Abbreviations
ADC Amplitude damping channel
EAM Environment-assisted measurement
MQGTP Maximized quantum gate-assisted teleportation protocol
MR Quantum measurement and reversing operation
QGCU Quantum gate control unit
QGED Quantum gate-assisted entanglement distribution
QGTP Quantum gate-assisted teleportation protocol
QGTP-EAM Quantum gate-assisted teleportation protocol combined with environment-sssisted measurement
STNP Standard teleportation with no protection
1. Introduction
Quantum communication, an indispensable branch of quantum information, mainly involves quantum teleportation [1–5], quantum dense coding [6–8], quantum secret sharing [9–11] and quantum key distribution [12–14]. Quantum communication, leveraging the intrinsic characteristics of quantum mechanics, offers a significantly higher level of security than classical communication [15]. Quantum teleportation is the key strategy for transmitting quantum information without physically transferring the particle that contains the quantum information [16]. Scalable quantum networks depend greatly on quantum teleportation, which has attracted special attention and has made considerable breakthroughs both theoretically and experimentally (see, for example, [16–19] and references therein). Additionally, distributed quantum computing is established by using quantum teleportation circuits to transfer quantum information between distant quantum processors within a quantum network [20].
It is theoretically proven that in an ideal noiseless environment the state of an arbitrary unknown qubit can be ‘reproduced' in another location with both average fidelity and success probability equal to one by implementing specific quantum teleportation protocols such as the protocols proposed in [21–23]. However, in practical quantum information processing tasks, decoherence occurs due to the inevitable interaction of the quantum system with the environment [24, 25]; this is one of the major obstacles to establishing a quantum network or connecting components in a scalable quantum architecture. Likewise, decoherence of shared entanglement caused by environmental noise is an important source of imperfections in quantum teleportation [26, 27]. Amplitude damping is one of the most important decoherence mechanisms; it describes the energy dissipation of a quantum system and is a major cause of imperfections in practical quantum communication in quantum networks and quantum computing [28–31]. To improve the performance of quantum teleportation in the presence of noise, two interesting decoherence suppression schemes—weak measurement and quantum measurement reversal [32–34] and quantum feed-forward control and its reversal [35, 36]—have been applied to the protection of shared entanglement. In fact, these schemes follow a similar strategy, namely the application of control operations both before and after the noisy channel. Before the noisy channel, the state of the system is transferred to a state that is less vulnerable to the effects of the noise, and after the noisy channel reversal operations are applied to restore the system to its initial state. In addition, environment-assisted measurement (EAM) is a powerful technique to overcome the effects of the decoherence channel [37–40, 53]. This technique performs a measurement on the noisy channel to select the system states that correspond to the desired measurement outcome of the environment; therefore, restoration operations are applied on the system conditioned on the results of the measurement on the environment. Moreover, quantum error correction shows bright prospects for application in enhancing quantum entanglement [26, 41], although it requires redundant qubits in entanglement. In all the above schemes the focus is on the entanglement distribution process, while it has been shown that the effects of the decoherence channel can be effectively suppressed by modifying the quantum teleportation protocol itself [42–44]. In [43, 44], a generalized concept of quantum teleportation in the framework of a quantum measurement and reversing operation (MR) was introduced, where the joint Bell state measurement and a corresponding single-qubit reversal operation were suggested. In the MR framework, a well-designed weak measurement operator takes the place of standard unitary operators in the final step of the original quantum teleportation protocol. However, utilizing time-consuming weak measurements (such as monitoring cavities by photon counting devices) slows the whole teleportation process. In fact, the rapidity of quantum teleportation is of vital significance and a necessity for advanced quantum networks and quantum computing. Hence, how to develop a more rapid protection scheme suitable for quantum teleportation protocols remains an open question. Zubairy's group [45–47] has shown that quantum gates can replace weak measurements in retrieving quantum states and protecting entanglement. They applied a Hadamard gate on an ancilla qubit and subsequently a controlled NOT (CNOT) gate on the combined system consisting of the system qubit and the ancilla qubit to entirely reverse the effect of weak measurement. One of the most significant advantages of the quantum state protection scheme via quantum gates is that it can be implemented in a shorter time than weak measurement reversal. Hence, it is more suitable for quantum networks and quantum computing.
In this paper, we utilize quantum gates to enable noise-resistant quantum teleportation through noisy channels. Unlike previous decoherence control strategies, our proposed schemes neither require redundant qubits in entanglement nor employ time-consuming weak measurements. We apply a quantum gate control unit (QGCU) in the last step of teleportation after the completion of the standard teleportation procedure. Specifically, we apply an appropriate rotation transformation to an ancilla qubit prepared in the ground state, interact the ancilla qubit with the receiver qubit and then apply a CNOT gate on the combined system. The proposed quantum gate-assisted teleportation protocol (QGTP) is considered to be successful if the ancilla qubit is measured in a certain state. To further improve the fidelity of teleportation, we demonstrate that it is possible to achieve unit average teleportation fidelity independent of the degree of decoherence and the input state parameters by applying EAM in the entanglement distribution process and a QGCU at the location of the receiver, and deduce the specific formula of the optimum rotation gate angle to gain unit average teleportation fidelity. Since the proposed teleportation protocols via QGCU are probabilistic, we also consider applying QGCU only in the entanglement distribution process followed by a deterministic standard teleportation protocol. The proposed entanglement distribution process aims to transfer the shared entanglement to a state that is robust to decoherence via a QGCU before the noisy channel, and recover the shared entanglement via another QGCU after the noisy channel. We should point out that the noisy channel is assumed to be fully characterized, i.e. the magnitude of decoherence is known and quantum gates can be implemented perfectly, hence the fault of quantum gates is not taken into account [48–50]. For comparison, we study standard teleportation with no protection (STNP) and teleportation in the MR framework, and show the improvement of the average teleportation fidelity of our proposed teleportation schemes by numerical simulations.
The remainder of this paper is organized as follows. In section 2 , we present the proposed teleportation protocols via quantum gates and EAM through a noisy channel, and analyze their performance in detail. In section 3 , we investigate the entanglement distribution via quantum gates, where we apply a QGCU before the noisy channel at the sender's location and another QGCU after the noisy channel at the receiver's location. Finally, our conclusion is drawn in section 4 .
2. Teleportation protocols through an amplitude damping channel via quantum gates
In this section we give details of the two proposed QGTPs through an amplitude damping channel (ADC). We consider the scenario in which Alice prepares the maximally shared entangled state and then distributes one particle to Bob through an ADC. In the first protocol, we only employ the designed QGCU in the last step of teleportation to overcome the effects of the ADC and retrieve the input state at Bob's location. In the second protocol, to further improve the teleportation fidelity, we combine the EAM technique in the entanglement distribution process with the first proposed protocol. Afterwards, we compare the average teleportation fidelity and the total teleportation success probability of the two proposed QGTPs.
