Eliminating the Unruh effect of relativistic Dirac fields by partial measurements
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N A Khan,1,∗, Syed Tahir Amin2,3,4, Munsif Jan,5,∗1Centro de Física das Universidades do Minho e Porto, Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal 2Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 3CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 4Instituto de Telecomunicações, ŁAv. Rovisco Pais, 1049-001 Lisboa, Portugal 5CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
First author contact: Authors to whom any correspondence should be addressed. Received:2020-07-14Revised:2020-09-10Accepted:2020-09-14Online:2020-12-01
Abstract The retrieval of lost entanglement for relatively accelerated fermionic observers of a tripartite system by a partial measurement technique has been investigated. From the prospective of the negativities of one-tangles and the π-tangle, we show that the degraded entanglement in noninertial frames with single-mode approximation is completely retrieved by an optimal strength of the partial measurement or the partial measurement reversal. In addition, we find that the optimal one-tangle with respect to inertial and noninertial observers turns out to be the same for an optimal strength of partial measurements at q0=0 when two accelerated observers move with infinite acceleration. Keywords：entanglement retrieval;π-tangle;Unruh decoherence
PDF (406KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article N A Khan, Syed Tahir Amin, Munsif Jan. Eliminating the Unruh effect of relativistic Dirac fields by partial measurements. Communications in Theoretical Physics, 2020, 72(12): 125103- doi:10.1088/1572-9494/abbccf
1. Introduction
Quantum entanglement is one of the most valuable physical resources for quantum information related tasks, including quantum-cryptography [1], quantum-teleportation [2, 3], quantum-algorithms [4] and quantum-computer technology [5]. It has also been actively pursued to attack condensed-matter problems [6].
The theory of quantum information within the relativistic framework has received significant attention in recent years. The main goal is to improve the quantum information tasks [7, 8] by incorporating relativistic effects. The relativistic quantum information scenario with a single-mode approximation can also be employed to relatively accelerated observers of entangled states [8-12]. In this case, the entanglement between parties—Dirac or Scalar fields—degrades from the perspective of an accelerated observer moving with a uniform acceleration. Similarly, the nonmaximal entanglement degradation of Dirac and Scalar fields in the relativistic frame has also been investigated [13, 14]. It is found that the initial nonmaximal entanglement between parties degrades along two different trajectories in noninertial frames. Bruschi et al [15] investigated the entanglement degradation of Dirac and Scalar fields beyond the single-mode approximation in the relativistic frame.
Quantum decoherence [8, 9, 16], as a ubiquitous physical process, appears due to the system-environment interaction, leading to the loss of coherence and degradation of quantum correlation. To eliminate the effect of environmental noise, Xiao et al [17] proposed a partial measurement technique, which protected or enhanced quantum Fisher information (QFI) teleportation from Unruh decoherence. It was also observed that the prior and post partial measurement can protect the QFI from amplitude damping noise. Recently, the recovery of lost entanglement due to Unruh decoherence by the partial measurement technique was investigated [18]. It was found that the lost entanglement between two modes of a free Dirac field in noninertial frames is amplified or retrieved completely by the optimal strength of the partial measurement. Furthermore, Jafarzadeh et al [19] investigated the effects of the partial measurement and partial measurement reversal on the quantum resources and QFI of a single and two-qubit quantum teleported states in noninertial frames. It was found that the encoded information in the weight parameter for the two-qubit quantum teleported states is better protected against Unruh decoherence.
The partial measurement procedure is an extremely important issue for protecting entanglement from decoherence in real systems. It has been successfully utilized to protect quantum entanglement from amplitude damping decoherence in non-relativistic frameworks [20, 21]. Moreover, in a relativistic scenario, the QFI teleportation under amplitude damping noise has been enhanced by the technique of partial measurements [17]. In addition, the teleportation fidelity has been enhanced by eliminating the Hawking effect using the partial measurements technique in the background of Schwarzschild space-time [22].
In this paper, we investigate the recovery of lost entanglement of a tripartite fermionic system in a noninertial frame by employing partial measurements and a partial measurement reversal technique. We show that the entanglement from the perspective of the one-tangle and π-tangle may be retrieved by the optimal strength of the partial measurements p0, or partial measurement reversal q0. In particular, we find that the lost negativities of one-tangles and the π-tangle of one or two accelerated observers may be completely retrieved for the optimal value of p0(q0) in the limit ${q}_{0}\to 1({p}_{0}\to 1)$. Interestingly, we obtain an equal optimal one-tangle for the optimal strength p0, when two observers move with uniform infinite acceleration at q0=0.
