Generation of quantum coherence for continuous variables between causally disconnected regions in dilaton spacetime

We study the dynamics of Gaussian quantum coherence under the background of a Garfinkle-Horowitz-Strominger dilaton black hole. It is shown that the dilaton field has evident effects on the degree of coherence for all the bipartite subsystems. The bipartite Gaussian coherence is not affected by the frequency of the scalar field for an uncharged or an extreme dilaton black hole. It is found that the initial coherence is not completely destroyed even for an extreme dilaton black hole, which is quite different from the behavior of quantum steering because the latter suffers from a"sudden death"under the same conditions. This is nontrivial because one can employ quantum coherence as a resource for quantum information processing tasks even if quantum correlations have been destroyed by the strong gravitational field. In addition, it is demonstrated that the generation of quantum coherence between the initial separable modes is easier for low-frequency scalar fields. At the same time, quantum coherence is smoothly generated betweenone pair of partners and exhibits a"sudden birth"behavior between another pairs in the curved spacetime.


I. INTRODUCTION
String theory is regarded as one of the most promising candidates for a consistent understanding of quantum mechanics and the theory of gravity. Different from general relativity, it is predicted in string theory that the presence of dilaton fields can change the properties of black holes [1][2][3][4].
A solution of static spacetime, i.e., the Garfinkle-Horowitz-Strominger (GHS) dilaton black hole [3,4], can be obtained by choosing the invariant to be the Lagrangian of the electromagnetic field. The Hawking temperature [5] of the GHS black hole not only depends on its mass, but also on the dilaton field because the latter is also a resource of gravity. On the other hand, the behavior of quantum information near the event horizon of black holes has received considerable attention in recent years [6][7][8][9][10][11][12][13][14][15][16][17]. It is believed that related studies can contribute to a deeper understanding of nonlocality between causally disconnected spacetime regions, as well as to a further understanding of the entropy and the problem of information loss for black holes [18][19][20].
Quantum coherence [21][22][23][24], as a key aspect of quantum physics, is not only an embodiment of the superposition principle of states but also the basis of many unique phenomena in quantum physics, such as quantum entanglement and steering. With the help of quantum coherence, one can implement various quantum information processing tasks which cannot be accomplished classically, such as quantum computing [25,26], quantum information storage [27,28] , quantum communication [29,30], and quantum metrology [31][32][33]. Therefore, understanding quantum coherence in a more general framework is important both for the fundamental research of physics and the development of modern quantum technology. Despite its fundamental role in physics, the study of coherence receives increasing attention until Baumgratz et al. introduced an architecture for the measurement of coherence [21]. In 2016, a quantification of coherence for continuous variables was proposed by Xu [34], which provides a framework of quantum resource theory for quantum superposition in infinite-dimensional quantum systems [37].
In this paper, we study the distribution and generation of continuous-variable coherence in the background of a GHS dilaton black hole. The considered initial state is a two-mode squeezed Gaussian state, which can be employed to define quantum vacuum when the spacetime has at least two asymptotically flat regions. This state has a special role in the quantum field theory because the values of squeezing parameter between causally disconnected regions are changed according to the spacetime structure. We find that Gaussian quantum coherence is more robust than the steering-type quantum correlations under the influence of gravity in the GHS spacetime.
The structure of the paper is as follows. In Sec. II, we discuss the dynamics and second quantization of the scalar field near the GHS dilaton black hole. In Sec. III and IV, we discuss the method of measuring the coherence of continuous variables and the behavior of Gaussian quantum coherence in the GHS spacetime, respectively. In the final section, we make a brief summary.
Throughout the paper, the units G = c = = κ B = 1 are used.

II. VACUUM STRUCTURE OF THE SCALAR FIELD
In this section, we review the quantum field theory in the GHS black hole spacetime. The equations describing gravity in the context of string theory can be approximated by Einstein's equations in the regions near the horizon of a black hole. In this scenario, the Schwarzschild solution is a good approximation to describe a static and uncharged black hole within string theory.
