Enhancement of teleportation average fidelity via photon addition operation

In this paper, using the linear entropy criterion to quantify the degree of entanglement, we demonstrate that the addition of photons can enhance the entanglement degree of the photon-added two-mode squeezed vacuum states (PATMSVSs). Besides, the entanglement degree can approach ideal in case of squeezing parameter or number of added photons is high. When using PATMSVSs as entanglement resources to teleport a coherent state via a canonical protocol, the results show that the average fidelity of the quantum teleportation process is enhanced by the photon addition operation. The average fidelity increases as the number of photons added to the two-mode increases. In addition, we show that an enhancement in average fidelity can also be achieved by increasing the squeezing parameter of the TMSVS.


Introduction
Recent research results have shown that entangled states play a central role in the tasks of quantum information, such as quantum teleportation, quantum key distribution, quantum coding [1], remote state preparation [2], joint remote state preparation [3], quantum secret sharing [4], quantum loss sensing [5], and so on.It is worth noting that the quality of these quantum tasks often depends on the degree of entanglement in the exploited states.Therefore, constructing new quantum states with high degree of entanglement and/or nonclassicality is a work of great significance in quantum optics research.
The construction of new quantum states is mainly done in the following two ways.First, new definitions of quantum states are introduced, such as the two-mode squeezed vacuum state (TMSVS) [6], the pair coherent state [7], the symmetric superposition coherent state [8].Second, based on the existing states, transformations are used on these states by actions such as photon addition, photon subtraction, superposition of photons addition and subtraction [9], photon catalysis [10] as well as exploiting the nonlinear effect [11].With the above two methods, the second method is often studied more widely for a deeper purpose, which is to improve the quantum properties of the original states.Among them, the addition of photons is guaranteed to be the safest and most feasible because photon addition to the coherent state has been experimentally proven [12].
With respect to the photon addition, many new states have been introduced, such as the photon-added two-mode coherent state [13], the photon-added pair coherent state [14], the photon-added two-mode displaced-squeezed state [15], the photon-added two-mode squeezed vacuum states (PATMSVSs) [16], the photon-added trio coherent state [17], and so on (see  [18,19,20,21]).Studies have shown that these new states have many enhanced quantum properties compared to the original states, for example, the negativity of Wigner function, squeezing, and entanglement.In order to evaluate the contribution of these new states to the field of quantum information, they have been used as resources of entanglement to perform various quantum tasks, the most prominent of which is quantum teleportation.However, contrary to expectations, all results indicate that the addition of photons reduces the quality of the quantum teleportation process, i.e. reduces the average fidelity, when using the traditional protocol of measuring orthogonal quadrature components.This has reduced the interest of theoretical physicists in studying the contribution of photon addition to the construction of quantum internet systems.Unsatisfied with the above results, we still expect to find another quantum teleportation protocol where, if used, the addition of photon could make a positive contribution.
To achieve our expectation, we focus our attention on the PATMSVSs introduced by Wang et al in 2015 [16].According to the results obtained by these authors, adding photons to both modes of the TMSVS can enhance but has not yet quantified the entanglement in the PATMSVSs using von Neumann entropy.To further study these states, we first use the linear entropy criterion to quantify the entanglement degree in the PATMSVSs, which helps us evaluate when the entanglement degree is maximum while that the von Neumann entropy cannot be clarified.Additionally, this criterion clearly indicates the role of simultaneously adding photon pairs to the TMSVS in enhancing the entanglement degree in the PATMSVSs.Besides, also in [16], they have indicated that the average fidelity of the quantum teleportation of a coherent state via the protocol for measuring the orthogonal quadrature components is below 0.85 when the squeezing parameter is less than 1.0.In this paper, exploiting the canonical quantum teleportation protocol, we use the PATMSVSs as resources of entanglement to quantum teleport a coherent state.We theoretically clarify this quantum teleportation process and calculate the analytical expression of the average fidelity, which allows us to evaluate the quality of the quantum teleportation.Thereby, we clarify whether using the photon addition effect to improve the quality of the quantum teleportation through the measurement of photon number sum and phase difference is possible.
This paper is structured as follows: Section 2 briefly presents the PATMSVSs and evaluates their entanglement via the linear entropy.In Section 3, we clarify the quantum teleportation process of a coherent state using the photon number sum and phase difference measurement protocol.Some results and discussions are listed in Section 4. Finally, some conclusions are given in Section 5.

