Effects of partial measurements on quantum resources and quantum Fisher information of a teleported state in a relativistic scenario

Abstract

We address the teleportation of single- and two-qubit quantum states, parametrized by weight θ and phase ϕ parameters, in the presence of the Unruh effect experienced by a mode of a free Dirac field. We investigate the effects of the partial measurement (PM) and partial measurement reversal (PMR) on the quantum resources and quantum Fisher information (QFI) of the teleported states. In particular, we discuss the optimal behaviour of the QFI, quantum coherence (QC) as well as fidelity with respect to the PM and PMR strength and examine the effect of the Unruh noise on optimal estimation. It is found that, in the single-qubit scenario, the PM (PMR) strength at which the optimal estimation of the phase parameter occurs is the same as the PM (PMR) strength with which the teleportation fidelity and the QC of the teleported single-qubit state reaches its maximum value. On the other hand, generalizing the results to two-qubit teleportation, we find that the encoded information in the weight parameter is better protected against the Unruh noise in two-qubit teleportation than in the one-qubit scenario. However, extraction of information encoded in the phase parameter is more efficient in single-qubit teleportation than in the two-qubit version.

1. Introduction

Quantum teleportation [1] is undoubtedly one of the most striking implications predicted by quantum mechanics, and it is an important ingredient for quantum communication and quantum information processing (QIP) [2,3]. In recent decades, the theoretical and experimental consideration of quantum teleportation has attracted many researchers’ attention [4–18]. Quantum teleportation is described as a process by which an arbitrary unknown quantum state can be transmitted faithfully from one object to another, without physical travelling of the object itself. The system is isolated from the external forces in the original form of the teleportation [1], and a maximally entangled pair is used as the resource. However, decoherence [19,20] is an inevitable phenomenon in open quantum systems; this takes place as a result of the interaction between the system and the environment. This leads to the degradation of quantum correlations, a fundamental resource for QIP, and therefore influences the fidelity in quantum state teleportation [21–23].

Relativistic quantum information (RQI) [24,25] aims to realize the relationship between relativity and quantum information and to combine relativistic effects to amend quantum information tasks, e.g. quantum teleportation. Moreover, in RQI we try to understand how these protocols may be realized in curved space–time. The Unruh effect [26,27], a significant prediction in quantum field theory, proposes that a uniformly accelerated observer in Minkowski space–time (Rindler observer) associates a thermal bath of Rindler particles with the no particle state of the inertial observer (called the Minkowski vacuum). The decoherence effect, produced by the Unruh effect, suppresses the quantum resources (QRs) such as quantum coherence (QC) [28], quantum discord (QD) [29,30] and entanglement [30] in the case of bosonic or Dirac field modes. The degradation of QRs unavoidably decreases the confidence of some quantum information tasks like quantum teleportation. In this context, it is really important to preserve QRs from decoherence during the teleportation process. Here we investigate the teleportation of single- and two-qubit quantum states in which the resource state of the teleportation is affected by the Unruh effect experienced by a mode of a free Dirac field, as seen by a relativistically accelerated observer. This type of teleportation channel discussed in this paper is called the Unruh noise channel throughout the text.

In addition to the teleportation of the whole quantum state, we also investigate the teleportation of the information encoded into a particular parameter. In contrast to quantum state teleportation where the quality of teleportation is characterized by fidelity, the credibility of the teleportation of specific information is usually determined by the quantum Fisher information (QFI) [31–34]. Representing the sensitivity of the state with respect to changes in a parameter, it plays an important role in parameter estimation theory and is extensively employed in QIP. In particular, the QFI has many applications in quantum information tasks such as entanglement detection [35,36], specifying the non-Markovianity [37–39], quantum thermometry [40] and consideration of uncertainty relations [41–43]. Hence it is of interest to study the QFI in a relativistic framework. Nevertheless, it is shown that the QFI is fragile and can be broken easily because of unavoidable decoherence effects [44–48]. This is the most restricting factor in QFI applications for quantum teleportation. Therefore, protecting the QFI from decoherence is a fundamental subject.

In partial measurement (PM), associated with a positive operator-valued measure (POVM), the system state does not completely collapse such that the initial state could be reversed with some operations. Recently, PM together with partial measurement reversal (PMR) have been exploited as a practical method to protect quantum correlations of two qubits as well as two qutrits and the fidelity of a single qubit, from amplitude damping (AD) decoherence [49–53]. In [54], the effect of PMs (hereafter, we use PMs to indicate both PM and PMR) on QFI of a teleported single-qubit state under the AD noise has been studied, and it was shown that the combination of PM and PMR could totally eliminate the influence of decoherence. The effects of PM and PMR on the enhancement of quantum coherence and QFI, transmitted under a quantum spin-chain channel, have been considered in [55]. Moreover, it has been shown that PM and PMR are able to improve the fidelity of teleportation when one or both qubits of the maximally entangled state shared between Alice and Bob suffer from the AD decoherence [56]. It was also shown in [56] that this protocol works for the Werner states. However, limited attention has been paid to protect the QRs and QFI against Unruh decoherence during the procedure of teleportation. Motivated by this, we study the enhancement effect of PM and PMR on QRs and QFI of the teleported state through the Unruh noise channel for both single- and two-qubit input quantum states.

In this paper, we have investigated the following scenario: the system consists of an inertial observer Alice and a uniformly accelerated observer Rob. Two PMs are performed before and after Rob’s acceleration, which are called PM and PMR, respectively. Then we use the above-mentioned system as a resource in order to teleport single- and two-qubit states, and consider how the degradation effect of the Unruh channel on QRs and QFI of the teleported state as well as teleportation fidelity can be improved by PM or PMR. According to our results, the combined effect of PM and PMR with the same strengths (p = q) may improve QRs and QFI with respect to phase parameter φ, of the teleported state, and also teleportation fidelity in both single-qubit and two-qubit scenarios. Our study differs from [57], in which the performance of the QFI of an arbitrary two-qubit state under the Unruh effect has been addressed. Here the two-qubit state exposed to the Unruh noise in the Dirac field and affected by PMs is used as a resource to teleport single- and two-qubit unknown quantum states. Our work also differs from [13], in which the two-qubit state exposed to the Unruh noise in the scalar field has been used as a resource to teleport a two-qubit state.

This paper is organized as follows: in §2, we give a brief description about teleportation, PM, PMR, QRs and QFI. The physical model is presented in §3. The probability of preparing the resource state of the quantum channel in a teleportation protocol is discussed in §4. We study the single-qubit teleportation as well as two-qubit teleportation under the Unruh noise channel in §§5 and 6, respectively. Finally, §7 is devoted to our conclusion.

