1. Introduction

Quantum entanglement, which relies on a fundamental principle of quantum mechanics, serves as a critical component in the field of current quantum information technologies (QITs) [1–3], such as trenchant quantum computation [4–7], quantum cryptography [8–10], entangled state analysis [11–13], and quantum secure direct communication [14–19], and enables their workable applications that markedly overpower their classical ones. Furthermore, quantum entanglement serves as a vital tool in experimental setting to test the underlying principles of quantum mechanics, providing a deeper comprehension of the intricacies at the quantum level [3,20]. Recently, people have witnessed extensive research focusing on the characteristics, applications, and conversions between entangled states due to their unique properties and potential utility. There are three common multipartite entangled states, i.e., Knill-Laflamme-Milburn (KLM) state [21–27], Greenberger-Horne-Zeilinger (GHZ) state [28–34], and W state [35,36]. These entangled state are highly valued in the realm of QITs and have presented the subject of many experimental and theoretical studies. On the one hand, the GHZ state demonstrates flawless correlation among its individual particles, such that the loss of any single qubit leads to its complete destruction [28,29]. Conversely, the KLM state named by Knill, Laflamme, and Milburn in 2001 [21], has shown to be robust against qubit-error rate. The multi-qubit KLM state can maintain entanglement even if one of their qubit is lost [23].

The valuable characteristics of KLM states have spurred their application in various protocols of QITs, sparking extensive research into the preparation and analysis of KLM states utilizing different physical systems [21–27,37]. To date, researchers have developed efficient methodologies for generating GHZ states for photon systems, including both deterministically using nonlinear platforms [30,31] and probabilistically employing linear optics [32–34]. Beyond entangled state preparation, state conversion represents another prominent area of investigation in Ref. [38–43]. Numerous schemes for converting between entangled states have been proposed and explored [44,45]. Notably, directly converting GHZ states into KLM states, or vice versa, is not feasible through stochastic local operations and simple classical communications [40,46]. Consequently, conversions from two- (or multi-) qubit KLM states to Bell (or GHZ) states hold paramount importance in the realm of QITs [40]. Besides, the conversion process not only enhances the entanglement of the quantum system but also furnishes valuable resources for quantum computing. Furthermore, it simplifies the experimental implementation of quantum protocols, thus serving as a crucial building block in the field of QITs.

Nowadays, photon-encoded qubits are gaining widespread acceptance for their compatibility with low-decoherence applications and their role as carriers for QITs. Significant progress have already been made in showcasing the capabilities of quantum logic gates in photon systems [47–52]. Furthermore, the polarization degree of freedom in photon system can be readily encoded and manipulated using state-of-the-art optical techniques. However, the quantum system inevitably interacts with its electromagnetic radiation, thermal fluctuations, mechanical fluctuations, or other particles involved in environment [53,54], which can introduce errors in the transmission of polarized states of photons, highlight the importance of finding effective strategies to mitigate their impact on the QIT. Fortunately, the decoherence-free subspace (DFS) [43,55–60] approach has garnered significant attention for mitigating the detrimental impact of environmental noise factors on quantum information transmission. In DFS, a specific subspace is defined within the unitary evolution. By using a pair of photons to encode a single logic qubit within a DFS, that is, $|\bar {0}\rangle =|RL\rangle$ and $|\bar {1}\rangle =|LR\rangle$ ($|R\rangle$ or $|L\rangle$ represent right-circularly or left-circularly polarized states of photons, respectively in Ref. [55], the initial quantum state $|\phi \rangle =\sin \theta |\bar {0}\rangle +\cos \theta |\bar {1}\rangle$ remains unchanged despite encountering symmetric decoherence. This allows for perfect protection against the adverse effects of decoherence. Up to date, DFS has found widespread applications in QITs, e.g., fault-tolerant quantum teleportation [56,57], the valid preparation of the entangled states, the complete mutual conversion between the entangled states [43,58], and the establishment of the remote quantum network [59].

Based on the aforementioned studies, our manuscript propose two protocols, i.e., one method involves converting the two-logic-qubit KLM state into two-logic-qubit Bell states, while another method transforms the three-logical-qubit KLM state into three-logical-qubit GHZ states, leveraging on the state-selected reflection property of a nitrogen-vacancy (NV) center confined within a cavity in the context of DFS, which effectively protects the quantum states from decoherence and other types of errors. The KLM states are grouped into different categories according the electron-spin state of the NV$^{-}$, so the required states will be identified. Compared to prior auxiliary-based approaches, two schemes achieves near-unit fidelities in principle and higher efficiencies with fewer nonlinear interactions and linear optical elements, as the error-detection efficacy containing potential failures and seamless interaction can be effectively heralded through the utilization of single-photon detectors. Moreover, it is possible to circumvent the limitations associated with strong coupling, thereby minimizing the requirements and constraints of the experimental conditions.