Photon-added two-mode squeezed vacuum states and entanglement degree
One of the most prominent and significant two-mode states in quantum physics is the TMSVS [6].It is defined as follow where ŝ = e r(â †b † +â b) is the squeezing operator of the two modes a and b, x † (x) is the photon creation (annihilation) operator of the mode x, x = (a, b), r is the squeezing parameter, |0, 0⟩ ab is the vacuum states.This state has been generated in the lab to quantum teleport a coherent state [22].However, at the small squeezing parameter values, the efficiency of performing the quantum tasks of the TMSVS is not high.Therefore, the effect of adding photons on this state was studied, from which the PATMSVSs were proposed [16].It is defined by acting k times the pair of creation operators â † b † on the modes a and b of the TMSVS where k is a positive integer, N k is the normalization coefficient.
where C k is determined as follows where 2 H 1 is denoted as the hypergeometric function.For the convenience of calculations, we rewrite the PATMSVSs as where The results studied in [16] show that the PATMSVSs exhibit quantum properties such as entanglement, sum squeezing, and EPR correlation.The effect of adding photons increases the entanglement degree according to the von Neumann entropy but reduces the sum squeezing, the EPR correlation as well as the average fidelity of the quantum teleportation process according to the orthogonal quadrature components measurement protocol.
In what follows, we use the linear entropy criterion to re-evaluate the entanglement degree in the PATMSVSs before using them as entanglement resources for quantum teleportation.The linear entropy coefficient is defined as [23] where Tr is denoted for the trace of the matrix, ρ is the reduced density operator.In a certain state, it is entangled when E > 0, the larger the E coefficient, the higher the entanglement degree.The ideal entanglement corresponds to E = 1.With the PATMSVSs in Eq. ( 5), we get where c n,k is given in Eq. ( 6).We discuss the degree of entanglement in the PATMSVSs in Section 4.

Quantum teleportation via canonical protocol
Quantum teleportation is a method of transferring quantum states from one place (A) to another (B) with an arbitrary distance between two places by exploiting the entanglement between A and B along with a classical communication channel.The quantum teleportation protocol was first proposed by Bennett et al in 1993 [24].Subsequently, many different quantum teleportation protocols for continuous or discrete entangled resources have been proposed, such as the measurement of orthogonal quadrature components [25], the measurement of photon number difference and phase sum [26], the measurement of photon number sum and phase difference [27], or photon counting [28].In this section, we use the PATMSVSs as resources to teleport a coherent state via the canonical protocol, that is, the measuring the photon number sum and the phase difference.The system is prepared as follows: The mode a of the PATMSVSs is held by party A, while the mode b is owned by party B. Party A is responsible for sending to party B the coherent state |α⟩ c of mode c, in the form of Fock states, the coherent state is given by where α is a complex number and d m = e −|α| 2 /2 α m / √ m!.To begin the quantum teleportation process, party A combines the system into a three-mode state |ψ in ⟩ abc = |r, k⟩ ab |α⟩ c .Next, party A performs the measurement of the photon number sum and the phase difference on the two modes a and c.After this measurement, the state of the system is reduced to the single-mode state, namely where P is the probability of obtaining values including the photon number sum N and the phase difference ϕ − , |ϕ − N ⟩ ac is the eigenstate of the photon number sum and the phase difference operators, it has the following form We can explicitly write the state in Eq. ( 10) as follows The probability of the measurement is now determined by After the measurement, party A sends party B the values of N and ϕ − via a traditional classical information channel.After receiving these two values, party B performs a phase shift by applying the unitary operator Ûb = e i Nb ϕ − on its state.The party B's state now becomes Next, party B converts the number of photons in its state from n + k to N − n − k to reconstruct the sent state.The process of quantum teleportation is now complete.The output state is determined as To evaluate the quality of the quantum teleportation process, we use the average fidelity coefficient.This is the coefficient representing the overlap between the output state at party B and the input state at party A. It is defined as follows 48th Vietnam Conference on Theoretical Physics (VCTP-48) Journal of Physics: Conference Series 2744 (2024) 012002 The values of F av are always in the range [0, 1].The quantum teleportation process is successful when F av > 0.5, the larger the F av , the higher the quality of this process, F av = 1 corresponds to the ideal teleportation process.After some calculation steps, the average fidelity is written as