The structure of the paper is as follows. In section 2.1 we give a brief description of the Unruh effect for relatively accelerated fermionic observers moving with uniform acceleration. In particular, we review the transformations between the Minkowski and Rindler modes for a free Dirac field. In section 2.2 we discuss the partial measurement operator: a tool to recover lost entanglement due to Unruh decoherence. Section 2.3 introduces the criterion of entanglement measurement—the negativities of the one-tangle and π-tangle—for the tripartite system. In section 3 we present the recovery of lost entanglement by partial measurement and partial measurement reversal when one and two observers move with a uniform acceleration. In the last section, we sum up our conclusions.
2. Theory
2.1. Dirac fields
A free Minkowski Dirac field in 3+1 dimensions can be expressed in flat Minkowski space-time $\left({ct},x\right)$ from the perspective of an inertial observer. This field can be expanded in terms of a complete set of positive (particle) and negative (anti-particle) frequency modes [9].$\begin{eqnarray}{\rm{\Psi }}=\int {\rm{d}}{\boldsymbol{k}}\left({a}_{{\boldsymbol{k}}}{\psi }_{{\boldsymbol{k}}}^{+}+{b}_{{\boldsymbol{k}}}^{\dagger }{\psi }_{{\boldsymbol{k}}}^{-}\right),\end{eqnarray}$where the wave vector ${\boldsymbol{k}}$ denotes the modes of massive Dirac fields and ${\psi }_{{\boldsymbol{k}}}^{+}({\psi }_{{\boldsymbol{k}}}^{-})$ shows the positive (negative) energy Minkowski modes. The ${a}_{{\boldsymbol{k}}}({a}_{{\boldsymbol{k}}}^{\dagger })$ and ${b}_{{\boldsymbol{k}}}({b}_{{\boldsymbol{k}}}^{\dagger })$ are the creation (annihilation) operators for the positive and negative frequency modes of momentum ${\boldsymbol{k}}$, respectively. The creation and annihilation operator satisfy the anti-commutation relations$\begin{eqnarray}\left\{{a}_{i},{a}_{j}^{\dagger }\right\}=\left\{{b}_{i},{b}_{j}^{\dagger }\right\}={\delta }_{{ij}}.\end{eqnarray}$However, the Dirac field for an accelerated observer can be expressed in Rindler coordinates $\left(\tau ,\xi \right)$. The Rindler space-time for an accelerated observer splits into two regions, I (particles) and II (anti-particles), that are separated by the Rindler horizon and thus are causally disconnected from each other. The Rindler coordinates in region I are expressed in terms of the Minkowski coordinates as follows [9]$\begin{eqnarray}t=\displaystyle \frac{1}{a}{{\rm{e}}}^{a\xi }\sinh a\tau ,\qquad x=\displaystyle \frac{1}{a}{{\rm{e}}}^{a\xi }\cosh a\tau ,\end{eqnarray}$where the ξ=a−1 is the world line of a uniformly accelerated observer with an acceleration of magnitude a. The coordinates' transformations in region II can be obtained by replacing t=−t and $x=-x.$
The solutions of the Dirac field for the accelerated observer in Rindler coordinates can be expanded in terms of a complete set of positive and negative frequency modes [9, 18].$\begin{eqnarray}{\rm{\Psi }}=\int {\rm{d}}{\boldsymbol{k}}\left({c}_{{\boldsymbol{k}}}^{{\rm{I}}}{\psi }_{{\boldsymbol{k}}}^{{\rm{I}}+}+{d}_{{\boldsymbol{k}}}^{{\rm{I}}\dagger }{\psi }_{{\boldsymbol{k}}}^{{\rm{I}}-}+{c}_{{\boldsymbol{k}}}^{\mathrm{II}}{\psi }_{{\boldsymbol{k}}}^{\mathrm{II}+}+{d}_{{\boldsymbol{k}}}^{\mathrm{II}\dagger }{\psi }_{{\boldsymbol{k}}}^{\mathrm{II}-}\right),\end{eqnarray}$where ${c}_{{\boldsymbol{k}}}^{{\text{}}n}({c}_{{\boldsymbol{k}}}^{{\text{}}n\dagger })$ and ${d}_{{\boldsymbol{k}}}^{{\text{}}n}({d}_{{\boldsymbol{k}}}^{{\text{}}n\dagger })$ are the annihilation (creation) operators for the particle and anti-particle in the region n with ${\text{}}n=({\rm{I}},\ \mathrm{II})$, respectively.
