1. Introduction
Quantum entanglement provides a new way for more secure and efficient transmission of information. Without physically sending the particle itself, Bennett et al [1] originally put forward a famous protocol called quantum teleportation (QT) to transmit an unknown qubit state between a sender and a remote receiver by utilizing entanglement and classical communication. Karlsson et al [2] proposed controlled quantum teleportation (CQT), a variant of QT, where the transmission is supervised by one or more added controllers. CQT has attracted great attention [3, 4] due to its important applications in various contexts [5, 6]. In 1999, Hillery et al [7] first introduced quantum secret sharing (QSS) where the secret information is divided into many portions and distributed to different sharers. Only when a certain number of sharers cooperate can the secret be restored. If the shared secret is a quantum state, QSS is named as quantum information splitting (QIS) [8, 9]. It is worth noting that QIS can be regarded as the extension of CQT as the number of receivers is extended from one to multiple and part of the receivers play the role of controller.
Remote transmission of operations is also crucial as its wide applications in remote control and real-time monitoring systems. For example, it can be used for remote medical diagnosis, smart home devices and industrial automation systems, etc. In 2001, Huelga et al [10] first described quantum operation teleportation (QOT), where a complete unknown quantum operation can be implemented on an unknown state located at a distant position. QOT can be called ‘remote implementation of operations' (RIO). When the operation is completely unknown, its implementation is usually accomplished with the aid of bidirectional QT. In detail, the receiver teleports his state to the sender who then performs the local unitary operation on this state and returns the operated state to the receiver. Huelga et al [11] proposed RIO of partially unknown single-qubit operations belong to two restricted sets consisting of diagonal operations and anti-diagonal operations. Wang [12] provided the general constraint set of multi-qubit partially unknown quantum operations, which have only one non-vanishing element in every row and column. It is found that the implementation of partially unknown operations consumes less resources and can meet some practical needs such as encryption and privacy protection of transmitted content.
Inspired by CQT, more and more researchers have focused on controlled remote implementation of partially unknown operations [13–18]. For example, Fan et al [14] generalized the controlled remote implementation of operation (CRIO) to the multi-party control system based on shared Greenberger–Horne–Zeilinger state. Combining the ideas of RIO and bidirectional CQT, Liu et al [17] investigated the bidirectional CRIO of partially unknown operations via multi-particle entangled states. Relative to two-dimensional systems, the availability of the high-dimensional Hilbert space offers great advantages in terms of information capacity, flexibility for quantum information processing and noise resilience [19]. Furthermore, it is difficult for eavesdroppers to fully recover the original information due to the complexity of high-dimensional unitary operations. Therefore, the study of RIO or CRIO in high-dimensional systems is of great significance. Zhan et al [18] proposed a CRIO protocol for partially unknown operations belonging to some restricted sets in high-dimensional systems. Situ et al [20] used restricted sets of multi-qubit operations to implement on distant qubits and extended them to d-dimensional systems. The above RIO and CRIO schemes concerned on a symmetric manner that all the agents have the same status. As commonly demanded in practice, an important problem would be better resolved by several committees of different ranks. Hierarchy plays an important role in different areas and organizations, such as hospitals, companies, schools and government agencies. Recently, a hierarchical version has been described in the field of quantum information splitting [21, 22] which mainly consider the situation that the agents are divided into two levels: the high level and the low level. In 2019, Bich et al [23] first investigated three-layer CQT and designed a hierarchical scheme for arbitrary single-qubit states such that teleportation of the state of the lowest importance is controlled only by the overall supervisor, teleportation of the state of the middle importance is controlled by the overall supervisor and one agent, while teleportation of the state of the highest importance requires permission from the supervisor and two agents. In respect to RIO, Peng et al [24] put forward two schemes for quadripartite hierarchical sharing a single-qubit operation on any sharer's qubit and there is a hierarchy among the powers of receivers to reconstruct the conceivable state. To our knowledge, there are few studies on multi-layer RIO. As a novel functional network supporting quantum communication, quantum network offers higher security and stronger information processing ability and has been actively promoted [25–29]. For example, Jiang et al [29] explored the quantum communication capacity within general quantum entangled networks. Due to the presence of numerous nodes and complex hierarchical structures in the network, this triggers the study of asymmetric RIO for several distant agents with unequal authorities.
In real circumstances, quantum systems inevitably suffer from the interference of external noise. There have been many studies on the influence of different noises on QT or CQT [30–32]. However, the effect of noise on RIO or CRIO has received relatively less attention. Liu et al [33] discussed tripartite cyclic-controlled QOT in four types of noisy environments and used fidelity to evaluate how much information has been lost during the process. Weak measurement (WM) and measurement reversal (MR) are one of the effective strategies to resist noise [34–36]. The external interference can be balanced by WM, which makes the system less interfered with the quantum correlation while maintaining its reversibility, and the entanglement is restored by MR after passing through the noisy channel. Singh et al [35] utilized WM and MR as potential solutions to combat the noise-induced decoherence and improve the fidelity of the existing cyclic QT in amplitude-damping (AD) noise. In addition, environment-assisted measurement (EAM) and MR are powerful anti-noise techniques [37, 38], where EAM detects the noisy environment coupled to the system and MR compensates for the entanglement. Questions may be naturally raised: is it possible to devise universal multi-layer CRIO schemes for arbitrary single-qudit partially unknown operation and explore anti-noise strategies to effectively reduce the influence of quantum noise?
In this paper, we investigate implementing single-qudit partially unknown operations among multiple layers and take effective methods to suppress noise. The remainder of this paper is organized as follows. In section 2 , we propose a hierarchical controlled remote implementation of operations (HCRIO) scheme to perform partially unknown quantum operations on arbitrary single-qubit states with multiple layers. In section 3 , we extend the above scheme from a two-dimensional system to high-dimensional systems. Compared with the existing hierarchical quantum operation sharing schemes, we not only investigate the case of arbitrary multiple layers but also solve the problem of implementing quantum operations in arbitrary high-dimensional systems. In section 4 , we provide two concrete examples with two layers, one is a HCRIO scheme of single-qubit operations and the other is a HCRIO scheme of single-qutrit operations. In section 5 , we select AD noise as the model, consider its influence and make use of WM and MR to increase fidelity. The last section contains the discussions and conclusions.
2. The HCRIO scheme of single-qubit operations with n layers
There are 2n + 1 participants including the overall controller Charlie, the senders Alice1, Alice2, ⋯, Alicen and the corresponding receivers Bob1, Bob2, ⋯, Bobn, n ≥ 2. Alice1, Alice2, ⋯, Alicen locate at different levels and their levels decrease in sequence. The receiver Bobj has an arbitrary unknown single-qubit state
$\begin{eqnarray}| {\xi }_{j}{\rangle }_{{O}_{j}}={\left({\alpha }_{j0}| 0\rangle +{\alpha }_{j1}| 1\rangle \right)}_{{O}_{j}},\end{eqnarray}$
where complex numbers αj0, αj1 satisfy the normalization condition ∣αj0∣2 + ∣αj1∣2 = 1, j = 1, 2, ⋯ , n.The multi-layer CRIO model is shown in figure 1. The j-th layer sender Alicej wants to remotely manipulate the unitary operation in the restricted set Uj = {Uj0, Uj1} on Bobj's state ∣ξj⟩, where
$\begin{eqnarray}{U}_{j0}=\left(\begin{array}{cc}{u}_{j0} & 0\\ 0 & {u}_{j1}\end{array}\right),\quad {U}_{j1}=\left(\begin{array}{cc}0 & {u}_{j0}\\ {u}_{j1} & 0\end{array}\right).\end{eqnarray}$
They are called partially unknown as their structure is known while the values of their matrix elements are unknown. For concreteness, suppose that the operators have the form $\begin{eqnarray}{U}_{j0}({\theta }_{j})=\left(\begin{array}{cc}{{\rm{e}}}^{{\rm{i}}{\theta }_{j}} & 0\\ 0 & {{\rm{e}}}^{-{\rm{i}}{\theta }_{j}}\end{array}\right),\quad {U}_{j1}({\theta }_{j})=\left(\begin{array}{cc}0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}}\\ {{\rm{e}}}^{-{\rm{i}}{\theta }_{j}} & 0\end{array}\right).\end{eqnarray}$
To sum up, ${U}_{{jm}}({\theta }_{j})| k\rangle ={{\rm{e}}}^{{\left(-1\right)}^{k+m}{\rm{i}}{\theta }_{j}}| k\oplus m\rangle $, where m, k = 0, 1, ⊕ means plus mod 2. Alice1's operation can be performed on the Bob1's state ∣ξ1⟩ with the help of the supervisor Charlie. While the remote operation between Alicej and Bobj is achieved through getting the assistance from both the supervisor Charlie and the higher-layer senders Alice1, Alice2, ⋯, Alicej−1, j = 2, 3, ⋯ , n.
Figure 1. The hierarchical CRIO model with n layers. |
Before formally describing the scheme, let's explain how the elaborately selected quantum channel is generated. The generation process of the entangled channel is illustrated in figure 2.
Figure 2. The generalization circuit of the quantum channel. |
C1 Introduce a 3n-qubit initial state $| {{ \mathcal Q }}_{0}\rangle =| 000000\,\cdots 000{\rangle }_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}$.
