Holographic study of entanglement and complexity for mixed states
aa r X i v : . [ h e p - t h ] J a n Holographic study of entanglement and complexity formixed states
Ashis Saha ∗ a and Sunandan Gangopadhyay † ba Department of Physics, University of Kalyani, Kalyani 741235, India b Department of Theoretical Sciences, S.N. Bose National Centre for Basic Sciences, JD Block,Sector-III, Salt Lake, Kolkata 700106, India
Abstract
In this paper, we holographically quantify the entanglement and complexity for mixed states by followingthe prescription of purification. The bulk theory we consider in this work is a hyperscaling violatingsolution, characterized by two parameters, hyperscaling violating exponent θ and dynamical exponent z . This geometry is dual to a non-relativistic strongly coupled theory with hidden Fermi surfaces. Forentanglement, we compute the holographic analogy of entanglement of purification (EoP), denoted as theminimal area of the entanglement wedge cross section and observe the effects of z and θ . On the otherhand, in order to probe the mixed state complexity we compute the mutual complexity (∆ C ) for the BTZblack hole and the hyperscaling violating geometry by incorporating the holographic subregion complexityconjecture. Furthermore, various aspects of holographic entanglement entropy such as entanglement Smarrrelation and Fisher information metric has also been studied. The gauge/gravity duality [1–3] has been employed to holographically compute quantum information the-oretic quantitites and has thereby helped us to understand the bulk-boundary relations. Among variousobservables of quantum information theory, entanglement entropy (EE) has been the most fundamentalthing to study as it measures the correlation between two subsystems for a pure state. EE has a very simpledefinition yet sometimes it is notoriously difficult to compute. However, the holographic computation of en-tanglement entropy which can be denoted as the Ryu-Takayanagi (RT) prescription, is a remarkably simpletechnique which relates the area of a codimension-2 static minimal surface with the entanglement entropyof a subsystem [4–6]. The RT-prescription along with its modification for time-dependent scenario (HRTprescription [7]) has been playing a key role for holographic studies of information theoretic quantitites asthe perturbative calculations which could not be done on the field theoretic side due to its strongly couplednature, can now be performed in the bulk side since it is of weakly coupled nature.Another important information theoretic quantity which has gained much attention recently is the computa-tional complexity. The complexity of a quantum state represents the minimum number of simple operationswhich takes the unentangled product state to a target state [8–10]. There are several proposals to computecomplexity holographically. Recently, several interesting attempts has been made to define complexity inQFT [11–13]. In context of holographic computation, initially, it was suggested that the complexity of astate (measured in gates) is proportional to the volume of the Einstein-Rosen bridge (ERB) which connectstwo boundaries of an eternal black hole [14, 15] C V ( t L , t R ) = V ERB ( t L , t R )8 πRG d +1 (1) ∗ [email protected] † [email protected] R is the AdS radius and V ERB ( t L , t R ) is the co-dimension one extremal volume of ERB which isbounded by the two spatial slices at times t L and t R of two CFTs that live on the two boundaries of theeternal black hole. Another conjecture states that complexity can be obtained from the bulk action evaluatedon the Wheeler-DeWitt patch [16–18] C A = I W DW π ~ . (2)The above two conjectures depends on the whole state of the physical system at the boundary. In additionto these proposals, there is another conjecture which depends on the reduced state of the system. This statesthat the co-dimension one volume enclosed by the co-dimension two extremal RT surface is proportional tothe complexity C V = V (Γ minA )8 πRG d +1 . (3)This proposal is known as the holographic subregion complexity (HSC) conjecture in the literature [19–21].Recently, in [22], it was shown that there exists a relation between the universal pieces of HEE and HSC.Furthermore, the universal piece of HSC is proportional to the sphere free energy F S p for even dimensionaldual CFTs and proportional to the Weyl a -anomaly for odd dimensional dual CFTs.In recent times, much attention is being paid to the study of entanglement entropy and complexity for mixedstates. For the study of EE for mixed states, the entanglement of purification (EoP) [23] and entanglementnegativity E [24] has been the promising candidates. In the subsequent analysis, our focus will be on thecomputation of EoP. Consider a density matrix ρ AB corresponding to mixed state in Hilbert space H , where H = H A ⊗ H B . Now the process of purification states that one can construct a pure state | ψ i from ρ AB byadding auxillary degrees of freedom to the Hilbert space H ρ AB = tr A ′ B ′ | ψ i h ψ | ; ψ ∈ H AA ′ BB ′ = H AA ′ ⊗ H BB ′ . (4)Such states ψ are called purifications of ρ AB . It is to be noted that the process of purification is not uniqueand different procedures for purification for the same mixed state exists. In this set up, the definition ofEoP ( E P ) reads [23] E P ( ρ AB ) = min tr A ′ B ′ | ψ ih ψ | S ( ρ AA ′ ); ρ AA ′ = tr BB ′ | ψ i h ψ | . (5)In the above expression, the minimization is taken over any state ψ satisfying the condition ρ AB = tr A ′ B ′ | ψ i h ψ | , where A ′ B ′ are arbitrary. In this paper we will compute the holographic analogy of EoP,given by the minimal area of entanglement wedge cross section (EWCS) E W [25]. However, there is nodirect proof of E P = E W duality conjecture yet and it is mainly based on the following properties of E P which are also satisfied by E W [23, 25]. These properties are as follows.