Evaluation of continuous variable controlled quantum teleportation through quantum nonlocality

Abstract

Quantum nonlocality is an important resource for ensuring the security of quantum communication, which plays an indispensable role in building quantum networks. From bipartite to multipartite quantum systems, various inequalities are constructed as tools to unearth the quantum nonlocality. Control power is a crucial indicator for evaluating supervision, control, and collaboration in quantum networks. However, there is currently no exploration of the relevance between the control power and the quantum nonlocality in continuous variable quantum networks. In this work, based on the Mermin−Klyshko (MK) inequality with rotated quantum-number parity operators, we reveal the quantum nonlocality of general Greenberger−Horne−Zeilinger-type (GHZ-type) entangled coherent states, which can be extended to arbitrary multi-mode states. And then, we evaluate the control power through quantum nonlocality in continuous variable controlled quantum teleportation via a 3-mode GHZ-type entangled coherent state as channel. The results show that the control power of continuous variable quantum networks is a monotonically increasing function of genuine tripartite nonlocality. This quantitative dependence of control power on quantum nonlocality remains robust even with decoherence. The advantages of the MK inequality are simplicity in computation and scalability into arbitrary multi-mode channels. In addition, maximal violation of the MK inequality can be achieved more easily in the experiment. Our work proves that, compared with the Bell−Mermin inequality, the MK inequality performs better in experimental measurements of complete control power; compared with the Svetlichny inequality, the MK inequality avoids negative and unreasonable control power when inequalities exceed the classical upper bound. Therefore, the MK inequality is found to be a more reliable tool to reveal the quantum nonlocality in multipartite systems. This work illuminates the important application of nonlocality in multipartite continuous variable quantum communications and is of significance for the construction of practical quantum protocols.

Graphical abstract

Keywords

continuous variable quantum information / quantum nonlocality

1 Introduction

Quantum nonlocality is a reliable and powerful resource for ensuring the security of quantum communication [1], such as device-independent quantum key distribution [2–4], quantum secret sharing [5–7], and quantum teleportation networks [8, 9]. Besides discrete variable (DV) quantum systems [10, 11], continuous variable (CV) quantum systems [12–15] are efficient physical carriers of quantum information processing, which can be utilized to construct bipartite or multipartite quantum communication networks. In DV quantum information processing systems, achieving maximum entanglement is possible, although typically in a probabilistic manner. Conversely, CV systems allow for deterministic entanglement generation at the cost of losing perfect entanglement [12, 14, 16]. In quantum networks, control power is a crucial indicator for evaluating supervision, control, and collaboration, which is widely discussed in DV quantum information processing [17, 18]. Although Ref. [8] has linked control power and quantum nonlocality for discrete variable quantum protocols, such relationships between these concepts remain unexplored for continuous variable quantum networks.

The quantum nonlocality in this work specifically refers to the quantum correlation based on the Bell inequality test, that is, Bell nonlocality. The definition of Bell nonlocality is following [19]. We consider two distant observers, Alice and Bob, performing measurements on a shared physical system, for instance a pair of entangled particles. Let p\left(ab|XY\right) denote the joint probability to obtain the output pair (a, b) given the input pair (X, Y), where X denotes Alice’s measurement and Y is Bob’s measurement. If the measurements are performed at spacelike separation, then Bell’s condition of local causality [20] implies that even if the particles have interacted in the past (or were produced together in the same source), they are now independent. Therefore, the probabilities can be written in the form

(1)p_{\lambda }\left(ab|XY\right)=p_{\lambda }\left(a|X\right)p_{\lambda }\left(b|Y\right),

where the state \lambda is conventionally referred to as a hidden state, since it is not part of the quantum description of the experiment. Any hidden state \lambda satisfies Eq. (1) is local. If the observed correlations can be explained by a locally causal theory, then they can be written as

(2)p\left(ab|XY\right)={\displaystyle \sum _{\lambda }{q}_{\lambda }{p}_{\lambda }\left(a|X\right)p_{\lambda }\left(b|Y\right),}

with q_{\lambda }\ge 0 and \sum _{\lambda }{q}_{\lambda }=1. In contrast, if correlations p\left(ab|XY\right) violate a Bell inequality [20], then they cannot be written in this form. Such correlations cannot be explained by a locally causal theory, and are referred to as nonlocal correlations.

In CV quantum systems, there are various inequalities constructed as tools to unearth the quantum nonlocality. In bipartite CV systems, the forms of inequality are widely studied and generalized based on different operators, consisting of Banaszek−Wödkiewicz (BW) operators [21, 22], pseudospin operators [23], and rotated quantum-number parity Bell operators [24]. Besides, the quantum nonlocality for multimode CV states plays an important role in the study of quantum networks, which has been addressed in several decades [25–28]. Recently, a novel version of the Mermin−Klyshko (MK) correlation operator has been constructed to reveal nonlocality for multimode entangled coherent states with rotated quantum-number parity operators [29], and the MK inequality violations have been developed. There are several advantages of the MK inequality. First, the MK inequality significantly simplifies the process of finding the global maximum value of the MK signal. Second, the MK signal is derived analytically, eliminating the need for multivariate optimization [25]. Third, the obtained MK signal can approach the maximal value even when the mean photon number of coherent states in each mode is small. Therefore, the MK inequality is a powerful candidate tool to evaluate the indicators and reveal the quantum nonlocality in continuous variable systems.

In this work, using the Mermin−Klyshko inequality with rotated quantum-number parity operators, we reveal the quantum nonlocality of general Greenberger−Horne−Zeilinger-type (GHZ-type) entangled coherent states. In addition, we evaluate the control power through quantum nonlocality in continuous variable controlled quantum teleportation via 3-mode GHZ-type entangled coherent state channels. When the mean photon number {\left|\alpha \right|}^2=0, the MK signal S_3 is beyond the classical upper bound 2. When {\left|\alpha \right|}^2=0.208, S_3 exceeds the lower bound 2\sqrt{2} for genuine tripartite nonlocality. When {\left|\alpha \right|}^2 increases further, it quickly approaches the quantum-mechanical upper bound 4, which implies that a nearly maximal violation of the MK inequality can be obtained with a small mean photon number. The control power is validated as a monotonically increasing function of the MK signal in the protocol. The robustness of the evaluation is analyzed with the decoherence caused by photon absorption. The results illuminate that the relation between control power and quantum nonlocality is robust against decoherence in continuous variable networks. Compared with the Bell−Mermin (BM) inequality and Svetlichny inequality, the Mermin−Klyshko inequality is proved in this work that it performs better in experimental measurements of complete control power and avoids the unreasonable negative control power. Therefore, the MK inequality is found a more reliable tool to reveal the quantum nonlocality in multipartite systems.

