Impact of the Hawking Effect on the Fully Entangled Fraction of Three-qubit States in Schwarzschild Spacetime

Quantum teleportation plays a vital role in quantum information processing, serving as a fundamental concept in various quantum tasks and contributing significantly to the advancement of quantum technologies such as quantum communication, quantum computing, and quantum networks Nielsen.2000 ; Braunstein.2005 ; Kimble.2008 ; Weedbrook.2012 ; Wilde.2013 . Bennett et al. first proposed quantum teleportation in 1993 Bennett.1993 . The core idea of a quantum teleportation protocol is to utilize the characteristics of quantum entanglement to transmit information to spatially separated receivers while achieving information concealment Pirandola.2015 . The basic scheme of quantum teleportation serves as a fundamental component in the advancement of various quantum technologies, including quantum repeaters Briegel.1998 , quantum gate teleportation Gottesman.1999 , measurement-based quantum computing Raussendorf.2001 , and port-based teleportation Ishizaka.2008 .

Let $H$ be a $d$-dimensional Hilbert space. If a bipartite quantum state $\rho\in H\otimes H$ is used as a physical resource in quantum teleportation, the fidelity of the teleportation reads $F(\rho)=\frac{f(\rho)d+1}{d+1}$, where $f(\rho)$ is the fully entangled fraction (FEF) spop , $f(\rho)=\mathop{\max}_{\phi}\langle\phi|\rho|\phi\rangle$, with the maximization over all maximally entangled states $|\phi\rangle$ cdjw .

The development of black-hole physics can be traced to the early 20th century. In 1915, Albert Einstein proposed the theory of general relativity, which describes the geometric nature of gravity. The equations of general relativity predicted that objects with mass concentrated in a very small region would form black holes. Although progress has been made in the field of black-hole physics, many questions remain unresolved, such as the black-hole information paradox Danielsson.1993 ; Anglin.1995 ; Giddings.2002 , black-hole singularity Alwis.1993 ; Anderson.1993 and event horizon structure Jensen.1995 ; Cvetic.1997 .

In recent decades, the theory of relativistic quantum information has emerged in an attempt to address the problem of unifying general relativity and quantum mechanics Asher.2004 . The Hawking effect of a black hole has a negative impact on quantum steering, entanglement, discord, coherence, and the fidelity of quantum teleportation for bosonic fields in the context of curved spacetime Fuentes-Schuller.2005 ; Alsing.2006 ; Pan.2008 ; Martn-Martnez.2010 ; Wang.2010 ; Esfahani.2011 ; Xu.2014 ; Wang.2014 ; Bruschi.2014 ; D.E.Bruschi.2014 ; Hosseinidehaj.2015 ; Wang.2016 ; He.2016 ; Huang.2018 ; Shahbazi.2020 ; Wu.2022 ; Bhattacharya.2022 ; Zeng.2022 ; Wushumin.2022 ; txsf ; J. Ariadna ; Qiang ; Dong19 ; Dong20 . For example, Torres-Arenas et al. presented the entanglement measures of tripartite W-states in a noninertial frame through the coordinate transformation between Minkowski and Rindler J. Ariadna . Qiang et al. presented analytical concurrences for bipartite and tripartite entanglements simultaneously of Dirac fields in noninertial frames Qiang . However, Wu et al. proposed a different viewpoint Wu.2023 . They found that as the Hawking temperature increases, the fidelity of quantum teleportation may increase monotonically, rather than necessarily decrease monotonically in a bipartite system. Naturally, we are curious about the impact of the Hawking effect of black holes on teleportation fidelity in tripartite systems. However, in many-body systems, the fidelity of quantum teleportation has not been fully established.