2.1. Quantum gate-assisted teleportation protocol through ADC
In this subsection, we suggest a QGTP to achieve high-fidelity quantum teleportation through an ADC, where the receiver applies a QGCU to his qubit after the standard teleportation protocol is accomplished. First, let us calculate the output state of the standard teleportation protocol through the ADC before the QGCU is employed.
The input state that Alice wishes to teleport to Bob is presented as
$\begin{eqnarray}{\rho }_{\mathrm{in}}=| {\psi }_{\mathrm{in}}\rangle \langle {\psi }_{\mathrm{in}}| =\left[\begin{array}{cc}| \alpha {| }^{2} & \alpha {\beta }^{* }\\ {\alpha }^{* }\beta & | \beta {| }^{2}\end{array}\right],\end{eqnarray}$
where ∣α∣2 + ∣β∣2 = 1 and * denotes complex conjugation.The maximally entangled state prepared by Alice is defined as
$\begin{eqnarray}| \psi {\rangle }_{{ab}}=\displaystyle \frac{1}{\sqrt{2}}(| 0{\rangle }_{a}| 0{\rangle }_{b}+| 1{\rangle }_{a}| 1{\rangle }_{b}),\end{eqnarray}$
where she keeps the first qubit and sends the second qubit of the entangled pair to Bob through a noisy channel. In this paper, we consider the ADC, which is described by the well-known Kraus operators, as [15] $\begin{eqnarray}{e}_{0}=\left[\begin{array}{cc}1 & 0\\ 0 & \sqrt{1-r}\end{array}\right],\,{e}_{1}=\left[\begin{array}{cc}0 & \sqrt{r}\\ 0 & 0\end{array}\right],\end{eqnarray}$
where 0 ≤ r ≤ 1 is the magnitude of the decoherence and represents the probability of decay from the upper level ∣1⟩ to the lower level ∣0⟩ with r = 1 − e−Γt, in which Γ is the energy relaxation rate and t is the evolving time.By following the standard teleportation protocol, Bob's non-normalized states, corresponding to Alice's different joint Bell state measurement results, are described as
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{B}_{i}}^{{U}_{i}}(i=1,2) & = & \displaystyle \frac{1}{4}\left[\begin{array}{cc}| \alpha {| }^{2}+| \beta {| }^{2}r & \alpha {\beta }^{* }\sqrt{1-r}\\ {\alpha }^{* }\beta \sqrt{1-r} & | \beta {| }^{2}(1-r)\end{array}\right],\\ {\rho }_{{B}_{i}}^{{U}_{i}}(i=3,4) & = & \displaystyle \frac{1}{4}\left[\begin{array}{cc}| \alpha {| }^{2}(1-r) & \alpha {\beta }^{* }\sqrt{1-r}\\ {\alpha }^{* }\beta \sqrt{1-r} & | \beta {| }^{2}+| \alpha {| }^{2}r\end{array}\right].\end{array}\end{eqnarray}$
The detailed procedure of the standard teleportation protocol is presented in the appendix .
According to equation (4 ), Bob's output state is severely damped due to the effects of the ADC, which causes a significant decrease in teleportation fidelity. In the following, we employ the QGCU in the last step of teleportation to compensate for the effects of the ADC and retrieve the input state at Bob's end. After finishing the standard teleportation process, Bob performs the following operations based on quantum gates on his damped output qubit in equation (4 ). Hereafter, we refer to these operations realized by quantum gates as the QGCU. A schematic diagram of our proposed QGTP is shown in figure 1.
Figure 1. Schematic diagram of the QGTP. The double lines indicate the classical communications. |
Apart from a CNOT and a rotation gate, the QGCU also contains an ancilla qubit in the state $\left|0\right\rangle $, for example an atom in the ground state that can interact with the cavity field [45]. A rotation gate ${H}_{\varphi }=[\cos \varphi ,-\sin \varphi ;\sin \varphi ,\cos \varphi ]$ with an angle φ is applied on the ancilla qubit. Hence, the state of the ancilla qubit after passing through the rotation gate is
$\begin{eqnarray}{\rho }_{A}=\left[\begin{array}{cc}{\cos }^{2}\varphi & \cos \varphi \sin \varphi \\ \cos \varphi \sin \varphi & {\sin }^{2}\varphi \end{array}\right].\end{eqnarray}$
Bob's combined state after he interacts the ancilla qubit with the damped state in equation (4 ) is given as
$\begin{eqnarray}{\rho }_{{B}_{i}}^{A}={\rho }_{{B}_{i}}^{{U}_{i}}\otimes {\rho }_{A}.\end{eqnarray}$
Afterwards, a CNOT gate is applied on the pair comprising Bob's qubit and the ancilla qubit, where Bob's qubit is the controlled qubit and the ancilla qubit is the target qubit. The CNOT gate is defined as
$\begin{eqnarray}C=\left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right].\end{eqnarray}$
The state of the whole system after applying the CNOT gate can be described in the systematic basis (∣00⟩, ∣01⟩, ∣10⟩ and ∣11⟩) by
$\begin{eqnarray}\begin{array}{l}{\rho }_{{B}_{i}}^{C}=C{\rho }_{{B}_{i}}^{A}{C}^{\dagger }\\ \,={\cos }^{2}\varphi \left[\begin{array}{cccc}{\rho }_{{B}_{i}}^{{U}_{i}}(1,1) & {\rho }_{{B}_{i}}^{{U}_{i}}(1,1)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(1,2)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(1,2)\\ {\rho }_{{B}_{i}}^{{U}_{i}}(1,1)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(1,1){\tan }^{2}\varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(1,2){\tan }^{2}\varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(1,2)\tan \varphi \\ {\rho }_{{B}_{i}}^{{U}_{i}}(2,1)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(2,1){\tan }^{2}\varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(2,2){\tan }^{2}\varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(2,2)\tan \varphi \\ {\rho }_{{B}_{i}}^{{U}_{i}}(2,1) & {\rho }_{{B}_{i}}^{{U}_{i}}(2,1)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(2,2)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(2,2)\end{array}\right].\end{array}\end{eqnarray}$