2. NV$^{-}$ center configuration

Establishing a connection between a single photon and individual spins is a fundamental requirement for the development of QITs [61]. A deterministic link can be achieved by employing a diamond NV$^{-}$ center configuration in a single-sided optical cavity as depicted in Fig. 1(a). This configuration involves a negatively charged NV$^{-}$ center, which consists of an adjacent vacancy, a substitutional nitrogen atom, and six electrons originating from the nitrogen atom and three carbon atoms. The energy-level structure of this system is highly intricate, arising from the complex interplay between electron-nuclear coupling and specific optical transitions.

 

Fig. 1. (a) The coupling system involved a NV$^{-}$ center and an optical cavity; (b) The structural arrangement of energy levels and optical transitions within the NV$^{-}$ center.

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As illustrated in Fig. 1(b), the ground states of the diamond NV$^{-}$ center experience the splitting caused by the spin-spin interaction, resulting in a zero-field splitting of 2.88 GHz between the states denoted as $|0\rangle (m_{s}=0)$ and $|\pm 1\rangle (m_{s}=\pm 1)$. Additionally, applying an external magnetic field along the symmetry axis degenerates the spin states $m_s=\pm 1$ and splits the energy levels for ground states $|0\rangle$ and $|+1\rangle$ (simplifying $|1\rangle$), as well as allowing spin-conserving optical transitions. The state $|0\rangle$ ($|1\rangle$) can be optically excited to excited state $|M_{3}\rangle$ ($|M_{5}\rangle$) of the NV$^{-}$ center by the absorption of the photon with resonant frequency $\omega _{0}$ ($\omega _{1}=\omega _{0} +\vartriangle$) [62–65], where $\vartriangle$ represents the energy difference between the states $|0\rangle$ and $|1\rangle$ caused by the applied magnetic field. In detail, if the NV$^{-}$ center is in state $|0\rangle$ and the frequency $\omega _{c}$ of the cavity mode $\hat {a}$ is near resonant with the transition $|0\rangle \leftrightarrow |M_{3}\rangle$, the reflection of the cavity will undergo substantial modification due to destructive interference. This interference effect will inhibit the photon from entering the cavity. On the contrary, when the state of the NV$^{-}$ center is $|1\rangle$, and its dipole-allowed transition is significantly detuned from the cavity mode $\hat {a}$ at frequency $\omega _{c}$, the cavity’s reflection remains mostly unchanged. Consequently, as an input photon with frequency $\omega$ interacts with the cavity-NV$^{-}$ system, it may scatter into the delivery port with a reflection coefficient relies on its state.

Applying the rotating wave approximation, the effective Hamiltonian [66] of the cavity-NV$^{-}$-center system is expressed as follows

The scattering behavior in a NV$^{-}$-cavity system closely resembles that of a dipole-cavity system, as two dipole-allowed transitions are both spin-conserving and decoupled. Moreover, we can determine the reflection coefficient of a dipole-cavity system by solving the system’s dynamic equations in conjunction with the standard input-output relation [36,67–69], which can be expressed as

With weak excitation limitation ($\langle \hat {\sigma }_{z}\rangle =-1$), when an photon enter and interact with the single-sided cavity-NV$^{-}$ system, the reflection coefficient $r_{s}(\omega )$ can be accurately expressed as [69]

3. Two converting protocols for two (three)-logic-qubit KLM states

In this segment, we propose two converting protocols for two (three)-logic-qubit KLM states to Bell (GHZ) states with near-unit fidelities in principle, assisted by heralded two-logic-qubit parity-check gates and control-not (CNOT) gates [50,52]. Firstly, we incorporate two types of heralded CNOT gates in Sec. 3.1, as they are proved to be instrumental in subsequent steps. Secondly, we focus on transitioning the two-logic-qubit KLM states into Bell states in Sec. 3.2. Eventually, we implement an advanced program to achieve the full conversion of three-logic-qubit KLM state into the GHZ states in Sec. 3.3.

