Testing Holographic Conjectures of Complexity with Born-Infeld Black Holes

In this paper, we use Born-Infeld black holes to test two recent holographic conjectures of complexity, the"Complexity = Action"(CA) duality and"Complexity = Volume 2.0"(CV) duality. The complexity of a boundary state is identified with the action of the Wheeler-deWitt patch in CA duality, while this complexity is identified with the spacetime volume of the WdW patch in CV duality. In particular, we check whether the Born-Infeld black holes violate the Lloyd bound: $\mathcal{\dot{C}\leq}\frac{2}{\pi\hbar}\left[ \left( M-Q\Phi\right) -\left( M-Q\Phi\right) _{\text{gs}}\right] $, where gs stands for the ground state for a given electrostatic potential. We find that the ground states are either some extremal black hole or regular spacetime with nonvanishing charges. Near extremality, the Lloyd bound is violated in both dualities. Near the charged regular spacetime, this bound is satisfied in CV duality but violated in CA duality. When moving away from the ground state on a constant potential curve, the Lloyd bound tend to be saturated from below in CA duality, while $\mathcal{\dot{C}}$ is $\pi/2$ times as large as the Lloyd bound in CV duality.


I. INTRODUCTION
Through gauge/gravity duality, concepts from quantum information theory have driven major advances in our understanding of quantum field theory and quantum gravity. For example, the holographic entanglement entropy [1,2] has been currently receiving considerable attentions in the ongoing research. Recently inspired by the observation that the size of the Einstein-Rosen bridge (ERB) grows linearly at late times, it was conjectured [3][4][5][6] that quantum complexity of a boundary state is dual to the volume of the maximal spatial slice crossing the ERB anchored at the boundary state. Roughly speaking, the complexity C of a state is the minimum number of quantum gates to prepare this state from a reference state [7][8][9]. However, one of unappealing features of this proposal is that there is an ambiguity in choosing a length scale in the bulk geometry, which provides some motivations to introduce the "Complexity = Action" (CA) duality [10,11].
In CA duality, the complexity of a boundary state is identified with the action of the Wheeler-DeWitt (WdW) patch in the bulk: where the WdW patch can be defined as the domain of dependence of any Cauchy surface anchored at the boundary state. After the original calculations of S WdW in [11], a detailed analysis was carried out in [12], of the contributions to the action of some subregion from a null segment and a joint at which a null segment is joined to another segment. It is interesting to note that although the two approaches used in [11] and [12] are different, the results for dS WdW /dt at late times of the AdS Schwarzschild and Reissner-Nordstrom (RN) AdS black holes turn out to be the same. A possible explanation was given in [12].
Similar to the holographic entanglement entropy, the holographic complexity in CA duality is divergent, which is related to the infinite volume near the boundary of AdS space. The divergent terms were considered in [13][14][15], which showed that these terms could be written as local integrals of boundary geometry. This implies that the divergence comes from the UV degrees of freedom in the field theory. On the other hand, there are two finite quantities associated with the complexity, which can be calculated without first obtaining these divergent terms. The first one is the "complexity of formation" [16], which is the difference of the complexity between a particular black hole and a vacuum AdS spacetime. The second one is the rate of complexity at late times,Ċ. If CA duality is correct,Ċ should saturate the Lloyd bound [17]. The Lloyd bound is the conjectured complexity growth bound, which states thatĊ should be bounded by the energy [11]: For a black hole, E is its mass M, and the Lloyd bound then readṡ As noted in [11], the rate of the complexity of a neutral black hole is faster than that of a charged black hole since the existence of conserved charges could put constraints on the system. That implies that the Lloyd bound can be generalized for a charged black hole with the charge Q and potential at the horizon Φ: where (M − QΦ) gs is M − QΦ calculated in the ground state. A similar bound can also be given for rotating black holes [11].
The rate of complexity in CA duality has been considered in several examples. In [11] [18]. The action growth was also discussed in case of massive gravities [20] and higher derivative gravities [21]. A general case was considered in [22], and it was proved that the action growth rate equals the difference of the generalized enthalpy at the outer and inner horizons. While this paper is in preparation, a preprint [23]