At this point, a projective measurement is made on the state of the ancilla qubit dependent on Alice's joint Bell state measurement result. Hence, the success of a QGCU as the final step of teleportation depends on a particular outcome in the measurement of the ancilla qubit. When the measurement results correspond to B1,2 (B3,4) as specified in equation (A4 ), the teleportation is considered successful if the measurement result of the ancilla qubit is $\left|0\right\rangle $ ($\left|1\right\rangle $). Hence, the state of the whole system is described as
$\begin{eqnarray}{\rho }_{{B}_{i}}^{{C}^{{\prime} }}={P}_{{B}_{i}}^{A}{\rho }_{{B}_{i}}^{C}{P}_{{B}_{i}}^{A\dagger },\end{eqnarray}$
where ${P}_{{B}_{\mathrm{1,2}}}^{A}={I}_{2}\otimes | 0\rangle \langle 0| $ and ${P}_{{B}_{\mathrm{3,4}}}^{A}={I}_{2}\otimes | 1\rangle \langle 1| $ are the projection operators corresponding to Alice's different measurement results with I2 being the 2 × 2 identity operator.Finally, Bob's reduced density matrix is calculated by tracing out the ancilla qubit as
$\begin{eqnarray}\begin{array}{l}{\rho }_{{B}_{i}}^{f}(i=1,2)=\displaystyle \frac{{\mathrm{Tr}}_{A}({\rho }_{{B}_{i}}^{C^{\prime} })}{\mathrm{Tr}\left[{\mathrm{Tr}}_{A}({\rho }_{{B}_{i}}^{C^{\prime} })\right]}\\ \,=\displaystyle \frac{1}{{\rho }_{{B}_{i}}^{{U}_{i}}(1,1)+{\rho }_{{B}_{i}}^{{U}_{i}}(2,2){\tan }^{2}\varphi }\\ \,\left[\begin{array}{cc}{\rho }_{{B}_{i}}^{{U}_{i}}(1,1) & {\rho }_{{B}_{i}}^{{U}_{i}}(1,2)\tan \varphi \\ {\rho }_{{B}_{i}}^{{U}_{i}}(2,1)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(2,2){\tan }^{2}\varphi \end{array}\right],\\ {\rho }_{{B}_{i}}^{f}(i=3,4)=\displaystyle \frac{{\mathrm{Tr}}_{A}({\rho }_{{B}_{i}}^{C^{\prime} })}{\mathrm{Tr}\left[{\mathrm{Tr}}_{A}({\rho }_{{B}_{i}}^{C^{\prime} })\right]}\\ \,=\displaystyle \frac{1}{{\rho }_{{B}_{i}}^{{U}_{i}}(1,1){\tan }^{2}\varphi +{\rho }_{{B}_{i}}^{{U}_{i}}(2,2)}\\ \,\left[\begin{array}{cc}{\rho }_{{B}_{i}}^{{U}_{i}}(1,1){\tan }^{2}\varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(1,2)\tan \varphi \\ {\rho }_{{B}_{i}}^{{U}_{i}}(2,1)\tan \varphi & {\rho }_{{B}_{i}}^{{U}_{i}}(2,2)\end{array}\right].\end{array}\end{eqnarray}$
Furthermore, the success probability of gaining the state ${\rho }_{{B}_{i}}^{f}$ is calculated as
$\begin{eqnarray}\begin{array}{l}{g}_{{B}_{i}}^{f}(i=1,2)=\mathrm{Tr}\left[{\mathrm{Tr}}_{A}({\rho }_{{B}_{i}}^{C^{\prime} })\right]\\ \,=\displaystyle \frac{1}{4}{\cos }^{2}\varphi (| \alpha {| }^{2}+| \beta {| }^{2}r+| \beta {| }^{2}(1-r){\tan }^{2}\varphi ),\\ {g}_{{B}_{i}}^{f}(i=3,4)=\mathrm{Tr}\left[{\mathrm{Tr}}_{A}({\rho }_{{B}_{i}}^{C^{\prime} })\right]\\ \,=\displaystyle \frac{1}{4}{\cos }^{2}\varphi (| \beta {| }^{2}+| \alpha {| }^{2}r+| \alpha {| }^{2}(1-r){\tan }^{2}\varphi ),\end{array}\end{eqnarray}$
where ${\mathrm{Tr}}_{A}(\bullet )$ is the partial trace over the ancilla qubit.To evaluate the performance of the proposed QGTP, we consider the fidelity between the input state in equation (1 ) and the final state received by Bob in equation (10 ) as
$\begin{eqnarray}\begin{array}{l}{\mathrm{fid}}_{i}(i=1,2)=\langle {\psi }_{\mathrm{in}}| {\rho }_{{B}_{i}}^{f}| {\psi }_{\mathrm{in}}\rangle \\ \,=\displaystyle \frac{| \alpha {| }^{4}+| \alpha {| }^{2}| \beta {| }^{2}r+2| \alpha {| }^{2}| \beta {| }^{2}\sqrt{1-r}\tan \varphi +| \beta {| }^{4}{\tan }^{2}\varphi (1-r)}{| \alpha {| }^{2}+| \beta {| }^{2}r+| \beta {| }^{2}(1-r){\tan }^{2}\varphi },\\ {\mathrm{fid}}_{i}(i=3,4)=\langle {\psi }_{\mathrm{in}}| {\rho }_{{B}_{i}}^{f}| {\psi }_{\mathrm{in}}\rangle \\ \,=\displaystyle \frac{| \beta {| }^{4}+| \alpha {| }^{2}| \beta {| }^{2}r+2| \alpha {| }^{2}| \beta {| }^{2}\sqrt{1-r}\tan \varphi +| \alpha {| }^{4}{\tan }^{2}\varphi (1-r)}{| \beta {| }^{2}+| \alpha {| }^{2}r+| \alpha {| }^{2}(1-r){\tan }^{2}\varphi }.\end{array}\end{eqnarray}$
It can be seen for equation (12 ) that a rotation gate angle φ such that the teleportation fidelity is always equal to one for all values of r in QGTP does not exist for all input states; nevertheless, for some specific input states, for example ∣0⟩ and ∣1⟩, a teleportation fidelity equal to one is achievable regardless of the value of φ and r.
Accordingly, by considering the probability of gaining different measurement results from Alice in equation (A6 ) and the corresponding fidelities in equation (12 ), the total teleportation fidelity of the proposed QGTP is described as
$\begin{eqnarray}{\mathrm{fid}}_{\mathrm{tot}}^{\mathrm{QGTP}}=\displaystyle \sum _{i=1,2,3,4}({P}_{{B}_{i}}{\mathrm{fid}}_{i})=\displaystyle \frac{1}{4}({\mathrm{fid}}_{1}+{\mathrm{fid}}_{2}+{\mathrm{fid}}_{3}+{\mathrm{fid}}_{4}).\end{eqnarray}$
According to equations (12 ) and (13 ), the total teleportation fidelity is dependent on the input state ρin parameters. Thus, to assess the performance of the proposed QGTP in a way that is independent of a certain input state, we utilize the average teleportation fidelity over all possible input states as
$\begin{eqnarray}\begin{array}{l}{\mathrm{Fid}}_{\mathrm{av}}^{\mathrm{QGTP}}=\displaystyle \int {\rm{d}}\rho {\mathrm{fid}}_{\mathrm{tot}}^{\mathrm{QGTP}}\\ \,=\displaystyle \int {\rm{d}}\rho \displaystyle \frac{1}{4}({\mathrm{fid}}_{1}+{\mathrm{fid}}_{2}+{\mathrm{fid}}_{3}+{\mathrm{fid}}_{4}).\end{array}\end{eqnarray}$
Moreover, the total teleportation success probability of the proposed QGTP is calculated as11 ).