( i ) E P ( ρ AB ) = S ( ρ A ) = S ( ρ B ); ρ AB = ρ AB ( i ) 12 I ( A : B ) ≤ E P ( ρ AB ) ≤ min [ S ( ρ A ) , S ( ρ B )] . In the above properties, I ( A : B ) = S ( A ) + S ( B ) − S ( A ∪ B ) is the mutual information between twosubsystems A and B . Some recent observations in this direction can be found in [26–34]. On the otherhand, recently the study of complexity for mixed states has gained appreciable amount of attention. Similarto the case of EE for mixed state, the concept of ‘purification’ is also being employed in this context [35,36].The purification complexity is defined as the minimal pure state complexity among all possible purificationsavailable for a mixed state. Preparing a mixed state on some Hilbert space H , starting from a reference(pure) state involves the extension of the Hilbert space H by introducing auxillary degrees of freedom [37,38].In this set up, a quantity denoted as the mutual complexity ∆ C has been prescribed in order to probe theconcept of purifiaction complexity. The mutual complexity ∆ C satisfies the following definition∆ C = C ( ρ A ) + + C ( ρ B ) − C ( ρ A ∪ B ) . (6)2n this paper, we will incorporate the HSC conjecture in order to compute the quantities C ( ρ A ), C ( ρ B ) and C ( ρ A ∪ B ).The paper is organized as follows. In Section 2, we briefly discuss the aspects of the bulk theorywhich in this case is a hyperscaling violating geometry. We then consider a single strip-like subsystem andholographically compute the EE in Section 3. We also make comments on the thermodynamical aspects ofthe computed HEE by computing the entanglement Smarr relation satisfied by the HEE. Furthermore, weholographically compute the relative entropy in order to obtain the Fisher information metric. In Section4, we consider two strip-like subsystems and holographically compute the EoP by using the E P = E W conjecture. We briefly study the temperature dependent behaviour of EWCS along with the effects of z and θ on the E W . We then compute the HSC corresponding to a single strip-like subsystem in Section 5. InSection 6, we holographically compute the mutual complexity ∆ C by incorporating the HSC conjecture forthe BTZ black hole and the hyperscaling violating geometry. We then conclude in Section 7. We also havean Appendix in the paper. We shall start our analysis with a bulk hyperscaling violating spacetime geometry. The solution correspondsto the following effective action of Einstein-Maxwell-scalar theory [39, 40] S bulk = 116 πG Z d d +1 x √− g (cid:20) ( R − − W ( φ ) F µν F µν − ∂ µ φ∂ µ φ − V ( φ ) (cid:21) (7)where F µν is the Faraday tensor associated with the gauge field A µ , φ is the scalar field associated withthe potential V ( φ ) and W ( φ ) is the coupling. Extremization of this action leads to the following black holesolution [40] ds = R r " − f ( r ) r d − z − d − θ − dt + r θd − θ − dr f ( r ) + d − X i =1 dx i . (8)The lapse function f ( r ) has the form f ( r ) = 1 − (cid:16) rr h (cid:17) ( d − zd − θ − ) where r H is the event horizon of theblack hole. The Hawking temperature of black hole is obtained to be T H = ( d − z + d − θ − π ( d − θ −
1) 1 r z ( d − / ( d − θ − h . (9)The above mentioned metric is holographic dual to a d -dimensional non-relativistic strongly coupled theorywith Fermi surfaces. The metric is associated with two independent exponents z and θ . The presence ofthese two exponents leads to the following scale transformations x i → ξ x i t → ξ z tds → ξ θd − ds . This non-trivial scale transformation of the proper spacetime interval ds is quite different from the usualAdS/CFT picture. The non-invariance of ds in the bulk theory implies violations of hyperscaling in theboundary theory. Keeping this in mind, θ is identified as the hyperscaling violation exponent and z isidentified as the dynamical exponent. In the limit z = 1, θ = 0, we recover the SAdS d +1 solution which isdual to a relativistic CFT in d -dimensions and in the limit z = 1, θ = 0, we obtain the ‘Lifshitz solutions’.The two independent exponents z and θ satisfy the following inequalities θ ≤ d − , z ≥ θd − . (10)3he ‘equalities’ of the above mentioned relations holds only for gauge theories of non-Fermi liquid states in d = 3 [41]. In this case, θ = 1 and z = 3 /
2. For general θ = d −
2, logarithmic violation of the ‘Area law’of entanglement entropy [42] is observed. This in turn means for θ = d −
2, the bulk theory holographicallydescribes a strongly coupled dual theory with hidden Fermi surfaces.