3.1 Heralded CNOT gates on two-photon system

In this subsection, we introduce two types of heralded CNOT gates applied to two-photon systems assisted by single-sided cavity-NV$^{-}$ center system. The CNOT$_{1}$ serves the purpose of flipping the target qubit exclusively when the control qubit is in the state $|L\rangle$. No operation is applied to the target qubit when the control qubit is in the state $|R\rangle$, as illustrated in Fig. 2. Conversely, CNOT$_{2}$ operates in the opposite manner, flipping the target qubit if and only if the control qubit is in the state $|R\rangle$, which is equivalent to perform a bit-flip operation on the control qubit of CNOT$_{1}$.

 

Fig. 2. Schematic diagram for the two-photon CNOT$_{1}$ gate assisted by cavity-NV$^{-}$ center system. The circularly polarized beam splitter CPBS$_k (k=1,2)$ reflects (transmits) $L$-($R$-) polarized photon. The half-wave plates (HWP$^{{22.5}^\circ }$ and HWP$^{{45}^\circ }$) fixed at $\theta =22.5^\circ$ and $\theta =45^\circ$, respectively. The single-photon detector is D. The wave-form corretor WFC makes the transformation $|R\rangle (|L\rangle )\rightarrow \dfrac {1}{2}(r_{1}-r_{0})|R\rangle (|L\rangle )$. The "Block" represents the gray rectangular box.

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The schematic diagram for the two-photon CNOT$_{1}$ gate shown in Fig. 2, the photon $a$ serves as the control qubit, while the $b$ photon acts as the target qubit. The circularly polarized beam spitters (CPBS$_{1}$ and CPBS$_{2}$) reflect (transmit) $L$-($R$-)polarized photon. The wave-form corretor (WFC) makes the transformation $|R\rangle (|L\rangle )\rightarrow \dfrac {1}{2}(r_{1}-r_{0})|R\rangle (|L\rangle )$. The half-wave plates (HWP$^{{22.5}^\circ }$ and HWP$^{{45}^\circ }$) fixed at $\theta =22.5^\circ$ and $\theta =45^\circ$, respectively, resulting in

Firstly, photon $a$ accesses the circuit from path $l_{1}$ and proceeds through the Block. In particular, before the photon $a$ interact with the NV$^{-}$ center, the Hadamard operations H$^{e}$ are acted on the electron-spin state of the NV$^{-}$ center by using a $\pi /2$ femtosecondlevel optical pulse [73], which can be descriped as $|1\rangle \rightarrow (|1\rangle +|0\rangle )/\sqrt {2}$ and $|0\rangle \rightarrow (|1\rangle -|0\rangle )/\sqrt {2}$. In this Block, the state $|R\rangle$ is transmited by CPBS$_{1}$, then goes through the HWP$^{22.5^{\circ }}\rightarrow$ NV$^{-}$ center $\rightarrow$ HWP$^{22.5^{\circ }}$ $\rightarrow$ HWP$^{45^{\circ }}$. While the state $|L\rangle$ is reflected directly to the WFC. Eventually, the two components $|L\rangle$ and $|R\rangle$ converge at the CPBS$_{2}$. If the output photon $|R\rangle$ is transformed into the left-circularly polarized photon $|L\rangle$, it will trigger the detector (D), and if the D is no response [74] the state of the system composed of photon $a$, photon $b$ and the NV$^{-}$ are changed from $|\Omega \rangle _{0}$ to $|\Omega \rangle _{1}$, here

Sequentially, allow photon $b$ to access path $l_{2}$. Prior to and following the interaction between photon $b$ and the NV$^{-}$ center, the operation H$^{e}$ is applied once more to the electron-spin state of the NV$^{-}$ center. Following photon $b$ traverses through the HWP$^{22.5^{\circ }}_{1}$, it also interacts with the Block and then passes through the HWP$^{22.5^{\circ }}_{2}$. If the detrctor D is not clicked, the quantum state $|\Omega \rangle _{1}$ is changed into

For the CNOT$_{2}$, it is only necessary to perform a bit-flip operation on the control photon $a$ based on the result of $|\Omega \rangle _{2}$ in Eq. (8), then the state of the system will be

3.2 Conversion from two-logic-qubit KLM state to Bell states in DFS

In sequence, we will extend the encoding method of qubits in DFS presented in Sec. 1, and showcase the conversion procedure from the two-logic-qubit KLM state to Bell state. The conversion procedure from the two-logic-qubit KLM state to Bell state entails two distinct steps in Fig. 3. Initially, a portion of the KLM state is directly transformed into the Bell state. Subsequently, the remaining part of the conversion necessitates the utilization of the CNOT gates. Ultimately, this scheme enables the complete transformation of the two-logic-qubit KLM state into Bell state in the DFS framework. Precisely, the configuration of photonic polarization KLM state in DFS can be expressed as [75]

 

Fig. 3. The diagram for converting the two-logic-qubit KLM state to Bell states. The time delay line (TDL) signifies an optical fiber with temporal lag.

