Fidelity Comparation of Two-Qubit Quantum Teleportation with Bit-Flip and Phase-Flip Channel

This study investigates the effect of noise on two-qubit quantum teleportation through a four-qubit entangled channel theoretically. Therefore, in this study a comparison of two states of teleportation through ideal and noisy channel was carried out. The noisy channel can have the effect of bit-flip channel and phase-flip channel. From these two effects, it is found that the channel transforms in-to a mixed state. If the noise is neglected, the channel is pure state. Then, this channel is measured and fidelity is calculated. The fidelity of an ideal environment is 1. This indicates that the teleportation was successfully sent and at the same time confirms the teleportation behavior under ideal conditions. In noisy environments, fidelity can be less than 1. Two factors in noisy environment that affect fidelity are noise factor (p) and the orginal parameter states (transmission coefficients). In this calculation, for the bit flip noise, we observed that as the values of the coefficient values of transmitted state increase, the fidelity value increases. Meanwhile, for the phase flip noise, we observed that as the values of the coefficient values of transmitted state increase, the fidelity value decreases.


Introduction
Entanglement is an intriguing phenomenon in quantum mechanics.This phenomenon is a consequence of the non-local nature of measurements in quantum mechanics.As a result, quantum mechanics, characterized by wave functions, does not govern an absolute representation of reality and locality.In 1935, Einstein and two colleagues (Podolsky and Rosen) were questioned about the incompleteness factors of quantum mechanics [1].Therefore, there is a need for another quantity called a hidden variable to describe the wave function more comprehensively.Subsequently, several theories and experiments regarding hidden variables were proposed to strengthen quantum mechanics [2].However, in 1964, John Bell proved through Bell inequalities that the proposed hidden variables did not exist [3].Hence, the concept of entanglement and its non-local behaviors were upheld.
Entanglement is one of the main factors in the improvement of quantum information theory.An application in communication with ongoing research until now is quantum teleportation.Quantum teleportation enables the occurrence of instant delivery over a long distance using entangled channel as a source.The first protocol was proposed by Benneth [4].In this process, an EPR channel is provided and distributed to Alice (sender) and Bob as (receiver).Then, the information sent by Alice becomes entangled with the EPR channel, and this entangled information is transmitted through classical communication.This entangled information is measured by Alice and Bob should executes associated unitary transformation on his qubit.
In addition to theoretical formulations, quantum teleportation has also been experimentally performed.For example, the transmission of a photon over a distance of 600 m via entangled photons shared between Alice and Bob [13], teleporting a photon from La Palma to Tenerife over 143 km [14], teleporting photon to solid-state qubit [15], and performs quantum teleportation in high dimension [16].One of the parameters measured in quantum teleportation experiments is fidelity.Fidelity is necessary to assess how closely a quantum state resembles another [13,14] which ranges from 0 to 1.If fidelity is equal to 1, it means that a state has been successfully transmitted.This can be achieved when quantum teleportation is performed in ideal conditions (noise-free).However, in quantum teleportation experiments, the occurrence of noise during the photon distribution process from the channel to the sender and receiver is highly possible.The occurrence of noise transforms a channel to mixed state.Thus, fidelity in quantum teleportation experiments is generally less than one [17,18].
Based on the fact that noise can affect the teleportation process, there have been several studies on noise-influenced quantum teleportation [19,20,21,22].The simplest example is studied two-qubit quantum teleportation with cluster channel in noisy environments [22].In a recent paper [22], the noise only affects Bob channel.In contrast, in actual situation, noise can also influence the channel distribution of both Alice (sender) and Bob (receiver).The novelty in this paper is we reconsider paper [22] with noise that affects both sender and receiver using the Bell state channel.Bell state channel is used rather than cluster channel because of the efficiency measurement.Efficiency means that we don't need to use controlled-Z gate in measurement.Then, we can improve measurement efficiency of quantum teleportation.
In this study, we examine the potential noise effects on the channel.To study the noise effects, it they can be reviewed first regarding the characteristics of photons.Naturally, photons can experience phase loss.Therefore, one of the effects we analyze is the phase-flip.In addition to phase, noise effects can also change the polarization direction of photons (for example, from vertical to horizontal).Thus, in this case, we also study the effects of bit-flip applied to the channel.The transmitted state is a two-qubit state through a four-qubit Bell channel.First, in section 2 we review the two-qubit state under ideal conditions and calculate its fidelity.Then, we derived mathematically quantum teleportation of the two-qubit state through the BF and the PF channel in section 3 and 4.Then, the fidelities of both noises are calculated.