2. Preliminaries

(a). Teleportation

The main idea of quantum teleportation is transferring quantum information about an unknown quantum state in one location (Alice) to another location (Bob) where it is spatially separated. An important factor in quantum teleportation is the channel connecting the sender and receiver. In a standard teleportation protocol T₀, local quantum operations, used to teleport the quantum state, include Bell measurements and Pauli rotations. According to the results of Bowen & Bose [58], the standard teleportation protocol T₀ with mixed states as a resource is tantamount to a generalized depolarizing channel.

(i). Single-qubit teleportation

As mentioned above, a teleportation protocol using a two-qubit mixed state as a resource acts as a generalized depolarizing channel Λ(ρ_ch). Therefore, for an arbitrary single-qubit state to be teleported (henceforth called the input state ρ_in), the output state ρ_out of the teleportation is obtained as follows [58]:

$$\begin{array}{rl}{\rho }_{\text{out}}& =\mathrm{\Lambda }({\rho }_{\text{ch}}){\rho }_{\text{in}}\\ \hspace{0.17em}& =\sum _{i=0}^3\text{Tr}({\mathcal{B}}_i{\rho }_{\text{ch}}){\sigma }_i{\rho }_{\text{in}}{\sigma }_i,\end{array}$$ 2.1

in which ${\mathcal{B}}_i$’s represent the Bell states associated with the Pauli matrices σ_i’s by

$${\mathcal{B}}_i=({\sigma }_0\otimes {\sigma }_i){\mathcal{B}}_0({\sigma }_0\otimes {\sigma }_i);i=1,2,3,$$ 2.2

where σ₀ = I, σ₁ = σ_x, σ₂ = σ_y, σ₃ = σ_z, and I is the 2 × 2 identity matrix. For any arbitrary two qubits, each described by basis $\{|0⟩,|1⟩\}$, we have ${\mathcal{B}}_0=\frac{1}{2}(|00⟩+|11⟩)(⟨00|+⟨11|)$, without loss of generality. In addition, ρ_ch indicates the resource state of the quantum channel shared between Alice and Bob. It should be noted that this resource state of the teleportation channel will be affected by the Unruh effect in our relativistic model.

(ii). Two-qubit teleportation

Teleportation of an unknown entangled state via two independent quantum channels has been studied by Lee & Kim [59]. Actually, their protocol may be carried out by doubling the standard teleportation protocol T₀. Figure 1 displays a schematic drawing of entanglement teleportation.

Figure 1.

Schematic drawing of entanglement teleportation. An unknown entangled state ρ_in is generated by source S, and its particles are dispensed into A₁ and A₂. In addition, two independent entangled pairs, numbered (3,5) and (4,6), are produced from source E. These pairs, each characterized by density matrix ρ_ch, play the role of the quantum channels Q₁ and Q₂. The measurement result at A_i (i = 1, 2) is transmitted through the classical channel C_i to B_i. Based on the information received by the classical communication, the unitary transformations are performed on particles 5 and 6 at B_i (i = 1, 2) to complete the teleportation. (Online version in colour.)

Generalizing equation (2.1), the output state of the entanglement teleportation is found as follows:

$${\rho }_{\text{out}}=\sum _{ij}{p}_{ij}({\sigma }_i\otimes {\sigma }_j){\rho }_{\text{in}}({\sigma }_i\otimes {\sigma }_j),i,j=0,x,y,z,$$ 2.3

where $p_{ij}=\text{Tr}({\mathcal{B}}_i{\rho }_{\text{ch}})\text{Tr}({\mathcal{B}}_j{\rho }_{\text{ch}})$ and $\sum p_{ij}=1$.

(b). Partial measurement and partial measurement reversal

We first give a brief introduction to PM and PMR. In contrast to the standard von Neumann projective measurement, which completely collapses the measured system, PM, as a generalization of standard von Neumann projective measurement, does not totally collapse the initial state into the eigenstates, and hence the measured state is reversible by proper PMR operations. For a single qubit, the PM is described by the following pair of measurement operators:

$$M_0=\sqrt{p}|0⟩⟨0|$$ 2.4

and

$$M_1=\sqrt{1-p}|0⟩⟨0|+|1⟩⟨1|,$$ 2.5

where p(0 ≤ p ≤ 1) is the strength of PM and $M_0^{†}{M}_0+M_1^{†}{M}_1=I$. M₀ is identical to the von Neumann projective measurement and is irreversible, while M₁ is a PM which could be reversed for the case p ≠ 1. On the other hand, the PMR can be described by a non-unitary operator

$$M_1^{-1}=\frac{1}{\sqrt{1-q}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}\sqrt{1-q}& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)=\frac{1}{\sqrt{1-q}}X{M}_{1}X,$$ 2.6

where X = |0〉〈1| + |1〉〈0| is the bit-flip operation. The last term of equation (2.6) implies that the PMR can be implemented physically by the sequence of a bit-flip operation, another PM with measurement strength q and a second bit-flip operation. Therefore, the PMR could exactly undo the PM by choosing q = p.

(c). Quantum Fisher information

QFI is an important concept in parameter estimation theory. QFI of an unknown parameter λ encoded in quantum state ρ(λ) is defined as [31,60]

$$F_Q(\lambda )=\text{Tr}[\rho (\lambda )L^2]=\text{Tr}[({\mathrm{\partial }}_{\lambda }\rho (\lambda ))L],$$ 2.7

where L, the symmetric logarithmic derivative (SLD), is given by ${\mathrm{\partial }}_{\lambda }\rho (\lambda )=\frac{1}{2}(L\rho (\lambda )+\rho (\lambda )L),$ with ∂_λ = ∂/∂λ.

A simple and explicit expression can be acquired for the single-qubit state. Any qubit state can be expressed in the Bloch sphere representation as $\rho =\frac{1}{2}(I+\mathit{\omega }\cdot \mathit{\sigma })$, where ω = (ω_x, ω_y, ω_z)^T is the Bloch vector and σ = (σ_x, σ_y, σ_z) indicates the Pauli matrices. Hence the QFI of the single-qubit state can be formulated as follows [61]:

$$F_Q(\lambda )=\{\begin{array}{ll}|{\mathrm{\partial }}_{\lambda }\mathit{\omega }{|}^2+\frac{{(\mathit{\omega }\cdot {\mathrm{\partial }}_{\lambda }\mathit{\omega })}^2}{1-|\mathit{\omega }{|}^2},& |\mathit{\omega }|<1,\\ |{\mathrm{\partial }}_{\lambda }\mathit{\omega }{|}^2,& |\mathit{\omega }|=1,\end{array}$$ 2.8

where |ω| < 1 is used for a mixed state while |ω| = 1 is applicable for a pure state.