The Minkowski and Rindler creation and annihilation operators are related by [9]$\begin{eqnarray}{a}_{{\boldsymbol{k}}}=\cos r\,{c}_{{\boldsymbol{k}}}^{{\rm{I}}}-\sin r\,{d}_{-{\boldsymbol{k}}}^{\mathrm{II}\dagger },\qquad {b}_{-{\boldsymbol{k}}}^{\dagger }=\sin r\,{c}_{{\boldsymbol{k}}}^{{\rm{I}}}-\cos r\,{d}_{-{\boldsymbol{k}}}^{\mathrm{II}\dagger }\end{eqnarray}$where r is the acceleration parameter given by$\begin{eqnarray}r=\arccos \displaystyle \frac{1}{\sqrt{1+{{\rm{e}}}^{-2\pi {wc}/a}}},\qquad 0\leqslant r\leqslant \pi /4.\end{eqnarray}$After some algebra (see [9] for further detail), the Minkowski particle vacuum for mode ${\boldsymbol{k}}$ in terms of Rindler Fock states in region I and II, with a single-mode approximation turns out$\begin{eqnarray}| {0}_{{\boldsymbol{k}}}{\rangle }_{M}=\cos r| {0}_{{\boldsymbol{k}}}{\rangle }_{{\rm{I}}}| {0}_{-{\boldsymbol{k}}}{\rangle }_{\mathrm{II}}+\sin r| {1}_{{\boldsymbol{k}}}{\rangle }_{{\rm{I}}}| {1}_{-{\boldsymbol{k}}}{\rangle }_{\mathrm{II}}.\end{eqnarray}$The other allowed state (excited state) of fermions is$\begin{eqnarray}| {1}_{{\boldsymbol{k}}}{\rangle }_{M}={a}_{{\boldsymbol{k}}}^{\dagger }| {0}_{{\boldsymbol{k}}}{\rangle }_{M}=| {1}_{{\boldsymbol{k}}}{\rangle }_{{\rm{I}}}| {0}_{-{\boldsymbol{k}}}{\rangle }_{\mathrm{II}}.\end{eqnarray}$For simplicity, we refer to the particle Minkowski mode $| {i}_{{\boldsymbol{k}}}{\rangle }_{M}=| i{\rangle }_{M}$ and particle Rindler mode $| {i}_{\sigma }{\rangle }_{\mu }=| i{\rangle }_{\mu }$ for $i\in \left(0,1\right)$, $\sigma =\pm {\boldsymbol{k}}$ and $\mu =\left({\rm{I}},\ \mathrm{II}\right)$.
2.2. Partial measurements and partial measurement reversal
In this subsection, we briefly introduce the partial measurement and the partial measurement reversal. Partial measurements are the generalizations of von Neumann measurements, and are associated with a positive-operator valued measure (POVM). However, unlike the von Neumann projective measurement, where the initial state of a quantum system is projected to one of the eigenstates of the measurement operator, the partial measurement does not collapse the initial state completely towards an eigenstate. Thus, the initial information can be recovered by some reverse measurement. For a single qubit, the partial measurement is described by the following operators [17, 18]:$\begin{eqnarray}{{ \mathcal M }}_{0}({p}_{0})=\sqrt{1-{p}_{0}}| 0\rangle \langle 0| +| 1\rangle \langle 1| ,\quad 0\leqslant {p}_{0}\leqslant 1,\end{eqnarray}$where p0 is known as the strength of the partial measurement, satisfying the relation$\begin{eqnarray}{{ \mathcal M }}_{0}^{\dagger }({p}_{0}){{ \mathcal M }}_{0}({p}_{0})+{{ \mathcal M }}_{1}^{\dagger }({p}_{0}){{ \mathcal M }}_{1}({p}_{0})=I,\end{eqnarray}$with ${{ \mathcal M }}_{1}({p}_{0})=\sqrt{{p}_{0}}| 0\rangle \langle 0| $. ${{ \mathcal M }}_{1}({p}_{0})$ is identical to the von Neumann projective measurement and is irreversible. Hence, ${{ \mathcal M }}_{0}({p}_{0})$ (equation (9)) is the only partial measurement operator that induces a partial collapse of quantum state. The initial information can be retrieved by reversing operations on the post measurement state [17, 18]. The reverse measurement procedure is described by a non-unitary operator,$\begin{eqnarray}{{ \mathcal M }}_{0}^{-1}({q}_{0})=\displaystyle \frac{1}{\sqrt{1-{q}_{0}}}{\sigma }_{x}{{ \mathcal M }}_{0}({q}_{0}){\sigma }_{x},\end{eqnarray}$where ${\sigma }_{x}=| 0\rangle \langle 1| +| 1\rangle \langle 0| $, is the bit-flip operator. The parameter q0 is the strength of the partial measurement reversal. The reverse measurement procedure, equation (11), can be implemented by applying the bit-flip, partial measurement with strength q0 and a second bit-flip operation sequentially.