C2 Let the qubits $(C,{A}_{1}^{{\prime} },{A}_{2}^{{\prime} },\cdots {A}_{n-1}^{{\prime} })$ pass through the Hadamard gate H and get
$\begin{eqnarray}| {{ \mathcal Q }}_{1}\rangle =\displaystyle \frac{1}{{2}^{\tfrac{n}{2}}}\displaystyle \sum _{{k}_{1},{k}_{2},\cdots ,\,{k}_{n}=0}^{1}| {k}_{1}00{k}_{2}00\cdots {k}_{n}00{\rangle }_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}.\end{eqnarray}$
C3 Carry out controlled-NOT (CNOT) gates ${C}_{{{CA}}_{1}}{C}_{{{CB}}_{1}}{\prod }_{j=1}^{n-1}{C}_{{A}_{j}^{{\prime} }{A}_{j+1}}{C}_{{A}_{j}^{{\prime} }{B}_{j+1}}$ on $| {{ \mathcal Q }}_{1}\rangle $ and obtain
$\begin{eqnarray}\begin{array}{l}| {{ \mathcal Q }}_{2}\rangle =\displaystyle \frac{1}{{2}^{\tfrac{n}{2}}}\\ \,\displaystyle \sum _{{k}_{1},{k}_{2},\cdots ,\,{k}_{n}=0}^{1}| {k}_{1}{k}_{1}{k}_{1}{k}_{2}{k}_{2}{k}_{2}\,\cdots \,\\ \,{k}_{n}{k}_{n}{k}_{n}{\rangle }_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}.\end{array}\end{eqnarray}$
Here, CNOT operation CAB acts on a control qubit A and a target qubit B in the manner: CAB∣pq⟩ = ∣p, p ⊕ q⟩AB.
C4 Perform CNOT operations ${\prod }_{j=1}^{n-1}{C}_{{A}_{j}^{{\prime} }{A}_{j}}{C}_{{A}_{j}^{{\prime} }{B}_{j}}$ and acquire a 3n-qubit entangled state
$\begin{eqnarray}\begin{array}{l}| { \mathcal Q }\rangle =\displaystyle \frac{1}{{2}^{\tfrac{n}{2}}}\\ \,\displaystyle \sum _{{k}_{1},{k}_{2},\cdots ,\,{k}_{n}=0}^{1}| {k}_{1}{\left[{k}_{1}\oplus {k}_{2}\right]}^{\otimes 2}{k}_{2}{\left[{k}_{2}\oplus {k}_{3}\right]}^{\otimes 2}\,\cdots \\ {\,k}_{n}^{\otimes 3}{\rangle }_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}.\end{array}\end{eqnarray}$
Distribute qubit C to Charlie, qubits $({A}_{j},{A}_{j}^{{\prime} })$ to Alicej, j = 1, 2, ⋯ , n − 1, qubit An to Alicen, qubit Bk to Bobk, k = 1, 2, ⋯ , n. Thus, the entanglement is established among the participants.
Now, we introduce the process of our HCRIO protocol in detail. The quantum circuit of the n-layer scheme is shown in figure 3.
Figure 3. The quantum circuit of our HCRIO scheme of single-qubit operations with n layers. Here, PMX and PMZ denote the projective measurements under the X and Z bases. The corresponding measurement results are placed in the dotted box. U1m(θ1), ⋯ , Unm(θn) are the partially unknown quantum operations, m=0,1. ${R}_{{B}_{1}},\cdots ,\,{R}_{{B}_{n}}$ respectively represent Bob1's, ⋯, Bobn's recovery operators based on the measurement results. |
Step 1 Alice1 performs the unitary operation in the set U1 on Bob1's qubit ∣ξ1⟩.
Step 1.1 The overall controller Charlie approves and executes a single-qubit projective measurement (PM) on his qubit C under the X basis
$\begin{eqnarray}| \widetilde{0}\rangle =\displaystyle \frac{| 0\rangle +| 1\rangle }{\sqrt{2}},\quad | \widetilde{1}\rangle =\displaystyle \frac{| 0\rangle -| 1\rangle }{\sqrt{2}}.\end{eqnarray}$
If the measurement result is $| \widetilde{c}\rangle $, he announces the classical bit c.The remaining particles collapse into
$\begin{eqnarray}\displaystyle \sum _{t,p=0}^{1}{\left(-1\right)}^{{pc}}{\alpha }_{1t}| {ppt}{\rangle }_{{A}_{1}{B}_{1}{O}_{1}}\otimes | {h}_{c}{\rangle }_{{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}},\end{eqnarray}$
where $| {h}_{c}\rangle ={\sum }_{{k}_{2},\cdots ,\,{k}_{n}=0}^{1}{\left(-1\right)}^{{k}_{2}c}| {k}_{2}{\left[{k}_{2}\oplus {k}_{3}\right]}^{\otimes 2}\cdots {k}_{n}^{\otimes 3}\rangle $ will serve as the quantum channel for Step 2.Step 1.2 Bob1 carries out operation ${C}_{{B}_{1}{O}_{1}}$ and the state becomes
$\begin{eqnarray}\displaystyle \sum _{t,p=0}^{1}{\left(-1\right)}^{{pc}}{\alpha }_{1t}| {pp}[p\oplus t]{\rangle }_{{A}_{1}{B}_{1}{O}_{1}}.\end{eqnarray}$
Then, he measures his qubit O1 under the Z basis {∣0⟩, ∣1⟩}. He transmits the cbit b1 to Alice1 if his measurement result is ∣b1⟩. The rest particles collapse into $\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{{tc}}{\alpha }_{1t}| {\left[t\oplus {b}_{1}\right]}^{\otimes 2}{\rangle }_{{A}_{1}{B}_{1}}\end{eqnarray}$
up to a global phase ${\left(-1\right)}^{{b}_{1}c}$. Notice that, we sometimes ignore the global phase factors of quantum states in the rest of this paper.Step 1.3 Alice1 first carries out ${X}^{{b}_{1}}$ and performs U1m(θ1) (m = 0, 1) on her particle A1. The transformed state is
$\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{{tc}}{\alpha }_{1t}{{\rm{e}}}^{{\left(-1\right)}^{m+t}{\rm{i}}{\theta }_{1}}| [t\oplus m][t\oplus {b}_{1}]{\rangle }_{{A}_{1}{B}_{1}}.\end{eqnarray}$
Then she executes a PM on her qubit A1 under the X basis and broadcasts a cbit a1 if her measurement outcome is $| \widetilde{{a}_{1}}\rangle $. The rest qubit B1 becomes $\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{t(c+{a}_{1})}{\alpha }_{1t}{{\rm{e}}}^{{\left(-1\right)}^{m+t}{\rm{i}}{\theta }_{1}}| t\oplus {b}_{1}{\rangle }_{{B}_{1}}.\end{eqnarray}$
Of course, the sender needs to transmit another classical message m to tell the receiver her implemented operation is diagonal or anti-diagonal unless they have reached a consensus in advance.Step 1.4 Bob1 applies the recovery operation
$\begin{eqnarray}{R}_{{B}_{1}}={X}^{{b}_{1}\oplus m}{Z}^{c\oplus {a}_{1}}\end{eqnarray}$
and reconstructs U1m∣ξ1⟩, where X, Z are the Pauli operations and X0, Z0 denote the identity operation.Step j Alicej performs the unitary operation in the set Uj on Bobj's qubit ∣ξj⟩, j = 2, ⋯ , n − 1.
