The influence of gravitational wave on tripartite quantum mutual information harvesting

Abstract

We examine quantum mutual information (QMI) extraction through local interactions of three Unruh–DeWitt detectors with the vacuum massless scalar field, comparing scenarios with and without gravitational wave perturbations in Minkowski spacetime. Our analysis reveals that gravitational waves can either enhance or diminish tripartite QMI compared to the flat spacetime case, demonstrating their dual capacity to amplify or suppress tripartite QMI harvesting. A significant resonance phenomenon emerges when detector energy gaps match the gravitational wave frequency. Furthermore, when harvesting a certain amount of tripartite QMI, gravitational wave modifies the spatial parameters for effective tripartite QMI harvesting: the achievable separation range undergoes extension or contraction depending on three critical parameters-detector energy gap, gravitational wave frequency, and the duration of the gravitational wave interaction.

Introduction

The study of quantum information theory has become central to advancing modern quantum technologies, particularly in the realm of multipartite systems. Practical applications such as multipartite quantum teleportation, quantum key distribution, and intricate quantum cryptographic protocols all rely fundamentally on the presence and manipulation of multipartite quantum correlations [1, 2, 3, 4–5]. Accurately characterizing and systematically quantifying these correlations is therefore not only theoretically significant but also operationally crucial [6, 7–8]. Among the various tools for this purpose, entropy-based measures have proven especially effective due to their theoretical consistency and practical relevance [9, 10, 11–12]. In this context, quantum mutual information (QMI) and its multipartite generalizations provide a unified framework for capturing the total correlations among multiple parties [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24–25]. The centrality of QMI extends across a wide range of quantum information processing tasks, from evaluating the capacities of quantum communication channels [26, 27], to quantifying representational fidelity in quantum machine learning algorithms [28, 29], and even to analyzing complex many-body quantum systems and emergent phenomena such as localization and objectivity [30]. These diverse applications underscore the indispensable role of multipartite mutual information as a fundamental diagnostic and computational tool in both foundational studies and applied quantum technologies.

Quantum information research is deeply intertwined with the fundamental properties of the quantum vacuum, whose characteristics are intrinsically shaped by the underlying spacetime structure [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89–90]. A powerful approach to probe these vacuum properties involves coupling quantum fields to first-quantized particle detectors, particularly through the Unruh–DeWitt (UDW) detector model [91, 92, 93, 94–95]. This framework has revealed that the vacuum state of quantum fields possesses rich nonlocal correlations that can be effectively extracted and quantified in terms of bipartite QMI, quantum discord, and entanglement between detectors in gravitational background [96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115–116]. In addition, recent investigations have shown a growing interest in exploring entanglement phenomena within gravitational wave backgrounds [117, 118, 119–120]. While the behavior of the quantum vacuum in curved spacetime has been extensively studied, the mechanisms by which QMI can be harvested in the presence of gravitational waves have yet to be fully understood. Given that QMI constitutes a fundamental measure of total correlations and serves as a critical resource in quantum information protocols, it is of great value to study the dynamics of extracting QMI from the quantum vacuum under gravitational wave perturbations. This constitutes one of the motivations for our work. As quantum information tasks become increasingly complex, bipartite QMI no longer meets the demands, making multipartite QMI a better choice. Furthermore, although the properties of bipartite QMI in various gravitational backgrounds are well understood, extending these properties to multipartite systems (especially harvesting tripartite QMI) remains a challenge, with important implications for our understanding of quantum vacuum in complex spacetime geometries. Consequently, we systematically investigate the extraction of multipartite QMI in gravitational wave backgrounds as an additional research motivation.

In this study, we establish a systematic framework for investigating gravitational wave assisted tripartite QMI harvesting through comparative analysis of correlation extraction mechanisms in Minkowski spacetime versus gravitational wave perturbed environments. Our research addresses two fundamental questions: (i) Quantitative amplification: whether gravitational waves enhance the magnitude of extractable tripartite QMI; (ii) Geometrical modification: whether they alter the optimal detector spatial configuration for tripartite QMI extraction compared to flat spacetime conditions. The analysis reveals a dual modulation mechanism whereby gravitational waves can either enhance or suppress tripartite QMI harvesting efficiency depending on parametric conditions. Notably, we find that the range of detector separations required to harvest a given amount of tripartite QMI follows a characteristic scaling determined by gravitational wave parameters. Three key factors govern this scaling: the detector energy gap, gravitational wave frequency, and the duration of the gravitational wave interaction.