$\begin{eqnarray}\begin{array}{l}{g}_{\mathrm{tot}}^{\mathrm{QGTP}}=\displaystyle \sum _{i=1,2,3,4}{g}_{{B}_{i}}^{f}\\ \,=\displaystyle \frac{1}{2}{\cos }^{2}\varphi (1+r+(1-r){\tan }^{2}\varphi ),\end{array}\end{eqnarray}$
where ${g}_{{B}_{i}}^{f}$ is the success probability of gaining the state ${\rho }_{{B}_{i}}^{f}$ defined in equation (For comparison, we consider STNP through an ADC. The total teleportation fidelity between the input state equation (1 ) and the final state received by Bob in equation (4 ) is
$\begin{eqnarray}\begin{array}{l}{\mathrm{fid}}_{\mathrm{tot}}^{\mathrm{STNP}}=\displaystyle \frac{1}{2}\left[(| \alpha {| }^{4}+| \beta {| }^{4})(2-r)\right.\\ \,+\left.| \alpha {| }^{2}| \beta {| }^{2}(4\sqrt{1-r}+2r)\right].\end{array}\end{eqnarray}$
Therefore, the average teleportation fidelity of the STNP over all possible input states is presented as
$\begin{eqnarray}\begin{array}{l}{\mathrm{Fid}}_{\mathrm{av}}^{\mathrm{SNTP}}=\displaystyle \int {\rm{d}}\rho \left\{\displaystyle \frac{1}{2}[(| \alpha {| }^{4}+| \beta {| }^{4})(2-r)\right.\\ \,+| \alpha {| }^{2}| \beta {| }^{2}(4\sqrt{1-r}+2r)]\}\\ \,=\displaystyle \frac{1}{15}(4\sqrt{1-r}-\displaystyle \frac{7}{2}r+11).\end{array}\end{eqnarray}$
For further comparison, we also consider another probabilistic teleportation protocol that is based on weak measurement, i.e. the MR framework of teleportation [43]. In the MR framework, Bob applies designed weak measurement reversal instead of unitary operations to suppress the effects of the ADC. By considering the maximally shared entangled state in equation (2 ), the total teleportation fidelity of the MR framework is presented as
$\begin{eqnarray}{\mathrm{fid}}_{\mathrm{tot}}^{\mathrm{MR}}=\displaystyle \frac{1}{2}\left(\displaystyle \frac{1+r| \alpha {| }^{2}| \beta {| }^{2}}{1+r| \beta {| }^{2}}+\displaystyle \frac{1+r| \alpha {| }^{2}| \beta {| }^{2}}{1+r| \alpha {| }^{2}}\right).\end{eqnarray}$
Consequently, the average teleportation fidelity of the MR framework over all possible input states is described as
$\begin{eqnarray}{\mathrm{Fid}}_{\mathrm{av}}^{\mathrm{MR}}=\int d\rho \left[\displaystyle \frac{1}{2}(\displaystyle \frac{1+r| \alpha {| }^{2}| \beta {| }^{2}}{1+r| \beta {| }^{2}}+\displaystyle \frac{1+r| \alpha {| }^{2}| \beta {| }^{2}}{1+r| \alpha {| }^{2}})\right].\end{eqnarray}$
The MR framework of teleportation is probabilistic due to the incompleteness of the weak measurement reversal employed in the last step of the teleportation procedure. The total teleportation success probability of the MR framework is defined as
$\begin{eqnarray}{g}_{\mathrm{tot}}^{\mathrm{MR}}=\displaystyle \frac{2-r-{r}^{2}}{2}.\end{eqnarray}$
In figure 2, we compare the total teleportation fidelity of our proposed QGTP with the MR framework and STNP for various input states in the presence of amplitude damping noise. In particular, we calculate the total teleportation fidelity by selecting specific input states in mutually unbiased bases on the Bloch sphere, i.e. $\left|0\right\rangle $, $\left|1\right\rangle $, $\left|\pm \right\rangle =\left(\left|0\right\rangle \pm \left|1\right\rangle \right)/\sqrt{2}$ and $\left|\pm i\right\rangle =\left(\left|0\right\rangle \pm i\left|1\right\rangle \right)/\sqrt{2}$ [51].
Figure 2. Total teleportation fidelity for specific input states. The dashed black line indicates the classical limit of 2/3. Here r = 0.5. |
Figure 2 shows that the proposed QGTP outperforms the MR framework and STNP with a total teleportation fidelity of more than 0.98 for various selected input states. Additionally, it can be seen that when the input state is $\left|0\right\rangle $ or $\left|1\right\rangle $, the effects of the ADC can be totally canceled and the unit teleportation fidelity is attainable by applying the optimum rotation gate angle φ.
Note here that while the total teleportation fidelity is reliant on the input state parameters, the average teleportation fidelity considers all possible input states and is independent of the input state parameters. From now on, we use the average teleportation fidelity as the performance indicator to evaluate the behavior of teleportation protocols.
In figure 3(a), we depict the average teleportation fidelity of the QGTP in equation (14 ), the average teleportation fidelity of the MR framework in equation (19 ) (the green plane) and the average teleportation fidelity of STNP in equation (17 ) (the gray plane) as a function of the rotation gate angle φ and the magnitude of the decoherence r are presented. Moreover, figure 3(b) is the total teleportation success probability of the QGTP in equation (15 ) and the total teleportation success probability of the MR framework in equation (20 ) (the green plane) versus φ and r. We highlight that STNP is a deterministic protocol, meaning the teleportation success probability is always equal to one. Therefore, we have excluded the success probability of STNP in figure 3(b).
Figure 3. Average teleportation fidelity and total teleportation success probability as a function of the rotation gate angle φ and magnitude of the decoherence r. The planes with a color map indicate the proposed QGTP, the green planes correspond to the MR framework and the gray plane corresponds to STNP. |
As figure 3(a) shows, for each r, the maximum teleportation fidelity of the QGTP is greater than that of the MR framework. Thus, by selecting the optimum rotation gate angle φ, the teleportation fidelity can be improved for each r compared with the MR framework. Additionally, as shown in figure 3(b), the total teleportation success probability of the QGTP is higher for smaller rotation gate angles and can be enhanced for intense r compared with that of the MR framework. However, for the optimum range of φ to maximize the average teleportation fidelity, the corresponding total teleportation success probability is lower than that of the MR framework. We shall discuss the maximized average teleportation fidelity and corresponding total teleportation success probability of the QGTP later in section 2.3 .
2.2. Further improving the teleportation fidelity of QGTP via environment-assisted measurement
In this section, we add the EAM technique to the entanglement distribution process of the QGTP to further improve the teleportation fidelity. We emphasize once more that the noisy channel is assumed to be fully characterized, i.e. Bob knows the value of r, which cannot be obtained freely. A schematic diagram of the QGTP combined with EAM (QGTP-EAM) is presented in figure 4.