At this time, the entangled system for qubits ${A}_{j-1}^{{\prime} },{A}_{j},{B}_{j},\cdots ,\,{A}_{n-1}^{{\prime} },{A}_{n},{B}_{n}$ is
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{{k}_{j},\cdots ,\,{k}_{n}=0}^{1}{\left(-1\right)}^{{k}_{j}[{\left(-1\right)}^{j}c+{\left(-1\right)}^{j-1}{a}_{1}^{{\prime} }+\cdots +{\left(-1\right)}^{2}{a}_{j-2}^{\prime} ]}| {k}_{j}\\ \quad \times {\left[{k}_{j}\oplus {k}_{j+1}\right]}^{\otimes 2}\cdots {k}_{n}^{\otimes 3}\rangle .\end{array}\end{eqnarray}$
Step j.1 Alicej−1 measures her qubit ${A}_{j-1}^{{\prime} }$ under the X basis and publishes her measurement outcome $| \widetilde{{a}_{j-1}^{{\prime} }}\rangle $ via a cbit ${a}_{j-1}^{{\prime} }$, the collapsed state is
$\begin{eqnarray}\displaystyle \sum _{t,q=0}^{1}{\left(-1\right)}^{{{qs}}_{j}}{\alpha }_{{jt}}| {qqt}{\rangle }_{{A}_{j}{B}_{j}{O}_{j}}\otimes | {h}_{{{ca}}_{1}^{{\prime} }\cdots {a}_{j-1}^{{\prime} }}{\rangle }_{{A}_{j}^{{\prime} }{A}_{j+1}{B}_{j+1}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}.\end{eqnarray}$
where $| {h}_{{{ca}}_{1}^{{\prime} }\cdots {a}_{j-1}^{{\prime} }}\rangle ={\sum }_{{k}_{j+1},\cdots ,\,{k}_{n}=0}^{1}{\left(-1\right)}^{{k}_{j+1}{s}_{j}}| {k}_{j+1}{\left[{k}_{j+1}\oplus {k}_{j+2}\right]}^{\otimes 2}\cdots {k}_{n}^{\otimes 3}\rangle ,{s}_{j}={\left(-1\right)}^{j}[c\oplus {\sum }_{u=1}^{j-1}{\left(-1\right)}^{u}{a}_{u}^{{\prime} }].$Step j.2 Bobj first performs operation ${C}_{{B}_{j}{O}_{j}}$ and then measures his qubit Oj under the Z basis. If the measurement result is ∣bj⟩, he tells Alicej the classical message bj. The state collapses into
$\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{{{ts}}_{j}}{\alpha }_{{jt}}| {\left[t\oplus {b}_{j}\right]}^{\otimes 2}{\rangle }_{{A}_{j}{B}_{j}}.\end{eqnarray}$
Step j.3 Alicej executes ${X}_{{A}_{j}}^{{b}_{j}}$ operation depending on Bobj's measurement outcome and operation Ujm(θj) on her particle Aj. The state is transformed into
$\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{{{ts}}_{j}}{\alpha }_{{jt}}{{\rm{e}}}^{{\left(-1\right)}^{m+t}{\rm{i}}{\theta }_{j}}| [t\oplus m][t\oplus {b}_{j}]{\rangle }_{{A}_{j}{B}_{j}}.\end{eqnarray}$
Then, she measures her qubit under the X basis and announces the measurement result $| \widetilde{{a}_{j}}\rangle $ in the form of cbit aj. The state collapses into $\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{t({s}_{j}+{a}_{j})}{\alpha }_{{jt}}{{\rm{e}}}^{{\left(-1\right)}^{m+t}{\rm{i}}{\theta }_{j}}| t\oplus {b}_{j}{\rangle }_{{B}_{j}}.\end{eqnarray}$
Step j.4 Bobj implements the recovery operation
$\begin{eqnarray}{R}_{{B}_{j}}={X}^{{b}_{j}\oplus m}{Z}^{{s}_{j}\oplus {a}_{j}},\end{eqnarray}$
and reconstructs Ujm∣ξj⟩.Step n Alicen performs the unitary operation in the set Un on Bobn's qubit ∣ξn⟩.
The entangled channel can be written as
$\begin{eqnarray}\displaystyle \sum _{{k}_{n}=0}^{1}{\left(-1\right)}^{{k}_{n}{s}_{n-1}}| {k}_{n}^{\otimes 3}{\rangle }_{{A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}.\end{eqnarray}$
Step n.1 Alicen−1 measures her qubit ${A}_{n-1}^{{\prime} }$ under the X basis and publishes the measurement result $| \widetilde{{a}_{n-1}^{{\prime} }}\rangle $ in the form of cbit ${a}_{n-1}^{{\prime} }$. The remaining state is
$\begin{eqnarray}\displaystyle \sum _{t,{k}_{n}=0}^{1}{\left(-1\right)}^{{k}_{n}{s}_{n}}{\alpha }_{{nt}}| {k}_{n}{k}_{n}t{\rangle }_{{A}_{n}{B}_{n}{O}_{n}},\end{eqnarray}$
where ${s}_{n}={\left(-1\right)}^{n}[c\oplus {\sum }_{u=1}^{n-1}{\left(-1\right)}^{u}{a}_{u}^{{\prime} }]$.Step n.2 Bobn first operates ${C}_{{B}_{n}{O}_{n}}$ and measures his qubit On under the Z basis. He tells Alicen a cbit bn if his measurement result is ∣bn⟩. The rest state collapses into
$\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{{{ts}}_{n}}{\alpha }_{{nt}}| {\left[t\oplus {b}_{n}\right]}^{\otimes 2}{\rangle }_{{A}_{n}{B}_{n}}.\end{eqnarray}$
Step n.3 Alicen first executes operations ${X}^{{b}_{n}}$ and Unm(θn) on her qubit An. The transformed state is
$\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{{{ts}}_{n}}{\alpha }_{{nt}}{{\rm{e}}}^{{\left(-1\right)}^{m+t}{\rm{i}}{\theta }_{n}}| [t\oplus m][t\oplus {b}_{n}]{\rangle }_{{A}_{n}{B}_{n}}.\end{eqnarray}$
Then she executes a PM on her qubit An under the X basis and broadcasts a cbit an if the measurement outcome is $| \widetilde{{a}_{n}}\rangle $. The rest qubit Bn becomes $\begin{eqnarray}\displaystyle \sum _{t=0}^{1}{\left(-1\right)}^{t({s}_{n}+{a}_{n})}{\alpha }_{{nt}}{{\rm{e}}}^{{\left(-1\right)}^{m+t}{\rm{i}}{\theta }_{n}}| t\oplus {b}_{n}{\rangle }_{{B}_{n}}.\end{eqnarray}$
Step n.4 Bobn applies the recovery operation
$\begin{eqnarray}{R}_{{B}_{n}}={X}^{{b}_{n}\oplus m}{Z}^{{s}_{n}\oplus {a}_{n}},\end{eqnarray}$
and reconstructs Unm∣ξn⟩.3. Generation to single-qudit operations
In this section, we extend the above scheme to high-dimensional systems. Denote ${\omega }_{d}={{\rm{e}}}^{\tfrac{2\pi {\rm{i}}}{d}}$ as the primitive d-th root of unity and the shorthand ${\left[z\right]}_{d}$ as $z\,\mathrm{mod}\,d$.
The single-qudit partially unknown operations to be remotely performed belong to the restricted set [20]
$\begin{eqnarray}\{U(f,\theta )=\displaystyle \sum _{x=0}^{d-1}{{\rm{e}}}^{{\rm{i}}\theta (f(x))}| f(x)\rangle \langle x| \},\end{eqnarray}$
where f is a permutation on the set {0, 1, ⋯ , d − 1}. It characterizes the kind of operators which only has one nonzero element in every row or every column.The sender Alicej wants to manipulate operation U(fj, θj) on Bobj's single-qudit state
$\begin{eqnarray}| {\psi }_{j}{\rangle }_{{O}_{j}}=\displaystyle \sum _{t=0}^{d-1}{\alpha }_{{jt}}| t{\rangle }_{{O}_{j}},\end{eqnarray}$
where complex coefficients satisfy the normalization condition ${\sum }_{t=0}^{d-1}| {\alpha }_{{jt}}{| }^{2}=1$, j = 1, 2, ⋯ , n. Alicej knows U(fj, θj) but Bobj only knows which restricted set the operation belongs to.Similar to the generation of the entangled state in equation (6 ), construct a quantum channel2 .
$\begin{eqnarray}\begin{array}{l}| {\rm{\Omega }}\rangle =\displaystyle \frac{1}{{d}^{\tfrac{n}{2}}}\displaystyle \sum _{{k}_{1},{k}_{2},\cdots ,\,{k}_{n}=0}^{d-1}| {k}_{1}{\left[{k}_{1}+{k}_{2}\right]}_{d}^{\otimes 2}{k}_{2}\\ \qquad \times {\left[{k}_{2}+{k}_{3}\right]}_{d}^{\otimes 2}\cdots {k}_{n}^{\otimes 3}{\rangle }_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}.\end{array}\end{eqnarray}$
The allocation of entangled particles is the same as section The d-dimensional HCRIO protocol is described as follows.
Step 1 Alice1 performs the operation U(f1, θ1) on Bob1's single-qudit state ∣ψ1⟩.