The violations of the MK inequality certify the quantum nature of the resources. Thus, violation of the MK inequality ensures the control power derives from genuine quantum resources. Without the verification of the nonlocality violations, the control power may be obtained by the classical strategies. In this scenario, quantum servers in the network may steal each other’s information or jointly steal the supervisor’s information without being discovered [17], undermining system security. Therefore, it is necessary to explore the relationship between the control power and nonlocality. This work illuminates the important application of nonlocality in multipartite continuous variable quantum communications, holding significance for the establishment of practical quantum protocols.

The present work is structured as follows. In Section 2, the scheme of using Mermin−Klyshko inequality with rotating quantum parity operators to calculate quantum nonlocality is reviewed, while the channel model is extended to general GHZ-type entangled coherent state channels. In particular, the MK signal of 3-mode maximally entangled GHZ-type coherent states is discussed. In Section 3, the control power of the supervisor in a CV controlled teleportation via a prior 3-mode GHZ-type entangled coherent state channel is reviewed. The function of the MK inequality and the control power is found to determine the efficiency of the proposed evaluation. In addition, the robustness of the evaluation is analyzed with the decoherence caused by photon absorption. Section 4 compares the method using MK inequality in this work with methods using BM inequality and Svetlichny inequality. Finally, the summary and outlook are drawn in Section 5.

2 Mermin−Klyshko inequality of general GHZ-type entangled coherent state channels

In this section, the Mermin−Klyshko inequality with rotated quantum-number parity operators [24, 29] of general GHZ-type entangled coherent states as channels is extended. The MK signal of a 3-mode maximally entangled GHZ-type coherent states as a channel is discussed as a specific example.

Without the loss of generality, the M-mode GHZ-type entangled coherent states under consideration are defined as follows [24, 29]:

(3)\begin{array}{rl}& |GHZ,(\alpha ,M)⟩\\ & =N_M({|{\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_M⟩}_{1,2,3,\dots ,M}\\ & +{\mathrm{e}}^{\mathrm{i}\xi }{|-{\alpha }_1,-{\alpha }_2,-{\alpha }_3,\dots ,-{\alpha }_M⟩}_{1,2,3,\dots ,M}),\end{array}

where N_M={\left\{2\left[1+\cos \left(\xi \right)\prod _{k=1}^M{x_k}^2\right]\right\}}^{-1/2} is the normalization factor, {\alpha }_k is the coherent state of amplitude denoted by {|{\alpha }_k⟩}_k=\exp \left(-{\left|{\alpha }_k\right|}^2/2\right){\sum }_{n=0}^{\mathrm{\infty }}\frac{{{\alpha }_k}^n}{\sqrt{n!}}|n⟩ for the k-th (k=1,2,3,...,M) harmonic oscillator. This state is defined on the tensor product of M-mode Fock Spaces, with each module described by the annihilation and generation operators, and x_k={\mathrm{e}}^{-{\left|{\alpha }_k\right|}^2}. The maximally entangled GHZ-type coherent states are obtained when \xi =\pi.

The MK operator O_M=\left\{O_1,O_2,\cdots ,O_k,\cdots ,O_M\right\} is a recursively constructed many-body observable, and its expected value S_M represents the global correlation. The MK operator for a multipartite entangled state is recursively expressed by [29]

(4)O_k={\displaystyle \frac{1}{\sqrt{2}}}\left[O_{k-1}\left({\sigma }_{k,a_k}+{\sigma }_{k,{a^{\prime }}_k}\right)+{O^{\prime }}_{k-1}\left({\sigma }_{k,a_k}-{\sigma }_{k,{a^{\prime }}_k}\right)\right],

starting with O_{k=1}={\sigma }_{1,a_1}. {\sigma }_{k,a_k} and {\sigma }_{k,{a^{\prime }}_k} denote two-valued observables for the kth party with the corresponding measurement settings a_k and a_k^{\prime }, respectively. O_k^{\prime } is generated from O_k by swapping all measurement settings a_j and a_j^{\prime } in O_k, for j\le k. The expectation value of the MK signal for the M-mode GHZ-type state is S_M=\left|{⟨O_M⟩}_{|GHZ,(\alpha ,M)⟩}\right|\le 2^{M-1}, which exceeds the bound 2^{\left(M-1\right)/2} imposed by the local realism [29]. For convenience, we use the notation {\sigma }_k\left({\varphi }_k\right) to replace {\sigma }_{k,a_k}, with {\varphi }_k being the angle between a unit vector a_k and the x axis. In the following discussion, we continue to use the parameter settings in Ref. [29] as

(5)\begin{array}{rl}& {\varphi }_1=0,{{\varphi }^{\prime }}_1=\pi /2,\\ & {\varphi }_k=-\pi /4,{{\varphi }^{\prime }}_k=\pi /4,k\ne 1.\end{array}

Similarly to Bell correlation operators for entangled coherent states [24], this choice in our work ensures that the MK signal reaches the maximum when \left|a_k\right|\to \mathrm{\infty }.

