GGravity dualities of quantum distances
Run-Qiu Yang
Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin Uni-versity, Yaguan Road 135, Jinnan District, 300350 Tianjin, P. R. China
E-mail: [email protected]
Abstract:
By choosing modular ground state as the reference state, this paper findsthat three most frequently-used distances and a quantum quasi-distance, i.e. the tracedistance, Fubini-Study distance, Bures distance and Rényi relative entropy, all have gravitydualities. Their gravity dualities have two equivalent descriptions: one is given by theintegration of the area of a cosmic brane, the other one is given by the Euclidian on-shellaction of dual theory and the area of the cosmic brane. It then applies these dualities intothe 2-dimensional conformal field theory as examples and finds the results match with thecomputations of field theory exactly. a r X i v : . [ h e p - t h ] F e b ontents (3.7)
8B Calculations from CFT 9
In recent years it has been suggested that quantum information theory and gravity theoryhave deep connection. The gauge/gravity duality, which shows an equivalence betweenstrongly coupled quantum field theories (QFTs) and weakly coupled gravitational theoriesin one higher dimensions [1–3], offers us a powerful tool towards such connection. Asa result, the quantum information theoretic considerations have provided various usefulviewpoints in the studies of gauge/gravity duality and quantum gravity. One example is theRyu-Takayanagi (RT) formula [4–6], which connects the area of a codimension-2 minimalsurface in the dual spacetime and the entanglement entropy of the boundary QFT. The RTformula has been generalized into the Rényi entropy [7, 8], higher order gravity theory [9–11]and the cases with quantum corrections [12, 13]. An other quantity in quantum informationnamed “complexity”, which measures the difference of two states according to the size ofquantum circuits in converting one state into the other, also has been studied widely ingravity and black hole physics [14–19].From a general viewpoint, the complexity is a kind of “distance” between quantumstates [20]. Except for complexity, there are other several different measures of the distancebetween states, which are widely used in quantum information [21, 22]. For example, giventwo density matrices ρ and σ in the same Hilbert space, two families of distance are widelyused in quantum information theory. The first one are based on the fidelityFi ( ρ, σ ) = Tr (cid:113) √ σρ √ σ . (1.1)The fidelity is not a distance but we can use it to define two kinds of distance, the Fubini-Study distance D F ( ρ, σ ) = arccos Fi ( ρ, σ ) and the Bures distance D B ( ρ, σ ) = (cid:112) − Fi ( ρ, σ ) .– 1 –he other family of distances, depending on a positive number n , is provided by the n -distances D n ( ρ, σ ) := 12 /n ( Tr | ρ − σ | n ) /n . (1.2)Here Tr | X | n := (cid:80) i λ ni and λ i is the i -th eigenvalue of √ X † X . When X is hermitian { λ i } arejust the the absolute values of eigenvalues of X . Two special choices are widely applied. Oneis Hilbert-Schmidt distance, which chooses n = 2 . This distance leads to some conveniencesin mathematics because the calculation is straightforward by its definition. The other choiceis n = 1 , which is called the “trace distance”.Though the trace distance and fidelity are complicated than Hilbert-Schmidt distance,several properties make them special [21]. Firstly, the trace distance and fidelity (andso Fubini-Study distance and Bures distance) are bounded by each others: − D ≤ Fi ≤ (cid:112) − D . Secondly, they offer us dimension-independent bounds on the differ-ence between the expected values of an operator O in different states: |(cid:104) O (cid:105) ρ − (cid:104) O (cid:105) σ | ≤ D ( ρ, σ ) (cid:112) Tr ( OO † ) ≤ (cid:112) − Fi ( ρ, σ ) (cid:112) Tr ( OO † ) . Thirdly, they supply lower bounds forthe relative entropy S ( ρ (cid:107) σ ) : [1 − Fi ( ρ, σ )] ≤ D ( ρ, σ ) ≤ (cid:112) S ( ρ (cid:107) σ ) / .Despite that trace distance and fidelity have these important properties, their compu-tations are high challenge in quantum field theory. The first breakthrough towards thisissue is achieved by Ref. [23], which develops a replica