(d). Quantum resources

(i). Quantum coherence

QC arising from the superposition principle is an important resource in quantum information and quantum computation processing. It plays a fundamental role in quantum mechanics. Various measures are expressed to quantify the coherence, such as trace norm distance coherence [62], l₁ norm and relative entropy of coherence [63]. For a quantum state with the density matrix ρ, the l₁ norm measure of quantum coherence [63] quantifying the coherence through the off-diagonal elements of the density matrix in the reference basis is given by

$${\mathcal{C}}_{l_1}(\rho )=\sum _{\genfrac{}{}{0ex}{}{i,j}{i\ne j}}|{\rho }_{ij}|.$$ 2.9

(ii). Entanglement

Entanglement is recognized as a resource in QIP and is accountable to the advantage of many quantum computation and communication tasks. Actually, entanglement indicates correlations regarding non-separability of the state of a composite quantum system. Entanglement of a bipartite system is quantified conveniently by concurrence [64], which can be computed analytically for an X state as follows:

$$C(\rho )=2\text{max}\{0,C_1(\rho ),C_2(\rho )\},$$ 2.10

where $C_1(\rho )=|{\rho }_{14}|-\sqrt{{\rho }_{22}{\rho }_{33}}$, $C_2(\rho )=|{\rho }_{23}|-\sqrt{{\rho }_{11}{\rho }_{44}}$ and ρ_ij’s are the elements of the density matrix. Concurrence equals unity for maximally entangled states and vanishes for separable states.

(iii). Quantum discord

QD representing quantumness of the state of a quantum system is a resource for certain quantum technologies. It can be preserved for a long time even when entanglement shows a sudden death. QD for any bipartite system is defined as the difference between the total correlations (i.e. quantum mutual information) and the classical correlations. Computation of QD for general states is not usually a convenient task since it involves the optimization of the classical correlations. However, for a two-qubit X-state system, the analytical expression of QD can be obtained as [65]

$$\text{QD}({\rho }_{AB})=\text{min}(Q_1,Q_2),$$ 2.11

where

$$\begin{array}{rl}\hspace{0.17em}& Q_j=H({\rho }_{11}+{\rho }_{33})+\sum _{i=1}^4{\lambda }_i{\text{log}}_2{\lambda }_i+D_j,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(j=1,2),\hspace{0.17em}\\ \hspace{0.17em}& D_1=H\left(\frac{1+\sqrt{{[1-2({\rho }_{33}+{\rho }_{44})]}^2+4{(|{\rho }_{14}|+|{\rho }_{23}|)}^2}}{2}\right),\\ \hspace{0.17em}& D_2=-\sum _i{\rho }_{ii}{\text{log}}_2{\rho }_{ii}-H({\rho }_{11}+{\rho }_{33}),\hspace{0.17em}\\ \hspace{0.17em}& H(x)=-x{\text{log}}_{2}x-(1-x){\text{log}}_2(1-x),\end{array}\}$$ 2.12

and λ_i’s denote the eigenvalues of density matrix ρ_AB.

3. Physical model

We consider a system including an inertial observer Alice (A) and a uniformly accelerated observer Rob (R). Let Alice and Rob initially share the following entangled state of two Dirac field modes:

$$|\mathrm{\Psi }(0)⟩=\text{sin}\hspace{0.17em}\frac{\vartheta }{2}{|0⟩}_A{|0⟩}_R+\text{cos}\hspace{0.17em}\frac{\vartheta }{2}{|1⟩}_A{|1⟩}_R,$$ 3.1

where $|0⟩$ and $|1⟩$ denote the Minkowski vacuum and excited states [57], respectively.

We assume that Rob first performs a PM of the form (2.5) on his particle. State (3.1) reduces to

$$|\mathrm{\Psi }(1)⟩=\frac{1}{\sqrt{N_1}}(\text{sin}\hspace{0.17em}\frac{\vartheta }{2}\hspace{0.17em}\sqrt{\overline{p}}{|0⟩}_{\text{A}}{|0⟩}_{\text{R}}+\text{cos}\hspace{0.17em}\frac{\vartheta }{2}{|1⟩}_{\text{A}}{|1⟩}_{\text{R}}),$$ 3.2

where $N_1={\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{p}+{\text{cos}}^2\frac{\vartheta }{2}$ represents the normalization factor and $\overline{p}=1-p$. If p approaches unity, state (3.2) is projected into state |1_A〉|1_R〉, which is not affected by the Unruh effect.

Now Rob begins to accelerate uniformly. Hence he sees that the Unruh effect is experienced by the Dirac mode which is in the vacuum state from an inertial perspective such that his detector perceives a Fermi–Dirac distribution of particles. Rindler coordinates (τ, ξ) is a better option to describe what he sees. Rindler space–time manifests two regions I and II, causally disconnected (figure 2). Because of the eternal acceleration, Rob travels on a hyperbola compelled in region I.

Figure 2.

Rindler space–time embedded into Minkowski space–time: a uniformly accelerated observer (Rob) with acceleration a travels on a hyperbola constrained to region I, while a fictitious observer (anti-Rob) travels on a corresponding hyperbola in region II. The straight line indicates the world line of an inertial observer (Alice). (Online version in colour.)

In Rob’s reference frame, the Minkowski vacuum state can be expressed in terms of the Rindler region I and II states [66],

$${|0⟩}_M=\text{cos}\hspace{0.17em}r{|0⟩}_{\text{I}}{|0⟩}_{\text{II}}+\text{sin}\hspace{0.17em}r{|1⟩}_{\text{I}}{|1⟩}_{\text{II}},$$ 3.3

while the excited state appears as a product state

$${|1⟩}_M={|1⟩}_{\text{I}}{|0⟩}_{\text{II}},$$ 3.4

where $r=\text{arccos}\sqrt{1+{\text{e}}^{-2\pi \omega /a}}$, ω is the Dirac particle frequency and a is the acceleration. Since 0 < a < ∞, therefore r ∈ [0, π/4]. Note that the observers in regions I and II are causally disconnected. Since the mode corresponding to II is not observable for Rob in region I, it should be traced out.