To begin our analysis, we consider our subsystem A to be a strip of volume V sub = L d − l , where − l < x < l and − L < x , ,..,d − < L . The amount of Hawking entropy captured by the above mentioned volume reads S BH = L d − l G d +1 r d − h . (11)It is to be noted that the thermal entropy of the dual field theory is related with the temperature as S th ∝ T d − − θz . For θ = d −
2, it reads S th ∝ T z . This result is observed for compressible states withfermionic excitations.We parametrize the co-dimension one static minimal surface as x = x ( r ) which leads to the followingarea of the extremal surface Γ minA A (Γ minA ) = 2 R d − L d − r (cid:16) pp − θ (cid:17) − pt ∞ X n =1 √ π Γ( n + )Γ( n + 1) α np (cid:16) zp − θ (cid:17) Z du u θp − θ − p + np (cid:16) zp − θ (cid:17) √ − u p ; u = rr t , α = r t r h (12)where r t is the turning point and p = ( d −
1) which we have introduced for the sake of simplicity. Bysubstituting the area functional (given in eq.(12)) in the RT formula, we obtain the HEE [4] S E = A (Γ minA )4 G d +1 = 2 L d − G d +1 (cid:16) pp − θ − p (cid:17) (cid:18) ǫ (cid:19) p − (cid:16) pp − θ (cid:17) + L d − r (cid:16) pp − θ (cid:17) − pt G d +1 ∞ X n =0 Γ( n + )Γ( n + 1) α np (cid:16) zp − θ (cid:17) p Γ pp − θ − p + np (cid:16) zp − θ (cid:17) p ! Γ pp − θ + np (cid:16) zp − θ (cid:17) p ! . (13)The relationship between the subsystem size l and turning point r t reads (with the AdS radius R = 1) l = r pp − θ t ∞ X n =0 Γ( n + )Γ( n + 1) α np (cid:16) zp − θ (cid:17) p Γ pp − θ + p + np (cid:16) zp − θ (cid:17) p ! Γ pp − θ +2 p + np (cid:16) zp − θ (cid:17) p ! . (14)We now proceed to probe the thermodynamical aspects of HEE. It can be observed from eq.(13) that theexprssion of S E contains a subsystem independent divergent piece which we intend to get rid by defining afinite quantity. We call this finite quantity as the renormalized holographic entanglement entropy ( S REE ).From the point of view of the dual field theory this divergence free quantity represents the change inentanglement entropy under an excitation. In order to obtain S REE holographically, firstly we need tocompute the HEE corresponding to the asymptotic form ( r h → ∞ ) of the hyperscaling violating black branesolution given in eq.(8). This yields the following expression S G = 2 L d − G d +1 (cid:16) pp − θ − p (cid:17) (cid:18) ǫ (cid:19) p − (cid:16) pp − θ (cid:17) − p − θ L d − G d +1 (cid:16) p − pp − θ (cid:17) (cid:18) p − θp (cid:19) p − θ − π p − θ l p − − θ Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) pp − θ p (cid:19) p − θ . (15) The thermal entropy of the dual field theory is basically the Hawking entropy of the black hole given in eq.(9).
4e now subtract the above expression (which represents the HEE corresponding to the vacuum of thedual field theory) from S E (given in eq.(13)) in order to get a finite quantity S REE . This can be formallyrepresented as S REE = S E − S G . (16)On the other hand, the internal energy E of the black hole can be obtained by using the Hawking entropy(given in eq.(11)) and the Hawking temperature (given in eq.(9)). The computed expression of E can berepresented as E = (cid:18) p − θz + p − θ (cid:19) S BH T H . (17)This is nothing but the classical Smarr relation of BH thermodynamics. In [43], it was shown that thequantity S REE and the internal energy E satisfies a Smarr-like thermodynamic relation corresponding to ageneralized temperature T g . In this set up, this relation reads [44] E = (cid:18) p − θz + p − θ (cid:19) S REE T g . (18)It is remarkable to observe that the relation given in eq.(18) has a striking similarity with the classical Smarrrelation of BH thermodynamics, given in eq.(17). In the limit r t → r h , the leading term of the generalizedtemperature T g produces the exact Hawking temperature T H whereas in the limit r t r h ≪
1, the leading termof T g reads [44] 1 T g = ∆ l z (19)where the detailed expression of ∆ reads∆ = 2 π / p pp − θ − p + p (cid:16) zp − θ (cid:17) p √ π Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) z Γ (cid:18) p + pp − θ + p (1+ zp − θ )2 p (cid:19) Γ (cid:18) p + pp − θ + p (1+ zp − θ )2 p (cid:19) . From eq.