The paper is structured as follows. In Sect. 2, we introduce the scalar field theory and a model of three Unruh–DeWitt (UDW) detectors in a gravitational wave background. Section 3 investigates the tripartite QMI of these detectors under gravitational wave influence. Finally, Sect. 4 presents a brief conclusion.

Unruh–DeWitt model in the background of gravitational waves

In the presence of a gravitational wave propagating along the z-direction, the spacetime metric can be expressed as

1

$$\begin{array}{cc}\hfill d{s}^2=& -d{t}^2+d{z}^2+(1+Acos[\omega (t-z)])d{x}^2\hfill \\ \hfill & +(1-Acos[\omega (t-z)])d{y}^2\hfill \\ \hfill =& -dudv+(1+Acos\omega u)d{x}^2+(1-Acos\omega u)d{y}^2.\hfill \\ \hfill \end{array}$$ Here, we introduce the light-cone coordinates $u:=t-z$ and $v:=t+z$, defined in terms of the Minkowski coordinates (t, x, y, z). This metric corresponds to a plane gravitational wave solution satisfying the linearized Einstein equations, valid to leading order in the perturbative parameter $A\ll 1$. Within this spacetime background, we consider a massless scalar field $\varphi (\mathsf{x})$ obeying the Klein–Gordon equation

2

$$\begin{array}{c}\hfill \square \varphi (\mathsf{x})=0,\end{array}$$where the d’Alembertian operator $\square$ is associated with the metric in Eq. (1). Solving this equation in light-cone coordinates $x=(u,v,x,y)$, we obtain a complete set of mode solutions as [121]

3

$$\begin{array}{cc}\hfill u_{\overrightarrow{k}}(\mathsf{x})=\frac{{\gamma }^{-1}(u)}{\sqrt{2k_{-}}{(2\pi )}^{\frac{3}{2}}}& e^{i{k}_a{x}^a-i{k}_{-}v-\frac{i}{4k_{-}}{\int }_0^{u}du(g^{\mathit{ab}}{k}_a{k}_b)},\hfill \end{array}$$where $k_{-}$ and $k_a$ (with indices $a,b\in \{x,y\}$) are separation constants that arise in solving the wave equation. The prefactor ${\gamma }^{-1}(u)={[det{g}_{\mathit{ab}}(u)]}^{\frac{1}{4}}$ encodes the influence of the gravitational wave on the mode functions. These solutions describe how the Minkowski vacuum is perturbed by the presence of the gravitational wave [122, 123].

Through detailed calculations, the vacuum Wightman function can be computed as

4

$$\begin{array}{cc}\hfill W(\mathsf{x},{\mathsf{x}}^{\prime }):=& ⟨0|\varphi (\mathsf{x})\varphi ({\mathsf{x}}^{\prime })|0⟩\hfill \\ \hfill =& \int d\mathsf{k}{u}_{\mathsf{k}}(\mathsf{x})u_{\mathsf{k}}^{\ast }({\mathsf{x}}^{\prime })=W_{\mathcal{M}}(\mathsf{x},{\mathsf{x}}^{\prime })\hfill \\ \hfill & +W_{\text{GW}}(\mathsf{x},{\mathsf{x}}^{\prime }),\hfill \end{array}$$where $W_{\mathcal{M}}(\mathsf{x},{\mathsf{x}}^{\prime })$ is the standard Minkowski Wightman function independent of the gravitational wave, and $W_{\mathit{GW}}(\mathsf{x},{\mathsf{x}}^{\prime })$ quantifies the first-order modification of the Minkowski Wightman function arising from variations in gravitational wave amplitude. Explicitly, these take the forms as

