Fidelity in quantum teleportation of 2 qubits with decoherent channel

This study discusses the one-way quantum teleportation of 2 qubits through a 4-qubit Bell state channel that is subject to decoherence. Decoherence is one of the unavoidable effects of noise in the transmission process. In the case of quantum teleportation, noise has an impact on the channel, causing changes in its state due to noise. In the absence of noise, teleportation occurs in an ideal state, resulting in a fidelity value of 1, signifying successful information transfer in the ideal scenario. However, when examined in the presence of noise, fidelity is not equal to 1 due to the influence of noise factors and the coefficient of the transmitted state. These noise factors indicate the strength of the noise. The stronger the noise, the lower the fidelity becomes.


Introduction
Quantum teleportation [1] is a method enabling the transmission of quantum information from a sender to a receiver, all without physically relocating the qubit.Its objective is to send an unknown quantum state of a qubit using two classical bits, enabling the receiver to precisely recreate the original qubit state.In 1993, Bennet et al. introduced the initial quantum teleportation method using the Einstein-Podolsky-Rosen (EPR) source [2].In quantum teleportation, teleportation fidelity measures how closely the quantum state produced at the receiver's location (Bob) matches the original quantum state sent by the sender (Alice).
In practical implementations of quantum teleportation, several environmental factors may cause disturbances to the quantum channel used for information transfer.These disturbances include bit flip (BF), phase flip (PF), bit-phase flip (BPF), phase damping (PD), amplitude damping (AD), and decoherence.Many papers have addressed several of these disturbances except for decoherence.Decoherence is a process in which the quantum channel loses its coherence, leading to a deterioration in the quality of quantum information transfer [3].
Several quantum teleportation schemes with noise, such as BF, PF, BPF, PD, and AD channels, have been proposed with varying fidelity values [4,5,6].High fidelity becomes a crucial criterion in evaluating the efficiency and reliability of a quantum teleportation protocol.[7,8,9].For a simple protocol, the research in [4] employs BF, PF, BPF, PD, and AD types of noise.In this research, our focus is on revisiting two-qubit quantum teleportation as presented in [4], incorporating a decoherence channel.However, we utilize a Bell-state four-qubit as the channel.The primary objective and main motivation of this research is to comprehend the influence of the decoherence channel on the fidelity of two-qubit quantum teleportation.We will examine various decoherence scenarios that may occur in Bob's channel and observe how the degradation of the channel's quality affects teleportation fidelity.By understanding the impact of these factors, we hope to enhance the understanding of quantum teleportation performance in realistic conditions.This paper is organize as follows: in Section 2, we will present a two-qubit quantum teleportation scheme in an ideal environment.The two-qubit quantum teleportation scheme with an environment experiencing decoherence will be provided in Section 3. Some conclusions will be presented in Section 4.

The Quantum Teleportation Scheme in The Ideal Environment
On Alice's side, she holds two qubits in an unknown state, with the values of ω 0 , ω 1 , ω 2 , and ω 3 being complex numbers satisfying normalization with the relationship as follows, She aims to transmit this state to Bob by leveraging an Einstein-Podolsky-Rosen (EPR) source.This source provides two qubits entangled in two pairs.There are four possible maximally entangled states for a single particle: When there are two particles, there are 16 possibilities.The state of these two pairs of qubits is EPR source and her own two particles |α⟩ a 1 a 2 , resulting in Afterward, Alice attempts to measure |ϕ⟩ using a specific Bell-State Measurement (BSM) When Alice makes this measurement, the state of Bob's particles from the EPR source collapses, and Alice gains information about the state of Bob's particles, |β⟩ B 1 B 2 .From this information, Alice can inform Bob through a classical channel that Bob needs to perform a specific Unitary Transformation to make his particles have the same state as the one Alice had previously.
The results of the BSM, the state of Bob's qubits, and the Unitary Transformation that Bob needs to perform are presented in table 1.With X, Y, and Z are

BSM
The state of Bob's qubits (|β⟩ with X(σ x ) and Z(σ z ) being the standard Pauli matrices that we are familiar with in quantum mechanics.Meanwhile, Y is represented as −iσ y , where i is an imaginary number, and σ y is one of the Pauli matrices.The Pauli matrices according to quantum mechanics are The use of Y instead of σ y in this research is due to the fact that the states defined in Eq. (1) do not explicitly involve imaginary numbers.