To construct the MK inequality, the rotated parity correlation is defined as

(6)\begin{array}{rl}& E_M\left({\varphi }_1,{\varphi }_2,{\varphi }_3,\dots ,{\varphi }_M,|\mathrm{\Psi }⟩\right)\\ & ={⟨P_1\left({\varphi }_1\right)\otimes P_2\left({\varphi }_2\right)\otimes \cdots \otimes P_M\left({\varphi }_M\right)⟩}_{|\mathrm{\Psi }⟩}\\ & =⟨\mathrm{\Psi }|P_1\left({\varphi }_1\right)\otimes P_2\left({\varphi }_2\right)\otimes \cdots \otimes P_M\left({\varphi }_M\right)|\mathrm{\Psi }⟩,\end{array}

where P_k\left({\varphi }_k\right) represents the rotated parity operator

(7)P_k\left({\varphi }_k\right)=R_{k,z}\left({\varphi }_k\right){\left(-1\right)}^{a_k^{†}{a}_k}{R}_{k,z}^{†}\left({\varphi }_k\right),

where

(8)\begin{array}{rl}& R_{k,z}\left({\varphi }_k\right)=D_k^{†}\left({\alpha }_k\right)G_k\left({\varphi }_k\right)D_k\left({\alpha }_k\right),\\ & G_k\left({\varphi }_k\right)={|0⟩}_k⟨0|{\mathrm{e}}^{\mathrm{i}{\varphi }_k}+\sum _{n=1}^{\mathrm{\infty }}{|n⟩}_k⟨n|,\\ & D_k\left({\alpha }_k\right)=\exp \left({\alpha }_k{a}_k^{†}-{\alpha }_k^{\ast }{a}_k\right),\end{array}

are the corresponding phase gate and the displacement operator, with a_k^{†} and a_k being the quantum-number rising and lowering operators, respectively. For the entangled coherent state of Eq. (3), the rotated parity correlation is calculated as

(9)\begin{array}{rl}& E_M\left({\varphi }_1,{\varphi }_2,{\varphi }_3,\dots ,{\varphi }_M,{|GHZ,\alpha ⟩}_M\right)\\ & =N_M^2\{{\mathrm{e}}^{-2\sum _{k=1}^M{\left|{\alpha }_k\right|}^2}+\prod _{k=1}^M{K}_{{\alpha }_k,{\alpha }_k}\left({\varphi }_k\right)\\ & +\left[{\mathrm{e}}^{\mathrm{i}\xi }\prod _{k=1}^M{K}_{{\alpha }_k,-{\alpha }_k}\left({\varphi }_k\right)+c.c.\right]\},\end{array}

with

(10)\begin{array}{rl}{K}_{{\alpha }_k,{\alpha }_k}\left({\varphi }_k\right)=& {\mathrm{e}}^{-2{\left|{\alpha }_k\right|}^2}\left(2\cos {\varphi }_k-1\right)\\ & +2{\mathrm{e}}^{-6{\left|{\alpha }_k\right|}^2}\left(1-\cos {\varphi }_k\right),\\ K_{{\alpha }_k,-{\alpha }_k}\left({\varphi }_k\right)=& {\mathrm{e}}^{-\mathrm{i}{\varphi }_k}+{\mathrm{e}}^{-4{\left|{\alpha }_k\right|}^2}\left(1-{\mathrm{e}}^{-\mathrm{i}{\varphi }_k}\right).\end{array}

For 3-mode GHZ-type entangled coherent states, the MK operator based on the effective rotated parity correlations among the three modes is given by

(11)\begin{array}{rl}{O}_3=& {\sigma }_1(0){\sigma }_2(-\pi /4){\sigma }_3(\pi /4)+{\sigma }_1(0){\sigma }_2(\pi /4){\sigma }_3(-\pi /4)\\ & +{\sigma }_1(\pi /2){\sigma }_2(-\pi /4){\sigma }_3(-\pi /4)-{\sigma }_1(\pi /2){\sigma }_2(\pi /4){\sigma }_3(\pi /4).\end{array}

Thus, the MK signal defined as the absolute value of the expectation value of O_3 can be following

(12)\begin{array}{rl}{S}_3\left(|\mathrm{\Psi }⟩\right)=& |E_3\left(0,-\pi /4,\pi /4,|\mathrm{\Psi }⟩\right)+E_3\left(0,\pi /4,-\pi /4,|\mathrm{\Psi }⟩\right)\\ & +E_3\left(\pi /2,-\pi /4,-\pi /4,|\mathrm{\Psi }⟩\right)\\ & -E_3\left(\pi /2,\pi /4,\pi /4,|\mathrm{\Psi }⟩\right)|.\end{array}

For the 3-mode maximally entangled GHZ-type coherent states, we set M=3, \xi =\pi and {\alpha }_1={\alpha }_2={\alpha }_3=\alpha. The four correlations E_3\left(0,-\pi /4,\pi /4,{|GHZ,\alpha ⟩}_3\right), E_3(0,\pi /4,-\pi /4,{|GHZ,\alpha ⟩}_3), E_3\left(\pi /2,-\pi /4,-\pi /4,{|GHZ,\alpha ⟩}_3\right) and E_3\left(\pi /2,\pi /4,\pi /4,{|GHZ,\alpha ⟩}_3\right) as functions of mean photon number {\left|\alpha \right|}^2, are displayed in Fig. 1(a). With the increases of {\left|\alpha \right|}^2, the parity of E_3\left(\pi /2,\pi /4,\pi /4,{|GHZ,\alpha ⟩}_3\right) increases and rapidly approaches +1, while the other correlations increase first and then drop, tending to -1. The blue solid line of Fig. 1(b) shows the MK signal S_3 for the 3-mode maximally entangled GHZ-type coherent states as a function of mean photon number {\left|\alpha \right|}^2.

When {\left|\alpha \right|}^2=0, the MK signal S_3 is beyond classical upper bound 2. This phenomenon is consistent with the results of Bell-type inequality tests of 3-mode maximally entangled GHZ-type coherent states [25]. The reason is that the GHZ-type entangled coherent states in the protocol, which can be generated from a single-mode coherent-state superposition using two beam splitters, will approach a highly nonlocal entangled state \frac{1}{\sqrt{3}}\left({|1,0,0⟩}_{2,3,4}+{|0,1,0⟩}_{2,3,4}+{|0,0,1⟩}_{2,3,4}\right) [25].