5

$$\begin{array}{c}\hfill W_{\mathcal{M}}(\mathsf{x},{\mathsf{x}}^{\prime })=\frac{1}{4\pi i\mathrm{\Delta }u}\delta (\frac{{\sigma }_{\mathcal{M}}(\mathsf{x},{\mathsf{x}}^{\prime })}{\mathrm{\Delta }u})+\frac{1}{4{\pi }^2{\sigma }_{\mathcal{M}}(\mathsf{x},{\mathsf{x}}^{\prime })},\\ \hfill \end{array}$$and

6

$$\begin{array}{cc}\hfill W_{\text{GW}}(\mathsf{x},{\mathsf{x}}^{\prime })=& -\frac{A}{4{\pi }^2}\text{sinc}\left(\frac{\omega }{2},\mathrm{\Delta },u\right)cos\left(\frac{\omega }{2},[u+u^{\prime }]\right)\hfill \\ \hfill & \times \frac{\mathrm{\Delta }{x}^2-\mathrm{\Delta }{y}^2}{\mathrm{\Delta }{u}^2}\left[i,\pi ,{\delta }^{\prime },,\left(\frac{{\sigma }_{\mathcal{M}}(\mathsf{x},{\mathsf{x}}^{\prime })}{\mathrm{\Delta }u}\right),,+,,\frac{\mathrm{\Delta }{u}^2}{{\sigma }_{\mathcal{M}}^2(\mathsf{x},{\mathsf{x}}^{\prime })}\right],\hfill \\ \hfill \end{array}$$where $\mathrm{\Delta }{\mathsf{x}}^{\mu }:={\mathsf{x}}^{\mu }-{\mathsf{x}}^{\prime \mu }$, and ${\sigma }_{\mathcal{M}}(\mathsf{x},{\mathsf{x}}^{\prime }):=-\mathrm{\Delta }u\mathrm{\Delta }v+\mathrm{\Delta }{x}^2+\mathrm{\Delta }{y}^2$ is the geodesic separation between two points in Minkowski spacetime. To explore the effect of gravitational waves on tripartite QMI, we consider a system of three Unruh–DeWitt detectors arranged in a linear configuration with a total length of $2L$. The central detector is equidistant from the other two, and we vary this separation. We label the detectors sequentially from left to right as A, B, and C, treating each as a two-level quantum system. The basis states are defined as the ground state $|0_D⟩$ and the excited state $|1_D⟩$ (where $D\in \{A,B,C\}$), with an energy gap of $2\mathrm{\Omega }$ between them. The detectors couple locally to a scalar field $\varphi (\mathsf{x})$ via trajectories ${\mathsf{x}}_D(t)$, with interactions described by the Hamiltonian

7

$$\begin{array}{c}\hfill H_D(t)=\lambda \chi (t)(e^{i\mathrm{\Omega }t}{\sigma }^{+}+e^{-i\mathrm{\Omega }t}{\sigma }^{-})\otimes \varphi [{\mathsf{x}}_D(t)],\end{array}$$where $\chi (t)=exp[-\frac{{(t-t_0)}^2}{2{\sigma }^2}]$ is the switching function of $t_0$ and $\sigma$, which correspond to the occurrence and duration of the interaction, respectively, $\lambda$ is the coupling constant and describes the strength of the interaction, and ${\sigma }^{+}=|1_D⟩⟨0_D|$ and ${\sigma }^{-}=|0_D⟩⟨1_D|$ denote SU(2) ladder operators acting on the detector Hilbert space.

Initially, let us consider the three detectors prepared as $(t\to -\infty )$ in their ground states ${|0⟩}_A{|0⟩}_B{|0⟩}_C$, while the field is assumed to be in an appropriately defined vacuum state $|0⟩$. Consequently, the entire initial state of the system comprising the detectors and the field is given by $|{\mathrm{\Psi }}_i{⟩=|0⟩}_A{|0⟩}_B{|0⟩}_C|0⟩$. Under the interaction Hamiltonian specified in Eq. (7), the final state at $(t\to \infty )$ evolves into $|{\mathrm{\Psi }}_f⟩=\mathcal{T}{e}^{-i{\int }_{\mathbb{R}}dt[H_A(t)+H_B(t)+H_C(t)]}|{\mathrm{\Psi }}_i⟩$, where $H_A(t)$, $H_B(t)$, and $H_C(t)$ are defined in Eq. (7), and $\mathcal{T}$ denotes the time-ordering operator. The reduced density matrix of the detectors is obtained by tracing out the field degrees of freedom and is expressed in the computational basis $\{|0_A{0}_B{0}_C⟩,|0_A{0}_B{1}_C⟩,|0_A{1}_B{0}_C⟩,|1_A{0}_B{0}_C⟩,|0_A{1}_B{1}_C⟩,|1_A{0}_B{1}_C⟩,|1_A{1}_B{0}_C⟩,|1_A{1}_B{1}_C⟩\}$ [124, 125] as