Step 1.1 Charlie approves executing a PM on his qudit C under the generalized X basis
$\begin{eqnarray}| \widetilde{l}\rangle =\displaystyle \frac{1}{\sqrt{d}}\displaystyle \sum _{k=0}^{d-1}{\omega }_{d}^{{kl}}| k\rangle ,\quad l=0,1,\cdots ,\,d-1.\end{eqnarray}$
If the measurement result is $| \widetilde{c}\rangle $, he announces a classical dit c. The state of the remaining particles turns into $\begin{eqnarray}\displaystyle \sum _{t,p=0}^{d-1}{\omega }_{d}^{-{pc}}{\alpha }_{1t}| {ppt}{\rangle }_{{A}_{1}{B}_{1}{O}_{1}}\otimes | {h}_{c}{\rangle }_{{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}},\end{eqnarray}$
where $| {h}_{c}{\rangle }_{{A}_{1}^{{\prime} }{A}_{2}{B}_{2}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}={\sum }_{{k}_{2},\cdots ,\,{k}_{n}=0}^{d-1}{\omega }_{d}^{{k}_{2}c}| {k}_{2}{\left[{k}_{2}+{k}_{3}\right]}_{d}^{\otimes 2}\cdots {k}_{n}^{\otimes 3}\rangle $.Step 1.2 Bob1 operates generalized XOR-gate ${\mathrm{GXOR}}_{{B}_{1}{O}_{1}}$ and measures his qudit O1 under the generalized Z basis {∣0⟩, ∣1⟩, ⋯ , ∣d − 1⟩}. He transmits the classical message b1 to Alice1 if his measurement result is ∣b1⟩. The rest of the state collapses into
$\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{-{tc}}{\alpha }_{1t}| {\left[t+{b}_{1}\right]}_{d}^{\otimes 2}{\rangle }_{{A}_{1}{B}_{1}}.\end{eqnarray}$
Here, ${\mathrm{GXOR}}_{{AB}}| {pq}\rangle =| p,{\left[p-q\right]}_{d}{\rangle }_{{AB}}$, which is CNOT gate in the two-dimensional system.Step 1.3 Alice1 executes operation ${X}^{-{b}_{1}}$ on her qudit A1, where ${X}^{-1}={\sum }_{j=0}^{d-1}| {\left[j-1\right]}_{d}\rangle \langle j| $ is the inverse operation of generalized X operation. Then, Alice1 carries out the operation U(f1, θ1) on her qudit A1. The transformed state is
$\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{-{tc}}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}({f}_{1}(t))}{\alpha }_{1t}| {f}_{1}(t){\left[t+{b}_{1}\right]}_{d}{\rangle }_{{A}_{1}{B}_{1}}.\end{eqnarray}$
She performs a PM on her qudit A1 under the generalized X basis and broadcasts the classical message a1 if her measurement result is $| \widetilde{{a}_{1}}\rangle $. The state turns into $\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{-[{tc}+{f}_{1}(t){a}_{1}]}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}({f}_{1}(t))}{\alpha }_{1t}| {\left[t+{b}_{1}\right]}_{d}{\rangle }_{{B}_{1}}.\end{eqnarray}$
Step 1.4 Bob1 executes the local unitary operation
$\begin{eqnarray}{R}_{{B}_{1}}={Z}^{{a}_{1}}V({f}_{1}){Z}^{c}{X}^{-{b}_{1}},\end{eqnarray}$
and reconstructs the conceivable state U(f1, θ1)∣ψ1⟩, where $\begin{eqnarray}V({f}_{j})=\displaystyle \sum _{x=0}^{d-1}| {f}_{j}(x)\rangle \langle x| ,\quad j=1,2,\cdots ,\,n.\end{eqnarray}$
Step j Alicej performs the operation U(fj, θj) on Bobj's single-qudit state ∣ψj⟩, j = 2, ⋯ , n − 1.
The entangled channel for qudits ${A}_{j-1}^{{\prime} },{A}_{j}\,,{B}_{j},\cdots ,\,{A}_{n-1}^{{\prime} },{A}_{n},{B}_{n}$ is
$\begin{eqnarray}\begin{array}{l}\displaystyle \sum _{{k}_{j},\cdots ,\,{k}_{n}=0}^{d-1}{\omega }_{d}^{{k}_{j}[{\left(-1\right)}^{j}c+{\left(-1\right)}^{j-1}{a}_{1}^{{\prime} }+\cdots +{\left(-1\right)}^{2}{a}_{j-2}^{\prime} )]}| {k}_{j}\\ \quad \times {\left[{k}_{j}+{k}_{j+1}\right]}_{d}^{\otimes 2}\cdots {k}_{n}^{\otimes 3}\rangle .\end{array}\end{eqnarray}$
Step j.1 Alicej−1 measures her qudit ${A}_{j-1}^{{\prime} }$ under the generalized X basis and publishes the classical message ${a}_{j-1}^{{\prime} }$ if her measurement result is $|\mathop{{a}_{j-1}^{{\rm{{\prime} }}}}\limits^{\sim }\rangle $. The collapsed state is
$\begin{eqnarray}\displaystyle \sum _{t,q=0}^{d-1}{\omega }_{d}^{{{qs}}_{j}}{\alpha }_{{jt}}| {qqt}{\rangle }_{{A}_{j}{B}_{j}{O}_{j}}\otimes | {h}_{{{ca}}_{1}^{{\prime} }\cdots {a}_{j-1}^{{\prime} }}{\rangle }_{{A}_{j}^{{\prime} }{A}_{j+1}{B}_{j+1}\cdots {A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}},\end{eqnarray}$
where $| {h}_{{{ca}}_{1}^{{\prime} }\cdots {a}_{j-1}^{{\prime} }}\rangle ={\sum }_{{k}_{j+1},\cdots ,\,{k}_{n}=0}^{d-1}{\omega }_{d}^{-{k}_{j+1}{s}_{j}}| {k}_{j+1}{\left[{k}_{j+1}+{k}_{j+2}\right]}_{d}^{\otimes 2}\cdots \,{k}_{n}^{\otimes 3}\rangle $, ${s}_{j}={\left(-1\right)}^{j}{\left[c+{\sum }_{u=1}^{j-1}{\left(-1\right)}^{u}{a}_{u}^{{\prime} }\right]}_{d}$.Step j.2 Bobj first operates ${\mathrm{GXOR}}_{{B}_{j}{O}_{j}}$ and then measures his qudit Oj under the generalized Z basis. He tells Alicej the classical message bj if his measurement result is ∣bj⟩. The remaining state collapses into
$\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{{{ts}}_{j}}{\alpha }_{{jt}}| {\left[t+{b}_{j}\right]}_{d}^{\otimes 2}{\rangle }_{{A}_{j}{B}_{j}}.\end{eqnarray}$
Step j.3 Alicej first carries out the operations ${X}^{-{b}_{j}}$ and U(fj, θj) on her qudit Aj. The transformed state is
$\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{{{ts}}_{j}}{{\rm{e}}}^{{\rm{i}}{\theta }_{j}({f}_{j}(t))}{\alpha }_{{jt}}| {f}_{j}(t){\left[t+{b}_{j}\right]}_{d}{\rangle }_{{A}_{j}{B}_{j}}.\end{eqnarray}$
Then, Alicej performs a PM on her qudit Aj under the generalized X basis and announces a classical dit aj if her measurement result is $| \widetilde{{a}_{j}}\rangle $. The remaining state collapses into $\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{{{ts}}_{j}-{f}_{j}(t){a}_{j}}{{\rm{e}}}^{{\rm{i}}{\theta }_{j}({f}_{j}(t))}{\alpha }_{{jt}}| {\left[t+{b}_{j}\right]}_{d}{\rangle }_{{B}_{j}}.\end{eqnarray}$
Step j.4 Bobj executes the local unitary operation
$\begin{eqnarray}{R}_{{B}_{j}}={Z}^{{a}_{j}}V({f}_{j}){Z}^{-{s}_{j}}{X}^{-{b}_{j}}\end{eqnarray}$
and reconstructs the conceivable state U(fj, θj)∣ψj⟩, where ${Z}^{-1}={\sum }_{j=0}^{d-1}{\omega }_{d}^{-j}| j\rangle \langle j| $ is the inverse operation of the generalized Z operation.Step n Alicen performs the operation U(fn, θn) on Bobn's single-qudit state ∣ψn⟩.
Now, the entangled system is
$\begin{eqnarray}\displaystyle \sum _{{k}_{n}=0}^{d-1}{\omega }_{d}^{-{k}_{n}{s}_{n-1}}| {k}_{n}^{\otimes 3}{\rangle }_{{A}_{n-1}^{{\prime} }{A}_{n}{B}_{n}}.\end{eqnarray}$
Step n.1 Alicen−1 measures her qudit ${A}_{n-1}^{{\prime} }$ under the generalized X basis and announces the classical message ${a}_{n-1}^{{\prime} }$ if her measurement result is $| \widetilde{{a}_{n-1}^{{\prime} }}\rangle $. The collapsed state is
$\begin{eqnarray}\displaystyle \sum _{t,{k}_{n}=0}^{d-1}{\omega }_{d}^{{k}_{n}{s}_{n}}{\alpha }_{{nt}}| {k}_{n}^{\otimes 2}t{\rangle }_{{A}_{n}{B}_{n}{O}_{n}},\end{eqnarray}$
where ${s}_{n}={\left(-1\right)}^{n}{\left[c+{\sum }_{u=1}^{n-1}{\left(-1\right)}^{u}{a}_{u}^{{\prime} }\right]}_{d}$.Step n.2 Bobn performs the ${\mathrm{GXOR}}_{{B}_{n}{O}_{n}}$ operation and measures his qudit On under the generalized Z basis. He transmits the classical message bn to Alicen if his measurement result is ∣bn⟩. The collapsed state becomes
$\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{{{ts}}_{n}}{\alpha }_{{nt}}| {\left[t+{b}_{n}\right]}_{d}^{\otimes 2}{\rangle }_{{A}_{n}{B}_{n}}.\end{eqnarray}$
Step n.3 Alicen executes ${X}^{-{b}_{n}}$ and the unitary operation U(fn, θn) on her qudit An. The transformed state is
$\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{{{ts}}_{n}}{{\rm{e}}}^{{\rm{i}}{\theta }_{n}({f}_{n}(t))}{\alpha }_{{nt}}| {f}_{n}(t){\left[t+{b}_{n}\right]}_{d}{\rangle }_{{A}_{n}{B}_{n}}.\end{eqnarray}$
Alicen performs a PM on her qudit An under the generalized X basis and announces the classical dit an if the measurement result is $| \widetilde{{a}_{n}}\rangle $. The remaining state collapses into $\begin{eqnarray}\displaystyle \sum _{t=0}^{d-1}{\omega }_{d}^{{{ts}}_{n}-{f}_{n}(t){a}_{n}}{{\rm{e}}}^{{\rm{i}}{\theta }_{n}({f}_{n}(t))}{\alpha }_{{nt}}| {\left[t+{b}_{n}\right]}_{d}{\rangle }_{{B}_{n}}.\end{eqnarray}$
Step n.4 Bobn executes the local unitary operation
$\begin{eqnarray}{R}_{{B}_{n}}={Z}^{{a}_{n}}V({f}_{n}){Z}^{-{s}_{n}}{X}^{-{b}_{n}},\end{eqnarray}$
and reconstructs the conceivable state U(fn, θn)∣ψn⟩.4. Examples of two-layer HCRIO
Two examples are offered to make it easier to understand our HCRIO schemes. One is the remote implementation of single-qubit operations, the other is of single-qutrit operations.