When {\left|\alpha \right|}^2=0.208, S_3 exceeds the lower bound 2\sqrt{2} for genuine tripartite nonlocality [30]. When {\left|\alpha \right|}^2 further increases, it quickly approaches the quantum-mechanical upper bound 4, which implies that a nearly maximal MK inequality violation can be obtained with a small cat state. When {\left|\alpha \right|}^2=1, for instance, the MK signal S_3\simeq 3.929, which is very close to upper bound 4. When {\left|\alpha \right|}^2\ge 3.424, the MK signal actually reaches the maximum theoretical value of 4. In this context, the reason is the same as the analysis of the 3-mode GHZ state in Ref. [29]. The two coherent states |\pm \alpha ⟩ of each bosonic mode are approximately orthogonal to each other and can perform the basis states of a logic qubit even for a moderate value of {\left|\alpha \right|}^2. Applying the effective cat state qubit rotation operator R_{k,z}\left({\varphi }_k\right), the bosonic mode is restricted in the logic space of the cat state qubit so that the rotated parity operator {\sigma }_k\left({\varphi }_k\right) is approximate to the corresponding Pauli operator. The 3-mode cat state is approximately an eigenstate of each of the four correlation operators in Eq. (6).

3 Evaluation of continuous variable control power through quantum nonlocality

In quantum networks, the control power [31] quantifies the authority of the supervisor to govern information transmission. Hence, it is an important measure to validate the effectiveness of the scheme. To evaluate the quality of continuous variable control power, the quantum nonlocality of the quantum channel used in the scheme is discussed. Quantum teleportation, as an essential role of quantum communication and a well-known scheme that highlights the characteristics of quantum singularity, can connect units of quantum information processing, contributing to the construction of quantum networks [32]. Controlled quantum teleportation (CQT) [33] is an important unit of the quantum networks, representing a sophisticated variation and extension of the conventional quantum teleportation protocols [31, 34]. In this section, the control power of the supervisor in continuous variable controlled quantum teleportation protocol [35] is reviewed. The MK inequality and the control power in the previous section are compared to determine the efficiency of the proposed evaluation. The protocol is a simple enough quantum teleportation network model to benefit the evaluation of CV controlled teleportation, as given in Fig. 2.

The continuous variable quantum networks (CVQN) considered in our work is a 3-node network connected in a GHZ entangled state, where Alice, Charlie, and Bob hold modes 2, 3, and 4, respectively. The basic steps of CV controlled quantum teleportation protocol are as follows:

Step 1. The sender (Alice) prepares an information state on mode 1 to be teleported to the receiver (Bob). Charlie, the supervisor, is brought in to monitor the quantum teleportation process until Bob is safe.

The information state can be encoded in term of even coherent states |EVEN⟩ and odd coherent states |ODD⟩ [36] as following:

(13)|I⟩=A_{+}|EVEN⟩+A_{-}|ODD⟩,

where

(14)\begin{array}{rl}& |EVEN⟩={\left[\sqrt{2\left(1+x^2\right)}\right]}^{-1}\left(|\alpha ⟩+|-\alpha ⟩\right),\\ & |ODD⟩={\left[\sqrt{2\left(1-x^2\right)}\right]}^{-1}\left(|\alpha ⟩-|-\alpha ⟩\right).\end{array}

Throughout the context, x={\mathrm{e}}^{-{\left|\alpha \right|}^2}. |EVEN⟩ and |ODD⟩ are well known Schrödinger cat states [36]. \theta and \phi are the polar angle (\theta \in [0,\pi ]) and the azimuthal angle (\phi \in [0,2\pi ]) in Bloch space. Without loss in generality, the relationship of A_{\pm } and \theta, \phi can be expressed by A_{+}=\cos \left(\theta /2\right),A_{-}={\mathrm{e}}^{\mathrm{i}\phi }\sin \left(\theta /2\right), satisfying the normalization condition {\left|A_{+}\right|}^2+{\left|A_{-}\right|}^2=1.

Step 2. A 3-mode maximally entangled GHZ-type coherent states channel is shared among the three partners of the network as following, where Alice has mode 2, Bob has mode 3, and Charlie has mode 4,

(15)\begin{array}{rl}|GHZ,\alpha ⟩=& {\left[2\left(1-x^6\right)\right]}^{-1/2}\cdot \\ & \left(|\alpha ,\alpha ,\alpha ⟩-|-\alpha ,-\alpha ,-\alpha ⟩\right).\end{array}

Step 3. The sender (Alice) mixes the information state with one mode of GHZ-type entangled coherent state using symmetric beam splitter (BS) and two phase shifters (PSs). Then Alice performs photon counting (PC) measurements on the output modes 5 and 6. In addition, Alice communicates her PC outcome to the receiver (Bob) using classical channel.

Step 4. The supervisor (Charlie) executes a PC measurement on mode 3 using detector, and if the supervisor agrees the Bob to get the quantum information Alice teleported to him, Charlie will communicate the outcome to Bob using classical channels.

Step 5. According to the received classical inputs from Alice and Charlie, Bob implements appropriate unitary operation on mode 4, which completes the teleportation protocol with the teleported state finally. Compared with the experimental protocol proposed by Furusawa et al. [37], the protocol in this work makes differences in the classification of the result of photon counting and in the planning for unitary transformations to be performed by Bob [38].

The control power CP in continuous variable systems is defined as the difference between two different fidelities [35],

(16)\begin{array}{rl}CP=& F_{CQT}-F_{NC}\\ =& \left[{\int }_0^{\pi }\left(f_{CQT}-f_{NC}\right)\mathrm{d}\theta \right]/\pi \\ =& \left[{\int }_0^{\pi }\left(C{P}_0\right)\mathrm{d}\theta \right]/\pi ,\end{array}

where F_{CQT} is the conditioned fidelity with the supervisor’s involvement, considering the cases that cause correct controlled teleportation. F_{NC} is the non-conditioned fidelity, which is the fidelity without the supervisor’s involvement. There are 15 cases of the protocol, where the PC counts of Alice are (0, 0) in 3 cases. In these 3 cases, the teleportation will definitely be wrong regardless of the results of Charlie so that the control power of the supervisor is zero. Therefore, in the discussion of control power, the cases in which Alice has successful counts are considered, which means that the PC counts of Alice are not (0, 0). {CP}_0, defined as the difference between conditioned fidelity f_{CQT} and non-conditioned fidelity f_{NC}, can be expressed as [35]