trick to compute the fidelity for2-dimensional conformal field theory. Refs. [24–26] then also develop replica method tocompute the trace distance for a class of special states for single short interval in 1+1dimensional CFTs. However, there are still huge difficulties in the calculations of tracedistance even for 1+1 dimensional CFTs, such as to compute the trace distance betweentwo thermal states or two large intervals in CFTs. There is also no compact method tocompute the trace distance in higher dimensional CFTs or general quantum field theories.On the other hand, the holographic descriptions of entanglement, relative entropy [27] andcomplexity have been found, however, the trace distance does not yet. Refs. [28, 29] proposeholographic duality to compute the fidelity between a state and its infinitesimal perturba-tional state, however, they cannot be used into the case when the difference between twostates are not infinitesimal.In this paper, it will develop holographic dualities to compute the trace distance, Fubini-Study distance, Bures distance and Rényi relative entropy. By choosing a characteristicreference state which will be called “modular ground state”, they all become the intrinsicproperties of the target state. Then this paper will show that they all have gravity dualities.Their gravity dualities have two equivalent descriptions. The one is given by the integrationof the area of a cosmic brane with respective to its tansion. The other one descriptioncontains two parts, one of which is the on-shell action of gravity theory and the otherone of which is the area of the cosmic brane. We then apply them to the calculations ofthe trace distance in 1+1 dimensional CFT and show our holographic calculations exactlymatch with the results of CFT’s. – 2 – Distances to modular ground state and holographic proposals
In the field theory two different density matrices will often be almost orthogonal Tr ( ρσ ) ≈ and so their trace distance will almost saturate the upper bound. In this case, it will bemore convenient to study a “refined trace distance” D T ( ρ, σ ) = − ln[1 − D ( ρ, σ )] . (2.1)Due to the monotonicity, the refined trace distance and trace distance contain same infor-mation. Similarly, we also defined a “refined Fubini-Study distance” D F ( ρ, σ ) and “refinedBures distance” D B ( ρ, σ ) as follows: D F ( ρ, σ ) = − ln cos D F ( ρ, σ ) , (2.2)and D B ( ρ, σ ) = − ln[1 − D B ( ρ, σ ) ] . (2.3)One can verify that D B ( ρ, σ ) = D F ( ρ, σ ) = − ln Fi ( ρ, σ ) .Differing from entanglement entropy which is an intrinsic property of target state ρ , theabove three quantum distances do not only depend on the target state ρ but also dependon a reference state σ . We can choose a characteristic reference state which belongs tothe target state so that these quantum distances also become intrinsic properties of targetstates. If ρ is a thermal state, one natural characteristic reference state is just the groundstate. Such choice has been widely used in studying quantum phase transitions, e.g. seeRefs. [30, 31]. This choice can be generalized into arbitrary target states. Assume that ρ is an arbitrary quantum state. As ρ is both hermitian and positive semi-definite, we canformally define ρ = e − K with a hermitian operator K . Here K is known as the modularHamiltonian in axiomatic quantum field theory [32] or entanglement Hamiltonian in someliteratures studying entanglement entropy [33, 34]. To discuss the quantum distance forstate ρ , we choose the reference state to be Ω( ρ ) = lim n →∞ Ω n ( ρ ) , Ω n ( ρ ) := ρ n Tr ( ρ n ) . (2.4)We call this special reference state to be “modular ground state” of state ρ , as it is the groundstate of modular Hamiltonian K . The quantum distance between ρ and its modular groundstate becomes an intrinsic quantity of quantum state ρ . We call such quantum distance tobe “intrinsic quantum distance” of ρ . If the state ρ is just a thermal state, then the modularground state is just the zero temperature state of the system. We denote the “intrinsicrefined quantum distances” to be ˆ D T ( ρ ) := D T ( ρ, Ω( ρ )) , ˆ D F ( ρ ) := D F ( ρ, Ω( ρ )) (2.5)and ˆ D B ( ρ ) := D B ( ρ, Ω( ρ )) . (2.6)Our main results are holographic formulas for above three intrinsic refined quantum dis-tances. They are given by two kinds of descriptions.