Assume that there are five legitimate participants: Charlie, Alice1, Alice2, Bob1 and Bob2. Charlie is the overall controller, Alice1 and Alice2 are the two senders who are respectively located at high level and low level, while Bob1 and Bob2 are two corresponding receivers. Alice1 wants to perform a partially unknown unitary operation on the Bob1's particle via the control of Charlie. In order to implement an operation on the Bob2's particle, Alice2 needs the help of Charlie and Alice1.
4.1. Example 1: two-layer HCRIO scheme of single-qubit operations
Suppose Bobj has a single-qubit state ∣ξj⟩ in equation (1 ) and the partially unknown unitary operations of Alicej are defined by equation (3 ), j = 1, 2.
The shared quantum channel is
$\begin{eqnarray}\begin{array}{rcl}| { \mathcal Q }\rangle & = & \displaystyle \frac{1}{2}\left(| 000000\rangle +| 011111\rangle +| 111000\rangle \right.\\ & & {\left.+\ | 100111\rangle \right)}_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}},\end{array}\end{eqnarray}$
where qubit C is held by Charlie, qubits ${A}_{1},{A}_{1}^{{\prime} }$ belong to Alice1, qubit A2 belongs to Alice2 and qubit Bj is possessed by Bobj.To achieve the task, the participants need to jointly perform the following actions.
Step 1 Alice1 performs the unitary operation in the set U1 on Bob1's qubit ∣ξ1⟩.
Charlie performs a PM on his qubit C under the X basis. Assume the measurement result is $| \widetilde{0}\rangle $, the state collapses into
$\begin{eqnarray}\begin{array}{l}{\left({\alpha }_{10}| 000\rangle +{\alpha }_{10}| 110\rangle +{\alpha }_{11}| 001\rangle +{\alpha }_{11}| 111\rangle \right)}_{{A}_{1}{B}_{1}{O}_{1}}\\ \quad \otimes \ | {h}_{0}{\rangle }_{{A}_{1}^{{\prime} }{A}_{2}{B}_{2}},\end{array}\end{eqnarray}$
where ∣h0⟩ = ∣000⟩ + ∣111⟩ will serve as the entanglement resource for Step 2.Bob1 operates ${C}_{{B}_{1}{O}_{1}}$ and measures qubit O1 under the Z basis. If his measurement result is ∣1⟩, the remaining state collapses into ${\left({\alpha }_{10}| 11\rangle +{\alpha }_{11}| 00\rangle \right)}_{{A}_{1}{B}_{1}}$.
Alice1 performs X operation and U10(θ1) on her qubit A1. The transformed state is ${\left({\alpha }_{10}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}}| 01\rangle +{\alpha }_{11}{{\rm{e}}}^{-{\rm{i}}{\theta }_{1}}| 10\rangle \right)}_{{A}_{1}{B}_{1}}$. Then, she executes a PM on her qubit A1 under the X basis. If the measurement outcome is $| \widetilde{0}\rangle $, the rest qubit B1 becomes ${\left({\alpha }_{10}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}}| 1\rangle +{\alpha }_{11}{{\rm{e}}}^{-{\rm{i}}{\theta }_{1}}| 0\rangle \right)}_{{B}_{1}}$.
Bob1 applies the recovery operation ${R}_{{B}_{1}}=X$ and reconstructs the conceivable state U10∣ξ1⟩.
Step 2 Alice2 performs the unitary operation in the set U2 on Bob2's qubit ∣ξ2⟩.
The assistance of Charlie and Alice1 is indispensable if Alice2 tries to complete the task. After Charlie's measurement, the composite system in this step is $| {h}_{0}{\rangle }_{{A}_{1}^{{\prime} }{A}_{2}{B}_{2}}\otimes | {\xi }_{2}\rangle $.
Alice1 agrees to supply the assistance and measures her qubit ${A}_{1}^{{\prime} }$ under the X basis. If the measurement result is $| \widetilde{1}\rangle $, the remaining state is
$\begin{eqnarray}{\left({\alpha }_{20}| 000\rangle -{\alpha }_{20}| 110\rangle +{\alpha }_{21}| 001\rangle -{\alpha }_{21}| 111\rangle \right)}_{{A}_{2}{B}_{2}{O}_{2}}.\end{eqnarray}$
Bob2 operates ${C}_{{B}_{2}{O}_{2}}$ and then measures his qubit O2 under the Z basis. If the measurement result is ∣0⟩, the residual state collapses into ${\left({\alpha }_{20}| 00\rangle -{\alpha }_{21}| 11\rangle \right)}_{{A}_{2}{B}_{2}}$.
Alice2 performs U21(θ2) on her qubit A2. The transformed state is ${\left({\alpha }_{20}{{\rm{e}}}^{-{\rm{i}}{\theta }_{2}}| 10\rangle -{\alpha }_{21}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}}| 01\rangle \right)}_{{A}_{2}{B}_{2}}$, then she executes a PM on her qubit A2 under the X basis. If the measurement outcome is $| \widetilde{0}\rangle $, the rest system becomes${\left({\alpha }_{20}{{\rm{e}}}^{-{\rm{i}}{\theta }_{2}}| 0\rangle -{\alpha }_{21}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}}| 1\rangle \right)}_{{B}_{2}}$.
Bob2 applies the recovery operation ${R}_{{B}_{2}}={XZ}$ and reconstructs the desired state U21∣ξ2⟩.
4.2. Example 2: two-layer HCRIO scheme of single-qutrit operations
The unitary operations the sender Alicej wants to remotely implement are in the restricted set Uj = {Uj0, Uj1, Uj2, Uj3, Uj4, Uj5}, j = 1, 2.
$\begin{eqnarray*}\begin{array}{rcl}{U}_{j0}({\theta }_{j}) & = & \left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}{\theta }_{j}(0)} & 0 & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(1)} & 0\\ 0 & 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(2)}\end{array}\right),\\ {U}_{j1}({\theta }_{j}) & = & \left(\begin{array}{ccc}{{\rm{e}}}^{{\rm{i}}{\theta }_{j}(0)} & 0 & 0\\ 0 & 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(1)}\\ 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(2)} & 0\end{array}\right),\end{array}\end{eqnarray*}$$\begin{eqnarray}\begin{array}{rcl}{U}_{j2}({\theta }_{j}) & = & \left(\begin{array}{ccc}0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(0)} & 0\\ {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(1)} & 0 & 0\\ 0 & 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(2)}\end{array}\right),\\ {U}_{j3}({\theta }_{j}) & = & \left(\begin{array}{ccc}0 & 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(0)}\\ {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(1)} & 0 & 0\\ 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(2)} & 0\end{array}\right),\end{array}\end{eqnarray}$$\begin{eqnarray*}\begin{array}{rcl}{U}_{j4}({\theta }_{j}) & = & \left(\begin{array}{ccc}0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(0)} & 0\\ 0 & 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(1)}\\ {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(2)} & 0 & 0\end{array}\right),\\ {U}_{j5}({\theta }_{j}) & = & \left(\begin{array}{ccc}0 & 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(0)}\\ 0 & {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(1)} & 0\\ {{\rm{e}}}^{{\rm{i}}{\theta }_{j}(2)} & 0 & 0\end{array}\right).\end{array}\end{eqnarray*}$
Assume that Bobj's state being executed is an arbitrary single-qutrit state
$\begin{eqnarray}| {\psi }_{j}{\rangle }_{{O}_{j}}={\left({\alpha }_{j0}| 0\rangle +{\alpha }_{j1}| 1\rangle +{\alpha }_{j2}| 2\rangle \right)}_{{O}_{j}}.\end{eqnarray}$
The shared quantum channel is
$\begin{eqnarray}\begin{array}{rcl}| {\rm{\Omega }}\rangle & = & \displaystyle \frac{1}{9}\left[(| 000\rangle +| 111\rangle +| 222\rangle )| 000\rangle \right.\\ & & +\ (| 011\rangle +| 122\rangle +| 200\rangle )| 111\rangle \\ & & {\left.+\ (| 022\rangle +| 100\rangle +| 211\rangle )| 222\rangle \right]}_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}},\end{array}\end{eqnarray}$
where qutrit C is in Charlie's site, qutrit Aj is held by Alicej, qutrit ${A}_{1}^{{\prime} }$ is in Alice1's location, qutrit Bj belongs to Bobj, j = 1, 2.The process of the protocol is described as follows.
Step 1 Alice1 performs the operation in the set U1 on Bob1's qutrit ∣ψ1⟩.