However, when we consider the solutions of the Einstein-Maxwell equations, the presence of a scalar field called dilaton should be considered. In this case, the added dilaton field couples with the Maxwell field, which changes the spacetime characters of the black hole. In the low energy limit of string theory, the static dilaton black hole solution was provided by Garfinkle, Horowitz, and Strominger [4]. The dilaton gravity in string theory is important because it is a good candidate for an eventual quantum theory of gravity [1,2]. Therefore, it is meaningful to study the behavior of Gaussian quantum coherence in the dilaton spacetime.
The line element for a GHS dilaton black hole can be written as [4] where dΩ 2 = dθ 2 + sin 2 θdϕ 2 , M is the mass of the black hole and D is the dilaton charge. The event horizon of the GHS black hole is located at r + = 2M .
The dynamics of massless scalar field in a general background is given by the Klein-Gordon Solving the Klein-Gordon equation near the event horizon of the black hole, one obtains the outgoing modes where v = t + r * , u = t − r * , and r * is the tortoise coordinates.
Employing the Schwarzschild modes given in Eqs. (3) and (4), the scalar field Φ near the event horizon can be expanded as [38] where b in,ωlm and b † in,ωlm are the annihilation and creation operators acting on the states of the interior region of the dilaton black hole. Similarly, b out,ωlm and b † out,ωlm are the operators acting on the vacuum of the exterior region, respectively. The Schwarzschild vacuum state for the scalar field can be defined as [38] b in,ωlm |0 in = b out,ωlm |0 out = 0.
On the other hand, the light-like Kruskal coordinates U and V are defined by [40,41], Then we can rewrite the field modes to Making an analytic continuation for Eqs. (8) and (9), we obtain [41] which shows that one can use the Kruskal coordinates to introduce new orthogonal basis for the scalar field.
By expanding the scalar field Φ in terms of φ I,ωlm and φ II,ωlm in the GHS spacetime, one where the annihilation operator a I,ωlm can be used to define the Kruskal vacuum outside the event horizon [41] a I,ωlm |0 K = 0.
It is seen that Eq. (5) is the expansion of the scalar field in Schwarzschild modes, while Eq. (12) corresponds to the decomposition of the field in Kruskal modes [40,41]. Then we can calculate the Bogoliubov transformations between the particle annihilation and creation operators which act on the Schwarzschild vacuum and Kruskal vacuum, respectively. After some calculation, the Bogoliubov transformations are found to be [40] where , a I,ωlm and a † I,ωlm are the annihilation and creation operators acting on the Kruskal vacuum of the exterior region, respectively. For the observer outside the black hole, the modes in the interior region should be traced over because a local observer has no access to the information in the causally disconnected region.
After normalizing the state vector, it is found that the Kruskal vacuum can be expressed as an entangled two-mode squeezed state where |n in and |n out are excited-states for Schwarzschild modes inside and outside the event horizon. This means that the observers in different coordinates will not agree on the particle content of each of these states. It is then interesting to investigate to what degree the coherence of the quantum state for continuous variables is changed when the state is described by the observers in different coordinates.

III. MEASUREMENT OF QUANTUM COHERENCE FOR CONTINUOUS VARIABLES
In this section, we review the measurement of quantum coherence for continuous variables [34].
A quantum system is called a continuous-variable system because it has an infinite-dimensional Hilbert space described by observables with continuous eigenspectra [35]. The prototype of a continuous-variable system is represented by N bosonic modes corresponding to the field opera- . These annihilation and creation operators can be arranged in a vectorial operator b := (â 1 ,â † 1 , · · · ,â N ,â † N ) T , which must satisfy the bosonic commutation relations where Ω ij are generic elements of known as the symplectic form. The Hilbert space of this system is infinite-dimensional because the single-mode Hilbert space H is spanned by a countable Fock basis {|n } ∞ n=0 . Besides the bosonic field operators, the bosonic system may be described by the quadrature field operators, formally arranged in the vectorR := (q 1 ,p 1 , . . . ,q N ,p N ) T [35], which are related to the annihilationâ i and creationâ † i operators of each mode, by the relationsq i = 2i . The quadrature operatorsq i andp i represent the canonical observables of the system. Similarly, the vector operator should satisfy the commutation relationship [R i ,R j ] = iΩ ij , which takes symplectic form. The most relevant quantities that characterize the nature of a two-mode Gaussian state ρ AB are the statistical moments. The first moment is the mean valuē R := R = Tr(Rρ), and the second moment is the covariance matrix σ, whose arbitrary el- The covariance matrix is a symmetric matrix which must satisfy the uncertainty principle σ + iΩ ≥ 0 [36], which implies the positive definiteness σ > 0.