Figure 4. Schematic diagram of QGTP-EAM. |
First, we give a brief explanation of the fundamental concept of EAM. The basic idea of EAM is to perform a measurement on the environment (decoherence channel) coupled to the system of interest; hence, the environment collapses to the eigenstates of the measured observable. As a result, the system will be projected into a state relative to each resulting environmental state. The aim is to keep the system states corresponding to invertible Kraus operators of the environment and discard other results [52, 53, 54]. To do so, a detector is added to monitor the excitation changes in the environment. If the measurement result is represented by 'no click', it means that the resulting environmental state is e0, so we keep the corresponding system states and discard other measurement results since they are non-invertible.
The scenario is almost the same as that in section 2 . Alice prepares the entangled state in equation (2 ) and sends one qubit through an ADC to Bob. Then, Bob applies EAM to the ADC and tells the result to Alice. If the ADC is in an unexcited state related to the invertible Kraus operator e0 in equation (3 ), entanglement distribution is successfully achieved, and the teleportation process can be started. Otherwise, he discards the entanglement distribution at this time and restarts the process. In other words, we only keep the damped entanglement corresponding to the invertible Kraus operator e0. Hence, the shared entangled state after a successful EAM is described as
$\begin{eqnarray}\begin{array}{l}{\rho }_{{ab}}^{\mathrm{EAM}}=\displaystyle \frac{{E}_{0}{\rho }_{{ab}}{E}_{0}^{\dagger }}{{g}^{\mathrm{ED}-\mathrm{EAM}}}=\displaystyle \frac{1}{2-r}\\ \,\times \left[\begin{array}{cccc}1 & 0 & 0 & \sqrt{1-r}\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \sqrt{1-r} & 0 & 0 & 1-r\end{array}\right],\end{array}\end{eqnarray}$
where ${g}^{\mathrm{ED}-\mathrm{EAM}}=\mathrm{Tr}\left({E}_{0}{\rho }_{{ab}}{E}_{0}^{\dagger }\right)=1-r/2$ is the success probability of the entanglement distribution via EAM.After the entanglement distribution via EAM has been done successfully, Alice starts the teleportation process by interacting the input qubit with her half of the entangled shared state and applies the joint Bell state measurement with measurement operators given in equation (A4 ). The non-normalized state of Bob's qubit corresponding to Alice's different measurement results Bi (i = 1, 2, 3, 4) after applying the unitary operations given in equation (A7 ) becomes
$\begin{eqnarray}\begin{array}{l}{\rho }_{{\mathrm{EAM}}_{{B}_{i}}}^{{U}_{i}}(i=1,2)=\displaystyle \frac{1}{4-2r}\\ \,\times \,\left[\begin{array}{cc}| \alpha {| }^{2} & \alpha {\beta }^{* }\sqrt{1-r}\\ {\alpha }^{* }\beta \sqrt{1-r} & | \beta {| }^{2}(1-r)\end{array}\right],\\ {\rho }_{{\mathrm{EAM}}_{{B}_{i}}}^{{U}_{i}}(i=3,4)=\displaystyle \frac{1}{4-2r}\\ \,\times \,\left[\begin{array}{cc}| \alpha {| }^{2}(1-r) & \alpha {\beta }^{* }\sqrt{1-r}\\ {\alpha }^{* }\beta \sqrt{1-r} & | \beta {| }^{2}\end{array}\right].\end{array}\end{eqnarray}$
Now, to recover the damped state in equation (22 ), Bob needs to apply the QGCU explained in section 2.2 . Hence, Bob's final density matrix after applying the QGCU is described as
$\begin{eqnarray}\begin{array}{l}{\rho }_{{\mathrm{EAM}}_{{B}_{i}}}^{f}(i=1,2)=\displaystyle \frac{1}{| \alpha {| }^{2}+| \beta {| }^{2}(1-r){\tan }^{2}\varphi }\\ \,\times \,\left[\begin{array}{cc}| \alpha {| }^{2} & \alpha {\beta }^{* }\sqrt{1-r}\tan \varphi \\ {\alpha }^{* }\beta \sqrt{1-r}\tan \varphi & | \beta {| }^{2}(1-r){\tan }^{2}\varphi \end{array}\right],\\ {\rho }_{{\mathrm{EAM}}_{{B}_{i}}}^{f}(i=3,4)=\displaystyle \frac{1}{| \alpha {| }^{2}(1-r){\tan }^{2}\varphi +| \beta {| }^{2}}\\ \,\times \,\left[\begin{array}{cc}| \alpha {| }^{2}(1-r){\tan }^{2}\varphi & \alpha {\beta }^{* }\sqrt{1-r}\tan \varphi \\ {\alpha }^{* }\beta \sqrt{1-r}\tan \varphi & | \beta {| }^{2}\end{array}\right].\end{array}\end{eqnarray}$
The success probability of gaining the state ${\rho }_{{\mathrm{EAM}}_{{B}_{i}}}^{f}$ is calculated as
$\begin{eqnarray}\begin{array}{l}{g}_{{\mathrm{EAM}}_{{B}_{i}}}^{f}(i=1,2)=\displaystyle \frac{1}{4-2r}\\ {\cos }^{2}\varphi (| \alpha {| }^{2}+| \beta {| }^{2}(1-r){\tan }^{2}\varphi ),\\ {g}_{{\mathrm{EAM}}_{{B}_{i}}}^{f}(i=3,4)=\displaystyle \frac{1}{4-2r}\\ {\cos }^{2}\varphi (| \beta {| }^{2}+| \alpha {| }^{2}(1-r){\tan }^{2}\varphi ).\end{array}\end{eqnarray}$
To achieve unit teleportation fidelity, Bob's output state in equation (23 ) must be the same as the input state in equation (1 ); hence, the optimum rotation gate angle is determined as
$\begin{eqnarray}{\varphi }_{\mathrm{opt}}^{\mathrm{QGTP}-\mathrm{EAM}}={\tan }^{-1}\left(\displaystyle \frac{1}{\sqrt{1-r}}\right).\end{eqnarray}$
By selecting the optimum rotation gate angle in equation (25 ) the teleportation fidelity is always equal to one, i.e. ${\mathrm{fid}}_{{\mathrm{EAM}}_{{B}_{i}}}=\langle {\psi }_{\mathrm{in}}| {\rho }_{{\mathrm{EAM}}_{{B}_{i}}}^{f}| {\psi }_{\mathrm{in}}\rangle =1$. Consequently, the average teleportation fidelity of the proposed QGTP-EAM over all possible input states becomes