Charlie approves executing a single-qutrit PM on the qutrit C under the generalized X basis
$\begin{eqnarray}\begin{array}{rcl}| \widetilde{0}\rangle & = & \displaystyle \frac{1}{\sqrt{3}}(| 0\rangle +| 1\rangle +| 2\rangle ),\\ | \widetilde{1}\rangle & = & \displaystyle \frac{1}{\sqrt{3}}(| 0\rangle +{\omega }_{3}| 1\rangle +{\omega }_{3}^{2}| 2\rangle ),\\ | \widetilde{2}\rangle & = & \displaystyle \frac{1}{\sqrt{3}}(| 0\rangle +{\omega }_{3}^{2}| 1\rangle +{\omega }_{3}| 2\rangle ).\end{array}\end{eqnarray}$
Suppose Charlie's measurement result is $| \widetilde{1}\rangle $. The remaining particles collapse into $\begin{eqnarray}{\left[\displaystyle \sum _{j=0}^{2}{\alpha }_{1j}(| 00j\rangle +{\omega }_{3}^{2}| 11j\rangle +{\omega }_{3}| 22j\rangle )\right]}_{{A}_{1}{B}_{1}{O}_{1}}\otimes | {h}_{1}\rangle ,\end{eqnarray}$
where $| {h}_{1}\rangle ={\left(| 000\rangle +{\omega }_{3}| 111\rangle +{\omega }_{3}^{2}| 222\rangle \right)}_{{A}_{1}^{{\prime} }{A}_{2}{B}_{2}}$ will be used as the entanglement resource for Step 2.Bob1 operates ${\mathrm{GXOR}}_{{B}_{1}{O}_{1}}$ and then measures his qutrit O1 under the generalized Z basis {∣0⟩, ∣1⟩, ∣2⟩}. Assume his measurement result is ∣2⟩. The state evolves into ${\left({\alpha }_{10}{\omega }_{3}| 22\rangle +{\alpha }_{11}| 00\rangle +{\alpha }_{12}{\omega }_{3}^{2}| 11\rangle \right)}_{{A}_{1}{B}_{1}}$.
Alice1 executes a generalized X operation on her qutrit A1, where X = ∣1⟩⟨0∣ + ∣2⟩⟨1∣ + ∣0⟩⟨2∣. After that, Alice1 carries out unitary operation ${U}_{12}({\theta }_{1})={{\rm{e}}}^{{\rm{i}}{\theta }_{1}(1)}| 1\rangle \langle 0| +{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(0)}| 0\rangle \langle 1| +{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(2)}| 2\rangle \langle 2| $ on her qutrit A1, it becomes${\left({\alpha }_{10}{\omega }_{3}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(1)}| 12\rangle +{\alpha }_{11}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(0)}| 00\rangle +{\alpha }_{12}{\omega }_{3}^{2}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(2)}| 21\rangle \right)}_{{A}_{1}{B}_{1}}$.
Alice1 performs a PM on her qutrit A1 under the basis $\{| \widetilde{0}\rangle ,| \widetilde{1}\rangle ,| \widetilde{2}\rangle \}$. Suppose the measurement result is $| \widetilde{1}\rangle $, the state collapses into ${\left({\alpha }_{10}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(1)}| 2\rangle +{\alpha }_{11}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(0)}| 0\rangle +{\alpha }_{12}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}(2)}| 1\rangle \right)}_{{B}_{1}}$.
According to the measurement results, Bob1 executes the local unitary operation ${R}_{{B}_{1}}=Z(| 1\rangle \langle 0| \,+| 0\rangle \langle 1| +| 2\rangle \langle 2| ){ZX}$ and reconstructs the conceivable state U12(θ1)∣ψ1⟩, where $Z=| 0\rangle \langle 0| +{\omega }_{3}| 1\rangle \langle 1| +{\omega }_{3}^{2}| 2\rangle \langle 2| $.
Step 2 Alice2 performs the operation in the set U2 on Bob2's qutrit ∣ψ2⟩.
Alice1 agrees to execute a single-qutrit measurement on her qutrit ${A}_{1}^{{\prime} }$ under the basis $\{| \widetilde{0}\rangle ,| \widetilde{1}\rangle ,| \widetilde{2}\rangle \}$. If her measurement result is $| \widetilde{0}\rangle $, the remaining particles collapse into
$\begin{eqnarray}{\left[\displaystyle \sum _{j=0}^{2}{\alpha }_{2j}(| 00j\rangle +{\omega }_{3}| 11j\rangle +{\omega }_{3}^{2}| 22j\rangle )\right]}_{{A}_{2}{B}_{2}{O}_{2}}.\end{eqnarray}$
Bob2 operates ${\mathrm{GXOR}}_{{B}_{2}{O}_{2}}$ and then measures his qutrit O2 under the basis {∣0⟩, ∣1⟩, ∣2⟩}. Assume the measurement result is ∣1⟩. The state evolves into${\left({\alpha }_{20}{\omega }_{3}| 11\rangle +{\alpha }_{21}{\omega }_{3}^{2}| 22\rangle +{\alpha }_{22}| 00\rangle \right)}_{{A}_{2}{B}_{2}}$.
Alice2 executes X2 on the qutrit A2. If Alice2 carries out the unitary operation ${U}_{24}({\theta }_{2})={{\rm{e}}}^{{\rm{i}}{\theta }_{2}(2)}| 2\rangle \langle 0| \,+{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(0)}| 0\rangle \langle 1| +{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(1)}| 1\rangle \langle 2| $ on her qutrit A2, the state becomes ${\left({\alpha }_{20}{\omega }_{3}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(2)}| 21\rangle +{\alpha }_{21}{\omega }_{3}^{2}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(0)}| 02\rangle +{\alpha }_{22}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(1)}| 10\rangle \right)}_{{A}_{2}{B}_{2}}$. Then she performs a PM on her qutrit A2 under the generalized X basis. Suppose her measurement result is $| \widetilde{2}\rangle $, the state evolves into ${\left({\alpha }_{20}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(2)}| 1\rangle +{\alpha }_{21}{\omega }_{3}^{2}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(0)}| 2\rangle +{\alpha }_{22}{\omega }_{3}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}(1)}| 0\rangle \right)}_{{B}_{2}}$.
According to the measurement outcomes, Bob2 executes the recovery operation ${R}_{{B}_{2}}={Z}^{2}(| 2\rangle \langle 0| \,+| 0\rangle \langle 1| +| 1\rangle \langle 2| ){Z}^{2}{X}^{2}$ and gets the conceivable state U24(θ2)∣ψ2⟩.
5. The effect of noise and anti-noise strategy
We have discussed the HCRIO under ideal situations. In reality, the quantum channel is inevitably disturbed by the outer environment. Thus, it is necessary to investigate the influence of noise on the proposed schemes. There are many types of quantum noise, such as amplitude-damping (AD) noise, phase-damping noise, bit-flip noise, phase-flip noise, and so on. In this section, we first take AD as the model and study its effect on specific scheme in Example 1. Then we adopt the strategy of weak measurement (WM) and measurement reversal (MR) to suppress the noise.
5.1. The influence of AD noise
AD noise is one of the most important decoherence noises. It describes the energy dissipation effect due to the loss of energy from a quantum system and can be characterized by the Kraus operators [39]:
$\begin{eqnarray}{E}_{0}^{\mathrm{AD}}=\left(\begin{array}{cc}1 & 0\\ 0 & \sqrt{1-\lambda }\end{array}\right),\quad {E}_{1}^{\mathrm{AD}}=\sqrt{\lambda }\left(\begin{array}{cc}0 & 1\\ 0 & 0\end{array}\right),\end{eqnarray}$
where λ (0 ≤ λ ≤ 1) represents the dissipation rate. Denote $\bar{\lambda }=1-\lambda $.Suppose the quantum channel $| { \mathcal Q }\rangle $ in equation (48 ) is prepared in the controller Charlie's laboratory. Charlie distributes entangled particles ${A}_{1},{B}_{1},{A}_{1}^{{\prime} },{A}_{2},{B}_{2}$ to residual participants under the AD noise. Assuming they are affected by the same Kraus operator.