(17)\begin{array}{rl}C{P}_0=& \left\{1/\left[8\left(1+x^2+x^4\right)\right]\right\}\cdot \{-2(-1+2x-2x^2\\ & +x^3-3x^4+x^5){\sin }^2\theta -4(-1+2x\\ & -3x^2+x^3){\left(-1+x^2\cos \theta \right)}^2/\left(1+x^2\right)\\ & +2(-1+2x-4x^2+3x^3-4x^4+5x^5-8x^6\\ & +7x^7-5x^8+x^9)\cdot {\sin }^2\theta /[{\left(1+x^2\right)}^2(1+x^4\\ & -2x^2\cos \theta )]\}.\end{array}

It has been theoretically proved that the control power increases with the mean photon number {\left|\alpha \right|}^2 and will approach an upper bound 1/2 under moderate values of coherent amplitudes ({\left|\alpha \right|}^2\sim 4 to 9) [35]. This protocol can achieve nearly perfect teleportation fidelity when the control power approaches the upper bound.

The control power for the 3-mode maximally entangled GHZ-type coherent states is depicted as a function of MK signal calculated by Eq. (12) in Fig. 3. One can see that the control power climbs slowly for most values of MK signal, while it rushes up to 1/2 when MK signal is extremely near the maximum value 4. The inserted graph corresponds to the region of the blue solid box. Control power is revealed as a monotonically increasing function of the MK signal, which is similar to the trends for control power and the nonlocality of the quantum channel in the DV CQT scheme [8]. Therefore, the result of control power in CV CQT protocol is reliable, and can be evaluated by quantum nonlocality.

In practical quantum networks, inevitable losses and noises in these channels result in decoherence of the quantum states. Decoherence is often caused by the interaction between system and the environment, which is a main factor limiting the development of the quantum information technology [39]. Here the robustness of the evaluation using MK inequality against the decoherence is discussed. A more general type of decoherence is caused by interactions with modes that are initially in some Gaussian state not equal to the vacuum, such as a thermal state at finite temperature. But for optical fields, the effective temperature is essentially zero, and the environment can be assumed to be in a vacuum [40]. The decoherence caused by photon absorption is considered here, which is a kind of passive loss. This loss can be modeled by a linear interaction of the mode of interest with one or more “environment” modes that are initially in the vacuum [40] as following:

(18)|\alpha ⟩\otimes {|0⟩}_e\to |\sqrt{\eta }\alpha ⟩\otimes {|\sqrt{1-\eta }\alpha ⟩}_e,

where the second state refers to the “environment” mode or modes, and \eta is the survived photon proportion that gives the fraction of photons surviving the absorption process. For simplicity, we assume that three modes are equally lost. Thus, the quantum channel becomes

(19)\begin{array}{rl}{|GHZ,\tilde{\alpha }⟩}_3=& {\left[2\left(1-{\tilde{x}}^6\right)\right]}^{-1/2}\\ & \cdot \left(|\tilde{\alpha },\tilde{\alpha },\tilde{\alpha }⟩-|-\tilde{\alpha },-\tilde{\alpha },-\tilde{\alpha }⟩\right),\end{array}

where \tilde{\alpha }=\sqrt{\eta }\alpha, \tilde{x}=e^{-{\left|\tilde{\alpha }\right|}^2}. In the presence of decoherence, the information states are of the form

(20)|\tilde{I}⟩=A_{+}|EVEN,\tilde{\alpha }⟩+A_{-}|ODD,\tilde{\alpha }⟩,

where

(21)\begin{array}{rl}& |EVEN,\tilde{\alpha }⟩={\left[\sqrt{2\left(1+{\tilde{x}}^2\right)}\right]}^{-1}\left(|\tilde{\alpha }⟩+|-\tilde{\alpha }⟩\right),\\ & |ODD,\tilde{\alpha }⟩={\left[\sqrt{2\left(1-{\tilde{x}}^2\right)}\right]}^{-1}\left(|\tilde{\alpha }⟩-|-\tilde{\alpha }⟩\right).\end{array}

The following steps are the same as Section 3. The results of the MK signal for different values of the survived photon proportion \eta are plotted in Fig. 4(a) as functions of mean photon number {\left|\alpha \right|}^2. One can see from Fig. 4(a) that the decreasing survived photon proportion will reduce the improving ability of the MK signal. With a large {\left|\alpha \right|}^2, the MK signal is approaching the maximum value 4. This maximum value is independent of the decoherence caused by photon absorption. Consequently, the relationships between control power and the MK signal are shown in Fig. 4(b). The inserted graph corresponds to the region of the blue solid box. One can see that the increasing loss will not influence the evaluation of control power using quantum nonlocality. Therefore, the evaluation method through quantum nonlocality by MK inequality can resist the decoherence caused by photon absorption, which shows the robustness.

We have also considered the cases with nonuniform decoherence. That means the photon losses of M-modes are unequal. In this case, the decreasing symmetry among the M modes will reduce the improving ability of the MK signal. The symmetry among the M modes can be reflected by the maximum difference of survived photon proportion. The maximum value is only effected by the bound 2^{M-1}, which is independent of the decoherence caused by photon absorption. The results make our decoherence analysis more complete.

4 Comparison with other evaluation methods

In this section, the evaluation methods using the MK inequality in this work is compared with the method using the Bell−Mermin (BM) inequality [25] and the method using Svetlichny inequality in discrete variable system [8].

4.1 Comparison with another continuous variable evaluation method

In order to further analyze which inequality can more effectively measure the control power in the quantum network, the continuous variable methods of evaluation using the MK inequality and the Bell−Mermin inequality [25] are compared. Figure 5(a) shows the MK and BM signals for the 3-mode maximally entangled GHZ-type coherent states as functions of mean photon number {\left|\alpha \right|}^2. Both the MK signal and the BM signal increase with the mean photon number {\left|\alpha \right|}^2 of the coherent state components. The functions of control power of the GHZ-type coherent states and the signals of different inequalities are plotted in Fig. 5(b). The control power is revealed as monotonically increasing functions of both the MK signal and the BM signal. Thus, the result of control power can be validly evaluated using both inequalities. However, there are limitations of the evaluation using the BM inequality and advantages using the MK inequality as follows.