II. BORN-INFELD ADS BLACK HOLES
In this section, we will consider the black hole solutions of Einstein-Born-Infeld action in The action of Einstein gravity and Born-Infeld field reads where we take 16πG = 1 for simplicity, L (F ) is given by and, β is the Born-Infeld parameter. When β → ∞, the Lagrangian of Born-Infeld field L (F ) becomes that of standard Maxwell field, L (F ) = −F µν F µν . The static black hole solution was obtained in [24,25]: where where Ω k,d−1 denotes the dimensionless volume of dΣ 2 k,d−1 . For k = 0 and −1, one needs to introduce an infrared regulator to produce a finite value of Ω k,d−1 .
For the sake of calculating the action growth and thermodynamic volume of the Born-Infeld black holes, we need to determine the number of their horizons. Depending on the values of the parameters q and m, the black holes could possess a naked singularity at r = 0, one, or two horizons. In fact, we could define a q-dependent function which does not depend on the parameter m. For a given value of m, one could solve m (r, q) = m for the position of the horizon. The derivative of m (r, q) with respect to r is which is a strictly increasing function. When r → ∞, dm (r, q) /dr goes to ∞. In the limit r → 0, we find that which shows that dm (r, q) /dr| r=0 ≥ 0 in the k = 1, d = 3, and βq ≤ 1/2 case, and dm (r, q) /dr| r=0 < 0 in the other cases. When dm(r,q) dr | r=0 < 0, the equation dm (r, q) /dr = 0 has one and only solution r e (q) > 0, such that dm (r, q) /dr| r=re(q) = 0. Thus, there is an extremal black hole solution with the parameter m = m (r e (q) , q) and the horizon being at r = r e (q). At r = r e (q), we obtain When k = 0 and 1, m (r e (q) , q) is always positive. However for k = −1, m (r e (q) , q) could be negative for some values of q. It is noteworthy that m (r e (q) , q) exists for q ≥ 1 2β in the k = 1, d = 3, and βq ≤ 1/2 case, while m (r e (q) , q) exists for all values of q in other cases.
1, we plot the function m (r, q) against r for different values of q, where we take L = 1 and β = 10.
With the above results, we can discuss when the Born-Infeld black hole solution (7) possesses a naked singularity, a single horizon, or two horizons: between the black holes with one horizon and these with two horizons are depicted as the black dashed lines, which are given by m = A (q). The colored lines (red and blue) are the boundaries between black holes and naked singularities. In FIG. 2(a), the red line divides the black holes with a single horizon and the spacetime with a naked singularity, and it meets the blue extremal line at the red dot, whose q coordinate is 1 2β = 0.05. To discuss the Lloyd bounds, we need to specify the electrostatic potential of the ground states, which are the colored lines in FIG. 2. The electrostatic potential at the black hole horizon, which is conjugate to the electric charge Q, is [24,25] where r h is the horizon's radius. When β → ∞, the Born-Infeld AdS black holes become the RN AdS black holes. When k = 1 and d = 3, it was found [11] that the boundary of RN AdS black holes in the phase diagram was the extremal line, and the potential Φ approached 16π as (q, m) → (0, 0) along the extremal line. Thus for a RN AdS black hole, the ground state of the geometry with the same electrostatic potential as this black hole is pure AdS spacetime for Φ 16π ≤ 1, but for Φ 16π > 1 it is some extremal black hole. Now we compute the asymptotic behavior of Φ as (q, m) → (0, 0) along the boundaries: • k = 0: The boundary is the extremal line, on which r e ∼ q 1 d−1 given by • k = −1 : The boundary is the extremal line, on which r e ∼ L as q → 0. One then It is interesting to note that If d = 3, the boundary line around (0, 0) is the red line in FIG. 2(a), on which r + = 0.
Again, we have

III. HOLOGRAPHIC CONJECTURES OF COMPLEXITY
In this section, we will discuss CA/CV dualities for the Born-Infeld AdS black holes.
In our appendix, the action growth of the Born-Infeld AdS black holes within the WdW patch at late-time approximation is calculated by following the approach in [12]. The action growth in the case with k = 1 and d = 3 was first calculated in [18]. The growth rate of the action dS/dt depends on the number of the horizons. In fact, we find that where Φ is the potential at the horion given by eqn. (16), Φ ± are Φ calculated at r = r ± , and r ± is the radius of the outer inner horizon. Furthermore, CA duality indicates that, in the late time regime,Ċ On the other hand, CV duality gives [19] that, in the late time regime, where P = d (d − 1) /L 2 is the pressure, and V is the volume of the WdW patch. For Born-Infeld AdS black holes, the rate of the complexity at late times is then given bẏ in one horizon case, The Lloyd bound for a charged black hole iṡ where (M − QΦ) gs is M − QΦ calculated in the ground state. The ground state is on the boundary between black hole region and no black hole region (colored lines in FIG. 2). If the system is treated as a grand canonical ensemble, the ground state has the same potential Φ as the black hole under consideration. Now we will calculate the rate of the complexity in the CA and CV dualities and check whether the Lloyd bound (26) is violated.

A. Around Extremal Line
We first consider a general static charged black hole with the line element where the radii of the outer and inner horizon are r + and r − , respectively. The first law of black hole thermodynamics reads dM = T dS + ΦdQ.
Since the entropy S is the function of r + , one finds At extremality where T = 0, we have where r e and Q e are the radius and charge, respectively, of the black hole at extremality.
The Lloyd bound then becomes Expanding r ± near extremality, we find that where From these we can expandĊ near extremality aṡ If c + 1 = c − 1 , the Lloyd bounds are violated near extremality under the two proposals. For the Born-Infeld AdS black holes with d = 3, we find that whereΦ = Φ 16π .