2.3. Quantification of entanglement in tripartite systems
To quantify entanglement in a tripartite system, there are different criteria of entanglement measurements that exist in the literature. These include the residual three tangle [23], the realignment criterion [24] and the π-tangle [25]. The three tangle depends on the concurrences of the bipartite and tripartite systems and is hard to calculate analytically for a few cases. The realignment criterion is comparatively easy for entanglement measurement but does not detect the entanglement of all states. On the other hand, the π-tangle is a good quantifier of entanglement for tripartite systems. It measures entanglement in terms of negativity. The π-tangle $\left({\pi }_{\alpha \beta \gamma }\right)$ for the tripartite system is given by [25]$\begin{eqnarray}{\pi }_{\alpha \beta \gamma }=\displaystyle \frac{1}{3}({\pi }_{\alpha }+{\pi }_{\beta }+{\pi }_{\gamma }),\end{eqnarray}$where πα is called the residual entanglement, quantified by the negativities of the one-tangle and two-tangle, defined as$\begin{eqnarray}{\pi }_{\alpha }={{ \mathcal N }}_{\alpha (\beta \gamma )}^{2}-{{ \mathcal N }}_{\alpha \beta }^{2}-{{ \mathcal N }}_{\alpha \gamma }^{2}.\end{eqnarray}$The ${{ \mathcal N }}_{\alpha (\beta \gamma )}$ is called the one-tangle, defined as [25]$\begin{eqnarray}{{ \mathcal N }}_{\alpha (\beta \gamma )}=\parallel {\rho }_{\alpha \beta \gamma }^{{{\rm{T}}}_{\alpha }}\parallel -1,\end{eqnarray}$where ${\rho }_{\alpha \beta \gamma }^{{{\rm{T}}}_{\alpha }}$ is the partial transpose of the density matrix ${\rho }_{\alpha \beta \gamma }$ over qubit α. The $\parallel { \mathcal O }\parallel =\mathrm{Tr}\sqrt{{ \mathcal O }{{ \mathcal O }}^{\dagger }}$ stands for the trace norm of an operator ${ \mathcal O }$. On the other hand, ${{ \mathcal N }}_{\alpha \beta }$ or ${{ \mathcal N }}_{\alpha \gamma }$ is called the two-tangle, and measures the entanglement of a bipartite system. The two-tangle ${{ \mathcal N }}_{\alpha \beta }$ is defined as$\begin{eqnarray}{{ \mathcal N }}_{\alpha \beta }=\parallel {\rho }_{\alpha \beta }^{{{\rm{T}}}_{\alpha }}\parallel -1,\end{eqnarray}$where ${\rho }_{\alpha \beta }={\mathrm{Tr}}_{\gamma }\left({\rho }_{\alpha \beta \gamma }\right)$ is the reduced density matrix. The entanglement quantified by one-tangle and two-tangle with respect to an observer, for instance α, satisfies the following Coffman-Kundu-Wootters (CKW) monogamy inequality [25]$\begin{eqnarray}{{ \mathcal N }}_{\alpha (\beta \gamma )}^{2}\geqslant {{ \mathcal N }}_{\alpha \beta }^{2}+{{ \mathcal N }}_{\alpha \gamma }^{2}.\end{eqnarray}$The residual tangles πβ and πγ are quantified by the negativities of the one-tangle and two-tangle with respect to the observers β and γ, respectively.
3. Retrieving the lost entanglement of tripartite fermionic systems
Let us consider three observers, Alice A, Bob B and Charlie C, that initially share the maximally entangled Greenberger-Horne-Zeilinger (GHZ) state$\begin{eqnarray}| {\rm{\Psi }}{\rangle }_{{ABC}}=\displaystyle \frac{| {0}_{{\omega }_{A}}{\rangle }_{A}| {0}_{{\omega }_{B}}{\rangle }_{B}| {0}_{{\omega }_{C}}{\rangle }_{C}+| {1}_{{\omega }_{A}}{\rangle }_{A}| {1}_{{\omega }_{B}}{\rangle }_{B}| {1}_{{\omega }_{C}}{\rangle }_{C}}{\sqrt{2}},\end{eqnarray}$where $| {z}_{{\omega }_{i}}{\rangle }_{i}$ for $z\in (0,1)$ are the states of the Dirac field in Minkowski space with modes specified by the subscript ωi (i=A, B, C). Under the single-mode approximation ${\omega }_{A}\sim {\omega }_{B}\sim {\omega }_{C}=\omega $, we can simply write $| {z}_{{\omega }_{i}}{\rangle }_{i}=| z{\rangle }_{i}$. Without loss of the generality, $| l{\rangle }_{A}| m{\rangle }_{B}| n{\rangle }_{C}=| {lmn}\rangle $ for $l,m,n\in (0,1)$.