– 3 –ssume that quantum state ρ is dual to a boundary region A in a time slice of anasymptotically AdS spacetime. In the first description, the intrinsic refined quantum dis-tances of state ρ are related to the area in Planck units of a bulk codimension-2 cosmicbrane C n which is homologous to the region A : ˆ D T ( ρ ) = 2 ˆ D F ( ρ ) = (cid:90) ∞ Area ( C n )4 G N n d n − Area ( C ∞ )4 G N . (2.7)Here G N is the Newton’s constant and we use the subscript n on the cosmic brane to denotethat its brane tension as a function of n is given by T n = ( n − / (4 nG N ) . The geometryof above asymptotically AdS spacetime with above cosmic brane can be regarded as thesolution an Euclidean gravity theory with the total action I ( n ) total = I bulk + I ( n ) brane . (2.8)Here I bulk = I gravity + I matters , I gravity is the Euclidean Hilbert-Einstein action with negativecosmological constant, I matters is the action of matter fields, I ( n ) brane = T n Area ( C n ) . One canobtain the classical solution for theory (2.8) by minimizing total action for a given boundarysubregion A and tension T n . Due to the nonzero tension, the brane will backreact on thebulk geometry. Thus, the bulk geometry and position of cosmic C n depend on subscript n .In the second description, we take M n ( ρ ) to be the bulk domina of which the equalEuclidean time hypersurface Σ satisfies Σ = A∪ C n . The intrinsic refined quantum distancesof state ρ are given by on-shell action of gravity theory and the area of cosmic brane infollowing way: ˆ D T ( ρ ) = 2 ˆ D F ( ρ ) = I bulk [ M ∞ ] − I bulk [ M ] − I ( ∞ ) brane . (2.9)By Eqs. (2.9) and (2.7), the calculations of intrinsic refined quantum distances becomesolving partial differential equations in an Euclidean gravity theory with cosmic brane.When the variation of I total admits more than one classical solutions, we have to choose theone which has the smallest bulk action I bulk rather than the one which has smallest totalaction (2.8). The reason is similar to the discussion in Ref. [8] and will be clarified brieflylater. We now present the holographic deviations on Eqs. (2.9) and (2.7). We first considerintrinsic refined trace distance. The even order n -distance between ρ and Ω m ( ρ ) satisfies D n ( ρ, Ω m ) n = 12 n (cid:88) k =0 C k n ( − k Tr ( ρ k + m (2 n − k ) ) Tr ( ρ ) k Tr ( ρ m ) n − k . (3.1)Here C k n := (2 n )! / [ k !(2 n − k )!] is the combinatorial number and Ω m is defined by Eq. (2.4).For convenience, we here do not assume ρ is normalized. The intrinsic refined trace dis-tance then is obtained by the limit m → ∞ and analytical continuation n = 1 / . Define exp( −F m,n,k ( ρ )) = Tr ( ρ k + m (2 n − k ) ) / [ Tr ( ρ ) k Tr ( ρ m ) n − k ] and we will have F m,n,k = (2 n − k ) ln Tr ( ρ m ) + k ln Tr ( ρ ) − ln Tr ( ρ k + m (2 n − k ) ) . (3.2)– 4 –ow we use the gravity replica method [8, 12] to compute the F m,n,k . Follows theusual holographic dictionary, the trace of ρ m is given by the partition function of the QFTon a branched cover M m . Here M m is m -fold cover branched over A , which is definedby taking m copies of the original Euclidean spacetime M where the QFT lives witha cut along the entangling region and gluing them along the cuts in a cyclic order. Inthe large N limit, the bulk physics is classical and we have Tr ( ρ m ) = e − I bulk ( M m ) . Thebranched cover M m has a manifest Z m symmetry, which is not spontaneously broken inthe dominant bulk solution [35]. Take this Z m replica symmetry into account and we candefine an orbifold M m := M m / Z m . As a result, we the bulk on-shell action becomes I bulk ( M m ) = mI bulk ( M m ) . Then we obtain F m,n,k =[ k + (2 n − k ) m ] I bulk ( M k +(2 n − k ) m ) − (2 n − k ) mI bulk ( M m ) − kI bulk ( M ) . (3.3)After the quotient of Z m , there are conical singularities which are the fixed point of Z m quotient. Such conical singularities are some co-dimensional 2 surfaces. The cosmic braneis added into the total action (2.8) so that the variation problem can just generate thesolutions with such conical singularities. The real