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It is noticeable that the spin state of NV$^{-}_{1}$ can distinguish between $|\phi \rangle _{1}^{+}$ associated with $|0\rangle _{1}$ and $|\bar {0}\bar {1}\rangle$ associated with $|1\rangle _{1}$. Consequently, the probability of obtaining the Bell state $|\phi \rangle _{1}^{+}$ is $2\xi ^{8}/3$.

Obviously, the function of the first step is equal to the two-logic-qubit heralded parity-check gate, where even parity $|\bar {0}\bar {0}\rangle$ (or $|\bar {1}\bar {1}\rangle$) and odd parity $|\bar {0}\bar {1}\rangle$ (or $|\bar {1}\bar {0}\rangle$) can be categorized according to the measuring result $|0\rangle _{1}$ and $|1\rangle _{1}$ of the NV$^{-}_{1}$, respectively.

To completely convert the two-logic-qubit KLM state into two-logic-qubit Bell state without photon wasted, certain associated feed-forward operations are carried out according to the measurement outcome of the NV$^{-}_{1}$ spin state $|0\rangle _{1}$. In the second step, the first and third photons pass through the HWP$^{22.5^{\circ }}$ on paths $l_{1}$ and $l_{3}$, and then the photons interact with the two CNOT gates, i.e., the CNOT$_{1}$ gate is placed on paths $l_{1}$ and $l_{2}$, and the CNOT$_{2}$ gate exists on paths $l_{3}$ and $l_{4}$, respectively. Following the traversal of photons through the quantum circuits of the initial stage once more, i.e, the heralded parity-check gate in Fig. 3(a), the photons from paths $l_{2}$ and $l_{4}$ interact with the Block$_{2}$ corresponding to the NV$_{2}^{-}$. The remaining item can be convert to two-logic-qubit Bell states according to the electron-spin state ($|0\rangle _{2}$ or $|1\rangle _{2}$) of the NV$_{2}^{-}$, the whole state of the system is evolved into

Table 1. The relationship between the measuring results and corresponding to the required operations, Bell state, and efficiency.

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3.3 Conversion from three-logic-qubit KLM state to GHZ states in DFS

A detailed procedure for transforming the three-logic-qubit KLM state to GHZ states in DFS below. This conversion process can be accomplished in two steps, similar to the above conversion from the two-logic-qubit KLM state to Bell states. Firstly, the GHZ states are identified from the KLM state by leveraging the electron-spin states of two NV$_{3}^{-}$ and NV$_{4}^{-}$ centers corresponding to Block$_{3} (i=3)$ and Block$_{4} (j=4)$, respectively. Subsequently, in the second step, the CNOT gates are utilized to seamlessly convert the remaining states into the GHZ states, where photons interact with the Block$_{5}$ and Block$_{6}$ corresponding to the NV$_{5}^{-}$ $(i=5)$ and NV$_{6}^{-}$ $(j=6)$, respectively. By skillfully performing these operations, the complete transformation from the three-logic-qubit KLM state into the desired GHZ states is completed.

The three-logic-qubit KLM state in the DFS can be described as

 

Fig. 4. The diagram for converting the three-logic-qubit KLM state to GHZ states.

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If both two electron-spin states are $|0\rangle _{3}|0\rangle _{4}$, the three-logic-qubit GHZ state $|\psi \rangle _{1}^{+}$ can be obtained with the efficiency $\frac {1}{2}\xi ^{16}$. Nevertheless, there are other potential outcomes apart from the mentioned state, the associated operations are shown in Table 2. As shown in Fig. 4(b), in the second step, we employ three CNOT gates in conjunction with appropriate quantum operations HWPs$^{22.5^{\circ }}$ added on paths $l_{1}$, $l_{3}$, and $l_{5}$. The control photons are assigned to the first, third, and fifth paths, while the target ones are allocated to the second, fourth, and sixth ones. After six photons pass through the quantum circuits of the first step again, i.e., using two heralded parity-check gates in Fig. 4(a), that is, the photons from paths $l_{2}$ and $l_{4}$ interacting with the Block$_{5}$ corresponding to the NV$_{5}^{-}$, then the photons from paths $l_{4}$ and $l_{6}$ interacting with the Block$_{6}$ $(j=6)$ corresponding to the NV$_{6}^{-}$, the remaining items both $|\bar {0}\bar {1}\bar {1}\rangle$ and $|\bar {0}\bar {0}\bar {1}\rangle$ can be converted to four kinds of three-logic-qubit GHZ states $|\psi \rangle ^{+}_{1}, |\psi \rangle _{2}^{+}, |\psi \rangle _{3}^{+}$, and $|\psi \rangle _{4}^{+}$ according to the measuring results of the two electron-spin states $|0\rangle _{5}|0\rangle _{6}$, $|1\rangle _{5}|1\rangle _{6}$, $|0\rangle _{5}|1\rangle _{6}$, and $|1\rangle _{5}|0\rangle _{6}$, respectively. For example,