Let a partial measurement with strength p0 be performed on the third qubit (Charlie); then, the normalized state of equation (17) becomes$\begin{eqnarray}| {\rm{\Phi }}{\rangle }_{{ABC}}=\displaystyle \frac{{{ \mathcal M }}_{0}({p}_{0})| {\rm{\Psi }}{\rangle }_{{ABC}}}{\sqrt{\langle {\rm{\Psi }}| {{ \mathcal M }}_{0}({p}_{0})| {\rm{\Psi }}{\rangle }_{{ABC}}}}=\displaystyle \frac{\sqrt{p}| 000\rangle +| 111\rangle }{\sqrt{1+p}},\end{eqnarray}$with $p=1-{p}_{0}$. Given that the Charlie undergoes uniform acceleration, the vacuum and excited states of Charlie in Minkowski space are transformed into two causally disconnected Rindler regions, I and II. In this case, the state of equation (18) is given by$\begin{eqnarray}| {\rm{\Psi }}{\rangle }_{{{ABC}}_{{\rm{I}}}{C}_{\mathrm{II}}}=\displaystyle \frac{\sqrt{p}\left(\cos r\left|0000\right\rangle +\sin r\left|0011\right\rangle \right)+\left|1110\right\rangle }{\sqrt{1+p}},\end{eqnarray}$where r is the acceleration parameter. It is easy to show that r=π/4 for the fermionic modes in the limit of infinite acceleration of the accelerated observer. To demise the Unruh effect and to recover entanglement we performed a partial measurement reversal on Charlie in region I. The normalized state of equation (19) after successfully performing partial measurement reversal is$\begin{eqnarray}\begin{array}{l}| {\rm{\Phi }}{\rangle }_{{{ABC}}_{{\rm{I}}}{C}_{\mathrm{II}}}\\ =\,\displaystyle \frac{\sqrt{p}\cos r\left|0000\right\rangle +\sqrt{{pq}}\sin r\left|0011\right\rangle +\sqrt{q}\left|1110\right\rangle }{\sqrt{{pq}{\sin }^{2}r+p{\cos }^{2}r+q}},\end{array}\end{eqnarray}$with $q=1-{q}_{0}$. In what follows, we study the entanglement among the Alice, Bob and Charlie modes in the Rindler basis (especially in region I). The reduced density matrix of Alice, Bob and Charlie in region I is obtained by tracing over region II. The resulting density matrix ${\rho }_{{{ABC}}_{{\rm{I}}}},$ is given by$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{{ABC}}_{{\rm{I}}}} & = & \displaystyle \frac{1}{{a}_{0}}[p{\cos }^{2}r| 000\rangle \langle 000| \\ & & +\sqrt{{pq}}\cos r\left(\left|000\right\rangle \left\langle 111\right|+\left|111\right\rangle \left\langle 000\right|\right)\\ & & +{pq}{\sin }^{2}r\left|001\right\rangle \left\langle 001\right|+q\left|111\right\rangle \left\langle 111\right|],\end{array}\end{eqnarray}$where ${a}_{0}={pq}{\sin }^{2}r+p{\cos }^{2}r+q.$ One can get the reduced density matrix of Alice, Bob and Charlie in region II by tracing over region I.
We employ equations (12)-(14) to calculate the one-tangle, two-tangle and π-tangle of the system. The corresponding analytical expression for the one-tangle (${{ \mathcal N }}_{A({{BC}}_{{\rm{I}}})}$, ${{ \mathcal N }}_{{C}_{{\rm{I}}}({AB})}$) and π-tangle are$\begin{eqnarray}{{ \mathcal N }}_{A({{BC}}_{{\rm{I}}})}={{ \mathcal N }}_{B({{AC}}_{{\rm{I}}})}=\displaystyle \frac{2\sqrt{{pq}}\cos r}{{a}_{0}},\end{eqnarray}$$\begin{eqnarray}{{ \mathcal N }}_{{C}_{{\rm{I}}}({AB})}=\displaystyle \frac{-{pq}{\sin }^{2}r+\sqrt{{pq}\left({pq}{\sin }^{4}r+4{\cos }^{2}r\right)}}{{a}_{0}},\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{\pi }_{{{ABC}}_{{\rm{I}}}}\\ =\,\displaystyle \frac{2{pq}[6{\cos }^{2}r+{pq}{\sin }^{4}r-{\sin }^{2}r\sqrt{{pq}({pq}{\sin }^{4}r+4{\cos }^{2}r)}]}{3{a}_{0}^{2}},\end{array}\end{eqnarray}$where all the possible two-tangles are zero. It is straightforward to show that the one-tangles and π-tangle reduce to the pure Unruh decoherence form in the limit p0=q0=0, as reported in the literature [26, 27]. In this limit, the entanglement depends only on the acceleration parameter of the noninertial observer. It degrades with the increasing acceleration parameter, reaching a finite value in the infinite-acceleration limit.