Results and discussions
For the entanglement degree, we use the analytical expression in Eq. ( 8) to investigate the entanglement degree according to the linear entropy criterion of the PATMSVSs is shown in Fig. 1, in which, we increase the number of photons addition k = 0, k = 1, k = 2 and k = 4; it should be noted that the case k = 0 corresponds to the original TMSVS.The results show that the linear entropy, i.e., the entanglement degree, is always higher in the PATMSVSs than in the TMSVS.The higher the number of photons addition is, the stronger the entanglement degree becomes.The results also show that the linear entropy rapidly approaches ideal when r > 2 or k ≥ 3 with r > 1.Thus, the above indications show that the addition of photon significantly improves the entanglement degree of the TMSVS.Thereby, we can completely believe that using the PATMSVS as an entanglement resource for quantum teleportation is successful.With respect to the average fidelity of quantum teleportation, using the analytical expression in Eq. ( 17), we clarify the average fidelity of the quantum teleportation process graphically as shown in Fig. 2. By fixing the teleported state amplitude |α| in each graph of Fig. 2, we gradually increase the number of added photons k = 0, 1, 2, 4, where k = 0 corresponds to the initial TMSVS (the no-photon-added state), the remaining cases correspond to the PATMSVSs.The results show that the quantum teleportation is successful (F av > 0.5).In particular, increasing the number of added photons increase the average fidelity.In case |α| ≤ 0.5, F av > 0.85 even k ≥ 1 and r < 0.6.This is exactly what we expect when studying the PATMSVSs.It is worth noting that the enhancement of the squeezing parameter r also increases the average fidelity.In addition, when examining the average fidelity coefficient according to the coherent amplitude |α|, it is easy to see that the F av coefficient decreases with increasing |α|.
In summary in this paper, we have first quantified the degree of entanglement and determined the role of adding paired photons to increase the entanglement degree of the TMSVS.The second is that with the canonical teleportation protocol we studied, the average fidelity approaches a maximum (approaches 1.0) as the number of added photons increases even when r < 0.6, and the more photon pairs, the average fidelity is further improved.While in the traditional protocol, the average fidelity only reaches 0.85 when r < 1.Thus, we realize that the most important thing in this paper is that the quantum teleportation protocol we apply in this paper is suitable for the entangled resource generated by the photon addition operation.

Conclusion
In this paper, we have studied the PATMSVSs in terms of entanglement and quantum teleportation.The degree of entanglement in the PATMSVSs has been investigated according to the linear entropy criterion.The results show that these states are always entangled.The entanglement degree is enhanced as the number of added photons increases and can rapidly reach a unit value.When using the PATMSVSs as a quantum channel to teleport the coherent state via the measurement of the photon number sum and the phase difference, the results show that the quantum teleportation is successful.Interestingly, the average fidelity is enhanced as the number of added photons increases.It contrasts with the quantum teleportation process that uses the measurement of the orthogonal quadrature components, where the addition of photons reduces the average fidelity coefficient.In addition, the average fidelity can also reach unit value when k and r are chosen appropriately.Thus, our above results play an important role in demonstrating the positive effects of the addition photons in quantum tasks, especially quantum communication.

Figure 1 .Figure 2 .
Figure 1.The dependence of the linear entropy coefficient E in Eq. (8) on the squeezing parameter r and the number of photons added k, where k = 0 corresponds to the TMSVS, the remaining k values correspond to the PATMSVS.
In the term of two-mode Fock states |p⟩ a |q⟩ b = |p, q⟩ ab , the PATMSVSs are written as |r, k⟩ ab = 3