8

$$\begin{array}{cc}& {\rho }_{\mathit{ABC}}={\text{tr}}_{\varphi }(|{\mathrm{\Psi }}_f⟩⟨{\mathrm{\Psi }}_f|)\hfill \\ \hfill & =\left(,,\begin{array}{cccccccc}1-(P_A+P_B+P_C)& 0& 0& 0& X_{\mathit{BC}}^{\ast }& X_{\mathit{AC}}^{\ast }& X_{\mathit{AB}}^{\ast }& 0\\ 0& P_C& C_{\mathit{BC}}^{\ast }& C_{\mathit{AC}}^{\ast }& 0& 0& 0& 0\\ 0& C_{\mathit{BC}}& P_B& C_{\mathit{AB}}^{\ast }& 0& 0& 0& 0\\ 0& C_{\mathit{AC}}& C_{\mathit{AB}}& P_A& 0& 0& 0& 0\\ X_{\mathit{BC}}& 0& 0& 0& 0& 0& 0& 0\\ X_{\mathit{AC}}& 0& 0& 0& 0& 0& 0& 0\\ X_{\mathit{AB}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array},,\right)\hfill \\ \hfill & +\mathcal{O}({\lambda }^4).\hfill \end{array}$$The transition probability P represents the likelihood that a detector will be found in the excited state $|1_D⟩$ in the distant future $(t\to \infty )$, due to vacuum fluctuations and a finite interaction time. Here, we assume all detectors are identical, with the same energy gap and switching function $\chi (\tau )$, to reduce the complexity of the parameter space. Consequently, all three detectors share the same transition probability $P_A=P_B=P_C=P$. For a detector at rest in Minkowski spacetime, its trajectory is ${\mathsf{x}}_D(t)=(t,0,0,0)$, and the transition probability reduces to

9

$$\begin{array}{c}\hfill P=\frac{{\lambda }^2}{4\pi }[e^{-{\sigma }^2{\mathrm{\Omega }}^2}-\sqrt{\pi }\sigma \mathrm{\Omega }(1-\text{erf}[\sigma \mathrm{\Omega }])].\end{array}$$However, for this trajectory, the contribution of gravitational waves $W_{\text{GW}}(\mathsf{x},{\mathsf{x}}^{\prime })$ to vanishes since $\mathrm{\Delta }{x}^2=\mathrm{\Delta }{y}^2=0$. Thus, P remains unchanged even in a gravitational wave background, implying that a single detector cannot detect the presence of gravitational waves.