Expanding Minkowski particle states ${|0⟩}_R$ and ${|1⟩}_R$ by using, respectively, equations (3.3) and (3.4), we find that the state represented in (3.2) changes to

$$|\mathrm{\Psi }(2)⟩=\frac{1}{\sqrt{N_1}}[\text{sin}\hspace{0.17em}\frac{\vartheta }{2}\hspace{0.17em}\sqrt{\overline{p}}(\text{cos}\hspace{0.17em}r{|0⟩}_{\text{A}}{|0⟩}_{\text{I}}{|0⟩}_{\text{II}}+\text{sin}\hspace{0.17em}r{|0⟩}_{\text{A}}{|1⟩}_{\text{I}}{|1⟩}_{\text{II}})+\text{cos}\hspace{0.17em}\frac{\vartheta }{2}{|1⟩}_{\text{A}}{|1⟩}_{\text{I}}{|0⟩}_{\text{II}}].$$ 3.5

In the next step, a PMR is carried out by Rob in region I. If the PMR is successfully performed, we achieve

$$\begin{array}{rl}|\mathrm{\Psi }(3)⟩=\frac{1}{\sqrt{N_2}}[& \text{sin}\hspace{0.17em}\frac{\vartheta }{2}\hspace{0.17em}\sqrt{\overline{p}}(\text{cos}\hspace{0.17em}r{|0⟩}_{\text{A}}{|0⟩}_{\text{I}}{|0⟩}_{\text{II}}+\text{sin}\hspace{0.17em}r\sqrt{\overline{q}}{|0⟩}_{\text{A}}{|1⟩}_{\text{I}}{|1⟩}_{\text{II}})\\ \hspace{0.17em}& +\text{cos}\hspace{0.17em}\frac{\vartheta }{2}\sqrt{\overline{q}}{|1⟩}_{\text{A}}{|1⟩}_{\text{I}}{|0⟩}_{\text{II}}],\end{array}$$ 3.6

where $N_2={\text{sin}}^2(\vartheta /2)\hspace{0.17em}\overline{p}\hspace{0.17em}{\text{cos}}^{2}r+{\text{sin}}^2(\vartheta /2)\hspace{0.17em}\overline{pq}\hspace{0.17em}{\text{sin}}^{2}r+{\text{cos}}^2(\vartheta /2)\hspace{0.17em}\overline{q}$ is the normalization factor and $\overline{q}=1-q$, in which q represents the second PM strength. Since Rob is restricted to region I as a result of the causality condition, we trace the state over region II, resulting in the following mixed state between Alice and Rob [67]:

$${\rho }_{A,R}=\frac{1}{N_2}\left(\begin{array}{cccc}{\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{p}\hspace{0.17em}{\text{cos}}^{2}r& 0& 0& \text{sin}\hspace{0.17em}\frac{\vartheta }{2}\text{cos}\hspace{0.17em}\frac{\vartheta }{2}\hspace{0.17em}\sqrt{\overline{pq}}\hspace{0.17em}\text{cos}r\\ 0& {\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{pq}\hspace{0.17em}{\text{sin}}^{2}r& 0& 0\\ 0& 0& 0& 0\\ \text{sin}\hspace{0.17em}\frac{\vartheta }{2}\text{cos}\hspace{0.17em}\frac{\vartheta }{2}\hspace{0.17em}\sqrt{\overline{pq}}\hspace{0.17em}\text{cos}r& 0& 0& {\text{cos}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{q}\end{array}\right).$$ 3.7

Next we discuss the preparation probability of state (3.7).

4. Preparation probability

In this section, we address the probability of preparing the system in state (3.7). If the state of a quantum system is ρ immediately before the measurement, then the probability that result m occurs is given by $P_m=\text{tr}(M_m^{†}{M}_m\rho )$ [68], where m is the measurement outcome. In order to find the preparation probability, we start with the initial state (3.1) shared between Alice and Rob. If the state of the system is $|\mathrm{\Psi }(0)⟩$ immediately before the first PM then the probability that M₀ is carried out successfully is given by

$$\begin{array}{rl}{P}_1& =⟨\mathrm{\Psi }(0)|M_1^{†}{M}_1|\mathrm{\Psi }(0)⟩\\ \hspace{0.17em}& ={\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{p}+{\text{cos}}^2\frac{\vartheta }{2}.\end{array}$$ 4.1

Now we proceed with finding the probability that the PMR is carried out successfully. As we explained in §2b, the PMR can be implemented by concatenation of a bit-flip operation, second PM of strength q and a final bit-flip operation. Assuming that the first PM has been performed successfully, we find that the state of the system is $\rho ={\text{tr}}_{\text{II}}(X|\mathrm{\Psi }(2)⟩⟨\mathrm{\Psi }(2)|X)$ immediately before the second PM, and hence the success probability of the second PM is obtained as follows:

$$\begin{array}{rl}{P}_2& =\text{tr}(M_1^{†}{M}_1\rho )\\ \hspace{0.17em}& =\frac{{\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{p}\hspace{0.17em}{\text{cos}}^{2}r+{\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{pq}\hspace{0.17em}{\text{sin}}^{2}r+{\text{cos}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{q}}{{\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{p}+{\text{cos}}^2\frac{\vartheta }{2}}.\end{array}$$ 4.2

Now we are ready to find the preparation probability of the system in state (3.7), i.e. the probability that the first and the second PMs are carried out successfully,

$$\begin{array}{rl}P& =P_1\cdot P_2\\ \hspace{0.17em}& ={\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{p}\hspace{0.17em}{\text{cos}}^{2}r+\overline{q}({\text{sin}}^2\frac{\vartheta }{2}\overline{p}\hspace{0.17em}{\text{sin}}^{2}r+{\text{cos}}^2\frac{\vartheta }{2}).\end{array}$$ 4.3

In figure 3, the preparation probability is plotted versus the acceleration parameter r, for different values of p and q, fixing $\vartheta =\pi /2$. We see that the preparation probability decreases with increases in p or q. Moreover, it is more robust against the increase of r for p > q than the case q > p. The same results are obtained for $0<\vartheta <\frac{\pi }{2}$.

Figure 3.

The probability of preparing the state of the channel as a function of r for different values of p and q, fixing $\vartheta =\pi /2$. (Online version in colour.)

Next we discuss how PM and PMR affect the QRs and QFI of the state teleported through the Unruh noise channel.