Teleporting Two-Qubit in Ideal Channel
Suppose Alice sends two-qubit to Bob as follows which satisfies normalization condition The entangled channel between Alice and Bob is The first two qubits (A 1 , A 2 ) is belong to Alice, and the other two qubits (B 1 , B 2 ) belong to Bob.The joint state between channel and information in Eq.( 1) is This joint state is measured with Alice with basis First, if Alice do measurement |η 1 ⟩ a 1 A 1 to Eq.( 3), so the joint state become  3), then Eq.( 5) is Therefore, Bob needs to perform a unitary transformation σ 1 B 1 ⊗ σ 1 B 2 on his qubit to recreate Alice sent state.With 4 available measurements in Eq.( 4), there are 16 possible combination measurements and the corresponding unitary transformations for each measurement can be seen in the table 1.The possible unitary transformations are tensor product of two of four possible unitary transformations, These unitary transformations correspond to quantum gates which define the transformation of quantum bit.σ 1 is identity matrix to define transformation |0⟩ → |0⟩ and |1⟩ → |1⟩, and σ 4 is standar Pauli matrix σ x and σ z .However, σ 3 is −iσ y where i is imaginary number and σ y also Pauli matrix.In this work, we used σ 3 rather than σ y because we didn't explicitly involve imaginary number when we wrote Alice information in Eq. (1).
After measurement we can find the fidelity according Eq.( 8) where |χ⟩ ′ is Bob states after unitary transformations.Then using Eq.( 8) the fidelity of using all 16 possible transformations is 1.

Teleporting Two-Qubit in Bit-Flip Channel
The Kraus Operator defining BF channel is [22] The channel in Eq.( 2) transformed with Kraus operator become mixed state as follows 10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012029 The density matrix of output measurement is U is unitary operaton that defines the relationship between measurement basis and unitary transformation The density matrix of transmitted state is B 2 is unitary transformation Eq.( 7) correspond with measurement basis where i, j ∈ (1, 2, 3, 4).For example when we use measurement basis Therefore, Eq.( 11) becomes The matrix density of channel according the Kraus operator is The output density matrix of measurement according Eq.( 11) The fidelity of state |χ⟩ B 1 B 2 is

Comparison
We have derived mathematically the fidelity for BF and PF channel in Eq.( 19) and (15).From these equations, the fidelity is depends on a noise strength (noise factor) and original parameter states (transmitted states constant).The plot of fidelity against the noise factor (p) using Eq. ( 15) and ( 19) are shown in figures 1 and 2. In both figures, it is shown that when p = 0, the fidelity value is 1, indicating that teleportation is in the ideal state.
Where noise is considered, There are three regions of the noise factor value that can be analyzed physically.When the value of p < 0.5, the probability of a particle experiencing flipping is smaller compared to not experiencing flipping.Thus, the presence of particles not experiencing flipping is more dominant than those experiencing flipping.Therefore, physically, when noise is added, fidelity can continue to decrease when the channel is given noise.When p = 0.5, it indicates a symmetrical value.This shows that the probability for particles to   experiencing flipping is greater than the probability of not experiencing flipping.This indicates that the presence of particles experiencing flipping is less than that of those not experiencing flipping (noise is more dominant).Therefore, physically, when the noise factor is added in the case of p > 0.5, fidelity can increase because the percentage of ideal particles (not influenced by noise) is decreasing.Next, we examine the fidelity values in terms of the transmission coefficient.In figure 1, it is shown that increasing the transmission coefficient leads to higher fidelity.However, in figure 2, an increasing transmission coefficient leads to lower fidelity.Thus, the transmission coefficient can affect fidelity.The fidelity comparison between the effect of BF and PF channel can be seen in figure 3. It is clear that effect of BF is stronger than PF noise.This is indicated by the lower fidelity value in the BF channel compared to the PF channel.

Conclusion
The comparison of fidelity between the BF channel and the PF channel has been conducted.Fidelity in the ideal state (p = 0) is 1.Meanwhile, the fidelity of both BF and PF noise depends on the noise factor (p) and the transmission coefficients (µ 00 , µ 01 , µ 10 , µ 11 ).The fidelity of the measurement affects the delivery efficiency.The effect of the BF channel is stronger than PF channel, so the efficiency of transmitting information influenced by BF is lower than PF channel.Therefore, quantum teleportation can be comprehensively studied in both ideal and noisy environment.

Figure 1 :
Figure 1: Plot of fidelity versus noise factor (p) in the BF channel

Figure 2 :Figure 3 :
Figure 2: Plot of fidelity versus noise factor (p) in the PF channel