1) As shown in Fig. 5(a), the maximal value of the BM signal approaches 3.4 when {\left|\alpha \right|}^2 is larger than 16, which is still lower than the quantum-mechanical upper bound of 4 for a three-partite system by 15%. Nevertheless, the MK signal can reach the theoretical global upper bound of 4. The reason of the advantageous range of the MK signal is benefited by the optimality of measurement angles in Eq. (5). The choices {\varphi }_k=\pi /4 and {{\varphi }^{\prime }}_k=-\pi /4 are optimized for maximizing parity contrast between |\alpha ⟩ and |-\alpha ⟩. For coherent states with amplitude \alpha, the overlap ⟨\alpha |-\alpha ⟩={\mathrm{e}}^{-2|\alpha {|}^2} approaches zero as |\alpha {|}^2 grows. The expectation of the rotated parity operator in Eq. (7) is function of {\varphi }_k, such as ⟨\alpha |P_k(\pi /4)|\alpha ⟩\propto \cos ({\varphi }_k),⟨-\alpha |P_k(\pi /4)|-\alpha ⟩\propto \cos (3{\varphi }_k). Setting {\varphi }_k=\pm \pi /4 maximizes the contrast ratio ⟨\alpha |P|\alpha ⟩-⟨-\alpha |P|-\alpha ⟩, which scales as \sqrt{2}{\mathrm{e}}^{-|\alpha {|}^2} for small |\alpha {|}^2. This ensures the MK signal S_3 rapidly approaches the quantum bound, as shown in Fig. 1(b).

2) From Fig. 5(b), both high MK signal and high control power can be obtained simultaneously using the MK inequality. The control power increases with mean photon number {\left|\alpha \right|}^2. In practical experiments, the superposition of coherent states of moderate value of coherent amplitudes ({\left|\alpha \right|}^2\sim 4 to 9) can be generated with high fidelity in various methods consisting of photon subtraction from squeezed vacuum state, Kerr-nonlinear interactions and cavity-assisted interactions [41]. When {\left|\alpha \right|}^2=4, the BM signal is 3.06 and the corresponding control power is 0.481; when {\left|\alpha \right|}^2=9, the BM signal is 3.32 and the corresponding control power is 0.498. Meanwhile, when {\left|\alpha \right|}^2=4, the MK signal has already reached 4 and the corresponding control power is 0.498. It is shown that, the high violation of BM inequality appears when {\left|\alpha \right|}^2 is quite large, which is hard to achieve in current experiments. The obtained MK signal reaches the upper bound 4 even when {\left|\alpha \right|}^2 in each mode is small. The superposition of coherent states of the moderate value of {\left|\alpha \right|}^2 can be generated more easily in experiments. Therefore, our work shows that a more complete control power will be easier to measure experimentally using the MK inequality.

3) The recursive construction of the MK operator in Eq. (4) inherently reduces the optimization complexity for multi-mode systems. With the method using BM inequality, there are 12 variables to be optimized to find the global maximum value of the BM signal in 3-mode GHZ-type coherent states. In addition, the number of the variables to be optimized increases with the number of entangled bosonic modes. It is difficult to obtain the optimal BM signal with three modes. When more bosonic modes are involved, the task will be extremely challenging. In contrast, the MK operator recursively incorporates prior correlations via local steps. Concretely, the recursive relation O_k=f(O_{k-1},{\sigma }_k) in Eq. (4) divides global optimization into M sequential steps, reducing the optimization complexity of variables from O(M^2) to O(M). For instance, in a 3-mode system, the MK operator requires optimizing only {\varphi }_1,{\varphi }_2,{\varphi }_3 sequentially with symmetry constraints Eq. (5), whereas BM requires searching all 12 angle parameters independently. The method using MK inequality significantly simplifies the process of finding the global maximum value of the MK signal, avoiding a number of variables to be optimized. This method using MK inequality can be easily extended to evaluate the protocols with arbitrary multi-mode channels.

From the construction of inequality, both the MK inequality and the BM inequality are based on the assumption of local realism. However, the MK inequality is constructed in an iterative way to deal with complex multipartite correlations and can be generalized to multipartite quantum systems. From a theoretical perspective, the MK inequality bounds the classical correlations in such scenarios but yields a quantum violation that increases exponentially with the partner number in quantum communications [42–45]. This means that the MK inequality can more clearly reveal the nonlocality of quantum systems, especially in multipartite systems.

4.2 Comparison with a discrete variable evaluation method

The evaluation methods using the MK inequality in continuous variable systems and Svetlichny inequality in discrete variable systems [8] are compared.

From a theoretical perspective, the constructions of both the MK inequality and the Svetlichny inequality are also based on the same assumption, that is, local realism. In Fig. 6, the results of the control power as functions of maximum Svetlichny inequality violation [8] in DV systems and the MK signal in CV systems are plotted. When the value of the Svetlichny inequality exceeds its classical upper bound 4 but is less than a certain bound, the actual value of the control power is negative. This negative value is caused by the inconsistency between the hierarchical assumption of Svetlichny inequality and GHZ entanglement. The Svetlichny inequality assumes a bipartition of the system (e.g., 1 vs. 2+3), modeled by hybrid local-nonlocal correlations: p(a,b,c)=\sum _{\lambda }{q}_{\lambda }{p}_{\lambda }(a|b,c)p_{\lambda }(b)p_{\lambda }(c). That allows classical communication between two partners. However, genuine tripartite nonlocality in GHZ states violates this hybrid assumption so that the negative control power occurs. The negative value of control power means that the involvement of the supervisor will damage the fidelity of the protocol, which is undesirable and contrary to common sense.