For a two-mode Gaussian state, we can write its covariance matrix in the block form where α = α T , β = β T and γ are 2 × 2 real matrices. Then, the Williamson form is simply with ∆ := det α + det β + 2 det γ and det is the determinant [35].
As shown in [34], the definition of the continuous variable quantum coherence of a Gaussian state is where S(ρ||δ) = tr(ρ log 2 ρ) − tr(ρ log 2 δ) is the relative entropy, δ is an incoherent Gaussian state and the minimization runs over all incoherent Gaussian states. In addition, the entropy of ρ is defined by [42] S where , and ν i are symplectic eigenvalues of each modes. The mean occupation value can be expressed as In this equation, σ i are elements of the subsystem of mode i in a continuous variable matrix, and where Z 2 = It has been shown in Eq. (15) that the Kruskal vacuum is an entangled two-mode squeezed state in terms of Schwarzschild modes. The two mode squeezed transformation can be expressed by a symplectic operator in the phase-space After the action of the two-mode squeezed transformation, the entire system involves three subsystems: the subsystem A described by the global observer Alice, the subsystem B described by Bob, and the subsystemB described by the virtual observer anti-Bob. The covariance matrix σ ABB of the tripartite quantum system is given by [43] σ where A AB = cosh 2sI 2 , C AB = cosh u sinh 2sZ 2 , and B AB = [cosh 2 u cosh 2s + sinh 2 u]I 2 .
We know that the symplectic matrix σ AB (s, D) is a case of the smallest mixed Gaussian state according to the partially transposed symplectic matrix. From the formula above, we obtain The mean occupation numbers operator for each mode from the covariance matrix aren A = sinh 2 s andn B = cosh 2 u cosh 2 s − 1. Inserting them into Eq. (22), we obtain the quantum coherence of the Gaussian state Eq. (31). Then we can see that the mean occupation numbers as well as the Gaussian coherence depend not only on the dilaton charge D and the mass M of the black hole but also on the frequency ω and the squeezing parameter s. In Fig. (1), we plot the accessible quantum coherence between Alice and Bob as a function of the initial squeezing parameter s and the frequency ω of the scalar field. It is illustrated that the Gaussian quantum coherence is a monotonic increasing function of the frequency. This indicates that one can obtain more coherence by choosing a field with a higher frequency. It is worth noting that the coherence is almost unaffected to the change of frequency if the squeezing parameter is very small. On the other hand, it becomes more sensitive to the change of the frequency when the squeezing parameter becomes larger. That is, the frequency of the field produces positive effects on the storage of quantum coherence in the GHS black hole.
In Fig. 2(a), we plot the quantum coherence between the initial correlated Alice and Bob as a function of the ratio between the dilaton charge and the mass of the black hole. We find that Gaussian quantum coherence in the state ρ AB is not affected by the frequency ω when the ratio is D/M = 0. In addition, the coherence is independent of the frequency ω in the limit of D → M , which corresponds to an extreme dilaton black hole. That is to say, the coherence is not affected by the frequency of the scalar field for an uncharged black hole or an extreme dilaton black hole.
It is found that the Gaussian coherence between Alice and Bob decreases with the growth of the ratio D/M , which means that the gravitational field induced by dilaton will destroy the quantum coherence between the initially correlated modes. This verifies the fact that the gravitational field induced by dilaton plays a key role in the dynamics of Gaussian coherence in the dilaton spacetime.