The quantum channel becomes
$\begin{eqnarray}\begin{array}{l}\varepsilon {\left(\rho \right)}^{\mathrm{AD}}=\displaystyle \sum _{i=0}^{1}[({E}_{i}^{{A}_{1}})({E}_{i}^{{B}_{1}})({E}_{i}^{{A}_{1}^{{\prime} }})({E}_{i}^{{A}_{2}})({E}_{i}^{{B}_{2}})]\rho \\ \,\times \ {\left[({E}_{i}^{{A}_{1}})({E}_{i}^{{B}_{1}})({E}_{i}^{{A}_{1}^{{\prime} }})({E}_{i}^{{A}_{2}})({E}_{i}^{{B}_{2}})\right]}^{\dagger },\end{array}\end{eqnarray}$
where the subscript indicates which Kraus operator is to be executed, the superscript represents which particle the operator E acts on, and, $\rho =| { \mathcal Q }\rangle \langle { \mathcal Q }| $. Through calculation, $\begin{eqnarray}\begin{array}{rcl}\varepsilon {\left(\rho \right)}^{\mathrm{AD}} & = & \displaystyle \frac{1}{4}\{[(| 000\rangle +\bar{\lambda }| 111\rangle )| 000\rangle \\ & & +\ {\bar{\lambda }}^{\tfrac{3}{2}}(| 100\rangle +\bar{\lambda }| 011\rangle )| 111\rangle ][(\langle 000| \\ & & +\ \bar{\lambda }\langle 111| )\langle 000| \\ & & +\ {\bar{\lambda }}^{\tfrac{3}{2}}(\langle 100| +\bar{\lambda }\langle 011| )\langle 111| ]\\ & & +\ {\lambda }^{5}| 000000\rangle \langle 000000| \}{}_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}}.\end{array}\end{eqnarray}$
Based on the measurement results, Bob1 and Bob2 execute the corresponding recovery operations on their particles B1 and B2 respectively and get the output state
$\begin{eqnarray}{\rho }_{\mathrm{out}}^{\mathrm{AD}}={\mathrm{Tr}}_{{O}_{1}{{CA}}_{1}{O}_{2}{A}_{1}^{{\prime} }{A}_{2}}[U(| {\xi }_{1}\rangle | {\xi }_{2}\rangle \langle {\xi }_{2}| \langle {\xi }_{1}| \otimes \varepsilon {\left(\rho \right)}^{\mathrm{AD}}){U}^{\dagger }],\end{eqnarray}$
where ${\mathrm{Tr}}_{{O}_{1}{{CA}}_{1}{O}_{2}{A}_{1}^{{\prime} }{A}_{2}}$ is the partial trace over qubits ${O}_{1},C,{A}_{1},{O}_{2},{A}_{1}^{{\prime} },{A}_{2}$, U is the transformation during the HCRIO process. In detail, $\begin{eqnarray}\begin{array}{ccc}U & = & {R}_{{B}_{2}}{M}_{{A}_{2}}{U}_{2m}({\theta }_{2}){X}_{{A}_{2}}^{{b}_{2}}{M}_{{O}_{2}}{C}_{{B}_{2}{O}_{2}}{M}_{{A}_{1}^{{\rm{{\prime} }}}}{R}_{{B}_{1}}{M}_{{A}_{1}}\\ & & \times \unicode{x000A0}{U}_{1m}({\theta }_{1}){X}_{{A}_{1}}^{{b}_{1}}{M}_{{O}_{1}}{C}_{{B}_{1}{O}_{1}}{M}_{C},\end{array}\end{eqnarray}$
where ${M}_{C}={M}_{{A}_{1}}={M}_{{A}_{1}^{{\prime} }}={M}_{{A}_{2}}=\{| \widetilde{0}\rangle \langle \widetilde{0}| ,| \widetilde{1}\rangle \langle \widetilde{1}| \}$ denote Charlie's, Alice1's and Alice2's measurement operators, ${M}_{{O}_{1}}={M}_{{O}_{2}}=\{| 0\rangle \langle 0| ,| 1\rangle \langle 1| \}$ denote Bob1's and Bob2's measurement operators, b1 and b2 denote the classical bit corresponding to Bob1's and Bob2's measurement results, U1m(θ1) and U2m(θ2) are the unitary operations Alice1 and Alice2 wish to implement, ${R}_{{B}_{1}}$ and ${R}_{{B}_{2}}$ are Bob1's and Bob2's recovery operations.Fidelity is used to assess the approximation between the output state ${\rho }_{\mathrm{out}}^{\mathrm{AD}}$ and the target state T = U1∣ξ1⟩ ⨂ U2∣ξ2⟩, which can be defined as
$\begin{eqnarray}{F}^{\mathrm{AD}}=\langle T| {\rho }_{\mathrm{out}}^{\mathrm{AD}}| T\rangle .\end{eqnarray}$
Under the assumption that the measurement results are $| \widetilde{0}{\rangle }_{C}$, $| \widetilde{0}{\rangle }_{{A}_{1}}$, $| \widetilde{1}{\rangle }_{{A}_{1}^{{\prime} }}$, $| \widetilde{0}{\rangle }_{{A}_{2}}$, $| 1{\rangle }_{{O}_{1}}$ and $| 0{\rangle }_{{O}_{2}}$. Alice1 performs Pauli X operation on her qubit A1. Suppose Alice1 and Alice2 want to remotely implement the unitary operations U10(θ1) and U21(θ2). We can derive the output state
$\begin{eqnarray}\begin{array}{cll}{\rho }_{{\rm{o}}{\rm{u}}{\rm{t}}}^{{\rm{A}}{\rm{D}}} & = & ({\alpha }_{10}\bar{\lambda }{{\rm{e}}}^{{\rm{i}}{\theta }_{1}}|0\rangle +{\alpha }_{11}{{\rm{e}}}^{-{\rm{i}}{\theta }_{1}}|1\rangle )({\alpha }_{20}{{\rm{e}}}^{-{\rm{i}}{\theta }_{2}}|1\rangle \\ & & +\unicode{x000A0}{\alpha }_{21}{\bar{\lambda }}^{{\textstyle \tfrac{3}{2}}}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}}|0\rangle )({\alpha }_{10}^{\ast }\bar{\lambda }{{\rm{e}}}^{-{\rm{i}}{\theta }_{1}}\langle 0|+{\alpha }_{11}^{\ast }{{\rm{e}}}^{{\rm{i}}{\theta }_{1}}\langle 1|)({\alpha }_{20}^{\ast }{{\rm{e}}}^{{\rm{i}}{\theta }_{2}}\langle 1|\\ & & +\unicode{x000A0}{\alpha }_{21}^{\ast }{\bar{\lambda }}^{{\textstyle \tfrac{3}{2}}}{{\rm{e}}}^{-{\rm{i}}{\theta }_{2}}\langle 0|)+|{\alpha }_{11}{\alpha }_{20}{|}^{2}{\lambda }^{5}|11\rangle \langle 11|\end{array}\end{eqnarray}$
and the fidelity $\begin{eqnarray}{F}^{\mathrm{AD}}=\displaystyle \frac{{\left(1-\lambda | {\alpha }_{10}{| }^{2}\right)}^{2}{\left[| {\alpha }_{20}{| }^{2}+(1-\lambda )\sqrt{1-\lambda }| {\alpha }_{21}{| }^{2}\right]}^{2}+{\lambda }^{5}| {\alpha }_{11}{\alpha }_{20}{| }^{4}}{[{\left(1-\lambda \right)}^{2}| {\alpha }_{10}{| }^{2}+| {\alpha }_{11}{| }^{2}][| {\alpha }_{20}{| }^{2}+{\left(1-\lambda \right)}^{3}| {\alpha }_{21}{| }^{2}]+{\lambda }^{5}| {\alpha }_{11}{\alpha }_{20}{| }^{2}}.\end{eqnarray}$
It is shown that the fidelity depends on the coefficients of the implemented states α10, α11, α20, α21 and the noise parameter λ.For the specific state $\tfrac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle )\otimes \tfrac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle )({\alpha }_{10}\,={\alpha }_{11}={\alpha }_{20}={\alpha }_{21}=\tfrac{1}{\sqrt{2}})$, the fidelity
$\begin{eqnarray}{F}^{\mathrm{AD}* }=\displaystyle \frac{1}{4}\displaystyle \frac{{\lambda }^{5}+{\left(2-\lambda \right)}^{2}{\left[1+(1-\lambda )\sqrt{1-\lambda }\right]}^{2}}{5{\lambda }^{4}-11{\lambda }^{3}+14{\lambda }^{2}-10\lambda +4}.\end{eqnarray}$
When λ = 0, FAD* = 1, which means perfect communication. When λ = 1, ${F}^{\mathrm{AD}* }=\tfrac{1}{4}$.5.2. WM and MR against AD noise
WM and MR have been widely applied to various aspects of quantum information processing to protect entanglement and suppress noise. WM and MR can be expressed as [34]
$\begin{eqnarray}\begin{array}{rcl}{M}_{0}^{\mathrm{WM}} & = & \left(\begin{array}{cc}1 & 0\\ 0 & \sqrt{1-{p}_{w}}\end{array}\right),\\ {M}_{1}^{\mathrm{MR}} & = & \left(\begin{array}{cc}\sqrt{1-{p}_{r}} & 0\\ 0 & 1\end{array}\right),\end{array}\end{eqnarray}$
where pw, pr are the strengths of WM and MR. Denote $\bar{{p}_{w}}=1-{p}_{w},\bar{{p}_{r}}=1-{p}_{r}$.In order to reduce the impact of noise, the controller executes WM on qubits ${A}_{1},{B}_{1},{A}_{1}^{{\prime} },{A}_{2},{B}_{2}$ before entanglement distribution. The quantum channel converts to
$\begin{eqnarray}\begin{array}{rcl}{\rho }^{\mathrm{WM}} & = & \displaystyle \frac{1}{4}[(| 000\rangle +\bar{{p}_{w}}| 111\rangle )| 000\rangle \\ & & +{\bar{{p}_{w}}}^{\tfrac{3}{2}}(| 100\rangle +\bar{{p}_{w}}| 011\rangle )| 111\rangle ]\\ & & \times \left[(\langle 000| +\bar{{p}_{w}}\langle 111| )\langle 000| +{\bar{{p}_{w}}}^{\tfrac{3}{2}}(\langle 100| \right.\\ & & {\left.+\bar{{p}_{w}}\langle 011| )\langle 111| \right]}_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}}.\end{array}\end{eqnarray}$