(19) it can be observed that in the UV limit, T g shows the similar behaviour as entanglementtemperature T ent (proportional to the inverse of subsystem size l ). We now proceed to compute the Fisher information metric for the hyperscaling violating geometry using theholographic proposal. The Fisher information metric measures the distance between two quantum statesand is given by [45] G F,λλ = h δρ δρ i ( σ ) λλ = 12 T r (cid:18) δρ dd ( δλ ) log( σ + δλδρ ) | δλ =0 (cid:19) (20)where σ is the density matrix and δρ is a small deviation from the density matrix. On the other hand, thereexists a relation between the Fisher information metric and the relative entropy S rel [46]. This reads G F,mm = ∂ ∂m S rel ( ρ m || ρ ); S rel ( ρ m || ρ ) = ∆ h H ρ i − ∆ S . (21)In the above expression, ∆ S is the change in the entanglement entropy from vacuum state, ∆ h H ρ i is thechange in the modular Hamiltonian and m is the perturbation parameter. In this set up, we holographicallycompute the relative entropy S rel ( ρ m || ρ ). We consider the background is slightly perturbed from pure5yperscaling violating spacetime while the subsystem volume L d − l is fixed. Then the inverse of the lapsefunction f ( r ) (given in eq.(8)) can be expressed as1 f ( r ) = 1 + mr p (cid:16) zp − θ (cid:17) + m r p (cid:16) zp − θ (cid:17) (22)where m = (cid:16) r H (cid:17) p (cid:16) zp − θ (cid:17) is the holographic perturbation parameter. Since we consider a perturbation tothe background geometry and also consider that the subsystem size l has not changed, we can express theturning point in the following perturbed form r t = r (0) t + mr (1) t + m r (2) t (23)where r (0) t is the turning point for the pure hyperscaling violating geometry and r (1) t , r (2) t are the first andsecond order corrections to the turning point. We now write down the subsystem length l upto second orderin perturbation as lr pp − θ t = a + ma r p (cid:16) zp − θ (cid:17) t + m a r p (cid:16) zp − θ (cid:17) t (24)where a = √ πp Γ (cid:18) pp − θ + p p (cid:19) Γ (cid:18) pp − θ p (cid:19) , a = √ π p Γ pp − θ + p + p (cid:16) zp − θ (cid:17) p ! Γ pp − θ + p (cid:16) zp − θ (cid:17) p ! , a = 3 √ π p Γ pp − θ + p +2 p (cid:16) zp − θ (cid:17) p ! Γ pp − θ +2 p (cid:16) zp − θ (cid:17) p ! . Using eq.(23) in eq.(24) and keeping in mind the consideration that l has not changed, we obtain the formsof r (0) t , r (1) t and r (2) t r (0) t = (cid:18) la (cid:19) p − θp , r (1) t = − (cid:18) p − θp (cid:19) (cid:18) a a (cid:19) (cid:16) r (0) t (cid:17) p (cid:16) zp − θ (cid:17) , r (2) t = ξ (cid:16) r (0) t (cid:17) p (cid:16) zp − θ (cid:17) (25)where ξ = (cid:18) p − θp (cid:19) "(cid:18) p − θ p (cid:19) (cid:18) a a (cid:19) + ( p − θ ) (cid:18) zp − θ (cid:19) (cid:18) a a (cid:19) − (cid:18) a a (cid:19) . (26)On a similar note, the expression for area of the static minimal surface upto second order in perturbationparameter m can be obtained from eq.(12). We then use eq.(23) to recast the expression for the area of theminimal surface in the form A (Γ minA ) = A (Γ minA ) (0) + mA (Γ minA ) (1) + m A (Γ minA ) (2) . (27)It has been observed that at first order in m , S rel vanishes [46] and in second order in m it reads S rel = − ∆ S .In this set up, it yields S rel = − m A (Γ minA ) (2) G d +1 = m L d − G d +1 ∆ (cid:18) la (cid:19) z +( p − θ ) (28)where∆ = 2 p (cid:18) p − θp (cid:19) (cid:18) a a (cid:19) + p (cid:18) p − θp − θ (cid:19) (cid:18) p − θp (cid:19) (cid:18) a a (cid:19) − pa p (1 + zp − θ ) − p + pp − θ ! − pa ξ . By substituting the above expression in eq.(21), the Fisher information is obtained to be G F,mm = L d − G d +1 ∆ (cid:18) la (cid:19) z +( p − θ ) ∝ l d +2 z − θ (29)In the limit z = 1 and θ = 0, the above equation reads G F,mm ∝ l d +2 which agrees with the result obtainedin [47]. The Fisher information corresponding to the Lifshitz type solutions can be found in [48].6 Entanglement wedge cross-section and the E P = E W duality We now proceed to compute the holographic entanglement of purification by considering two subsystems,namely, A and B of length l on the boundary ∂M . From the bulk point of view, ∂M is the boundaryof a canonical time-slice M made in the static gravity dual. Furthermore, A and B are separated by adistance D so that the subsystems does not have an overlap of non-zero size ( A ∩ B = 0). Following theRT prescription, we denote Γ minA , Γ minB and Γ minAB as the static minimal surfaces corresponding to A , B and AB respectively. In this set up, the domain of entanglement wedge M AB is the region in the bulk with thefollowing boundary ∂M AB = A ∪ B ∪ Γ minAB . (30)It is also to be noted that if the separation D is effectively large then the codimension-0 bulk region M AB will be disconnected. We now divide Γ minAB into two partsΓ minAB = Γ AAB ∪ Γ BAB (31)such that the boundary ∂M AB of the canonical time-slice of the full spacetime M AB can be represented as ∂M AB = ¯Γ A ∪ ¯Γ B (32)where ¯Γ A = A ∪ Γ AAB and ¯Γ B = B ∪ Γ BAB . In this set up, it is now possible to define the holographicentanglement entropies S ( ρ A ∪ Γ AAB ) and S ( ρ B ∪ Γ BAB ). These quantities can be computed by finding a staticminimal surface Σ minAB such that ∂ Σ minAB = ∂ ¯Γ A = ∂ ¯Γ B . (33)There can be infinite number possible choices for the spliting given in eq.