In contrast, the continuous variable method using the MK inequality directly enforces full local realism (p(a,b,c)=\sum _{\lambda }{q}_{\lambda }{p}_{\lambda }(a)p_{\lambda }(b)p_{\lambda }(c)). It ensures that when the MK inequality exceeds the classical upper bound, the control power is monotonically increasing positive values, as shown in Fig. 6. The MK inequality in this work avoids the negative and unreasonable control power, which causes the range of control power to be compressed. Then the growth rate of control power with inequality signals increasing slows down. Therefore, the MK inequality in this work is more advantaged and can more effectively evaluate the control power in the quantum network.

5 Summary and outlook

In summary, the quantum nonlocality of general GHZ-type entangled coherent state channels is unearthed by the Mermin-Klyshko inequality with rotated quantum-number parity operators, and the control power in continuous variable controlled quantum teleportation via a 3-mode maximally entangled GHZ-type coherent state channel is evaluated by the MK inequality. The MK signal S_3 of 3-mode maximally entangled GHZ-type coherent states increases with the mean photon number {\left|\alpha \right|}^2. When the mean photon number {\left|\alpha \right|}^2=0, the MK signal S_3 is beyond the classical upper bound 2. When {\left|\alpha \right|}^2=0.208, S_3 exceeds the lower bound 2\sqrt{2} for genuine tripartite nonlocality. When {\left|\alpha \right|}^2 increases further, it quickly approaches the quantum-mechanical upper bound 4. It means that a nearly maximal MK inequality violation can be obtained with a small mean photon number. A monotonically increasing function between the control power and genuine tripartite nonlocality in the protocol is validated, which is similar to the trends in the DV CQT scheme. Therefore, the solidity of the results of control power in the CV CQT protocol can be confirmed. Leveraging this intrinsic correlation, the MK signal serves as a reliable metric for the evaluation of the control power. In addition, the robustness of the evaluation is discussed with decoherence. When considering the decoherence in the protocol, the evaluation method through quantum nonlocality by MK inequality can resist the decoherence caused by photon absorption. The MK inequality can reach the theoretic upper bound 4 under limited experiments, avoiding the complicated process of optimizing lots of variables. Compared with the method using the BM inequality, our work proves that the MK inequality performs better in experimental measurements of complete control power. The advantage range of MK inequality is due to the optimality of measurement angles of the rotated parity operator. The optimization complexity for multi-mode systems is reduced inherently by the recursive construction of the MK operator. Compared with the method using Svetlichny inequality in discrete variable systems, the method using MK inequality has advantages in terms of the protocol measurement to obtain reasonable evaluation indicators. The usage of MK inequality avoids negative control power, which is caused by the inconsistency between the hierarchical assumption of Svetlichny inequality and GHZ entanglement. Therefore, the MK inequality is proved as a more reliable tool to reveal the quantum nonlocality in multipartite systems.

In practical CV CQT nonlocality verification experiments, the attainment of experimental verifiability requires a thorough consideration of two fundamental parts. One is the CV CQT experimental implementation system using coherent states with appropriate mean photon numbers. The other is the detection of the MK signal of the quantum channel. To offset the control power damage due to the decoherence, the enlargement of optical cat states can be used, which has been developed in several experimental schemes, including the synthesis method using a pre-prepared coherent state as a seed [46], and a recent method for the deterministic preparation of optical coherent states with arbitrarily large mean photon numbers [47]. As for the detection of the MK signal, the required phase gate G_k\left({\varphi }_k\right) can be realized using a two-level coupler to couple multiple cavities based on circuit quantum electrodynamics [48]. In this method, the gate operation time is independent of the number of qubits, and there is no measurement needed.

As for the extensibility, the method in this work can be extended in larger CV systems with more parties. The upper bound of the control power in the system remains unchanged compared with tripartite systems. The increasing of the violation value of nonlocality in larger CV teleportation systems can enhance the security in other aspects, such as providing backup receivers and enhancing fault tolerance. The results in this work illuminate the role of nonlocality in multipartite continuous variable quantum information processing and are important for constructing practical quantum communication protocols.

References

[1] A. K. Ekert , Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67(6), 661 (1991)

[2] S. Camalet , Quantifying nonlocality as a resource for device-independent quantum key distribution, Phys. Rev. A 102(1), 012617 (2020)

[3] Y. Wang, Y. Zheng, C. Fang, H. Shi, W. Pan , Light-injection attack against practical continuous-variable measurement-device-independent quantum key distribution systems, Opt. Express 32(19), 33656 (2024)

[4] C. J. Liu, Y. Chao, L. Wang, Q. S. Li , Continuous-variable measurement-device-independent quantum key distribution with multi-ring discrete modulation, Opt. Express 32(18), 31549 (2024)

[5] M. Moreno, S. Brito, R. V. Nery, R. Chaves , Device-independent secret sharing and a stronger form of Bell nonlocality, Phys. Rev. A 101(5), 052339 (2020)

[6] J. Gu, Y. M. Xie, W. B. Liu, Y. Fu, H. L. Yin, Z. B. Chen , Secure quantum secret sharing without signal disturbance monitoring, Opt. Express 29(20), 32244 (2021)

[7] Q. Liao, H. Liu, Y. Gong, Z. Wang, Q. Peng, Y. Guo , Practical continuous-variable quantum secret sharing using plug-and-play dual-phase modulation, Opt. Express 30(3), 3876 (2022)

[8] S. Gangopadhyay, T. Wang, A. Mashatan, S. Ghose , Controlled quantum teleportation in the presence of an adversary, Phys. Rev. A 106(5), 052433 (2022)

[9] T. Zhang, N. Jing, S. M. Fei , Sharing quantum non-locality in star network scenarios, Front. Phys. (Beijing) 18(3), 31302 (2023)

[10] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters , Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70(13), 1895 (1993)

[11] K. Wang, X. T. Yu, Z. C. Zhang , Two-qubit entangled state teleportation via optimal POVM and partially entangled GHZ state, Front. Phys. (Beijing) 13(5), 130320 (2018)

[12] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, S. Lloyd , Gaussian quantum information, Rev. Mod. Phys. 84(2), 621 (2012)