It is interesting to note that the Gaussian coherence between Alice and Bob is not completely destroyed even in the limit of D → M , which corresponds to an extreme black hole. This is quite different from the behavior of quantum steering in the GHS spacetime because the latter suffers from a "sudden death" [17]. This is also different from the behavior of quantum entanglement in the GHS spacetime because the entanglement decays to zero only in the limit of D → M [40]. That is to say, quantum coherence is more robust than entanglement and steering under the influence of spacetime effects near the event horizon of the dilaton black hole. This is reasonable because quantum resources are hierarchic. It is known that quantum coherence can be defined for the integral system, while quantum correlations characterize the quantum features of a bipartite or a multipartite system [23]. The results in the present paper, as well as those in Refs. [17,40], verify the fact that the quantum resources are hierarchic and quantum coherence is more robust than quantum correlations in the dilaton spacetime. On the other hand, this result indicates the possibility of Bob being capable of performing quantum information tasks even in the case of an extreme dilaton black hole because quantum coherence is a usable resource for the tasks. This is nontrivial because one can employ quantum coherence as the resource of quantum information processing tasks even if the quantum correlations have been destroyed by strong gravitational effects. Fig. 2(b) shows a contour diagram of Gaussian coherence versus the squeezing parameter s and the ratio D/M . We can see that, if the initial squeezing parameter is close to 1, the gravitational field of the spacetime has significant influence on the Gaussian quantum coherence only near the D → M limit. The coherence between Alice and Bob is nonzero even in the limit of an extreme dilaton black hole. This means that one can perform quantum information processing tasks in the GHS spacetime if sufficient resource is prepared in the initial state. In addition, the coherence becomes more sensitive with the change of the squeezing parameter s for larger dilaton parameters.

B. Generating quantum coherence between the initially uncorrelated modes
In this subsection, we study the dynamics of quantum coherence among the initially uncorrelated modes. The covariance matrix between Alice and the observer anti-Bob is obtained by The corresponding symplectic value of the covariance matrix σ AB is found to be ν + = cosh 2 u cosh 2s + sinh 2 u and ν − = 1. Similarly, we find ∆ (AB) = 1+[cosh 2 u cosh 2s+sinh 2 u] 2 .
Then we calculate the quantum coherence between the Alice and anti-Bob and plot it in Fig (3). Gaussian quantum steering is zero for any dilaton charge. In addition, the lower the frequency of the mode in the state ρ AB , the stronger the Gaussian quantum coherence. This indicates the generation of quantum coherence between Alice and Bob is easier for low-frequency fields. In Fig. 3(b), it is shown that the Gaussian coherence between Alice and anti-Bob exhibits a "sudden birth" behavior under the influence of the dilaton field.
The covariance matrix between Bob and anti-Bob inside the event horizon is obtained by tracing where A BB = [cosh 2 u cosh 2s + sinh 2 u]I 2 , and B BB = [sinh 2 u cosh 2s + cosh 2 u]I 2 .
We can also calculate the Gaussian quantum coherence between Bob and anti-Bob.
In Fig. 4, we show that the quantum coherence between Bob and anti-Bob is a monotonically increasing function of D/M . It changes slowly at first and becomes more sensitive with the increase of D/M . It is also found that the quantum coherence is independent of the frequency ω for the non-dilaton and extreme dilaton black hole. As the increase of the ratio D/M , the quantum coherence is smoothly generated between Bob and anti-Bob. This is different from the generation of Gaussian coherence between Alice and anti-Bob because the latter exhibits a "sudden birth" behavior. It is shown that the gravitational field induced by dilaton generates Gaussian quantum coherence between the causally disconnected regions. In other words, Bob and anti-Bob can perform quantum information processing tasks by local measurements even though they are separated by the event horizon.

V. SUMMARY
In this work, we study the behavior of quantum coherence for Gaussian states in the background of a GHS dilaton black hole. It is shown that the dilaton field has evident effect on the degree of coherence for all the bipartite subsystems. However, the Gaussian coherence is not affected by the frequency of the scalar field for an uncharged or an extreme dilaton black hole. This verifies the fact that the gravity induced by dilaton field plays a key role in the dynamics of Gaussian coherence in the GHS spacetime. The Gaussian coherence between Alice and Bob is not completely destroyed even for an extreme dilaton black hole. This is quite different from the behavior of quantum steering because the latter suffers from a "sudden death" under the same condition. The coherence between Alice and Bob is nonzero even in the limit of an extreme dilaton black hole.
This means that one can perform quantum information processing tasks in the GHS spacetime if sufficient resources are prepared in the initial state.