(31) and this in turn means therecan be infinite number of choices for the surface Σ minAB . The entanglement wedge cross section (EWCS) isobtained by minimizing the area of Σ minAB over all possible choices for Σ minAB . This can be formally writtendown as E W ( ρ AB ) = min ¯Γ A ⊂ ∂M AB " A (cid:0) Σ minAB (cid:1) G d +1 . (34)We now proceed to compute E W for the holographic dual considered in this paper. As we have mentionedearlier, EWCS is the surface with minimal area which splits the entanglement wedge into two domainscorresponding to A and B . This can be identified as a vertical, constant x hypersurface. The time inducedmetric on this constant x hypersurface reads ds ind = R r " r θp − θ dr f ( r ) + d − X i =1 dx i . (35)By using this above mentioned induced metric, the EWCS is obtained to be E W = L d − G d +1 Z r t (2 l + D ) r t ( D ) drr d − p f ( r )= L d − G d +1 ∞ X n =0 √ π Γ( n + )Γ( n + 1) " r t (2 l + D ) np (1+ zp − θ ) − p +1 − r t ( D ) np (1+ zp − θ ) − p +1 np (1 + zp − θ ) − p + 1 r h (cid:19) np (1+ zp − θ ) . As mentioned earlier, the above expression of E W always maintains the following bound E W ≥ I ( A : B ); I ( A : B ) = S ( A ) + S ( B ) − S ( A ∪ B ) (36)where I ( A : B ) is the mutual information between two subsystems A and B . On the other hand, in [49] itwas shown that there exists a critical separation between A and B beyond which there is no connected phase7or any l . This in turn means that at the critical separation length D c , E W probes the phase transition ofthe RT surface Γ minAB between connected and disconnected phases. The disconnected phase is characterisedby the fact that the mutual information I ( A : B ) vanishes, which in this case reads2 S ( l ) − S ( D ) − S (2 l + D ) = 0 . (37)The above condition together with the bound given in eq.(36) leads to the critical separation length D c .We now write down the expression for E W (given in eq.(36)) in terms of the parameters of the boundarytheory. This we do for the small temperature case ( r t ( D ) r h ≪ r t (2 l + D ) r h ≪
1) and for the high temperature case( r t ( D ) r h ≪ , r t (2 l + D ) r h ≈ E W in the low temperature limit In the limit r t ( D ) r h ≪ r t (2 l + D ) r h ≪
1, it is reasonable to consider terms upto order m (where m = r p (1+ zp − θ ) h )in the expression for E W (given in eq.(36)). On the other hand for low temeprature considerations, it ispossible to perturbatively solve eq.(14) which leads to the following relationship between a subsystem size l and its corresponding turning point r t r t ( l ) = l p − θp p √ π Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) p − θp × (cid:20) − ∆ l ( p − θ ) (cid:16) zp − θ (cid:17) T ( p − θz ) (cid:21) (38)where∆ = (cid:18) p − θ p (cid:19) πp (1 + zp − θ ) ! p − θz Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ + p (1+ zp − θ )2 p (cid:19) Γ (cid:18) p + pp − θ + p (1+ zp − θ )2 p (cid:19) p √ π Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) ( p − θ ) (cid:16) zp − θ (cid:17) . Now by using eq.(38) for r t (2 l + D ) and r t ( D ), we obtain the expression for E W in the low temperaturelimit to be E W = E T =0 W − L d − G d +1 ∆ (cid:20) (2 l + D ) (cid:16) p − θp (cid:17)(cid:16) pzp − θ (cid:17) − D (cid:16) p − θp (cid:17)(cid:16) pzp − θ (cid:17) (cid:21) T ( p − θz ) + ... (39)where the detailed expression for ∆ is given in the Appendix. The first term in eq.(39) is the EWCS at T = 0. This reads E T =0 W = L d − p − G d +1 √ πp Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) ( p − θ ) − (cid:16) p − θp (cid:17) (cid:18) D (cid:19) ( p − θ ) − (cid:16) p − θp (cid:17) − (cid:18) l + D (cid:19) ( p − θ ) − (cid:16) p − θp (cid:17) . (40)It can be observed from eq.(39) that the EWCS is a monotonically decreasing function of temperature T (as1 + p − θz > D c at which E W probes the phase transitionof the RT surface Γ minAB between the connected and disconnected phases. This to be obtained from thecondition 2 S ( l ) − S ( D ) − S (2 l + D ) = 0 . (41)8he general expression for the HEE of a strip of length l is given in eq.(13). Now similar to the abovecomputation, in the limit r t ( D ) r h ≪ r t (2 l + D ) r h ≪