5. Single-qubit teleportation under the Unruh noise channel

In this section, the QFIs and QC of the teleported single-qubit state, through the Unruh noise channel, are investigated. We consider |ψ_in〉 = cos~θ/2 |0〉 + e^iφ~sin~θ/2 |1〉, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π as the input state in the process of teleportation, where θ and φ are the weight and phase parameters, respectively. We use the shared state between Alice and Rob, equation (3.7), as the resource (ρ_A,I = ρ_ch) to teleport the single-qubit input state. Using equation (2.1), the output state can be obtained as follows:

$${\rho }_{\text{out}}^{\text{PM}}=\frac{1}{N_2}\left(\begin{array}{cc}\mathcal{A}{\text{cos}}^2\frac{\theta }{2}+\mathcal{D}{\text{sin}}^2\frac{\theta }{2}& \mathcal{F}{\text{e}}^{-\text{i}\phi }\text{sin}\hspace{0.17em}\theta \\ \mathcal{F}{\text{e}}^{\text{i}\phi }\text{sin}\hspace{0.17em}\theta & \mathcal{A}{\text{sin}}^2\frac{\theta }{2}+\mathcal{D}{\text{cos}}^2\frac{\theta }{2}\end{array}\right),$$ 5.1

where

$$\begin{array}{rl}\hspace{0.17em}& \mathcal{A}={\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{p}\hspace{0.17em}{\text{cos}}^{2}r+{\text{cos}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{q},\\ \hspace{0.17em}& \mathcal{D}={\text{sin}}^2\frac{\vartheta }{2}\hspace{0.17em}\overline{pq}\hspace{0.17em}{\text{sin}}^{2}r,\\ \hspace{0.17em}& \mathcal{F}=\text{sin}\frac{\vartheta }{2}\text{cos}\frac{\vartheta }{2}\sqrt{\overline{pq}}\hspace{0.17em}\text{cos}\hspace{0.17em}r\end{array}\}$$ 5.2

and $N_2=\mathcal{A}+\mathcal{D}$.

The QFIs of the input state |ψ_in〉 = cos~θ/2 |0〉 + e^iφ~sin~θ/2 |1〉, with respect to parameters θ and φ, is easily found to be F_in(θ) = 1 and F_in(φ) = sin²θ, respectively. It is seen that F_in(φ) is dependent on θ and is maximized for $\theta =\frac{\pi }{2}$ while F_in(θ) is independent of weight parameter θ and has a constant value. Therefore, if the input state is not exposed to the Unruh noise channel, it is preferred to be balance-weighted to achieve the best estimation of the phase parameter.

We use equation (2.8) to calculate the QFIs of the output state (the teleported state) (5.1). The corresponding Bloch vector is given by

$$\mathit{\omega }=\frac{1}{N_2}(2\mathcal{F}\text{sin}\theta \hspace{0.17em}\text{cos}\phi ,2\mathcal{F}\text{sin}\theta \hspace{0.17em}\text{sin}\phi ,(\mathcal{A}-\mathcal{D})\hspace{0.17em}\text{cos}\theta ),$$ 5.3

where its components are calculated by ${\omega }_i=\text{Tr}({\rho }_{\text{out}}^{\text{PM}}{\sigma }_i),\hspace{0.17em}i=1,2,3$. Therefore, the QFIs with respect to weight and phase parameters are found, respectively, as follows:

$$F_{\text{out}}^{\text{PM}}(\theta )=\frac{1}{N_2^2}({(\mathcal{A}-\mathcal{D})}^2{\text{sin}}^2\theta +4{\mathcal{F}}^2{\text{cos}}^2\theta +\frac{\frac{1}{4}{({(\mathcal{A}-\mathcal{D})}^2-4{\mathcal{F}}^2)}^2{\text{sin}}^{2}2\theta }{N_2^2-{(\mathcal{A}-\mathcal{D})}^2{\text{cos}}^2\theta -4{\mathcal{F}}^2{\text{sin}}^2\theta })$$ 5.4

and

$$F_{\text{out}}^{\text{PM}}(\phi )=4{\left|\frac{\mathcal{F}\text{sin}\hspace{0.17em}\theta }{N_2}\right|}^2.$$ 5.5

In figure 4, the QFI with respect to the weight parameter, $F_{\text{out}}^{\text{PM}}(\theta )$, for single-qubit state teleportation through the pure Unruh decoherence and for the case that the combination of PM and PMR have been applied is plotted as a function of acceleration parameter r. It can be seen that, after teleportation under the pure Unruh channel (i.e. p = q = 0), when the acceleration increases the QFI decays monotonously for all values of the initial parameter $\vartheta$. Studying the behaviour of $F_{\text{out}}^{\text{PM}}(\theta )$, when the PM and PMR are applied on the channel, we observe that applying either PM (i.e. q = 0) or PMR (i.e. p = 0) may improve $F_{\text{out}}^{\text{PM}}(\theta )$ for all initial states of the channel (figure 4a,b). For sufficiently strong measurement strength (p → 1 or q → 1), the precision of estimating the weight parameter can be enhanced remarkably and it is almost robust against the Unruh decoherence.

Figure 4.

QFI of the single-qubit teleported state with respect to the weight parameter, θ, as a function of acceleration parameter r by fixing $\theta =\frac{\pi }{2}$ and for $0<\vartheta <\pi$ for (a) q = 0, (b) p = 0. (Online version in colour.)

The important question that comes up is: If the acceleration is constant, how can one control the QFI by applying PM and PMR? In figure 5, we consider the $F_{\text{out}}^{\text{PM}}(\theta )$ behaviour versus p. It is observed that, in the absence of PMR (i.e. q = 0), $F_{\text{out}}^{\text{PM}}(\theta )$ enhances with increases in p (dashed purple line) for all values of the channel parameter $\vartheta$. It is also seen that, with the combined effect of PM and PMR, estimation precision of the weight parameter is also improved. We obtain the same results investigating the behaviour of $F_{\text{out}}^{\text{PM}}(\theta )$ versus q.

Figure 5.

QFI of the single-qubit teleported state with respect to the weight parameter, θ, as a function of PM strength, p, for r = 0.6, and different values of PMR strength, q. (Online version in colour.)

In figure 6, $F_{\text{out}}^{\text{PM}}(\phi )$, for single-qubit state teleportation, is plotted as functions of PM as well as PMR strength for a fixed value of the acceleration parameter r = 0.6 and the maximally entangled input state (θ = π/2, φ = 0). From figure 6a, it is seen that, for $\frac{\pi }{2}\le \vartheta <\pi$, with increases in PM strength $F_{\text{out}}^{\text{PM}}(\phi )$ increases to reach a maximum value and then decreases with more increases in p. Moreover, comparing the behaviour of $F_{\text{out}}^{\text{PM}}(\phi )$ for different values of PMR strength, we see that, with an increase in q, the optimal estimation of the phase parameter occurs for larger values of p. Nevertheless, the increase in the PMR strength interestingly raises the optimal value of the QFI, leading to enhancement of the phase parameter estimation. We also see that in this range of θ, while for small values of p the QFI may fall with an increase in q, it can enhance as q increases for larger values of p. We obtain the same results when investigating the behaviour of $F_{\text{out}}^{\text{PM}}(\phi )$ versus q for $0<\vartheta \le \frac{\pi }{2}$. In particular, in this range, the QFI may decrease with an increase in p for small values of q, while it can exhibit increasing behaviour as p increases for large values of q (figure 6b).