To investigate the optimal behavior of the one-tangle ${{ \mathcal N }}_{A({{BC}}_{I})}$, we calculate the optimal strength of the partial measurement reversal qopt, in terms of the partial measurement strength p, as follows:$\begin{eqnarray}\displaystyle \frac{\partial {{ \mathcal N }}_{A({{BC}}_{I})}}{\partial q}{| }_{q={q}_{\mathrm{opt}}}=0\Rightarrow {q}_{\mathrm{opt}}=\displaystyle \frac{p}{p{\tan }^{2}r+{\sec }^{2}r}.\end{eqnarray}$Similarly, the optimal strength of the partial measurements popt, in terms of the partial measurement reversal strength q, is calculated as$\begin{eqnarray}\displaystyle \frac{\partial {{ \mathcal N }}_{A({{BC}}_{I})}}{\partial p}{| }_{p={p}_{\mathrm{opt}}}=0\Rightarrow {p}_{\mathrm{opt}}=\displaystyle \frac{q}{q{\sin }^{2}r+{\cos }^{2}r}.\end{eqnarray}$The optimal values of q and p for investigating the optimal behavior of ${{ \mathcal N }}_{{C}_{I}({AB})}$ read as:$\begin{eqnarray}{q}_{\mathrm{opt}}=\displaystyle \frac{p\left(p{\sin }^{2}2r(1-\sqrt{p{\sin }^{2}r+1})+4{\cos }^{2}r\right)}{4\left(-{p}^{3}{\sin }^{6}r+2p{\sin }^{2}r+1\right)}.\end{eqnarray}$$\begin{eqnarray}{p}_{\mathrm{opt}}=\displaystyle \frac{q\cos r}{q{\sin }^{2}r\left(\cos r+\sqrt{q{\sin }^{2}r+{\cos }^{2}r}\right)+{\cos }^{3}r}.\end{eqnarray}$
The optimal behavior of the one-tangle with respect to the inertial observer, ${{ \mathcal N }}_{A({{BC}}_{I})},$ as a function of p0 (left panel) with the optimal value of q0, and q0 (right panel) with the optimal value of p0 for different values of the acceleration parameter is depicted in figure 1. Note that ${{ \mathcal N }}_{A({{BC}}_{I})}={{ \mathcal N }}_{B({{AC}}_{I})}$ shows that the two subsystems of the inertial frames are symmetrical for any values of the acceleration parameter. In the absence of measurements, the one-tangle ${{ \mathcal N }}_{A({{BC}}_{I})},$ will have a finite minimum value in the infinite-acceleration limit, as reported in the literature [26, 27]. However, the implementation of partial measurements and partial measurement reversal leads to the retrieval of a degraded one-tangle in the noninertial frame. In fact, the Unruh effect may be completely eliminated by employing the partial measurements and partial measurement reversal in the limit of ${p}_{0}\to 1$ (left panel) and ${q}_{0}\to 1$ (right panel). Figure 2 shows the optimal behavior of the one-tangle with respect to an accelerated observer, ${{ \mathcal N }}_{{C}_{I}({AB})}$, as a function of p0 (left panel) with an optimal value of q0 (equation (27)), and q0 (right panel) with an optimal value of p0 (equation (28)) for different values of the acceleration parameter. The optimal one-tangle, ${{ \mathcal N }}_{{C}_{I}({AB})}$, starts to recover with increasing strength of the partial measurements and partial measurement reversal. For sufficiently strong measurement strength, the optimal ${{ \mathcal N }}_{{C}_{I}({AB})}$ may be completely recovered.
Figure 1.
New window|Download| PPT slide Figure 1.The optimal behavior of one-tangle $({{ \mathcal N }}_{A({{BC}}_{{\rm{I}}})})$ of fermionic modes as a function of the strength of partial measurements p0 (left panel) and partial measurement reversal q0 (right panel) for different values of the acceleration parameter r. The degraded one-tangle due to Unruh decoherence is completely recovered for the optimal value of q0 (left panel) and p0 (right panel) in the limit of ${p}_{0}\to 1$ and ${q}_{0}\to 1$, respectively.
Figure 2.
New window|Download| PPT slide Figure 2.The optimal behavior of one-tangle $({{ \mathcal N }}_{{C}_{{\rm{I}}}({AB})})$ of fermionic modes as a function of the strength of partial measurements p0 (left panel) and partial measurement reversal q0 (right panel) for different values of the acceleration parameter r. The degraded one-tangle due to Unruh decoherence is completely recovered for the optimal values of q0 (left panel) and p0 (right panel) in the limit of ${p}_{0}\to 1$ and ${q}_{0}\to 1$, respectively.