1, we consider terms upto O ( m ) in eq.(13). By using thisconsideration, eq.(41) can be expressed as β (cid:18) l p − θ − − D p − θ − − l + D ) p − θ − (cid:19) + β r p (1+ zp − θ ) h (cid:0) l z − D z − (2 l + D ) z (cid:1) = 0 (42)where β = √ π p Γ (cid:18) pp − θ − p p (cid:19) Γ (cid:18) pp − θ p (cid:19) √ πp Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) p − θ − β = ∆ (cid:18) p − pp − θ (cid:19) β + √ π p p √ π Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) z Γ (cid:18) p + pp − θ + p (1+ zp − θ )2 p (cid:19) Γ (cid:18) p + pp − θ + p (1+ zp − θ )2 p (cid:19) . By solving the above equation we can find out the critical separation length D c (where we substitute Dl = k = constant ). It is worth mentioning that in the above computations of E W and I ( A : B ), we haveconsidered terms upto O ( m ) which is the leading order term for thermal correction. Similarly, one canincorporate next-to leading order terms or more to get a more accurate result. E W l E W at z=1, θ =0.0I(A:B)/2 at z=1, θ =0.0E W at z=1, θ =0.2I(A:B)/2 at z=1, θ =0.2E W at z=1, θ =0.4I(A:B)/2 at z=1, θ =0.4 Effect of hyperscaling violating exponent θ (we set z = 1) E W l E W at z=1, θ =0.2I(A:B)/2 at z=1, θ =0.2E W at z=2, θ =0.2I(A:B)/2 at z=2, θ =0.2E W at z=3, θ =0.2I(A:B)/2 at z=3, θ =0.2 Effect of dynamical exponent z (we set θ = 0 . Figure 1: Effects of θ and z on E W and I ( A : B ) at low temperature (with d = 3, k = 0 . L = 1 and G d +1 = 1) E W in the high temperature limit We now consider the limit r t (2 l + D ) → r h and r t ( D ) r h ≪
1. This in turn means the static minimal surfaceassociated with the turning point r t (2 l + D ) wraps a portion of the event horizon r h . In the large n limit,the infinite sum associated with the turning point r t (2 l + D ) goes as ≈ √ π (cid:0) n (cid:1) / (cid:16) r t (2 l + D ) r h (cid:17) np (1+ zp − θ ) whichmeans it is convergent ( P ∞ n =1 1 n / = ξ ( )). Further, we are considering r t ( D ) r h ≪ m in the infinite sum associated with r t ( D ). In this set up, theexpression for EWCS reads E W ( T ) = E T =0 W + L d − G d +1 ∆ D z + (cid:16) p − θp (cid:17) T ( p − θz ) − L d − G d +1 ∆ T ( p − θz ) − (cid:16) p − θpz (cid:17) + ... (43)9here the temperature independent term (EWCS at T = 0) is being given by E T =0 W = L d − p − G d +1 D ( p − θp )( p − √ πp Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) ( p − θ ) − (cid:16) p − θp (cid:17) . (44)The expressions for ∆ and ∆ are given in the Appendix. Now we procced to compute the critical separationlength D c in the high temperature configuration. In the limit r t → r h , the computed result of S E (HEE ofa strip like subsystem with length l ), given in eq.(13) can be rearranged in the following form S E = L d − l G d +1 r ph + L d − G d +1 r p − pp − θ h ∞ X n =0 P n ; P n = pp − θ − p + np (1 + zp − θ ) ! Γ (cid:0) n + (cid:1) Γ ( n + 1) Γ (cid:18) p + pp − θ + np (1+ zp − θ )2 p (cid:19) Γ (cid:18) p + pp − θ + np (1+ zp − θ )2 p (cid:19) . (45)By using the above form of HEE, we can write down eq.(41) in the following form P ∞ n =0 P n r p − pp − θ h − D r p − pp − θ h − β D p − θ − − β r p (1+ zp − θ ) h D z = 0 . (46)In Fig(s).(1) and (2), we have graphically represented the effects of z and θ on the EWCS and holographic E W , I ( A : B ) / D E W at z=1, θ =0.0I(A:B)/2 at z=1, θ =0.0E W at z=1, θ =0.2I(A:B)/2 at z=1, θ =0.2E W at z=1, θ =0.4I(A:B)/2 at z=1, θ =0.4 Effect of hyperscaling violating exponent θ (we set z = 1) E W , I ( A : B ) / D E W at z=1, θ =0.2I(A:B)/2 at z=1, θ =0.2E W at z=2, θ =0.2I(A:B)/2 at z=2, θ =0.2E W at z=3, θ =0.2I(A:B)/2 at z=3, θ =0.2 Effect of dynamical exponent z (we set θ = 0 . Figure 2: Effects of θ and z on E W and I ( A : B ) at high temperature (with d = 3, L = 1 and G d +1 = 1)mutual information (HMI) for both low and high temperature case respectively. For the low temperaturecase (Fig.(1)) we have chose the separation length D between the subsystems to be D = 0 . l . Fromthe above plots it can be observed that the EWCS always maintains the bound E W > I ( A : B ). TheHMI continously decays and approaches zero at a particular critical separation length D c . This criticalseparataion D c decreases with increasing z and θ . On the other hand E W shows a discontinous jump at D c .Upto D c , E W has a finite cross-section due to the connected phase whereas beyond this critical separationlength E W vanishes due to the disconnected phase of the RT surface Γ minAB . In this section we study the holographic subregion complexity (HSC) proposal [19]. The subregion complexityis proportional to the volume enclosed by the co-dimension one static minimal surface with the boundary The first term of the expression is nothing but the thermal entropy of the boundary subsystem given in eq.