The matrix elements $X_{D{D}^{\prime }}$ and $C_{D{D}^{\prime }}$$(D^{\prime }\in \{A,B,C\},D\ne D^{\prime })$ encode nonclassical correlations and decompose into Minkowski and gravitational-wave contributions: $X_{D{D}^{\prime }}=X_{D{D}^{\prime }\mathcal{M}}+X_{D{D}^{\prime }\text{GW}}$ and $C_{D{D}^{\prime }}=C_{D{D}^{\prime }\mathcal{M}}+C_{D{D}^{\prime }\text{GW}}$. Specifically, $X_{D{D}^{\prime }\mathcal{M}}$ and $C_{D{D}^{\prime }\mathcal{M}}$ denote the values of $X_{D{D}^{\prime }}$ and $C_{D{D}^{\prime }}$ in Minkowski spacetime, whereas $X_{D{D}^{\prime }\text{GW}}$ and $C_{D{D}^{\prime }\text{GW}}$ quantify gravitational-wave-induced modifications. Notably, these elements depend explicitly on the spatial separations between detectors. For instance, when the detectors are arranged in an equidistant configuration such that the separation between adjacent pairs is $d=L$, the off-diagonal elements characterizing quantum correlations satisfy $X_{\mathit{BC}}=X_{\mathit{AB}}\equiv X_L$ and $C_{\mathit{AB}}=C_{\mathit{BC}}\equiv C_L$. In contrast, for the non-adjacent, outermost detector pair separated by $d=2L$, the corresponding terms become $X_{\mathit{AC}}\equiv X_{2L}$ and $C_{\mathit{AC}}\equiv C_{2L}$. This spatial dependence reflects the nonlocal nature of vacuum-induced correlations and is essential for understanding how quantum information is distributed in space. To investigate the extraction of tripartite QMI across spacetime regions, we consider the following trajectories for three detectors A, B, and C, delineated in Minkowski coordinates ${\mathsf{x}}_A(t)=(t,0,0,0)$, ${\mathsf{x}}_B(t)=(t,L,0,0)$, ${\mathsf{x}}_C(t)=(t,2L,0,0)$. The spatial separation between any two detectors D and $D^{\prime }$ is given by $d:=|{\mathsf{x}}_D-{\mathsf{x}}_{D^{\prime }}|$. Since the detectors interact with the field for an approximate proper time $\sigma$, their interactions can be regarded as approximately spacelike when $d>\sigma$, and timelike when $d<\sigma$ [122]. Using the Wightman function, the correlation terms $X_{D{D}^{\prime }\mathcal{M}}$, $C_{D{D}^{\prime }\mathcal{M}}$, $X_{D{D}^{\prime }\text{GW}}$, and $C_{D{D}^{\prime }\text{GW}}$ can be explicitly written as

10

$$\begin{array}{cc}& X_{D{D}^{\prime }\mathcal{M}}:=i\frac{\sigma {\lambda }^2}{4d\sqrt{\pi }}{e}^{-{\sigma }^2{\mathrm{\Omega }}^2-2i\mathrm{\Omega }{t}_0-\frac{d^2}{4{\sigma }^2}}\hfill \\ \hfill & \left[\text{erf},\left(\frac{\mathit{id}}{2\sigma }\right),-,1\right],\hfill \end{array}$$

11

$$\begin{array}{cc}& C_{D{D}^{\prime }\mathcal{M}}:=\frac{\sigma {\lambda }^2}{4d\sqrt{\pi }}{e}^{-\frac{d^2}{4{\sigma }^2}}\hfill \\ \hfill & \times \left(\text{Im},\left[e^{id\mathrm{\Omega }},\text{erf},\left(i,\frac{d}{2\sigma },+,\sigma ,\mathrm{\Omega }\right)\right],-,sin,\mathrm{\Omega },d\right),\hfill \end{array}$$

12

$$\begin{array}{cc}& X_{D{D}^{\prime }\text{GW}}:=\frac{A\sigma {\lambda }^2}{4d^2{\pi }^{3/2}}f(\omega ,\mathrm{\Omega },\sigma ,t_0)\left(I_{D{D}^{\prime }1},+,I_{D{D}^{\prime }2}\right),\hfill \\ \hfill \end{array}$$and

13

$$\begin{array}{c}\hfill C_{D{D}^{\prime }\text{GW}}:=-\frac{A\sigma {\lambda }^2}{4d^2{\pi }^{3/2}}{e}^{-\frac{{\sigma }^2{\omega }^2}{4}}cos\left(\omega ,t_0\right)\left(I_{D{D}^{\prime }3},+,I_{D{D}^{\prime }4}\right),\\ \hfill \end{array}$$with