The Quantum Teleportation Behaviour in The Decoherence Channel
In this section, we mathematically derive the fidelity of teleportation influenced by the nature of decoherence.After the mathematical derivation, we proceed to analyze this fidelity.To formulate this fidelity, several steps are required, as follows,

Step 1: Formulating the Channel Subject to Decoherence
The steps carried out do not differ significantly from the scheme in the ideal environmental scenario.The difference lies only in the mathematical operations performed on a state using its density matrix.If using the |B 1 ⟩ |B 1 ⟩ channel from the EPR source, the density matrix of that channel is In simple terms, decoherence can transform a channel into a mixed state.If defined in terms of density matrices in quantum mechanics, the state after undergoing decoherence is 10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012030 With λ being the decoherence factor, we can examine the equation for λ → 0 and λ → 1.When λ → 0, the density matrix of the channel becomes a pure state.On the other hand, if λ → 1, the channel becomes a completely mixed state.Moreover, Eq. ( 8) can also be interpreted as a superposition of a mixed state and a pure state, where 1−λ represents the probability of the channel transforming the state into a mixed state, and λ is interpreted as the probability of the channel maintaining itself in a pure state.Thus, the decoherence factor can also be interpreted as a probability.
The density matrix of the state can be reformulated in the form of a combination of Kraus operators in the case of two qubits, defined as follows: with M i and M j being the Kraus operators for the decoherence channel as follows [1] The indices B 1 and B 2 on the Kraus operators indicate that the noise operators operate on Bob's qubits in the channel given in Eq. (7).The operators are assumed to only act on Bob's qubits for mathematical simplicity.If the noise operators also acted on Alice's qubit (channel), there would be 256 terms.
In the equation, there are terms indexed by † indicating the conjugate form of the Kraus operators.When Eq. ( 10) is substituted into Eq.( 9), the equation for the channel after applying the Kraus operators is Eq. ( 11).This equation contains 16 terms resulting from the expansion of indices i, j.Additionally, there is a factor λ that arises from the Kraus operators.The first term in Eq. ( 11) is the result of the operator M 1 M 1 and its conjugate in Eq. (7).Meanwhile, the term containing the coefficient λ(4 − 3λ)/64 is a combination of the operators M 1 with M 2 , M 3 , M 4 and their conjugates.The term λ 2 /16 is a combination of two operators among M 2 , M 3 , and M 4 .
10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012030 In formulating the density matrix, it is evident that there are four qubits: A 1 , B 1 , A 2 , B 2 .Then, from this formulation, Alice will perform measurements on qubits a 1 A 1 and a 2 A 2 , which mathematically are performed on the qubits a 1 and a 2 , which belong to the information sent by Alice.

Step 2: Formulating the Channel Subject to Alice's Measurement
In this step, we formulate the state held by Bob in the form of a density matrix resulting from the measurement process carried out by Alice.First, Alice creates a joint state between the state of her two qubits ρ a 1 a 2 = |α⟩ ⟨α| and the ρ After define joint state, Alice then performs a measurement using BSM to Eq. ( 12), resulting in the state of Bob's qubits.The measurement is carried out as follows, (13) with, where i, j ∈ {1, 2, 3, 4}.Meanwhile, Therefore, we can calculate ρ out B 1 B 2 using Eq. ( 13).Thus, Bob's density matrix is Now, we can see from the equation that the density matrix state of the channel depends only on Bob's qubits (B 1 B 2 ).To compare the similarity of states between Alice and Bob, we need to calculate the fidelity.

Step 3: Calculating Fidelity
The fidelity of a state is mathematically defined as follows [1]: Here, |ψ⟩ represents Eq. ( 1).If we use Eq. ( 1) and the density matrix in Eq. ( 15), the fidelity can be expressed as follows In the equation, it is evident that the fidelity depends on the decoherence factor (λ) and the original parameter state (ω 0 , ω 1 , ω 2 , and ω 3 ).

Step 4: Analysis
The study provides fidelity (F ) plots in figures 1, 2, and 3. Notably, figures 1 and 2 reveal that when the noise factor (λ) is set to 0, the fidelity reaches its maximum value of 1.This signifies that information transmission to Bob remains unchanged, akin to the scenario of quantum teleportation in an ideal environment.Conversely, as the λ value approaches 1, the fidelity progressively tends toward 0. This indicates that the qubit received by Bob becomes significantly different from the one Alice initially intended to transmit.

Conclusion
The study involved calculating the fidelity of quantum teleportation in scenarios where one of the channels experiences decoherence.The results revealed that as the level of decoherence in Bob's channel increases, the fidelity of the transmitted quantum state progressively decreases.This highlights the vulnerability of quantum teleportation to environmental factors that induce decoherence.Secondly, in scenarios where the parameters ω 0 and ω 3 are subject to variation, it was observed that the fidelity exhibits a modest improvement as either ω 0 or ω 3 values increase.This suggests that optimizing the values of these parameters can enhance the reliability of quantum teleportation processes.
The main point that serves as the new message or novelty in this paper is that we can mathematically explain the physical phenomenon of teleportation influenced by disturbance factors.Moreover, it can also address the fidelity values less than 1 in teleportation experiments [10].Thus, there is a possibility that the photon transmission in the experiment [10] may undergo decoherence around the source.In summary, the research underscores the critical influence of decoherence in quantum teleportation and the potential for fidelity improvement by strategically adjusting the entangled channel parameters.These insights are valuable for the practical implementation of quantum teleportation in real-world applications.

Table 1 .
16Bell-State Measurements (BSM), the state of Bob's qubits, and the unitary transformation.