secret state and measure the amplitude and phase quadrature variances for the input secret state (blue traces) and the reconstructed output state (orange traces), as shown in Figs. 2(c) and 2(d), respectively. The corresponding fidelity of the reconstruction structure is . This means that we almost retrieve the secret state, which is consistent with the theory.
Figure 2.The typical noise power results for
The typical noise power results for the deterministic AOQSS with reconstruction structure are shown in Fig. 3. The noise spectra with modulations for amplitude and phase quadratures are shown in Figs. 3(a) and 3(b), respectively. The blue dashed trace in Figs. 3(a) and 3(b) shows the amplitude (phase) quadrature variance of the input state. The amplitude (phase) quadrature variance of the output state without the EPR entangled source is represented by the orange solid trace in Figs. 3(a) and 3(b). The overlap of the peaks on the blue dashed traces and the orange solid traces shows that the gains and of the reconstruction structure are almost unity. To quantify the fidelity of the AOQSS with the reconstruction structure, we also turn off the modulation signals and measure the amplitude and phase quadrature variances for the input secret state and the reconstructed output state , as shown in Figs. 3(c) and 3(d), respectively. In Figs. 3(c) and 3(d), the blue traces show the amplitude and phase quadrature variances of the input secret state, respectively. With the help of the entangled source, the variances of the amplitude and phase quadratures of are shown as the green traces in Figs. 3(c) and 3(d), respectively. The relative phase between the two EPR entangled beams is scanned by . When the variance of the amplitude (phase) quadrature of the output state reaches the minima of the green trace in Figs. 3(c) and 3(d), the relative phase between and corresponds to (). Therefore, we can treat the minima of green traces in Figs. 3(c) and 3(d) as the variances of and of the reconstructed output state, respectively. Consequently, the fidelity of the AOQSS with the structure is , as the variances of and are and above the corresponding variances of and , respectively. The orange traces show the variances of the output state without the help of the EPR entangled source, which are referred to the corresponding classical structure. We can see that, without the EPR entangled source, the amplitude (phase) variance of the output state is () above the input state . It can be calculated that the fidelity of the classical structure (corresponding classical limit) is . Figures 3(e) and 3(f) show the noise spectra for the adversary structure . The peaks of orange traces (quadrature variances of with modulation) are about 3 dB below the peak of the blue traces (quadrature variances of with modulation), which means that and are almost . Then, based on the red traces (quadrature variances of without modulation) and the black traces (quadrature variances of without modulation) in Figs. 3(e) and 3(f), the obtained fidelity for the structure is only . In other words, player2 gets almost nothing. The typical results for the AOQSS with a reconstruction structure are shown in the Sec. 3 of the Supplemental Material, which are similar to Fig. 3. The obtained fidelity for the AOQSS with a structure is . In this sense, the average fidelity for the (2, 3) threshold deterministic AOQSS is , which beats the corresponding theoretical (experimental) classical limit of 2/3 (about 0.66). In other words, we successfully experimentally implement the (2, 3) threshold deterministic AOQSS. It is worth noting that the reconstructed state of our AOQSS is in the same form as the secret state, which is distinct from the experiments in Refs. [37] and [38], whose reconstructed state has a different form from the secret state. This results in the directly measured fidelity of the experiments in Refs. [37] and [38] being very low, and a posteriori must be applied to the reconstructed state to obtain a meaningful fidelity. Therefore, our AOQSS without a posteriori is essentially different from the experiments in Refs. [37] and [38].
Figure 3.The typical noise power results for
To show the advantage of our deterministic AOQSS protocol, which avoids the optic-electro and electro-optic conversions, we scan the analysis frequency from 1.4 to 2.4 MHz and measure the fidelities for AOQSS with the structure, as shown in Fig. 4. The fidelity of AOQSS versus analysis frequency is shown as the blue trace, while the corresponding classical limit is shown as the orange trace. It can be seen that as the analysis frequency ranges from 1.4 to 2.4 MHz, the fidelities of AOQSS with the structure can beat the corresponding classical limit, which shows that the AOQSS can be successfully implemented within this sideband frequency range. When the analysis frequency is larger than 2 MHz, the slight decrease of fidelity is due to the decrease of the squeezing of the EPR entangled source.
Figure 4.The experimental fidelities versus the sideband frequency. The fidelities of deterministic AOQSS with a
To implement a scaled-up version of the (2, 3) threshold deterministic AOQSS, there is no need to increase the number of active elements (PIAs) required by the players for retrieving the secret state, while the number of EPR entangled sources (two-mode squeezed states) required by the dealer should be increased.37,48 In other words, in a scaled-up AOQSS, the dealer uses more EPR entangled sources and more passive elements (BSs) to encode a secret state. Correspondingly, a PIA and more passive elements (BSs) are utilized to retrieve the secret state. For example, a (3, 5) threshold deterministic AOQSS can be implemented by increasing one set of EPR entanglements and some passive elements, as shown in Sec. 4 of the Supplemental Material.
3 Conclusion
We have experimentally implemented a (2, 3) threshold deterministic AOQSS without a posteriori and the optic-electro and electro-optic conversions. We demonstrate that any two players can cooperate to retrieve the secret state, while the other player gets nothing. The average fidelity of all reconstruction structures is . More importantly, we have shown that AOQSS can be successfully implemented for the bandwidth ranging from 1.4 to 2.4 MHz. Our results provide a promising scheme to construct an all-optical broadband multipartite quantum network.
Yingxuan Chen is a PhD candidate in optics at the State Key Laboratory of Precision Spectroscopy, East China Normal University under the supervision of Professor Jietai Jing. Her current research interests include quantum optics and quantum information, especially quantum communication protocols.
Qiqi Zhu is a PhD candidate in optics at the State Key Laboratory of Precision Spectroscopy, East China Normal University under the supervision of Professor Jietai Jing. His current research interests include quantum optics and quantum communication.
Xutong Wang is a PhD candidate in optics at the State Key Laboratory of Precision Spectroscopy, East China Normal University under the supervision of Professor Jietai Jing. His current research interests include quantum optics and quantum information, especially the generation and distribution of high-capacity and multi-beam quantum light sources.
Yanbo Lou is an associate professor at the State Key Laboratory of Precision Spectroscopy of East China Normal University, Shanghai, China. He received his PhD in optics from the East China Normal University in 2021. His current research interests include high-capacity quantum information protocols, high-precision quantum metrology, and high-resolution quantum imaging.
Shengshuai Liu is a young professor at the State Key Laboratory of Precision Spectroscopy of East China Normal University, Shanghai, China. He received his PhD in optics from the East China Normal University in 2020. His current research interests include the generation of high-quality quantum squeezing and their applications in quantum information.
Jietai Jing is a professor at the State Key Laboratory of Precision Spectroscopy of East China Normal University, Shanghai, China. He received his PhD in optics from the Shanxi University in 2004. He did his postdoctoral work at the University of Virginia and the University of Maryland, College Park. His main research field is quantum optics and quantum information, especially the generation of high-capacity, multi-beam, low-noise quantum light sources, and their applications in quantum information.
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