14

$$\begin{array}{cc}\hfill f(\omega ,\mathrm{\Omega },\sigma ,t_0):=& e^{-\frac{{\sigma }^2}{4}{(\omega -2\mathrm{\Omega })}^2-i{t}_0(\omega +2\mathrm{\Omega })}\hfill \\ \hfill & +e^{-\frac{{\sigma }^2}{4}{(\omega +2\mathrm{\Omega })}^2+i{t}_0(\omega -2\mathrm{\Omega })}.\hfill \end{array}$$Here, the terms $I_{D{D}^{\prime }1}$, $I_{D{D}^{\prime }2}$, $I_{D{D}^{\prime }3}$ and $I_{D{D}^{\prime }4}$ involve intricate dependencies on $\omega$, d, $\sigma$ and $\mathrm{\Omega }$, with explicit forms provided in the subsequent equations$$\begin{array}{cc}& I_{D{D}^{\prime }1}=i\frac{\pi e^{-\frac{d^2}{4{\sigma }^2}}}{\omega }\left[\left(\frac{d^2}{4{\sigma }^2},+,1\right),sin,\left(\frac{\omega }{2},d\right),-,\frac{d\omega }{4},cos,\left(\frac{\omega }{2},d\right)\right],\hfill \\ \hfill & I_{D{D}^{\prime }2}=\frac{\pi }{\omega }\left(\text{erf},\left(\frac{\sigma \omega }{2}\right)\right)\hfill \\ \hfill & \left(-,e^{-\frac{d^2}{4{\sigma }^2}},\text{Re},\left[e^{i\frac{\omega }{2}d},\left(1,+,\frac{d^2}{4{\sigma }^2},-,i,\frac{d\omega }{4}\right),\text{erf},\left(\frac{\omega }{2},\sigma ,+,\frac{i\text{d}}{2\sigma }\right)\right]\right),\hfill \\ \hfill & I_{D{D}^{\prime }3}=\frac{\pi e^{-\frac{d^2}{4{\sigma }^2}}}{4\omega }[\left(d,\omega ,+,2,d,\mathrm{\Omega }\right)sin\left(d,\left[\frac{\omega }{2},+,\mathrm{\Omega }\right]\right)\hfill \\ \hfill & +\left(d,\omega ,-,2,d,\mathrm{\Omega }\right)sin\left(d,\left[\frac{\omega }{2},-,\mathrm{\Omega }\right]\right)\hfill \\ \hfill & +\left(\frac{d^2}{{\sigma }^2},+,4\right)\left(cos,\left(d,\left[\frac{\omega }{2},+,\mathrm{\Omega }\right]\right),-,cos,\left(d,\left[\frac{\omega }{2},-,\mathrm{\Omega }\right]\right)\right)],\hfill \\ \hfill & I_{D{D}^{\prime }4}=\frac{\pi }{\omega }\left(\text{erf},\left[\sigma ,\left(\frac{\omega }{2},-,\mathrm{\Omega }\right)\right],+,\text{erf},\left[\sigma ,\left(\frac{\omega }{2},+,\mathrm{\Omega }\right)\right]\right)\hfill \\ \hfill & \left(-,e^{-\frac{d^2}{4{\sigma }^2}},\text{Re},\left[Q_{+},R_{+},+,Q_{-},R_{-}\right]\right),\hfill \end{array}$$with $Q_{\pm }:=-i{e}^{id\left(\frac{\omega }{2},\pm ,\mathrm{\Omega }\right)}\text{erf}\left[i,\frac{d}{2\sigma },+,\sigma ,\left(\frac{\omega }{2},\pm ,\mathrm{\Omega }\right)\right]$ and $R_{\pm }:=\frac{d}{2}\left(\frac{\omega }{2},\pm ,\mathrm{\Omega }\right)+i\left(1,+,\frac{d^2}{4{\sigma }^2}\right)$. Therefore, the density matrix Eq. (8) in this linear configuration can be rephrased as

15

$$\begin{array}{c}\hfill {\rho }_{\mathit{ABC}}=\left(,,\begin{array}{cccccccc}1-3P& 0& 0& 0& X_L^{\ast }& X_{2L}^{\ast }& X_L^{\ast }& 0\\ 0& P& C_L^{\ast }& C_{2L}^{\ast }& 0& 0& 0& 0\\ 0& C_L& P& C_L^{\ast }& 0& 0& 0& 0\\ 0& C_{2L}& C_L& P& 0& 0& 0& 0\\ X_L& 0& 0& 0& 0& 0& 0& 0\\ X_{2L}& 0& 0& 0& 0& 0& 0& 0\\ X_L& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array},,\right)+\mathcal{O}({\lambda }^4).\\ \hfill \end{array}$$