B. Around Regular Charged Spacetime
As shown in FIG. 2(a), there is a red boundary, which is m = A (q) for q ≤ 1 2β , in the case with d = 3 and k = 1. Above this boundary, one has a black hole with a single horizon, whose radius goes to zero as approaching the boundary. When r ≪ 1, we find which means that the metric is regular at r = 0 for m = A (q). Therefore, one has some regular spacetime with nonvanishing charges on the red boundary. The potential Φ = 16πΦ of the ground states on the boundary can be obtained from finding the limit of eqn. (16) as A little bit above the boundary, the radius of a black hole with the potential Φ is given by where δm = m−m 0 , and m 0 is the m parameter of the ground state with the same potential Φ. Since r + ≪ 1 implied by eqn. (40), eqn. (37) gives that the temperature of the black hole is which goes to zero as approaching the ground state. For this black hole, we find that the Lloyd bound is On the other hand, we can expandĊ aṡ It appears that the bound is satisfied in CV duality although far from saturated near the boundary. However, the bound is violated in CA duality.
Since eqn. (44) implies that r + ≫ 1 when q ≫ 1, the parameter m is The Lloyd bound for q ≫ 1 (r + ≫ 1) is then given by From eqns. (45) and (46), it follows that Since A (q) ∼ q d d−1 , the Born-Infeld AdS black holes with fixed potential Φ always lie above the m = A (q) line for large enough q, which means that these black holes always possess a single horizon for q ≫ 1 with fixed Φ. Therefore, eqns. (22) and (25) give thaṫ where (50) We see immediately that the Lloyd bound is satisfied in CA duality for sufficiently large q and tends to be saturated as q → ∞. However in CV duality,Ċ is π/2 times as large as the Lloyd bound for q ≫ 1.

D. Numerical Results
Here we consider two curves of constant potential,Φ = 1 andΦ = 1.5, in the case with Note that theΦ = 1 curve (green) starts from some regular spacetime, while theΦ = 1.5 curve (purple) starts from some extremal black hole. Both curves enter the "Single Horizon" region for large enough q, which is in agreement with the argument below eqn. (48).
To check whether the Lloyd bound is violated on the curves, we define   Along theΦ = 1.5 curve, FIG. 4(a) shows that the Lloyd bound is satisfied in CA duality for large enough q, while FIG. 4(b) shows that the Lloyd bound is violated in CV duality.
Note that the kinks in the R A curves in FIG. 4(a) are where theΦ = 1.5 curve enter the "Single Horizon" region from the "Two Horizons" region. Along theΦ = 1 curve, FIG. 4(d) shows that the Lloyd bound is only satisfied in CV duality for small q. It is interesting to see that the R A curves in FIG. 4(c) start to oscillate for small q when β is large enough (β = 5, 10, and 100). Even for β = 10 and 100, there is a range of q over whichĊ A < 0.
In summary, the Lloyd bound is violated in CA duality as we approach the ground states, but this bound tend to be saturated as we go away from the ground states with fixed potential. As noted in [11], the violations near the ground states have something to do with hair. In CV duality, the Lloyd bound is violated everywhere along the constant potential curves, except near the ground states on the red line.  and only vary it on the left boundary. There is a divergence appearing when calculating the action near the boundary r = ∞. So a surface of constant r = r max is defined to regulate the action. In [23], the action was regulated by defining the boundaries of the WdW patch originate slightly inside the AdS boundary. It turns out that these two choices for the regulator yield the same results. We also introduce a spacelike surface r = ε near the future singularities and let ε → 0 at the end of calculations. Note that we have an affine parametrization for each null surface, and these make no contribution to the action. To calculate δS, we introduce the null coordinates u and v in the metric (7): where r * = f −1 (r) dr. (A2)

Single Horizon Case
We calculate δS for a Born-Infeld AdS black hole with a single horizon, whose Penrose diagram is illustrated in FIG. 5(a). Due to time translation, the joint contributions from D and D ′ are identical, and they therefore make no contribution to δS. Similarly, the joint and surface contributions from MN cancel against these from M ′ N ′ on r = r max in calculating δS. Therefore, we have where we follow the conventions in [13].

Two Horizons Case
The Penrose diagram for a Born-Infeld AdS black hole with two horizons is illustrated in FIG. 5(b). Thus, we have While the volume contribution S V 2 is also given by eqn. (A8), we find that, in this case, where Hence the volume contribution to δS is where the portion of V 1 below the future horizon cancels against the portion of V 2 above the past horizon. The joint contributions from B and B ′ are the same as in the case with a single horizon. Analogously to calculating the joint contributions from B and B ′ , we find where r − is the inner horizon radius. Summing up all the contributions, we obtains that where Φ ± is the potential Φ evaluated at r = r ± . When approaching the boundary between the "Single Horizon" and "Two Horizons" regions, we have r − → 0 and QΦ − → A (q) dΩ k,d−1 . Since m = A (q) on this boundary, eqn. (A28) becomes Comparing with eqn. (A21), we find that dS/dt is continuos when crossing this boundary.