Figure 6.

QFI of the single-qubit teleported state with respect to the phase parameter, φ, as a function of (a) PM strength, p, fixing $\vartheta =\frac{3\pi }{4}$ and (b) PMR strength, q, fixing $\vartheta =\frac{\pi }{4}$. We have chosen the acceleration parameter r = 0.6. (Online version in colour.)

Considering the optimal behaviour of QFI of the single-qubit teleported state with respect to the phase parameter as functions of p and q, we obtain p_opt and q_opt as follows:

$$\begin{array}{rl}\hspace{0.17em}& p_{\text{opt}}=\frac{q{\text{cos}}^{2}r\hspace{0.17em}{\text{sin}}^2\frac{\vartheta }{2}-\text{cos}\vartheta (1-q)}{{\text{sin}}^2\frac{\vartheta }{2}(1-q{\text{sin}}^{2}r)}\\ \text{and}& q_{\text{opt}}=\frac{{\text{sin}}^2\frac{\vartheta }{2}(p+2(1-p){\text{cos}}^{2}r)-1}{{\text{sin}}^2\frac{\vartheta }{2}(p+(1-p){\text{cos}}^{2}r)-1}.\end{array}\}$$ 5.6

Figure 7 shows the variation in the optimal values of p and q, at which the optimal estimation of the phase paramater happens, in terms of the acceleration parameter r. We see that, when the acceleration increases, the optimal value of p, p_opt, decreases, i.e. the optimal estimation is achieved by a weaker PM (figure 7a). However, figure 7b shows that a stronger PMR is required for attaining the optimal value of $F_{\text{out}}^{\text{PM}}(\phi )$ when the accelerated observer moves with larger acceleration.

Figure 7.

The optimal value of PM and PMR strengths as a function of acceleration parameter r for (a) q = 0.6 and $\vartheta =\frac{3\pi }{4}$ and (b) p = 0.6 and $\vartheta =\frac{\pi }{4}$. (Online version in colour.)

The behaviour of the QFI with respect to the phase parameter, $F_{\text{out}}^{\text{PM}}(\phi )$, as a function of the acceleration parameter, r, for different ranges of the channel parameter, $\vartheta$, is investigated in figure 8. It is seen that for teleportation under a pure Unruh channel (i.e. p = q = 0) there is monotonous degradation in $F_{\text{out}}^{\text{PM}}(\phi )$ with an increase in r. However, we find that the combined effect of PM and PMR with the same strength, (p = q), leads to a partial improvement in the estimation precision of the phase parameter. In addition, when this common measurement strength increases $F_{\text{out}}^{\text{PM}}(\phi )$ is protected much better for $\frac{\pi }{2}\le \vartheta <\pi$; it even increases surprisingly with an increase in acceleration for $\frac{\pi }{2}<\vartheta <\pi$, in the limit p → 1 and q → 1. In addition, our numerical calculation shows that, in order to protect the QFI with respect to φ and QRs of the teleported state against the Unruh effect, we can use the following special choice for PMR strength [67]:

$$q_{\text{s}}=1-(1-p){\text{cos}}^{2}r.$$ 5.7

In fact, the Unruh noise may be approximately eliminated provided that the PM strength is sufficiently strong (p → 1) and the above choice for the PMR is applied (see dashed lines in figure 8).

Figure 8.

QFI of the single-qubit teleported state with respect to the phase parameter, φ, as a function of the acceleration parameter r by fixing $\theta =\frac{\pi }{2}$ for (a) $0<\vartheta <\frac{\pi }{2}$, (b) $\vartheta =\frac{\pi }{2}$ and (c) $\frac{\pi }{2}<\vartheta <\pi$. (Online version in colour.)

Finally, if we intend to teleport only the information encoded into the phase parameter, we can manage the input state by choosing the weight parameter as $\theta =\frac{\pi }{2}$ to estimate the phase parameter with the best precision, i.e. the best estimation of the phase parameter is obtained if the input state is maximally entangled (see equation (5.5)).

In the following, the effects of PM or PMR on QC of the teleported state of a single qubit are studied. Using the l₁-norm measure (equation (2.9)), QC for the density matrix (5.1) can be obtained as follows:

$${\mathcal{C}}_{l_1}({\rho }_{\text{out}}^{\text{PM}})=\left|\frac{\text{sin}\vartheta \sqrt{\overline{pq}}\text{cos}\hspace{0.17em}r\text{sin}\hspace{0.17em}\theta }{N_2}\right|.$$ 5.8

In the case of teleportation without application of PM or PMR on the Unruh channel, i.e. p = q = 0 and then N₂ = 1, we find

$${\mathcal{C}}_{l_1}({\rho }_{\text{out}})=|\text{sin}\vartheta \hspace{0.17em}\text{cos}\hspace{0.17em}r\text{sin}\hspace{0.17em}\theta |,$$ 5.9

which is QC of the teleported state under the pure Unruh decoherence.

Investigating QC of the single-qubit teleported state as a function of r or studying its behaviour versus PM and PMR strength for a fixed value of the acceleration parameter, one can see that the results, qualitatively similar to $F_{\text{out}}^{\text{PM}}(\phi )$, are observed.

In order to determine the quality of teleportation, i.e. the closeness of the teleported state to the input state, the fidelity [69] between ρ_in and ρ_out defined as $f({\rho }_{\text{in}},{\rho }_{\text{out}})={\left\{\text{Tr}\sqrt{{({\rho }_{\text{in}})}^{\frac{1}{2}}{\rho }_{\text{out}}{({\rho }_{\text{in}})}^{\frac{1}{2}}}\right\}}^2=⟨{\psi }_{\text{in}}|{\rho }_{\text{out}}|{\psi }_{\text{in}}⟩$ should be computed. Therefore, we obtain

$$f=\frac{1}{N_2}[(\frac{\mathcal{A}-\mathcal{D}}{2}+\mathcal{F}\text{cos}\hspace{0.17em}2\phi ){\text{sin}}^2\theta +\mathcal{D}].$$ 5.10