QMI of three Unruh–DeWitt detectors

The QMI of a bipartite state ${\rho }_{\mathit{AB}}$ quantifies the total correlations between subsystems A and B, defined as

16

$$\begin{array}{c}\hfill I({\rho }_{\mathit{AB}})=S({\rho }_A)+S({\rho }_B)-S({\rho }_{\mathit{AB}}),\end{array}$$where ${\rho }_A={\text{tr}}_B({\rho }_{\mathit{AB}})$ and ${\rho }_B={\text{tr}}_A({\rho }_{\mathit{AB}})$ are the reduced density matrix obtained by tracing out subsystem B and A, respectively, and $S(\rho )=-\text{tr}(\rho {log}_2\rho )$ is the von Neumann entropy. The non-negativity of QMI establishes it as a proper measure of quantum correlations. From a physical perspective, the quantum mutual information (QMI) quantifies the thermodynamic cost in terms of work or noise-required to completely erase correlations between subsystems. While bipartite QMI provides insight into the total correlations between two subsystems, realistic quantum systems often involve multiple parties. In such scenarios, it becomes essential to consider a generalization of QMI that captures the full spectrum of correlations present in multipartite systems. For n-partite systems, this concept generalizes to [126, 127]

17

$$\begin{array}{c}\hfill I_x(A_1:A_2:...:A_n)=\sum _{k=1}^{n}S({\rho }_{A_k})-S({\rho }_{A_1{A}_2...A_n}),\end{array}$$where $S({\rho }_{A_k})$ denotes the von Neumann entropy of the reduced density matrix for subsystem $A_k$. This multipartite QMI captures the total correlations shared among all n subsystems.

To derive the tripartite QMI analytically in the presence of gravitational waves, we first trace out the modes of two detectors from the system described by Eq. (15), obtaining the reduced density matrix for each detector

18

$$\begin{array}{c}\hfill {\rho }_D=\left(,,\begin{array}{cc}1-P& 0\\ 0& P\end{array},,\right).\end{array}$$Combining Eq. (15), Eq. (17), and Eq. (18), we obtain the analytical expression for calculating tripartite QMI as

19

$$\begin{array}{cc}\hfill I({\rho }_{\mathit{ABC}})=& \lambda {log}_2\lambda +{\lambda }_{+}{log}_2{\lambda }_{+}+{\lambda }_{-}{log}_2{\lambda }_{-}-3P{log}_{2}P\hfill \\ \hfill & +\mathcal{O}({\lambda }^4),\hfill \\ \hfill \end{array}$$with $\lambda =P-C_{2L}$ and ${\lambda }_{\pm }=\frac{1}{2}(C_{2L}\pm \sqrt{8C_L^2+C_{2L}^2}+2P)$. We examine how gravitational waves affect the harvesting of tripartite QMI and whether they expand the range over which such correlations can be extracted. To directly assess their impact, we compare the harvested tripartite QMI with that in flat Minkowski spacetime through parallel plotting and analysis.

[See PDF for image]

Fig. 1

Tripartite QMI in the background of gravitational waves and in Minkowski spacetime as a function of the gravitational wave frequency $\omega \sigma$ for various values of the energy gap $\mathrm{\Omega }\sigma$ and $t_0/\sigma$ with $L/\sigma =0.4$

Figure 1 illustrates the tripartite QMI harvested by three detectors in both Minkowski spacetime and a gravitational wave background, plotted as a function of the gravitational wave frequency $\omega \sigma$ for varying detector energy gaps $\mathrm{\Omega }\sigma$. Both switching scenarios, $t_0=0$ and $t_0\ne 0$, are considered. Note that, as the gravitational wave frequency $\omega \sigma$ increases, the correlation term $C_{D{D}^{\prime }\text{GW}}$ approaches zero, and the transition probability P does not depend on gravitational waves. Therefore, in the high-frequency regime $(\omega \sigma \gg \mathrm{\Omega }\sigma )$, the harvested QMI in the presence of the gravitational wave asymptotically approaches its flat-spacetime counterpart, suggesting that the gravitational wave background exerts minimal influence on the harvesting process in this limit.