Table 2. The relationship between the measuring results, the finally converted GHZ state and the required operations.

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As a result, the efficiencies of all the GHZ states $|\psi \rangle ^{+}_{1}, |\psi \rangle _{2}^{+}, |\psi \rangle _{3}^{+}$, and $|\psi \rangle _{4}^{+}$ are equal to ${\xi ^{44}}/{8}$. We acknowledge that the enhancement strategies have the potential to fully maximize the utilization of the remaining states.

4. Discussion and conclusion

For the input photon $|R\rangle$ with a frequency close to resonance, $\omega \simeq \omega _{c}$ (i.e., $\Delta _{c} \simeq 0$) condition without detunings, the reflection coefficient $r_{s}(\omega )$ is simplified to $r_{s}=(C-i\Delta _{s}-1)/(C+i\Delta _{s}+1)$. We can get $r_{0} \simeq 1$ in case of $C \gg$ max $(\Delta _{s}, 1)$ met, but get $r_{1} \simeq -1$ in case of $\Delta _{s}\gg$ max($C, 1$) or $C \simeq 0$ met. In the context of the NV$^{-}$-cavity system, when the input $R$-polarized photon with a frequency close to resonant $\omega _{0}\simeq \omega _{c}$, it is theoretically possible to achieve substantial scattering into the output port. Specifically, if the NV$^{-}$ center is in state $|0\rangle (|1\rangle )$, corresponding to a state-based phase shift of 0 ($\pi$), the reflection coefficients are approximately $r_{0} \simeq 1$ ($r_{1} \simeq -1$). In other words, the state-based $r_{s}(\omega )$ of the NV$^{-}$-cavity system is affected by the spin state of the NV$^{-}$-center. Nonetheless, the finite cooperativity $C$ and detunings invariably results in a deviation from ideal single-photon scattering and affect the fidelities of our protocols [76,77]. Fortunately, the obstacles presented by photon dissipation loss and frequency detuning are converted into heralded failure events through the response of detectors from Blocks. Consequently, two CNOT gates and parity-check gates have the potential to achieve near-unit fidelities in principle, so do the fidelities of complete conversion from two-logic-qubit (three-logic-qubit) KLM state to two-logic-qubit Bell (three-logic-qubit GHZ) state.