The aim of this study was to generalize the FEF of bipartite systems to the multipartite case. We explored the FEF of multipartite states for Dirac fields in Schwarzschild spacetime. We suppose that Alice, Bob, and Charlie initially share an X-type state or a W-like state. Alice and Bob remain stationary in an asymptotically flat region, while Charlie positions himself near the event horizon of the black hole. For an X-type state, we discovered that the Hawking effect of the black hole has both positive and negative impacts on the FEF of Dirac fields, depending on the selection of the initial states. For a W-like state, the Hawking effect of the black hole has only a positive impact on the FEF of Dirac fields, independent of the selection of the initial states. Consequently, the Hawking effect of the black hole can not only decrease but also increase the FEF in Schwarzschild spacetime.

The rest of this paper is organized as follows. Section II discusses the calculation of the multipartite FEF and the lower bounds of $N$-qubit states. In particular, we introduce the FEF for X-type states of tripartite systems. Section III presents our investigation of the influence of the Hawking effect on the FEF with the X-type state in Schwarzschild spacetime. As discussed in Section IV, we investigated the influence of the Hawking effect on the FEF with the W-like state in Schwarzschild spacetime. Conclusions are presented in Section V.

We consider the $N$-qubit $(N\geq 2)$ systems $H^{\otimes N}$ with $d=2$. Denote $\{|0\rangle,|1\rangle\}$ the computational basis of $H$. Then, the multipartite FEF of the $N$-qubit state $\rho$ in $H^{\otimes N}$ is given by Xujianwei.2016

where max runs over all $2\times 2$ unitary matrices $U_{1},U_{2},$ $\ldots,U_{N}$ and $|\phi\rangle$ is the GHZ state,

Generally, it is challenging to calculate $f(\rho)$ analytically. We can calculate the lower bounds given by

where $m_{k}\in\{0,1,2,3\}$, $\sigma_{0}=I$, and $\sigma_{1}=X$, $\sigma_{2}=Y$ and $\sigma_{3}=Z$ are the standard Pauli matrices.

From (3), we obtain $f_{l}(\rho_{X})=\mathop{\max}\langle\phi|(\sigma_{m_{1}}\otimes\sigma_{m_{2}})\rho(\sigma_{m_{1}}\otimes\sigma_{m_{2}})|\phi\rangle$, where $|\phi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ and max runs over all Pauli matrices $\sigma_{i}$, $i=0,1,2,3$. We obtain (see Appendix A)

In this case, $m_{1}=0$ and $m_{2}=2$. In fact, the lower bound $f_{l}$ obtained here is the same as the real $f$ in Badziag.2000 .

From (3), the FEF is estimated by

with $\rho_{11}+\rho_{88}\geq\frac{1}{2}$ and $\rho_{18}\geq\frac{1}{2}(1-\rho_{11}-\rho_{88})$; see the detailed analysis in Appendix B.

We considered the Hawking effect on the FEF of the X-type state in Schwarzschild spacetime. The X-type states are of particular significance in quantum information, especially in the research of quantum entanglement. Rau has made excellent contributions to the study of X-type states and provided the algebraic characterization of X-type states Rau.2009 . Moreover, in 2010 Rau studied the generalized $N$-qubit X-type states and their symmetries Rau.2010 , giving rise to a comprehensive understanding of the properties of the X-type states.

Assume that Alice, Bob, and Charlie initially share an X-type state for three Unruh modes at an asymptotically flat region of a Schwarzschild black hole; then, Alice and Bob remain in the asymptotically flat region, while Charlie lingers near the event horizon of the black hole. Charlie uses his excited detector to probe the thermal Fermi-Dirac particle distribution. The Unruh vacuum state and the excited state of the fermionic mode in the Schwarzschild spacetime can be written as Wu.2023

and

where $T=\frac{1}{8\pi M}$ is the Hawking temperature with $M$ the mass of the black hole. $\omega$ is frequency. $|mm^{{}^{\prime}}n^{{}^{\prime}}n\rangle=|m_{k}\rangle^{+}_{out}|m^{{}^{\prime}}_{-k}\rangle^{-}_{in}\\ |n^{{}^{\prime}}_{-k}\rangle^{-}_{out}|n_{k}\rangle^{+}_{in}$, where $\{|m_{k}\rangle^{+}_{out}\}$ and $\{|m_{-k}\rangle^{-}_{in}\}$ are the orthonormal bases for the exterior and interior regions (denoted by the subscripts $\{out,in\}$) of the Schwarzschild black hole, respectively. The superscripts $\{+,-\}$ represent fermions and anti-fermions, respectively. The coefficients $q_{R}$ and $q_{L}$ in (20) satisfy $|q_{R}|^{2}+|q_{L}|^{2}=1$.