Figure 1a–d demonstrate a pronounced resonance phenomenon when the gravitational wave frequency satisfies the condition $\omega \approx 2\mathrm{\Omega }$, where $2\mathrm{\Omega }$ denotes the energy gap between the ground and excited states of the Unruh–DeWitt detectors. This resonance is particularly evident in the case where the switching function is centered at $t_0=0$, leading to maximal suppression of the harvested tripartite QMI across a wide range of parameter values. In contrast, Fig. 1e–h show that for an asymmetric switching time $t_0\ne 0$, a strong resonance still occurs near $\omega \approx 2\mathrm{\Omega }$, but the effect is qualitatively different. In this case, the harvested tripartite QMI in the presence of the gravitational wave exceeds that of flat spacetime, indicating that the gravitational wave background can actually enhance the extraction of quantum correlations under certain temporal conditions. This enhancement is attributed to constructive interference between the gravitational wave-induced field modulation and the time-asymmetric detector switching, which amplifies the vacuum-induced correlations.

[See PDF for image]

Fig. 2

Tripartite QMI among three detectors in the background of gravitational waves and in Minkowski spacetime as a function of the detector separation $L/\sigma$

Figure 2 illustrates tripartite QMI shared among three detectors in both gravitational wave backgrounds and Minkowski spacetime, plotted as a function of the dimensionless detector separation $L/\sigma$. In the general scenario, the harvestable tripartite QMI gradually decreases with increasing $L/\sigma$, which is consistent with the expected degradation of quantum correlations due to spatial separation. Specifically, the left panel of Figure 2 shows that, to harvest a certain amount of QMI, the achievable separation range for harvesting QMI in the gravitational wave background is narrower than that in Minkowski spacetime, indicating that gravitational waves shorten the viable detector separation range. Conversely, when harvesting a certain amount of tripartite QMI, the right panel of Fig. 2 reveals that the gravitational wave-induced range is broader than the Minkowski case, suggesting that gravitational waves can widen the effective separation range. Overall, the ability to extract a specific amount of tripartite QMI is highly sensitive to the system parameters $t_0/\sigma$, $\mathrm{\Omega }\sigma$, and $\omega \sigma$. Whether gravitational waves enhance or inhibit QMI harvesting depends intricately on the interplay among these parameters.

Conclusions

We have investigated the influence of gravitational waves on the harvesting of tripartite quantum mutual information (QMI) among three linearly aligned Unruh–DeWitt (UDW) detectors. By allowing the detectors to interact locally with a massless scalar field in the presence of a passing gravitational wave, we explore how gravitational perturbations affect the structure of multipartite quantum correlations harvesting in such a configuration. Our analysis revealed that gravitational waves can induce a pronounced resonance effect when their frequency approaches the detector energy gap, $\omega \approx 2\mathrm{\Omega }$. In particular, when the interaction is centered at $t_0=0$, this resonance leads to a notable suppression of the harvested tripartite QMI compared to the Minkowski vacuum, indicating destructive interference with the field’s vacuum correlations. In contrast, when $t_0\ne 0$, the same resonance can enhance the harvested tripartite QMI, highlighting the critical role played by the temporal alignment between the detector switching function and the gravitational wave.

Additionally, we explore the dependence of harvested QMI on the dimensionless detector separation $L/\sigma$. For a fixed amount of harvested QMI, we find that gravitational waves can either reduce or extend the spatial range over which significant correlations can be extracted, depending sensitively on the values of $t_0/\sigma$, $\mathrm{\Omega }\sigma$, and $\omega \sigma$. In certain parameter regimes, for a given amount of harvested QMI, gravitational waves shorten the effective harvesting range compared to flat spacetime, whereas in others, they enable the extraction of tripartite QMI over a broader spatial domain.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant nos. 12205133) and the Special Fund for Basic Scientific Research of Provincial Universities in Liaoning under Grant no. LS2024Q002.

Data Availability Statement

This manuscript has no associated data. [Authors’ comment: I would like to emphasize that all relevant physical and mathematical calculations are explicitly presented in this paper].

Code Availability Statement

This manuscript has no associated code/software. [Authors’ comment: Code/Software sharing not applicable to this article as no code/software was generated or analysed during the current study].

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