The efficiency, denoted as $\eta =\eta _{out}/\eta _{in}$, where $\eta _{in}$ ($\eta _{out}$) represents the number of the input (output) photons. In the process of converting two-logic-qubit (three-logic-qubit) KLM state to Bell state (GHZ state), the presence of finite cooperativity denoted as $C$ and effective detunings invariably leads to deviations from ideal single-photon scattering, influencing the efficiencies of our schemes. The Blocks of the two CNOT gates and the Blocks$_{p}(p=1,2,\ldots 6)$ of conversion process exert a significant influence on the entire procedure and are instrumental in selecting efficacious overcomes. Figure 5(a) illustrates that the undetected efficiency $\eta _{\mathrm {Block}}$ and the detected efficiency $\eta _{\mathrm {D}}$ of the block corresponding to the detector response or not, are $99.58{\%}$ and $0.42{\%}$ with resonant frequency $\omega = \omega _{c}=\omega _{s}$, respectively, in a typical operating condition characterized by $C=480$. Notably, any errors stemming from the operation of detector can be safely disregarded, attesting to the robustness of our protocols. Similar to the earlier discussion on efficiency in Sec. 3.1, it is worth noting that the efficiencies of both the CNOT$_{1}$ and the CNOT$_{2}$ gates under the same condition are mathematically expressed as $\eta _{\mathrm {CNOT}}=\xi ^{4}$. The correlation between the efficiency $\eta _{\mathrm {CNOT}}$ and the parameter $C$ shown in Fig. 5(b). The efficiency of the heralded parity-check gate of the first step is $\eta _{\mathrm {P}}=\xi ^{8}$ shown in Fig. 5(c). Obviously, as $C$ increases, the efficiencies $\eta _{\mathrm {CNOT}}$ and $\eta _{\mathrm {P}}$ tends to approach to unit, signifying remarkable performance and reliability of two CNOT gates and the parity-check gate. Finally, the efficiency $\eta _{1}$ of complete conversion from two-logic-qubit KLM state to two-logic-qubit Bell state and the efficiency $\eta _{2}$ of that from three-logic-qubit KLM state to three-logic-qubit GHZ state are $\eta _{1}=\frac {2}{3}\xi ^{8}+\frac {1}{3}\xi ^{24}$ and $\eta _{2}=\frac {1}{2}\xi ^{16}+\frac {1}{2}\xi ^{44}$ vs the coefficient $C$, respectively, as shown in Fig. 5(d), For example, the $\eta _{1}$ and $\eta _{2}$ increases from 94.03${\%}$ and 87.12${\%}$ to 98.29${\%}$ and 96.20${\%}$ as the parameter $C$ increases from $C=85$ to $C=308$ [78]. It is clear that our approaches demonstrate effectiveness not only in scenarios with strong coupling rates but also in those with weak coupling rates, particularly when the resonant frequencies are aligned, denoted as $\omega = \omega _{c}=\omega _{s}$ (i.e., $\Delta _{c} =\Delta _{s} =0$).

 

Fig. 5. (a) The undetected efficiency $\eta _{\mathrm {Block}}$ and the detected efficiency $\eta _{\mathrm {D}}$ of the block corresponding to the detector response or not; (b) The effeciency $\eta _{\mathrm {CNOT}}$ of the CNOT$_{1}$; (c) The efficiency $\eta _{\mathrm {P}}$ of the heralded parity-check gate of the first step; (d) The efficiency $\eta _{1}$ of complete conversion from two-logic-qubit KLM state to Bell state and the efficiency $\eta _{2}$ of that from three-logic-qubit KLM state to GHZ state vs the cooperativity $C$ with $\gamma = 0.01\kappa$ and $\Delta _{c} = \Delta _{s}=0$.

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Next, we will delve into a comprehensive discussion on the experimental implementations of converting from the KLM state to Bell state (or GHZ state) in DFS. The entire conversion processes, encompassing both KLM to Bell state and GHZ state, require only a few simple optical elements, such as HWPs, CPBSs, and WFCs, and so on. This streamlined setup significantly reduces experimental complexity. Notably, the heralded Block, as a significant unit of our protocols, can facilitate immediately reflection of errors during the processes, herald experiment repeatability, and contribute to ensuring overall experimental success. Moreover, the interaction among photons is achieved through the photon-electron-spin interaction in the NV$^{-}$ center. This interaction is substantially enhanced when the NV$^{-}$ center is coupled to an optical cavity, a fiber-based microcavity, or a microring resonator [78–80]. Further, our proposed protocols demonstrate remarkable resilience against spectral diffusion and charge fluctuations, owing to the narrow linewidth of the state. Numerous techniques have been explored to mitigate and eliminate the influence of spectral diffusion in Ref. [81].

In summary, we have put forth two procedures, i.e., one successfully converts two-logic-qubit KLM state to Bell states, and another converts three-logic-qubit KLM state to GHZ states, based on robust-fidelity quantum gates containing heralded CNOT gates and parity-check gates. Photon loss in DFS may effectively protects the quantum states from decoherence and have minimal impact on our proposed protocols. The KLM states are grouped into different categories with the two-logic-qubit heralded parity-check gates according to the quantum state of the NV$^{-}_{k}(k=1,2,\ldots 6)$ centers shown in Tables 1–2, and the required Bell and GHZ states can be identified. Compared with prior auxiliary-based approaches [58], the fidelities of two schemes are near unit in principle and their efficiencies also are high with fewer nonlinear interactions and linear optical elements, as the error-detection efficacy containing potential failures and incomplete interaction can be effectively heralded through the response of detectors. Moreover, it is possible to circumvent the limitations associated with strong coupling, thereby minimizing the requirements and constraints of the experimental conditions. Further, The conversions of KLM entangled states to entangled Bell and GHZ states can not only realize the information transmission between quantum states, but also play a significant role for QITs, including increasing scalability and computational power, and reducing error rates.