(11). V (Γ minA ) = 2 L d − r p + θp − theta − pt Z ǫ/r t du u θ − pp − θ − p p f ( u ) Z u dk k p + θp − θ p f ( k ) √ − k p = L d − l (cid:16) p − θp − θ (cid:17) ǫ (cid:16) p − θp − θ (cid:17) + L d − r p + θp − θ − pt ∞ X n =0 ∞ X m =0 V (cid:18) r t r h (cid:19) ( m + n ) p (cid:16) zp − θ (cid:17) . (47)We now use the above volume to obtain the HSC. This is given by (setting AdS radius R = 1) C V ( A ) ≡ V (Γ minA )8 πG d +1 = L d − l π (cid:16) p − θp − θ (cid:17) G d +1 ǫ (cid:16) p − θp − θ (cid:17) + L d − r p + θp − θ − pt πG d +1 ∞ X n =0 ∞ X m =0 V (cid:18) r t r h (cid:19) ( m + n ) p (cid:16) zp − θ (cid:17) (48)where V = Γ( n + )Γ( m + )Γ (cid:18) ( m + n ) p (1+ zp − θ )+1+ θp − θ p (cid:19) √ π Γ( n + 1)Γ( m + 1)Γ (cid:18) ( m + n ) p (1+ zp − θ )+1+ p + θp − θ p (cid:19) p (cid:16) mp (1 + zp − θ ) + θo − θ − p (cid:17) . ∆ C ) The complexity for mixed states (purification complexity) is defined as the minimal (pure state) complexityamong all possible purifications of the mixed state. This in turn means that one has to optimize over thecircuits which take the reference state to a target state | ψ AB i (a purification of the desired mixed state ρ A )and also need to optimize over the possible purifications of ρ A . This can be expressed as C ( ρ A ) = min B C ( | ψ AB i ); ρ A = Tr B | ψ AB i h ψ AB | (49)where A c = B . Recently a quantity denoted as the ‘mutual complexity (∆ C )’ has been defined in orderto compute the above mentioned mixed state complexity. The computation of ∆ C starts with a pure state ρ AB in an extended Hilbert space (including auxillary degrees of freedom), then by tracing out the degreesof freedom of B , one gets the mixed state ρ A . On the other hand, tracing out the degrees of freedom of A yields ρ B . These computed results then can then be used in the following formula to compute the mutualcomplexity ∆ C ∆ C = C ( ρ A ) + + C ( ρ B ) − C ( ρ A ∪ B ) . (50)The mutual complexity ∆ C is said to be subadditive if ∆ C > C < C = V conjecture to compute the quantities C ( ρ A ) , C ( ρ B ) and C ( ρ A ∪ B ). Similary one can follow the C = A conjecture or C = V . l (lengths corresponding to rest of the directions are fixed), namely, A and B with zero overlapping( A c = B and A ∩ B = 0). We first compute the HSC corresponding to a single subsystem A for the BTZ black hole. The mentionedblack hole geometry is characterized by the following metric [51, 52] ds = R z (cid:20) − f ( z ) dt + dz f ( z ) + dx (cid:21) ; f ( z ) = 1 − z z h . (51)11ollowing the prescription of C = V conjecture, the HSC corresponding to a single subsystem A of length l in the dual field theory is obtained to be C ( ρ A ) = 2 R πRG Z z t x ( z ) z p f ( z ) dz = 18 π (cid:18) lǫ − π (cid:19) ( setting G = 1) (52)where ǫ is the cut-off introduced to prevent the UV divergence. It is to be observed that the computedresult of HSC in this case is independent of the black hole parameter (that is, event horizon z h ). This is aunique feature of AdS . We now consider two subsystems A and B of equal length l in order to computethe mutual complexity ∆ C . This reads∆ C = C ( ρ A ) + + C ( ρ B ) − C ( ρ A ∪ B ) = 18 π (cid:20) lǫ − π + lǫ − π − lǫ + π (cid:21) = − . (53)It can be observed that the mutual complexity is less than zero that is the complexity is superadditive. Thisin turn means that the complexity of the state corresponding to the full system is greater than the sum ofthe complexities of the states in the two subsystems. We now proceed to compute the mixed state complexity for the hyperscaling violating geometry. By usingthe holographic prescription given in eq.(48), ∆ C in this set up reads∆ C = 18 πG d +1 (cid:2) V (Γ minA ) + V (Γ minB ) − V (Γ minA ∪ B ) (cid:3) = L d − πG d +1 ∞ X n =0 ∞ X m =0 V r t ( l ) (cid:16) p + θp − θ (cid:17) − p (cid:18) r t ( l ) r h (cid:19) ( m + n ) p (cid:16) zp − θ (cid:17) − r t (2 l ) (cid:16) p + θp − θ (cid:17) − p (cid:18) r t (2 l ) r h (cid:19) ( m + n ) p (cid:16) zp − θ (cid:17) . (54)It is to be noted that the mutual complexity ∆ C in this set up is a divergence free finite quantity as theleading divergent piece ≈ L d − lǫ ( p − θp − θ ) of HSC cancels out.We now try to recast the above expression in terms of the field theoretic parameters. In order to executethis, we consider the limit r t ( l ) r h ≪ r t (2 l ) r h ≪