Examining the behaviour of the teleportation fidelity, we again see that the results obtained for fidelity are the same as the obtained results for QC and $F_{\text{out}}^{\text{PM}}(\phi )$, i.e. the fidelity degrades with increases in r under a pure Unruh effect. However, the combined effects of PM and PMR for a channel parameter lying in the region $\frac{\pi }{2}\le \vartheta <\pi$ can improve the quality of teleportation and it may even enhance with an increase in acceleration for $\frac{\pi }{2}<\vartheta <\pi$ in the limit p, q → 1. Moreover, Unruh decoherence is approximately eliminated for all values of $\vartheta$, with q = q_opt and in the limit p → 1; consequently, the teleportation process may be implemented with better quality. On the other hand, investigating the fidelity of the single-qubit teleportation as functions of PM as well as PMR strength, we find that, similar to $F_{\text{out}}^{\text{PM}}(\phi )$ and QC, with proper selection of the channel parameter $\vartheta$ the quality of teleportation may be enhanced with increases in p or q to reach a maximum value. In addition, in the range $0<\vartheta \le \frac{\pi }{2}$ ($\frac{\pi }{2}<\vartheta <\pi$), analysing the fidelity behaviour as a function of q (p), we see that it may be decreased (improved) with an increase in p (q) for small values of q (p), while it can exhibit increasing behaviour as p (q) increases for large values of q(p). In addition, optimal teleportation fidelity becomes greater with increases in q or q; hence, the teleportation process is done more successfully.

Finally, in figure 9a,b, we compare and illustrate the harmonic behaviour of $F_{\text{out}}^{\text{PM}}(\phi )$, QC and teleportation fidelity as functions of PM strength for $\frac{\pi }{2}<\vartheta <\pi$, and PMR strength for $0<\vartheta \le \frac{\pi }{2}$, in the case of single-qubit teleportation. We conclude that, for both q = 0 and q ≠ 0, the PM strength which optimizes the estimation precision of the phase parameter is the strength at which the quality of teleportation is the best and the coherence of the output single-qubit state reaches its maximum value. Investigating the behaviour of the above-mentioned quantities as functions of PMR strength, we achieve the same results (figure 9c,d).

Figure 9.

Comparing $F_{\text{out}}^{\text{PM}}(\phi )$ and QC of the teleported state as well as the teleportation fidelity in a single-qubit scenario by fixing $\theta =\frac{\pi }{2},\phi =0$, (a) versus p in the absence of PMR (i.e. q = 0), (b) versus p in the presence of PMR with q = 0.6, (c) versus q in the absence of PM (i.e. p = 0), and (d) versus q in the presence of PM with p = 0.6. (Online version in colour.)

6. Two-qubit teleportation under the Unruh noise channel

In order to study the QRs and QFIs of the teleported two-qubit state through the Unruh channel, |ψ_in〉 = cos~θ/2 |10〉 + e^iφ~sin~θ/2 |01〉, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π is considered to be the input state in the teleportation process. We follow Lee & Kim’s [59] two-qubit teleportation protocol, and use two copies of shared state (3.7) between Alice and Rob as the quantum channel, i.e. again ρ_A,I = ρ_ch. Using equation (2.3), we obtain the output state as

$${\rho }_{\text{out}}^{\text{PM}}=\frac{1}{N_2^2}\left(\begin{array}{cccc}\mathcal{A}\mathcal{D}& 0& 0& 0\\ 0& {\mathcal{A}}^2{\text{sin}}^2\frac{\theta }{2}+{\mathcal{D}}^2{\text{cos}}^2\frac{\theta }{2}& 2{\mathcal{F}}^2{\text{e}}^{\text{i}\phi }\text{sin}\hspace{0.17em}\theta & 0\\ 0& 2{\mathcal{F}}^2{\text{e}}^{-\text{i}\phi }\text{sin}\hspace{0.17em}\theta & {\mathcal{A}}^2{\text{cos}}^2\frac{\theta }{2}+{\mathcal{D}}^2{\text{sin}}^2\frac{\theta }{2}& 0\\ 0& 0& 0& \mathcal{A}\mathcal{D}\end{array}\right),$$ 6.1

where $\mathcal{A}$, $\mathcal{D}$ and $\mathcal{F}$ are determined by equation (5.2).

We study how the PM and PMR help us to enhance the QRs and QFIs of the teleported state in the presence of the Unruh effect. Using density matrix (6.1), the corresponding QC is obtained as follows:

$${\mathcal{C}}_{l_1}({\rho }_{\text{out}}^{\text{PM}})=4\left|\frac{{\mathcal{F}}^2\text{sin}\hspace{0.17em}\theta }{N_2^2}\right|.$$ 6.2

The results, obtained for QC of the teleported two-qubit state under the Unruh noise channel, are similar to single-qubit teleportation.

According to equations (2.10) and (6.1), entanglement of the teleported two-qubit state is obtained as

$$C({\rho }_{\text{out}}^{\text{PM}})=2\hspace{0.17em}\max \{0,2\left|\frac{{\mathcal{F}}^2\text{sin}\hspace{0.17em}\theta }{N_2^2}\right|-\left|\frac{\mathcal{A}\mathcal{D}}{N_2^2}\right|\}.$$ 6.3

In figure 10, we plot the concurrence of the teleported two-qubit state as a function of acceleration parameter r for different strengths of PM and PMR. It is clear that the entanglement absolutely decreases with the increase in acceleration under the pure Unruh decoherence. However, it can be amplified with combined action of PM and PMR for all values of initial channel parameter $\vartheta$. In fact, when the strength of PM increases, the entanglement degradation decreases; especially in the limit p = q → 1, entanglement is approximately protected against the Unruh decoherence. Surprisingly, as shown in figure 10, in that limit, entanglement of the teleported state may increase under the Unruh effect for an initial channel parameter lying in the region $\frac{\pi }{2}<\vartheta <\pi$. In addition, it is seen that the entanglement is also improved by applying q_s even without the first PM (i.e. p = 0) for $0<\vartheta \le \frac{\pi }{2}$.

Figure 10.

Entanglement of the teleported two-qubit state as a function of acceleration parameter r by fixing $\theta =\frac{\pi }{2}$ for (a) $0<\vartheta \le \frac{\pi }{2}$ and (b) $\frac{\pi }{2}<\vartheta <\pi$. (Online version in colour.)

Considering the behaviour of QD as a function of acceleration parameter r for different PM strengths, we see that the combined action of PM and PMR can raise QD for $\vartheta$ lying in the range $\frac{\pi }{2}\le \vartheta <\pi$ (figure 11). In particular, in the limit p, q → 1, QD may increase with applying PMs for $\frac{\pi }{2}<\vartheta <\pi$. Moreover, if we choose q = q_opt, QD can increase even in the absence of the first PM (i.e. p = 0) for $0<\vartheta \le \frac{\pi }{2}$.

Figure 11.