$\begin{eqnarray}{\mathrm{Fid}}_{\mathrm{av}}^{\mathrm{QGTP}\,-\,\mathrm{EAM}}=\int d\rho \displaystyle \sum _{i=1,2,3,4}({P}_{{\mathrm{EAM}}_{{B}_{i}}}{\mathrm{fid}}_{{\mathrm{EAM}}_{{B}_{i}}})=1,\end{eqnarray}$
where ${P}_{{\mathrm{EAM}}_{{B}_{i}}}$ is the probability of gaining the measurement outcome corresponding to the measurement operator Bi in the QGTP-EAM as $\begin{eqnarray}\begin{array}{l}{P}_{{\mathrm{EAM}}_{{B}_{i}}}(i=1,2)=\displaystyle \frac{| \alpha {| }^{2}+| \beta {| }^{2}(1-r)}{2(2-r)},\\ \,{P}_{{\mathrm{EAM}}_{{B}_{i}}}(i=3,4)=\displaystyle \frac{| \beta {| }^{2}+| \alpha {| }^{2}(1-r)}{2(2-r)}.\end{array}\end{eqnarray}$
Similarly, by considering the ${\varphi }_{\mathrm{opt}}^{\mathrm{QGTP}-\mathrm{EAM}}$ in equation (25 ), the total teleportation success probability of QGTP-EAM is described as
$\begin{eqnarray}\begin{array}{c}{g}_{\mathrm{tot}}^{\mathrm{QGTP}\,-\,\mathrm{EAM}}=\displaystyle \sum _{i=1,2,3,4}{g}_{{\mathrm{EAM}}_{{B}_{i}}}^{f}\\ =\displaystyle \frac{4}{4-2r}{\cos }^{2}{\varphi }_{\mathrm{opt}}^{\mathrm{QGTP}-\mathrm{EAM}}=\displaystyle \frac{2(1-r)}{{\left(2-r\right)}^{2}}.\end{array}\end{eqnarray}$
2.3. Performance comparison
We now proceed to numerically compare our proposed teleportation protocols and analyze their performance for different values of r. In the QGTP and QGTP-EAM, the aim is to maximize the average teleportation fidelity for each scheme by finding the optimum rotation gate angles and calculating the corresponding total teleportation success probability. A comparison of the results of three schemes [maximized QGTP(MQGTP), QGTP-EAM and the MR framework] is shown in figure 5. In figure 5(a), we plot the following: (1) the maximum average teleportation fidelity of the MQGTP in equation (14 ) by selecting the optimum rotation gate angle; (2) the average teleportation fidelity of the QGTP-EAM in equation (26 ) with the optimum rotation gate angle ${\varphi }_{\mathrm{opt}}^{\mathrm{QGTP}-\mathrm{EAM}}$ in equation (25 ) to gain unit average teleportation fidelity; and (3) the average teleportation fidelity of the MR framework in equation (19 ), which uses weak measurement reversal. Moreover, figure 5(b) shows the result of comparing the following: (1) the corresponding total teleportation success probability of the MQGTP in equation (15 ); (2) the total teleportation success probability of the QGTP-EAM in equation (28 ); and (3) the total teleportation success probability of the MR framework in equation (20 ). Additionally, the optimum rotation gate angles of the two QGTPs are shown in figure 5(c).
Figure 5. Comparison of the MQGTP, QGTP-EAM and MR frameworks. |
As shown in figure 5(a), the two proposed teleportation protocols significantly improve the average teleportation fidelity compared with the MR framework. As we mentioned before, QGTP-EAM can obtain an average teleportation fidelity of unity independent of the magnitude of decoherence. The interesting point, according to figure 5(b), is that the total teleportation success probability of QGTP-EAM is higher than that of MQGTP for most values of r. Therefore, by adding EAM to QGTP, both the teleportation fidelity and the total teleportation success probability are improved. Moreover, as illustrated by figure 5(c), the optimum rotation gate angles in the MQGTP and QGTP-EAM are in the range of 0.8 to π/2. Therefore, assuming that EAM is available, it can be concluded that the QGTP-EAM is absolutely preferable.
3. Quantum gate-assisted entanglement distribution through amplitude damping channel
In the previous section, to overcome the effects of the decoherence channel we recovered the damped output state by modifying the teleportation process via a QGCU. In fact, as mentioned in section 1 , the degradation of shared entangled states is the fundamental reason for the decrease in quantum teleportation fidelity. The proposed teleportation protocols are probabilistic, but in some cases its preferable to first distribute the entangled state between Alice and Bob and then implement a quantum teleportation protocol that works in a deterministic manner. The main motivation of this section is to use a QGCU in the entanglement distribution process and then employ a standard teleportation protocol. Hence, in this section, we consider applying QGCUs only in the entanglement distribution process. Before the decoherence channel, Alice employs a QGCU on the second qubit of the entangled pair to transfer it to a state that is less vulnerable to the effects of the ADC; after the decoherence channel, Bob recovers his half of the entangled pair by applying another QGCU. After a successful entanglement distribution process, Alice and Bob proceed to the standard teleportation process. A schematic diagram of the proposed quantum gate-assisted entanglement distribution process (QGED) is depicted in figure 6.
Figure 6. Schematic diagram of the quantum gate-assisted entanglement distribution process. |
Alice applies the QGCU explained in section 2.1 to the second qubit of the entangled pair that will be sent to Bob. As a result, after applying the QGCU and tracing out the ancilla qubit, the non-normalized density matrix of the entangled state is obtained as7 ) and ${P}_{{A}_{1}}={I}_{2}\otimes {I}_{2}\otimes | 0\rangle \langle 0| $ is the measurement operator applied on A1.