Figure 3, illustrates the retrieval of the π-tangle of the tripartite system when one observer is moving with a constant acceleration. Note that the π-tangle decreases with the increasing acceleration parameter, which has been successfully examined [26, 27]. However, the degraded π-tangle due to Unruh decoherence may be recovered by partial measurements for any acceleration parameter. It is interesting to note that the recovery of lost entanglement by the partial measurement technique is rapid for higher acceleration. Hence, the π-tangle in the ${p}_{0}\to 1$ (left panel) and ${q}_{0}\to 1$ (right panel) limit have an equal and coherent entanglement. We have noticed that the entanglement of the tripartite system in the noninertial frame with respect to the inertial or accelerated observer can be completely recovered by the employing partial measurement technique.
Figure 3.
New window|Download| PPT slide Figure 3.The optimal behavior of the π-tangle of fermionic modes as a function of the strength of partial measurements (left panel) and partial measurement reversal (right panel) for different values of the acceleration parameter r.
We are now going to study the retrieval of lost entanglement of a tripartite fermionic system when two observers are accelerated. Let Alice stay stationary while Bob and Charlie move with a uniform acceleration rb and rc, respectively.
To protect the entanglement, we apply a similar approach for both Bob and Charlie, as follows: Perform a partial measurement on the observers Bob and Charlie with strength ${p}_{0,b}$ and ${p}_{0,c},$ respectively. Bob and Charlie undergo uniform acceleration, therefore rewrite the Rindler modes for Bob and Charlie. Perform a partial measurement reversal on Bob and Charlie in region I with strength ${q}_{0,b}$ and ${q}_{0,c},$ respectively. Following section 2.3 we find that the analytical expressions of one-tangles are given by$\begin{eqnarray}\begin{array}{l}{{ \mathcal N }}_{A({B}_{{\rm{I}}}{C}_{{\rm{I}}})}\,=\,\displaystyle \frac{\sqrt{{p}_{b}{p}_{c}{q}_{b}{q}_{c}\left({p}_{b}{p}_{c}{q}_{b}{q}_{c}{\sin }^{4}{r}_{b}{\sin }^{4}{r}_{c}+4{\cos }^{2}{r}_{b}{\cos }^{2}{r}_{c}\right)}-{p}_{b}{p}_{c}{q}_{b}{q}_{c}{\sin }^{2}{r}_{b}{\sin }^{2}{r}_{c}}{{p}_{b}{p}_{c}\left({q}_{b}{\sin }^{2}{r}_{b}+{\cos }^{2}{r}_{b}\right)\left({q}_{c}{\sin }^{2}{r}_{c}+{\cos }^{2}{r}_{c}\right)+{q}_{b}{q}_{c}},\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}{{ \mathcal N }}_{{B}_{{\rm{I}}}({{AC}}_{{\rm{I}}})}\,=\,\displaystyle \frac{\cos {r}_{c}\sqrt{{p}_{b}{p}_{c}{q}_{b}\left({p}_{b}{p}_{c}{q}_{b}{\sin }^{4}{r}_{b}{\cos }^{2}{r}_{c}+4{q}_{c}{\cos }^{2}{r}_{b}\right)}-{p}_{b}{p}_{c}{q}_{b}{\sin }^{2}{r}_{b}{\cos }^{2}{r}_{c}}{{p}_{b}{p}_{c}\left({q}_{b}{\sin }^{2}{r}_{b}+{\cos }^{2}{r}_{b}\right)\left({q}_{c}{\sin }^{2}{r}_{c}+{\cos }^{2}{r}_{c}\right)+{q}_{b}{q}_{c}},\end{array}\end{eqnarray}$and$\begin{eqnarray}\begin{array}{l}{{ \mathcal N }}_{{C}_{{\rm{I}}}({{AB}}_{{\rm{I}}})}\,=\,\displaystyle \frac{\cos {r}_{b}\sqrt{{p}_{b}{p}_{c}{q}_{c}\left({p}_{b}{p}_{c}{q}_{c}{\sin }^{4}{r}_{c}{\cos }^{2}{r}_{b}+4{q}_{b}{\cos }^{2}{r}_{c}\right)}-{p}_{b}{p}_{c}{q}_{c}{\sin }^{2}{r}_{c}{\cos }^{2}{r}_{b}}{{p}_{b}{p}_{c}\left({q}_{b}{\sin }^{2}{r}_{b}+{\cos }^{2}{r}_{b}\right)\left({q}_{c}{\sin }^{2}{r}_{c}+{\cos }^{2}{r}_{c}\right)+{q}_{b}{q}_{c}}.\end{array}\end{eqnarray}$where we use ${p}_{\mu }=1-{p}_{0,\mu }$ and ${q}_{\mu }=1-{q}_{0,\mu }$ for μ=(b, c). It can easily be checked that all the one-tangles reduce to the pure Unruh decoherence form [26, 27] in the limit ${p}_{0,\mu }={q}_{0,\mu }=0$ for μ=(b, c). In addition, if Bob and Charlie move with the same acceleration (rb=rc), also qb=qc, then the one-tangles with respect to the accelerated observers are equal, ${{ \mathcal N }}_{{B}_{{\rm{I}}}({{AC}}_{{\rm{I}}})}={{ \mathcal N }}_{{C}_{{\rm{I}}}({{AB}}_{{\rm{I}}})}$. In this case, the Bob and Charlie subsystems are symmetrical.