Because Charlie is unable to access the modes within the event horizon of the black hole, we trace out the inaccessible modes. The reduced density matrix $\rho^{ABC_{out}}_{X}$ is of the type (17), with all the nonzero entries given by

Assume that Charlie’s detector exclusively detects fermionic modes, i.e., the antifermionic modes remain unexcited in a single detector upon fermion detection. Consequently, by tracing out the antifermionic mode $\{|n^{{}^{\prime}}_{-k}\rangle^{-}_{out}\}$ beyond the event horizon of the Schwarzschild black hole, we obtain

where

Assuming that

we obtain from (18???)

The difference of $f(\rho^{S}_{X})$ between Hawking temperatures $T=T_{0}$ and $T=0$ is given by

$\Delta f_{l}(\rho^{S}_{X}(T_{0}))>0$ ($<0$) implies that $\frac{\partial f_{l}(\rho^{S}_{X})}{\partial T}|_{T_{0}}>0$ ($<0$).

Next, we investigated the variation of $f_{l}(\rho^{S}_{X})$ with respect to the Hawking temperature $T$ through several examples. As shown in Fig. 1 (GHZ state) and Fig. 2, it is evident that the fully entangled fraction $f_{l}(\rho^{S}_{X})$ exhibits a monotonic decrease with increasing Hawking temperature $T$. Furthermore, we observe that $f_{l}(\rho^{S}_{X})$ is a monotonically increasing function of the frequency $\omega$ and $q_{R}$. Note that the fully entangled fraction $f_{l}(\rho^{S}_{X})$ depends on the selection of the Unruh modes. An Unruh mode with $q_{R}=1$ is optimal. Consequently, the FEF can be preserved by selecting the high-frequency mode for maximally entangled states in Schwarzschild spacetime.

As shown in Fig. 3, $f_{l}(\rho^{S}_{X})$ increases monotonically as the Hawking temperature $T$ increases. This result implies that the Hawking effect of the black hole has a beneficial impact on the FEF. Consequently, the Hawking effect may result in a positive impact on the overall fidelity of quantum teleportation. Additionally, we observe that $f_{l}(\rho^{S}_{X})$ increases with increasing $q_{R}$. This result again demonstrates that the FEF depends on the choice of the Unruh modes, and the Unruh mode with $q_{R}=1$ is always optimal for the FEF. However, an increase in the frequency $\omega$ has a by-effect on the FEF.

As shown in Fig. 4, for $q_{R}=1$, $f_{l}(\rho^{S}_{X})$ initially rises to its peak value and subsequently decreases consistently as the Hawking temperature $T$ increases. This result indicates that the FEF for the single-mode approximation is both positively and negatively affected by the Hawking effect. It is evident that the maximum fidelity is contingent upon the Hawking temperature $T$ and the frequency $\omega$. Interestingly, for $q_{R}=0.9$ and $q_{R}=0.8$, $f_{l}(\rho^{S}_{X})$ exhibits a monotonic increase as the Hawking temperature $T$ increases. Therefore, we can conclude that for different types of Unruh modes, the FEF exhibits entirely distinct properties with increasing Hawking temperature $T$. That is, the Hawking effect of the black hole has both positive and negative impacts on the FEF of Dirac fields for the X-type state.

We considered the following W-like states of the tripartite systems:

The density matrix is given by

where $\rho_{22}+\rho_{33}+\rho_{55}=1$.

For the state (39), the lower bound of FEF is given by

with $\rho_{22}\geq\frac{1}{2}$; see the detailed analysis in Appendix C.