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{ab}}^{{\prime} } & = & {\mathrm{Tr}}_{{A}_{1}}[{P}_{{A}_{1}}{C}_{1}({\rho }_{{ab}}\otimes {\rho }_{{A}_{1}}){C}_{1}^{\dagger }{P}_{{A}_{1}}^{\dagger }]\\ & = & \displaystyle \frac{1}{2}\left[\begin{array}{cccc}{\cos }^{2}{\varphi }_{1} & 0 & 0 & \cos {\varphi }_{1}\sin {\varphi }_{1}\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \cos {\varphi }_{1}\sin {\varphi }_{1} & 0 & 0 & {\sin }^{2}{\varphi }_{1}\end{array}\right],\end{array}\end{eqnarray}$
where ${\rho }_{{A}_{1}}=[{\cos }^{2}{\varphi }_{1},\cos {\varphi }_{1}\sin {\varphi }_{1};\cos {\varphi }_{1}\sin {\varphi }_{1},{\sin }^{2}{\varphi }_{1}]$ is the density matrix of the ancilla qubit A1 after it passes through a rotation gate with an angle φ1, C1 = I2 ⨂ C is the CNOT gate with C in equation (Afterwards, the second qubit of the entangled pair passes through an ADC defined by equation (A1 ), and the non-normalized density matrix of the shared entangled state evolves to
$\begin{eqnarray}\begin{array}{l}{\rho }_{{ab}}^{\mathrm{damp}^{\prime} }={E}_{0}{\rho }_{{ab}}^{{\prime} }{E}_{0}^{\dagger }+{E}_{1}{\rho }_{{ab}}^{{\prime} }{E}_{1}^{\dagger }\\ \,=\displaystyle \frac{1}{2}\left[\begin{array}{cccc}{\cos }^{2}{\varphi }_{1} & 0 & 0 & \cos {\varphi }_{1}\sin {\varphi }_{1}\sqrt{1-r}\\ 0 & 0 & 0 & 0\\ 0 & 0 & r{\sin }^{2}{\varphi }_{1} & 0\\ \cos {\varphi }_{1}\sin {\varphi }_{1}\sqrt{1-r} & 0 & 0 & {\sin }^{2}{\varphi }_{1}(1-r)\end{array}\right].\end{array}\end{eqnarray}$
Following the receipt of the second qubit of the entangled pair at Bob's location, he applies the QGCU on his qubit. Hence, the normalized density matrix of the shared entanglement is obtained as
$\begin{eqnarray}\begin{array}{l}{\rho }_{{ab}}^{\mathrm{ED}}=\displaystyle \frac{{\mathrm{Tr}}_{{A}_{2}}[{P}_{{A}_{2}}{C}_{1}({\rho }_{{ab}}^{\mathrm{damp}^{\prime} }\otimes {\rho }_{{A}_{2}}){C}_{1}^{\dagger }{P}_{{A}_{2}}^{\dagger }]}{\mathrm{Tr}\{\,{\mathrm{Tr}}_{{A}_{2}}[{P}_{{A}_{2}}{C}_{1}({\rho }_{{ab}}^{\mathrm{damp}^{\prime} }\otimes {\rho }_{{A}_{2}}){C}_{1}^{\dagger }{P}_{{A}_{2}}^{\dagger }]\}}\\ \,=\displaystyle \frac{1}{2{g}^{\mathrm{QGED}}}\left[\begin{array}{cccc}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2} & 0 & 0 & {\rho }_{14}\\ 0 & 0 & 0 & 0\\ 0 & 0 & r{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1} & 0\\ {\rho }_{41} & 0 & 0 & {\sin }^{2}{\varphi }_{1}{\sin }^{2}{\varphi }_{2}(1-r)\end{array}\right],\end{array}\end{eqnarray}$
where ${\rho }_{14}={\rho }_{41}=\cos {\varphi }_{1}\cos {\varphi }_{2}\sin {\varphi }_{1}\sin {\varphi }_{2}\sqrt{1-r}$ and ${\rho }_{{A}_{2}}=[{\cos }^{2}{\varphi }_{2},\cos {\varphi }_{2}\sin {\varphi }_{2};\cos {\varphi }_{2}\sin {\varphi }_{2},{\sin }^{2}{\varphi }_{2}]$ is the density matrix of the ancilla qubit A2 after it passes through a rotation gate with an angle φ2, ${P}_{{A}_{2}}={P}_{{A}_{1}}$ is the measurement operator applied on A2 and gQGED is the success probability of QGED calculated as $\begin{eqnarray}\begin{array}{l}{g}^{\mathrm{QGED}}=\mathrm{Tr}\{\,{\mathrm{Tr}}_{{A}_{2}}[{P}_{{A}_{2}}{C}_{1}({\rho }_{{ab}}^{\mathrm{damp}^{\prime} }\otimes {\rho }_{{A}_{2}}){C}_{1}^{\dagger }{P}_{{A}_{2}}^{\dagger }]\}\\ \,=\displaystyle \frac{1}{2}[{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+{\cos }^{2}{\varphi }_{2}].\end{array}\end{eqnarray}$
This completes the QGED process. After successful entanglement distribution, Alice and Bob proceed to the standard teleportation process by considering the shared entangled state in equation (31 ). It is worth noting that the probability of gaining the measurement outcome corresponding to the measurement operators Bi is different from the case presented previously because of QGED. In this case, these probabilities are calculated as
$\begin{eqnarray}\begin{array}{l}{P}_{{B}_{i}}^{\mathrm{ED}}(i=1,2)=\mathrm{Tr}[{B}_{i}({\rho }_{\mathrm{in}}\otimes {\rho }_{{ab}}^{\mathrm{ED}}){B}_{i}^{\dagger }]\\ \,=\displaystyle \frac{| \alpha {| }^{2}({\cos }^{2}{\varphi }_{1}-{\sin }^{2}{\varphi }_{2})+| \beta {| }^{2}r{\sin }^{2}{\varphi }_{1}(1-2{\sin }^{2}{\varphi }_{2})+{\sin }^{2}{\varphi }_{1}{\sin }^{2}{\varphi }_{2}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}},\\ {P}_{{B}_{i}}^{\mathrm{ED}}(i=3,4)=\mathrm{Tr}[{B}_{i}({\rho }_{\mathrm{in}}\otimes {\rho }_{{ab}}^{\mathrm{ED}}){B}_{i}^{\dagger }]\\ \,=\displaystyle \frac{| \beta {| }^{2}({\cos }^{2}{\varphi }_{1}-{\sin }^{2}{\varphi }_{2})+| \alpha {| }^{2}r{\sin }^{2}{\varphi }_{1}(1-2{\sin }^{2}{\varphi }_{2})+{\sin }^{2}{\varphi }_{1}{\sin }^{2}{\varphi }_{2}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}}.\end{array}\end{eqnarray}$
Since we used teleportation fidelity in the previous section to evaluate the performance of the proposed protocols, here we also investigate the average teleportation fidelity of the standard teleportation protocol by considering the shared entangled state from the QGED strategy in equation (31 ).