The optimal behavior of the one-tangles and π-tangle when two accelerated observers move with infinite acceleration $({r}_{b}={r}_{c}=\pi /4)$ is shown in figure 4. It is seen that the lost one-tangles and π-tangle at infinite acceleration start to recover with the increase in the strength of the partial measurements (left panel) and partial measurement reversal (right panel). It is interesting to note that the optimal values of p0 with respect to inertial and accelerated observers are equal (${{ \mathcal N }}_{A({B}_{{\rm{I}}}{C}_{{\rm{I}}})}={{ \mathcal N }}_{{B}_{{\rm{I}}}({{AC}}_{{\rm{I}}})}={{ \mathcal N }}_{{C}_{{\rm{I}}}({{AB}}_{{\rm{I}}})}=0.4$) at q0=0 for rb=rc=π/4, as shown in figure 4 (right panel). However, all the optimal one-tangles and the π-tangle under the influence of infinite acceleration of the two accelerated observers tends toward its maximum value in the limit of ${p}_{0}\to 1$ (left panel) and ${q}_{0}\to 1$ (right panel). It is observed that all the optimal one-tangles and the π-tangle under Unruh decoherence are completely protected in the presence of the maximum strength of the measurements. This shows that the Unruh effect of relativistic Dirac fields for the two accelerated observers is successfully eliminated by applying partial measurements.
Figure 4.
New window|Download| PPT slide Figure 4.The optimal behavior of the π-tangle of fermionic modes as a function of the strength of partial measurements ${p}_{0,b}={p}_{0,c}={p}_{0}$ (left panel) and partial measurement reversal ${q}_{0,b}={q}_{0,c}={q}_{0}$ (right panel) with infinite acceleration (rb=rc=π/4).
4. Conclusions
We have investigated the recovery of degraded entanglement due to the Unruh effect of a tripartite system when one or two observers are uniformly accelerated. It is found that the lost entanglement in the relativistic frame from the perspective of one-tangles and the π-tangle is successfully retrieved by partial measurements and partial measurement reversal. It is noted that the lost one-tangles and π-tangle for different acceleration turned out to be completely protected from the Unruh effect for the optimal strength of the partial measurements in the limit of ${q}_{0}\to 1$. Similar results are obtained for the optimal strength of the partial measurement reversal in the limit of ${p}_{0}\to 1$. Interestingly, the one-tangles for the two accelerated observers with uniform infinite acceleration turned out to be observer-independent, i.e.,${{ \mathcal N }}_{A({B}_{{\rm{I}}}{C}_{{\rm{I}}})}\,={{ \mathcal N }}_{{B}_{{\rm{I}}}({{AC}}_{{\rm{I}}})}={{ \mathcal N }}_{{C}_{{\rm{I}}}({{AB}}_{{\rm{I}}})}$ for the optimal strength of the partial measurements at q0=0. Moreover, it is found that the lost one-tangle with respect to the accelerated observer is recovered more rapidly compared to the inertial observer. As a result, the optimal one-tangles turned out to be fully protected in the limit of ${p}_{0}\to 1$ (with optimal q0) or ${q}_{0}\to 1$ (with optimal p0).
Acknowledgments
For this work, NAK was supported by the INTERWEAVE project, Erasmus Mundus Action 2 Strand 1 Lot 11, EACEA/42/11 Grant Agreement 2013-2538/001-001 EM Action 2 Partnership Asia-Europe, Fundação da Ciência e Tecnologia and COMPETE 2020 program in FEDER component (EU), through the projects POCI-01-0145-FEDER-028887 and UID/FIS/04650/2013. STA is grateful for the support from the Fundação da Ciência e Tecnologia (FCT) through a Doctoral Programme in the Physics and Mathematics of Information and the associated scholarship PD/ BD/113651/2015 and through the grant UID/CTM/04540/2013. STA also gratefully acknowledges the support of SQIG -- Security and Quantum Information Group, under the Fundação para a Ciência e a Tecnologia (FCT) project UID/EEA/50008/2019, and European funds, namely H2020 project SPARTA.
,1,∗, Syed Tahir Amin2,3,4, Munsif Jan
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