After qubits ${A}_{1},{B}_{1},{A}_{1}^{{\prime} },{A}_{2},{B}_{2}$ are transmitted through AD noise, ρWM becomes $\begin{eqnarray}\begin{array}{rcl}{\rho }^{\mathrm{WM}-\mathrm{AD}} & = & \displaystyle \frac{1}{4}\{[(| 000\rangle +\bar{{p}_{w}}\bar{\lambda }| 111\rangle )| 000\rangle \\ & & +{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}(| 100\rangle +\bar{{p}_{w}}\bar{\lambda }| 011\rangle )| 111\rangle ]\\ & & \times [(\langle 000| +\bar{{p}_{w}}\bar{\lambda }\langle 111| )\langle 000| +{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}(\langle 100| \\ & & +\bar{{p}_{w}}\bar{\lambda }\langle 011| )\langle 111| ]\\ & & +{\bar{{p}_{w}}}^{5}{\lambda }^{5}| 000000\rangle \langle 000000| \}{}_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}}.\end{array}\end{eqnarray}$
Then, performing MR on qubits ${A}_{1},{B}_{1},{A}_{1}^{{\prime} },{A}_{2},{B}_{2}$, one can get $\begin{eqnarray}\begin{array}{rcl}{\rho }^{\mathrm{WM}-\mathrm{AD}-\mathrm{MR}} & = & \displaystyle \frac{1}{4}\{[{\bar{{p}_{r}}}^{\tfrac{3}{2}}(\bar{{p}_{r}}| 000\rangle +\bar{{p}_{w}}\bar{\lambda }| 111\rangle )| 000\rangle +{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}\\ & & \times \ (\bar{{p}_{r}}| 100\rangle +\bar{{p}_{w}}\bar{\lambda }| 011\rangle )| 111\rangle ]\\ & & \times \ [{\bar{{p}_{r}}}^{\tfrac{3}{2}}(\bar{{p}_{r}}\langle 000| +\bar{{p}_{w}}\bar{\lambda }\langle 111| )\langle 000| +{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}(\bar{{p}_{r}}\langle 100| \\ & & +\ \bar{{p}_{w}}\bar{\lambda }\langle 011| )\langle 111| ]\\ & & +\ {\bar{{p}_{w}}}^{5}{\bar{{p}_{r}}}^{5}{\lambda }^{5}| 000000\rangle \langle 000000| \}{}_{{{CA}}_{1}{B}_{1}{A}_{1}^{{\prime} }{A}_{2}{B}_{2}}.\end{array}\end{eqnarray}$
Similar to the computation under AD noise, we can get the output state $\begin{eqnarray}\begin{array}{rcl}{\rho }_{\mathrm{out}}^{\mathrm{WM}-\mathrm{AD}-\mathrm{MR}} & = & \left[({\alpha }_{10}\bar{{p}_{w}}\bar{\lambda }{{\rm{e}}}^{{\rm{i}}{\theta }_{1}}| 0\rangle +{\alpha }_{11}\bar{{p}_{r}}{{\rm{e}}}^{-{\rm{i}}{\theta }_{1}}| 1\rangle )\right.\\ & & \times \ ({\alpha }_{20}{\bar{{p}_{r}}}^{\tfrac{3}{2}}{{\rm{e}}}^{-{\rm{i}}{\theta }_{2}}| 1\rangle \\ & & +\ {\alpha }_{21}{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}}| 0\rangle )\\ & & \times \ ({\alpha }_{10}^{* }\bar{{p}_{w}}\bar{\lambda }{{\rm{e}}}^{-{\rm{i}}{\theta }_{1}}\langle 0| +{\alpha }_{11}^{* }\bar{{p}_{r}}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}}\langle 1| )({\alpha }_{20}^{* }{\bar{{p}_{r}}}^{\tfrac{3}{2}}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}}\langle 1| \\ & & +\ {\alpha }_{21}^{* }{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}{{\rm{e}}}^{-{\rm{i}}{\theta }_{2}}\langle 0| )\\ & & {\left.+\ | {\alpha }_{11}{\alpha }_{20}{| }^{2}{\bar{{p}_{w}}}^{5}{\bar{{p}_{r}}}^{5}{\lambda }^{5}| 11\rangle \langle 11| \right]}_{{B}_{1}{B}_{2}}\end{array}\end{eqnarray}$
and the fidelity $\begin{eqnarray}\begin{array}{l}{F}^{\mathrm{WM}-\mathrm{AD}-\mathrm{MR}}\\ \quad =\ \displaystyle \frac{{\left(| {\alpha }_{10}{| }^{2}\bar{{p}_{w}}\bar{\lambda }+| {\alpha }_{11}{| }^{2}\bar{{p}_{r}}\right)}^{2}{\left(| {\alpha }_{20}{| }^{2}{\bar{{p}_{r}}}^{\tfrac{3}{2}}+| {\alpha }_{21}{| }^{2}{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}\right)}^{2}+| {\alpha }_{11}{\alpha }_{20}{| }^{4}{\bar{{p}_{w}}}^{5}{\bar{{p}_{r}}}^{5}{\lambda }^{5}}{(| {\alpha }_{10}{| }^{2}{\bar{{p}_{w}}}^{2}{\bar{\lambda }}^{2}+| {\alpha }_{11}{| }^{2}{\bar{{p}_{r}}}^{2})(| {\alpha }_{20}{| }^{2}{\bar{{p}_{r}}}^{3}+| {\alpha }_{21}{| }^{2}{\bar{{p}_{w}}}^{3}{\bar{\lambda }}^{3})+| {\alpha }_{11}{\alpha }_{20}{| }^{2}{\bar{{p}_{w}}}^{5}{\bar{{p}_{r}}}^{5}{\lambda }^{5}}.\end{array}\end{eqnarray}$
It is shown that the fidelity depends on the coefficients of the implemented states α10, α11, α20, α21, noise parameter λ, WM strength pw and MR strength pr. For the specific state $\tfrac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle )\otimes \tfrac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle )$, the fidelity
$\begin{eqnarray}{F}^{\mathrm{WM}-\mathrm{AD}-\mathrm{MR}* }=\displaystyle \frac{{\left(\bar{{p}_{w}}\bar{\lambda }+\bar{{p}_{r}}\right)}^{2}{\left({\bar{{p}_{r}}}^{\tfrac{3}{2}}+{\bar{{p}_{w}}}^{\tfrac{3}{2}}{\bar{\lambda }}^{\tfrac{3}{2}}\right)}^{2}+{\bar{{p}_{w}}}^{5}{\bar{{p}_{r}}}^{5}{\lambda }^{5}}{4[({\bar{{p}_{w}}}^{2}{\bar{\lambda }}^{2}+{\bar{{p}_{r}}}^{2})({\bar{{p}_{r}}}^{3}+{\bar{{p}_{w}}}^{3}{\bar{\lambda }}^{3})+{\bar{{p}_{w}}}^{5}{\bar{{p}_{r}}}^{5}{\lambda }^{5}]}.\end{eqnarray}$
If we use the optimal reversing condition [36]: pr = pw + λ(1 − pw), then $\begin{eqnarray}{F}^{\mathrm{WM}-\mathrm{AD}-\mathrm{MR}* }=\displaystyle \frac{16+{\left(1-{p}_{w}\right)}^{5}{\lambda }^{5}}{16+4{\left(1-{p}_{w}\right)}^{5}{\lambda }^{5}}.\end{eqnarray}$
Figure 4 shows the fidelity comparison in terms of whether WM and MR are used or not. The results show that WM and MR can improve the fidelities efficiently.
Figure 4. 3D comparison between the fidelities with and without WM and MR when the implemented state is $\tfrac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle )\otimes \tfrac{1}{\sqrt{2}}(| 0\rangle +| 1\rangle )$. |
6. Discussions and conclusions
In order to meet the practical needs of remote control, we study remotely implementing unitary operations on arbitrary single-particle states among multiple layers. To reduce the resource consumption, the implemented operations are partially unknown choosing from some restricted sets instead of completely unknown. We propose HCRIO schemes of partially unknown quantum operations in two-dimensional and high-dimensional systems. The receiver can recover the implemented quantum operation on his particle only if the task is approved by the controller and the higher-level senders. Two examples with two layers are given to elaborate the proposed schemes. One is a HCRIO scheme of single-qubit operations and the other is a HCRIO scheme of single-qutrit operations. Then, we study the effect of AD noise and the fidelity is adapted to characterize the closeness between the output state and the target state. WM and MR are utilized to reduce the impact of noise. Compared to previous RIO or CRIO schemes [11–18, 20], ours have the following advantages: (1) The asymmetric implementation of partially unknown quantum operations with multiple layers is investigated. (2) The teleported operations are extended from the two-dimensional system to arbitrary high-dimensional systems. (3) The general formula of the recovery operator is derived which clearly discloses the relationship with the measurement results. (4) We consider the HCRIO protocol in the noisy environment and utilize the anti-noise strategy to reduce the impact of noise. It is hoped that our work may enrich the theory of hierarchical quantum operation sharing and provide inspirations to explore its potential applications in quantum networks.