Therefore, after a successful QGED process, the output state received by Bob is described as
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{B}_{i}}^{f^{\prime} }(i=1,2) & = & \left[\begin{array}{cc}\tfrac{| \alpha {| }^{2}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}+| \beta {| }^{2}r{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}} & \tfrac{\alpha {\beta }^{* }\cos {\varphi }_{1}\cos {\varphi }_{2}\sin {\varphi }_{1}\sin {\varphi }_{2}\sqrt{1-r}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}}\\ \tfrac{{\alpha }^{* }\beta \cos {\varphi }_{1}\cos {\varphi }_{2}\sin {\varphi }_{1}\sin {\varphi }_{2}\sqrt{1-r}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}} & \tfrac{| \beta {| }^{2}{\sin }^{2}{\varphi }_{1}{\sin }^{2}{\varphi }_{2}(1-r)}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}}\end{array}\right],\\ {\rho }_{{B}_{i}}^{f^{\prime} }(i=3,4) & = & \left[\begin{array}{cc}\tfrac{| \alpha {| }^{2}{\sin }^{2}{\varphi }_{1}{\sin }^{2}{\varphi }_{2}(1-r)}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}} & \tfrac{\alpha {\beta }^{* }\cos {\varphi }_{1}\cos {\varphi }_{2}\sin {\varphi }_{1}\sin {\varphi }_{2}\sqrt{1-r}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}}\\ \tfrac{{\alpha }^{* }\beta \cos {\varphi }_{1}\cos {\varphi }_{2}\sin {\varphi }_{1}\sin {\varphi }_{2}\sqrt{1-r}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}} & \tfrac{| \beta {| }^{2}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}+| \alpha {| }^{2}r{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}}{2{\sin }^{2}{\varphi }_{1}(1-r)(2{\sin }^{2}{\varphi }_{2}-1)+2{\cos }^{2}{\varphi }_{2}}\end{array}\right].\end{array}\end{eqnarray}$
Thus, the teleportation fidelity between the input state in equation (1 ) and the final state received by Bob in equation (34 ) is calculated as
$\begin{eqnarray}\begin{array}{l}{\mathrm{fid}}_{{\mathrm{ED}}_{i}}(i=1,2)=\langle {\psi }_{\mathrm{in}}| {\rho }_{{B}_{i}}^{f^{\prime} }| {\psi }_{\mathrm{in}}\rangle \\ \,=\displaystyle \frac{| \beta {| }^{4}(1-r)(1-{\cos }^{2}{\varphi }_{1}-{\cos }^{2}{\varphi }_{2}+{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2})+| \alpha {| }^{4}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}}{| \beta {| }^{2}{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}+| \alpha {| }^{2}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}-| \beta {| }^{2}{\sin }^{2}{\varphi }_{1}(1-r)(1-2{\cos }^{2}{\varphi }_{2})}\\ \,+\displaystyle \frac{| \alpha {| }^{2}| \beta {| }^{2}r{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}+2| \alpha {| }^{2}| \beta {| }^{2}\cos {\varphi }_{1}\cos {\varphi }_{2}\sin {\varphi }_{1}\sin {\varphi }_{2}\sqrt{1-r}}{| \beta {| }^{2}{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}+| \alpha {| }^{2}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}-| \beta {| }^{2}{\sin }^{2}{\varphi }_{1}(1-r)(1-2{\cos }^{2}{\varphi }_{2})},\\ \,{\mathrm{fid}}_{{\mathrm{ED}}_{i}}(i=3,4)=\langle {\psi }_{\mathrm{in}}| {\rho }_{{B}_{i}}^{f^{\prime} }| {\psi }_{\mathrm{in}}\rangle \\ \,=\displaystyle \frac{| \alpha {| }^{4}(1-r)(1-{\cos }^{2}{\varphi }_{1}-{\cos }^{2}{\varphi }_{2}+{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2})+| \beta {| }^{4}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}}{| \alpha {| }^{2}{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}+| \beta {| }^{2}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}-| \alpha {| }^{2}{\sin }^{2}{\varphi }_{1}(1-r)(1-2{\cos }^{2}{\varphi }_{2})}\\ \,+\displaystyle \frac{| \alpha {| }^{2}| \beta {| }^{2}r{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}+2| \alpha {| }^{2}| \beta {| }^{2}\cos {\varphi }_{1}\cos {\varphi }_{2}\sin {\varphi }_{1}\sin {\varphi }_{2}\sqrt{1-r}}{| \alpha {| }^{2}{\cos }^{2}{\varphi }_{2}{\sin }^{2}{\varphi }_{1}+| \beta {| }^{2}{\cos }^{2}{\varphi }_{1}{\cos }^{2}{\varphi }_{2}-| \alpha {| }^{2}{\sin }^{2}{\varphi }_{1}(1-r)(1-2{\cos }^{2}{\varphi }_{2})}.\end{array}\end{eqnarray}$
Hence, the average teleportation fidelity of the standard teleportation protocol with QGED over all possible input states is
$\begin{eqnarray}{\mathrm{Fid}}_{\mathrm{av}}^{\mathrm{QGED}}=\int d\rho ({P}_{{B}_{i}}^{\mathrm{ED}}{\mathrm{fid}}_{{\mathrm{ED}}_{i}}).\end{eqnarray}$
For different values of r, each set of rotation gate angles (φ1, φ2) uniquely derives its corresponding entanglement distribution success probability and average teleportation fidelity, symbolized by a point (${\mathrm{Fid}}_{{av}}^{{QGED}}$—gQGED). In figure 7, the rotation gate angles (φ1, φ2) are separately collected over all real numbers in the range 0 ≤ φ1, φ2 ≤ π/2, and phase diagrams of corresponding average teleportation fidelity and entanglement distribution success probability (${\mathrm{Fid}}_{{av}}^{{QGED}}$—gQGED) for selected values of r = 0.5 and 0.8 are plotted. Each blue dot represents the (${\mathrm{Fid}}_{{av}}^{{QGED}}$—gQGED) of a selected rotation gate angle set (φ1, φ2). The red line represents the average teleportation fidelity of the MR framework, the black line represents the maximized average teleportation fidelity of the QGTP and the green line is the average teleportation fidelity of the STNP. It should be stressed that the success probability in figure 7 corresponds to the entanglement distribution process rather than the teleportation process. In other words, once the entangled qubits have been successfully distributed between Alice and Bob, unit total teleportation success probability is always achieved by implementing the standard teleportation protocol.
Figure 7. Phase diagram of (${\mathrm{Fid}}_{{av}}^{\mathrm{QGED}}$—gQGED). Each blue dot represents the value of (${\mathrm{Fid}}_{{av}}^{{QGED}}$—gQGED) for a selected set of rotation gate angles (φ1, φ2), collected over all real numbers within the range 0 ≤ φ1, φ2 ≤ π/2. The solid green line represents the average teleportation fidelity of the STNP, the dash-dotted red line represents the average teleportation fidelity of the MR framework and the dashed black line represents the average teleportation fidelity of the MQGTP. |
In figure 7, for a given average teleportation fidelity (entanglement distribution success probability) the point (${\mathrm{Fid}}_{{av}}^{{QGED}}$—gQGED) where the entanglement distribution success probability (average teleportation fidelity) is maximized is dispersed on the boundary line of the phase diagram. As figure 7 depicts, by selecting proper rotation gate angles (φ1, φ2), the QGED can improve the average teleportation fidelity compared with the QGTP and the MR framework. It is also possible to obtain an average teleportation fidelity of unity at the price of low success probability in the entanglement distribution process.
4. Conclusion
In this paper, we improved the fidelity of teleportation through noisy channels by utilizing a designed quantum gate control unit (QGCU). We proposed teleportation protocols and an entanglement distribution scheme via rotation and CNOT gates to teleport an unknown state through noisy channels with high fidelity. The proposed teleportation protocol applies a QGCU in the last step of teleportation to negate the effects of the noisy channel. The optimum rotation gate angles for maximizing the average teleportation fidelity were discovered by numerical simulations, and the corresponding total teleportation success probability was studied. Furthermore, unit average teleportation fidelity that is independent of the input state parameter and the magnitude of decoherence was achieved by integrating EAM with the quantum gate-assisted teleportation protocol. Moreover, to take advantage of a QGCU in a deterministic teleportation protocol, we considered employing QGCUs only in the entanglement distribution process followed by the standard teleportation protocol. It was demonstrated that our proposed schemes could greatly improve the average teleportation fidelity compared with the weak measurement-based teleportation in [43] and standard teleportation with no protection. The possible implementation schemes of the QGCU are given in a linear optics system [46] and in a cavity QED system where the quantum states are stored in superconducting cavities [55].