Consider that Alice, Bob, and Charlie initially share a W-like state for three Unruh modes at an asymptotically flat region of the Schwarzschild black hole. Afterwards, Alice and Bob stay in a region that approaches flatness, whereas Charlie remains close to the event horizon of the black hole. Charlie intends to utilize his excited detector to investigate the thermal Fermi-Dirac particle distribution. Because Charlie is unable to access the modes within the event horizon of the black hole, we trace out the inaccessible modes and derive the density matrix $\rho^{ABC_{out}}_{W}$ according to (19) and (20); see Appendix D).

Assume that Charlie’s detector exclusively detects fermionic modes, indicating that the antifermionic modes remain unexcited in a single detector upon fermion detection. Consequently, it is necessary for us to trace out the antifermionic mode $\{|n^{{}^{\prime}}_{-k}\rangle^{-}_{out}\}$ beyond the event horizon of the Schwarzschild black hole. Then, we obtain

where

Assume that $\rho^{S}_{W}$ satisfies the condition

We obtain

and

Therefore, $\Delta f_{l}(\rho^{S}_{W}(T_{0}))>0$ implies $\frac{\partial f_{l}(\rho^{S}_{W})}{\partial T}|_{T_{0}}>0$ and $\frac{\partial f_{l}(\rho^{S}_{W})}{\partial T}|_{T_{0}}<0$ implies $\Delta f_{l}(\rho^{S}_{W}(T_{0}))<0$.

We illustrate the fluctuation pattern of FEF $f_{l}(\rho^{S}_{W})$ with respect to the Hawking temperature $T$ by an example. Fig. 5 presents the relationship between the lower bound of FEF $f_{l}(\rho^{S}_{W})$ and the Hawking temperature $T$ for different $\omega$, $q_{R}$ and initial parameters. The result shows that for $q_{R}=1$, $f_{l}(\rho^{S}_{W})$ is a constant number $\frac{3}{8}$. For $q_{R}=0.9$ and $q_{R}=0.8$, $f_{l}(\rho^{S}_{W})$ monotonically increases as the Hawking temperature $T$ increases, regardless of the initial parameters. We also discovered that $f_{l}(\rho^{S}_{W})$ increases with increasing $q_{R}$, and the Unruh mode with $q_{R}=1$ is optimal. However, $f_{l}(\rho^{S}_{W})$ decreases with increasing $\omega$. Therefore, it is recommended to use low-frequency modes to improve the FEF. We found that the Hawking effect of the black hole has a positive impact on the FEF of Dirac fields for the W-like state.

In this study, we investigated the lower bound of FEF of Dirac fields among the users in Schwarzschild spacetime for tripartite systems. Alice, Bob, and Charlie initially share an X-type state or a W-like state. Alice and Bob stay still in an asymptotically flat region, while Charlie situates himself close to the event horizon of the black hole. For an X-type state, we found that the Hawking effect of the black hole has both positive and negative impacts on the FEF of Dirac fields, depending on the selection of the initial states. For a W-like state, the Hawking effect of the black hole has a positive impact on the FEF of Dirac fields, independent of the selection of the initial states. Hence, the Hawking effect of the black hole may both decrease and increase the FEF in Schwarzschild spacetime.

Furthermore, the choice of Unruh modes affects the lower bound of FEF. We found that the Unruh mode with $q_{R}=1$ is always optimal. Furthermore, from Fig. 4 we can conclude that for different types of Unruh modes, the FEF exhibits entirely distinct properties with increasing Hawking temperature $T$. Wu et al. Wu.2023 obtained some surprising results that overturned the belief that the Hawking effect of the black hole can only destroy the fidelity of quantum teleportation in a bipartite system. Our results show that the Hawking effect may either decrease or increase the FEF lower bound, which may provide an insightful view from the perspective of quantum information and general relativity, and highlight further studies on the fidelity of many-body quantum states under Hawking effects.