1. In this limit, we keep terms upto order O ( m ) in the expressionof ∆ C . This in turn leads to the following expression∆ C = L d − πG d +1 ∆ l (cid:16) p + θp (cid:17) − ( p − θ ) − L d − πG d +1 ∆ l z + (cid:16) p + θp (cid:17) T ( p − θz ) (55)where the expressions of ∆ and ∆ are given in the Appendix. We observe that similar to BTZ case, theabove result for ∆ C is less than zero. This in turn means that the mutual complexity computed using theHSC conjecture yields a superadditive result. In this paper, we compute the entanglement entropy and complexity for mixed states by using the gauge/gravitycorrespondence. We start our analysis by considering a hyperscaling violating solution as the bulk theory.This geometry is associated with two parameters, namely, hyperscaling violating exponent z and dynamicalexponent θ . It is dual to a non-relativistic, strongly coupled theory with hidden Fermi surfaces. We thenconsider a single strip-like subsystem in order to compute the HEE of this gravitational solution. We observethat the computed result of HEE along with the internal energy E , satisfies a Smarr-like thermodynamicsrelation associated with a generalized temeperature T g . This thermodynamic relation naturally emerges by12emanding that the generalized temperature T g reproduces the Hawking temperature T H as the leadingterm in the IR ( r t → r h ) limit. In UV limit ( r t r h ≪ T g ∝ l z , that is, T g is inverselyproportional to subsystem size l . This behaviour is compatible with the definition of entanglement temper-ature given in the literature. We then holographically compute the relative entropy S rel , by incorporatingthe perturbative approach. Using this the Fisher information metric is computed. We find that in this casethe power of l carries both the exponents z and θ . We then consider two strip-like subsystems A and B separated by a length D , in order to compute the EWCS ( E W ) which is the holographic analogy of EoP.We compute E W for both low and high temperature conditions. In both cases, there is a temperatureindependent term (denoted as E T =0 W ) which is independent of the hyperscaling violating exponent z butdepends on the dynamical exponent θ . On the other hand for a large enough value of D (critical separationlength D c ), the RT surface Γ minAB becomes disconnected and E W should vanish. This in turn means that E W probes the phase transition between the connected and disconnected phases of the RT surface Γ minAB .We evaluate these critical separation point D c by using the property that at D c the mutual informationbetween A and B becomes zero as they become disconnected. This behaviour for I ( A : B ) and E W isshown in Fig(s).(1, 2) for both low and high temperature cases. We observe that E W always satisfies theproperty E W > I ( A : B ). We then compute the HSC by considering again a single strip-like subsystem.The complexity for mixed state is computed by following the concept of mutual complexity ∆ C . We haveused the HSC conjecture to compute the ∆ C for both BTZ black hole and hyperscaling violating geometry.We observe that the computed result of mutual complexity is superadditive that is ∆ C <
0. This in turnmeans that the complexity of the state corresponding to the full system is greater than the sum of thecomplexities of the states in the two subsystems. We observe that for BTZ black hole ∆ C is independent oftemeprature however for hyperscaling violating solution it contains a temperature independent term as wellas a temperature dependent term. It is to be kept in mind that this nature of ∆ C is observed for the HSCconjecture and similarly one can use ‘Complexity=Action’ conjecture or ‘CV2.0’ conjecture to compute ∆ C in this context. Acknowledgements
A.S. would like to acknowledge the support by Council of Scientific and Industrial Research (CSIR, Govt.of India) for Senior Research Fellowship.
In this appendix, we give the expressions of quantities that appear in the main text. These are as follows.∆ = ∆ p √ π Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) ( p − θ ) (cid:16) − pp (cid:17) − (cid:18) πp (1+ zp − θ ) (cid:19) p − θz (cid:16) p (1 + zp − θ ) − p + 1 (cid:17) p √ π Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) (cid:16) p − θp (cid:17)(cid:16) pzp − θ (cid:17) ∆ = √ π ( p − πp (1 + zp − θ ) ! ( p − p − θpz ) Γ (cid:18) pzp − θ p (1+ zp − θ ) (cid:19) Γ (cid:18) − p + pzp − θ p (1+ zp − θ ) (cid:19) ∆ = 2 (cid:20) − (cid:16) p − θp (cid:17) − ( p − θ ) − (cid:21) λ λ (cid:16) p + θp (cid:17) − ( p − θ )2 ∆ = (cid:20) z + (cid:16) p + θp (cid:17) − (cid:21) (cid:18) p − p + θp − θ (cid:19) ∆ λ λ (cid:16) p + θp (cid:17) − ( p − θ )2 + λ (cid:16) p + θp (cid:17) + z πp (1 + zp − θ ) ! ( p − p − θpz ) λ λ = − Γ (cid:18) θp − θ p (cid:19) Γ (cid:18) p + θp − θ p (cid:19) √ πp (cid:16) p − θp − θ (cid:17) λ = p √ π Γ (cid:18) p + pp − θ p (cid:19) Γ (cid:18) p + pp − θ p (cid:19) λ = √ π p Γ p (cid:16) zp − θ (cid:17) +1+ θp − θ p ! Γ p (cid:16) zp − θ (cid:17) + p + θp − θ p ! (cid:16) pzp − θ + θp − θ (cid:17) − (cid:16) p − θp − θ (cid:17) References [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity”,
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