QD of the teleported two-qubit state as a function of acceleration parameter r by fixing $\theta =\frac{\pi }{2}$ and φ = 0 for (a) $0<\vartheta <\frac{\pi }{2}$, (b) $\vartheta =\frac{\pi }{2}$ and (c) $\frac{\pi }{2}<\vartheta <\pi$. (Online version in colour.)

Next we consider the QFI of the teleported two-qubit state. In order to calculate the QFIs associated with the weight and phase parameters encoded in the quantum state (6.1), we follow the method proposed in [70,71] to block diagonal states, because state (6.1) is of X type and hence becomes block diagonal after changing the order of the basis vectors. A block diagonal state can be written as $\rho =\underset{i=1}{\overset{n}{⨁}}{\rho }_i$, where ⊕ represents the direct sum. The SLD operator is block diagonal here, i.e. $L=\underset{i=1}{\overset{n}{⨁}}{L}_i$, where L_i indicates the corresponding SLD operator for ρ_i. For two-dimensional blocks, it can be proved that the SLD operator for the ith block is given by [70]

$$L_i=\frac{1}{{\mu }_i}[{\mathrm{\partial }}_x{\rho }_i+{\xi }_i{\rho }_i^{-1}-{\mathrm{\partial }}_x{\mu }_i],$$ 6.4

where ξ_i = 2μ_i∂_xμ_i − ∂_xP_i/4, in which μ_i = Trρ_i/2 and $P_i=\text{Tr}{\rho }_i^2$. If detρ_i = 0, ξ_i vanishes.

Using the above-mentioned method, we find the QFIs of the two-qubit teleported state with respect to weight and phase parameters as follows:

$$F_{\text{out}}^{\text{PM}}(\theta )=\frac{1}{N_2^2}[\zeta +\frac{8{\mathcal{A}}^2{\mathcal{D}}^2({\zeta }^2-16{\mathcal{F}}^4)}{\zeta [({({\mathcal{A}}^2-{\mathcal{D}}^2)}^2-16{\mathcal{F}}^4)\text{cos}\hspace{0.17em}2\theta -({\zeta }^2+4({\mathcal{A}}^2{\mathcal{D}}^2-4{\mathcal{F}}^4))]}]$$ 6.5

and

$$F_{\text{out}}^{\text{PM}}(\phi )=\frac{16{\mathcal{F}}^4{\text{sin}}^2\theta }{\zeta N_2^2},$$ 6.6

where $\zeta ={\mathcal{A}}^2+{\mathcal{D}}^2$. Surprisingly, investigating QFI of a teleported two-qubit state under the Unruh channel, we obtain results similar to single-qubit teleportation.

In figures 12 and 13, we compare the QFI of both single- and two-qubit teleported states (supposing that θ or φ carries the same information in both cases). In figure 12, we see that the information encoded in the weight parameter θ is better protected against the Unruh effect during teleportation of the two-qubit state, comparing it with the single-qubit scenario. Nevertheless, extraction of information encoded into phase parameter φ is more efficient in single-qubit teleportation than in two-qubit teleportation (figure 13). Therefore, depending on what parameter we want to teleport, we use a single- or two-qubit state to encode the required information.

Figure 12.

Comparing $F_{\text{out}}^{\text{PM}}\hspace{0.17em}(\theta )$ of the teleported single- and two-qubit states, fixing $\theta =\frac{\pi }{2}$ and $\vartheta =\frac{\pi }{2}$ (a) in the absence of measurements, (b) in the presence of measurements. (Online version in colour.)

Figure 13.

Comparing $F_{\text{out}}^{\text{PM}}(\phi )$ of the teleported single- and two-qubit states fixing $\theta =\frac{\pi }{2}$ and $\vartheta =\frac{\pi }{2}$ (a) in the absence of measurements, (b) in the presence of measurements. (Online version in colour.)

Finally, fidelity for the two-qubit teleportation under the Unruh channel is found to be

$$f=\frac{1}{N_2^2}[(\frac{{\mathcal{A}}^2-{\mathcal{D}}^2}{4}+{\mathcal{F}}^2\text{cos}\hspace{0.17em}2\phi )2{\text{sin}}^2\theta +{\mathcal{D}}^2].$$ 6.7

We obtain results similar to single-qubit teleportation fidelity by investigating the behaviour of two-qubit teleportation fidelity under the Unruh noise channel with and without applying the PMs.

Comparing the fidelity of single- and two-qubit teleportation in figure 14, we observe that the quality of teleportation is better in a single-qubit case than in a two-qubit one. This means that single-qubit teleportation is more robust against the Unruh decoherence.

Figure 14.

Comparing the fidelity of single- and two-qubit teleportation, fixing $\theta =\frac{\pi }{2},\phi =0$ and $\vartheta =\frac{\pi }{2}$ for (a) in the absence of measurements, (b) in the presence of measurements. (Online version in colour.)

7. Summary and conclusion

QRs and QFI of the teleported single- and two-qubit states, under the Unruh effect experienced by a mode of a free Dirac field, were discussed in this paper. We investigated the conditions under which the degradation of the Unruh effect on QRs and QFI of the teleported state can be improved by PMs, and found that the value of the initial parameter of the channel $\vartheta$ plays a key role in this scenario. Moreover, we examined how the PMs can be performed to eliminate the Unruh effect or how they may be designed such that the Unruh effect can be used to enhance the quantum communication. In addition, fixing the acceleration and considering the behaviour of the QFI, QC and teleportation fidelity as functions of PM strength (PMR strength), we found that $F_{\text{out}}^{\text{PM}}(\phi )$, QC and teleportation fidelity harmonically increase to reach a maximum value and then decrease with increasing PM (PMR) strength. We also analysed the optimal behaviour of the QFI with respect to the phase parameter. Finally, comparing the QFI of the teleported single- and two-qubit states as functions of acceleration, we showed that the information encoded in the weight parameter θ is better protected against the Unruh effect in the case of two-qubit teleportation. However, in the case of single-qubit teleportation, encoding information in the phase parameter φ is more efficient. Therefore, we encode the information into either the weight or phase parameter, depending on whether the two- or single-qubit scenario, respectively, is used for the teleportation.

Data accessibility

This paper does not have any experimental data.

Authors' contributions

All the authors conceived the work and agreed on the approach to pursue. H.R.J. with M.A.-T. planned and supervised the research. M.J. and H.R.J. carried out the theoretical calculations, discussed the results and wrote the manuscript. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

M.J. and M.A.-T. wish to acknowledge the financial support of Urmia University. H.R.J. acknowledges funding by grant no. 2471666442HRJ from Jahrom University.