[13] X. B. Wang, T. Hiroshima, A. Tomita, M. Hayashi , Quantum information with Gaussian states, Phys. Rep. 448(1−4), 1 (2007)

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

[15] L. Vaidman , Teleportation of quantum states, Phys. Rev. A 49(2), 1473 (1994)

[16] X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, K. Peng , Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states, Phys. Rev. Lett. 118(23), 230501 (2017)

[17] N. F. Gong, T. J. Wang, S. Ghose , Control power of a high-dimensional controlled nonlocal quantum computation, Phys. Rev. A 103(5), 052601 (2021)

[18] W. X. Duan and T. J. Wang , Control power of high-dimensional controlled dense coding, Phys. Rev. A 105(5), 052417 (2022)

[19] J. D. Bancal, J. Barrett, N. Gisin, S. Pironio , Definitions of multipartite nonlocality, Phys. Rev. A 88(1), 014102 (2013)

[20] J. S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge University Press, 2004

[21] K. Banaszek and K. Wódkiewicz , Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation, Phys. Rev. A 58(6), 4345 (1998)

[22] K. Banaszek and K. Wódkiewicz , Testing quantum non-locality in phase space, Phys. Rev. Lett. 82(10), 2009 (1999)

[23] Z. B. Chen, J. W. Pan, G. Hou, Y. D. Zhang , Maximal violation of Bell’s inequalities for continuous variable systems, Phys. Rev. Lett. 88(4), 040406 (2002)

[24] Z. R. Zhong, J. Q. Sheng, L. H. Lin, S. B. Zheng , Quantum nonlocality for entanglement of quasi-classical states, Opt. Lett. 44(7), 1726 (2019)

[25] H. Jeong and N. B. An , Greenberger-Horne-Zeilinger-type and W-type entangled coherent states: Generation and Bell-type inequality tests without photon counting, Phys. Rev. A 74(2), 022104 (2006)

[26] A. Acín, N. J. Cerf, A. Ferraro, J. Niset , Tests of multimode quantum nonlocality with homodyne measurements, Phys. Rev. A 79(1), 012112 (2009)

[27] G. McKeown, F. L. Semião, H. Jeong, M. Paternostro , Genuine multipartite nonlocality of entangled thermal states, Phys. Rev. A 82(2), 022315 (2010)

[28] J. Li, T. Fogarty, C. Cormick, J. Goold, T. Busch, M. Paternostro , Tripartite nonlocality and continuous-variable entanglement in thermal states of trapped ions, Phys. Rev. A 84(2), 022321 (2011)

[29] J. Q. Sheng, L. H. Lin, Z. R. Zhong, S. B. Zheng , Nearly maximal violation of the Mermin-Klyshko inequality with multimode entangled coherent states, Opt. Express 27(22), 31864 (2019)

[30] D. Collins, N. Gisin, S. Popescu, D. Roberts, V. Scarani , Bell-type inequalities to detect true n-body non-separability, Phys. Rev. Lett. 88(17), 170405 (2002)

[31] T. J. Wang, G. Q. Yang, C. Wang , Control power of high-dimensional controlled teleportation, Phys. Rev. A 101(1), 012323 (2020)

[32] J. W. Pan , Quantum information technology: Current status and prospects, Acta. Phys. Sin. 73(1), 010301 (2024)

[33] A. Karlsson and M. Bourennane , Quantum teleportation using three-particle entanglement, Phys. Rev. A 58(6), 4394 (1998)

[34] F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, Y. Wang , Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement, Phys. Rev. A 72(2), 022338 (2005)

[35] Y. Tian, D. Cai, N. Gong, Y. Li, L. Qian, R. Zhang, Z. Huang, T. Wang , Control power in continuous variable controlled quantum teleportation, Entropy (Basel) 26(12), 1017 (2024)

[36] V. V. Dodonov, I. A. Malkin, V. I. Man'Ko , Even and odd coherent states and excitations of a singular oscillator, Physica 72(3), 597 (1974)

[37] H. Yonezawa, T. Aoki, A. Furusawa , Demonstration of a quantum teleportation network for continuous variables, Nature 431(7007), 430 (2004)

[38] H. Prakash, N. Chandra, R. Prakash, Shivani , Effect of decoherence on fidelity in teleportation using entangled coherent states, J. Phys. At. Mol. Opt. Phys. 40(8), 1613 (2007)

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

[40] S. van Enk , Decoherence of multidimensional entangled coherent states, Phys. Rev. A 72(2), 022308 (2005)

[41] R. K. Pandey, R. Prakash, H. Prakash , Controlled quantum teleportation of superposed coherent state using GHZ entangled coherent state, Int. J. Theor. Phys. 58(10), 3342 (2019)

[42] N. D. Mermin , Extreme quantum entanglement in a superposition of macroscopically distinct states, Phys. Rev. Lett. 65(15), 1838 (1990)

[43] M. Ardehali , Bell inequalities with a magnitude of violation that grows exponentially with the number of particles, Phys. Rev. A 46(9), 5375 (1992)

[44] A. Belinsky and D. Klyshko, A modified n-particle bell theorem, the corresponding optical experiment and its classical model, Phys. Lett. A 176(6), 415 (1993)

[45] A. V. Belinskiĭ and D. N. Klyshko , Interference of light and Bell’s theorem, Phys. Uspekhi 36(8), 653 (1993)

[46] D. V. Sychev, A. E. Ulanov, A. A. Pushkina, M. W. Richards, I. A. Fedorov, A. I. Lvovsky , Enlargement of optical Schrödinger’s cat states, Nat. Photonics 11(6), 379 (2017)

[47] Z. H. Li, F. Yu, Z. Y. Li, M. Al-Amri, M. S. Zubairy , Method to deterministically generate large-amplitude optical cat states, Commun. Phys. 7(1), 134 (2024)

[48] Q. P. Su, Y. Zhang, L. Bin, C. P. Yang , Efficient scheme for realizing a multiplex-controlled phase gate with photonic qubits in circuit quantum electrodynamics, Front. Phys. (Beijing) 17(5), 53505 (2022)

RIGHTS & PERMISSIONS

Higher Education Press