action of theory in the branched cover M m only contains the bulk term. This explains why the we need to choose the classicalsolution minimizing I bulk when theory (2.8) has multiple classical solutions.Though the parameters m, n and k are assumed to be integers in the definition of F m,n,k , we will analytically continues them into nonnegative real numbers. For large m , wehave two cases: if k (cid:54) = 2 n we have F m,n,k = k [ I bulk ( M ∞ ) − I bulk ( M ) − m ∂ m I bulk ( M m )]+ (2 n − m ∂ m I bulk ( M m ) + O (1 /m ) (3.4)and if k = 2 n we have F m,n, n = 2 n [ I bulk ( M n ) − I bulk ( M )] (3.5)On the other hand, it has been shown [8] m ∂ m I bulk ( M m ) = Area ( C m )4 G N . (3.6)Using Eqs. (3.4) and (3.5), we find that the summation (3.1) can be computed analytically.Thus we find (see appendix A for mathematical details) D ( ρ, Ω ∞ ) = 1 − e −F ⇒ D T ( ρ ) = F . (3.7)where F := I bulk ( M ∞ ) − I bulk ( M ) − I ( ∞ ) brane . (3.8)Thus, we obtain the holographic formula (2.9) for intrinsic refined trace distance. Theholographic formula (2.7) can be obtained after we integrate Eq. (3.6) and use the fact I ( ∞ ) brane = Area ( C ∞ ) / (4 G N ) . – 5 –n order to compute the fidelity, we consider a more general expression, i.e. Rényirelative entropies of two states, which in general is defined as follow [23, 36, 37] S k ( ρ (cid:107) σ ) = 1 k − Tr [( σ − k k ρσ − k k ) k ] . (3.9)The Rényi relative entropy is just a quasi-distance function as it is not symmetric in general,i.e. S k ( ρ (cid:107) σ ) (cid:54) = S k ( σ (cid:107) ρ ) . It is clear that D F ( ρ, σ ) = S / ( ρ (cid:107) σ ) . The limit lim k → S k ( ρ (cid:107) σ ) = S ( ρ (cid:107) σ ) is just he relative entropy between ρ and σ . Take σ = Ω m ( ρ ) and we find S k ( ρ (cid:107) Ω m ) = 1 k − Tr ( ρ k Ω − km ) = F m, / ,k − k . (3.10)The intrinsic refined Fubini-Study distance and intrinsic refined Bures distance are obtainedby ˆ D F ( ρ ) = ˆ D B ( ρ ) = 12 S / ( ρ (cid:107) Ω) = F ∞ , / , / = F / . (3.11)Take Eq. (3.8) into Eq. (3.11) and we then obtain dualities of ˆ D F ( ρ ) and ˆ D B ( ρ ) .Using Eq. (3.8), we also find a holographic duality for the k -th Rényi relative entropybetween ρ and its modular ground state when k ∈ (0 , S k ( ρ (cid:107) Ω) = k − k (cid:110) I bulk [ M ∞ ] − I bulk [ M ] − I ( ∞ ) brane (cid:111) = k − k (cid:20)(cid:90) ∞ Area ( C n )4 G N n d n − Area ( C ∞ )4 G N (cid:21) . (3.12)The relative entropy S ( ρ (cid:107) Ω) is given by the limit k → , which is divergent. This agrees withthe analysis from quantum information theory. By definition of relative entropy, we have S ( ρ (cid:107) σ ) = Tr ( ρ ln ρ ) − Tr ( ρ ln σ ) . When the state σ = Ω , we can find that Tr ( ρ ln σ ) = ∞ andso S ( ρ (cid:107) Ω) = ∞ . Note the Rényi relative entropy is not symmetric about two states when k (cid:54) = 1 / . However, there is a special permutation symmetry (1 − k ) S k (Ω (cid:107) ρ ) = kS − k ( ρ (cid:107) Ω) .We then have S k (Ω (cid:107) ρ ) = I bulk [ M ∞ ] − I bulk [ M ] − I ( ∞ ) brane = (cid:90) ∞ Area ( C n )4 G N n d n − Area ( C ∞ )4 G N . (3.13)Thus, we see S k (Ω (cid:107) ρ ) = ˆ D T ( ρ ) is independent of k . As a self-consistent check, we cancompute S k (Ω (cid:107) ρ ) directly in qubit system. Assume that ρ is density matrix in a finitedimensional Hilbert space and λ is its largest eigenvalue. Then we can find S k (Ω (cid:107) ρ ) = k − ln Tr ( W k ρ − k ) = k − ln Tr ( W ρ − k ) = − ln λ , which is independent of k as expected. In following, we will show examples about how to use the holographic formula (2.7) to findthe intrinsic trace distance of some states in CFT . In the first example, we consider aspherical disk A with radius R in a d -dimensional vacuum state. In princple, we need to– 6 –olve Einstein’s equation with the cosmic brane. However, as the disk is spherical, the taskof finding the cosmic brane solution can be essentially simplified. In this case we can useconformal map of Ref. [7] to direct obtain the bulk geometry. After the conformal map, thebulk geometry is a d + 1 -dimensional Euclidean hyperbolic AdS black hole [8, 38]d s = d τ f n ( r ) + f n ( r ) d r + r [ d u + sinh ( u ) d Ω d − ] . (4.1)and f n ( r ) = r − − r d − n ( r n − /r d − . The cosmic brane is mapped into the horizon.The Euclidean time direction τ has the period π and so leads to the conical singularityat r = r n . To match with the cosmic brane, we have to set f (cid:48) n ( r n ) = 4 π/n . Then we find r n = [1 + (cid:112) n d ( d − / ( nd ) and r ∞ = (cid:112) ( d − /d . As the cosmic brane is just thehorizon, we find Area n ( C ) = V d − ( R ) r d − n with V d − ( R ) = Ω d − (cid:90) ln(2 R/(cid:15) )0 sinh ( d − ( u ) d u . (4.2)Here Ω d − = 2 π ( d − / / Γ( d − ) is the area of the unit ( d − -sphere. The upper limit ofintegration is the UV-cut off [7]. In the case d = 2 , we find r n = 1 /n and Eq. (2.7) gives usa simple result ˆ D T ( ρ A ) = ln(2 R/(cid:15) ) / (4 G N ) = c R/(cid:15) ) . (4.3)Here c = 3 / (2 G N ) is the central charge. In the appendix B we will give a calculation fromCFT side and show that two results match with each other exactly.In the second example, we assume that the subregion A contains two symmetric disjointintervals in 2D case, i.e. A = A ∪ A , where A = [0 , l ] and A = [1 , l ] . The cross ration x = l . We first consider the limit x (cid:28) , which means two cosmic brane will be separatedfar enough. See the subfigure (a) of Fig. 1. Though every brane will backreact on the bulk Figure 1 . The cosmic branes of two intervals in the limit x (cid:28) (subfigure (a)) and x → (subfigure (b)). geometry, the interaction between two branes will be suppressed. Thus, up to the leadingorder of x , the final result would be simply twice of a single brane. The intrinsic refinedtrace distance in this case becomes ˆ D T ( ρ A ) ≈ c l/(cid:15) ) × c x/(cid:15) ) . (4.4)For large cross ration x → , the configuration of two cosmic branes is shown in subfigure(b) of Fig. 1. The second cosmic brane C will shrink into the boundary and two brane will– 7 –lso decouple with each other. As the result, we can compute two branes separately andobtain ˆ D T ( ρ A ) ≈ c { ln[(1 + l ) /(cid:15) ] + ln[(1 − l ) /(cid:15) ] } = c − x ) /(cid:15) ] . (4.5)Comparing with Eq. (4.4), we see that two limits have a symmetry x → − x . Note that thecomputation here only involves the leading terms of x → or x → . It will be interestingto study what will happen if the interaction between two branes cannot be neglected in thefuture. To summary, this paper studies the holographic dualities of three most frequently-usedquantum distances and a quantum quasi-distance, i.e. the trace distance, Fubini-Studydistance, Bures distance and Rényi relative entropy. By choosing the modular vacuum asthe reference state, it finds that they all have holographic dualities. Then it applies theseholographic dualities into 2-dimensional CFTs and show that holographic results exactlymatch with the calculations of field theory.For the holographic formula (2.7), there is no difficulty to obtain generalizations totheories dual to higher derivative gravity. The basic idea is similar to the directions ofRefs. [9–11]. The area term in right-hand side of Eq. (2.7) should be replaced by the Waldentropy [39] evaluated on a cosmic brane. In addition, following the methods of [12, 13]we can also include quantum corrections into Eq. (2.9) by taking the bulk matters intoaccount. For 2 dimensional CFT with 2+1 dimensional gravity duality, it is also interestingto consider the perturbational expansion of small cross ration in gravity side by taking theinteraction of two cosmic branes and then compare it with the results of CFTs.
Acknowledgments
The work is supported by the Natural Science Foundation of China under Grant No.12005155.
A About Eq. (3.7)
We first note that, for large m , I bulk ( M m ) = I bulk ( M ∞ ) + I /m + O (1 /m ) . (A.1)If k (cid:54) = 2 n , then we also have I bulk ( M k +(2 n − k ) m ) = I bulk ( M ∞ ) + I k + (2 n − k ) m + O (1 /m ) . (A.2)This gives us Eq. (3.4) at large m limit when k (cid:54) = 2 n . We then take Eqs. (3.4) and (3.5)into Eq. (3.1) and take the limit m → ∞ . Then we find D n ( ρ, Ω) n = 12 n − (cid:88) k =0 C k n ( − k e − ka − (2 n − I + e − B n . (A.3)– 8 –ere a = I bulk ( M ∞ ) − I bulk ( M ) − Area ( C ∞ )4 G N , I = − m ∂ m I bulk ( M m ) , and B n = − n [ I bulk ( M n ) − I bulk ( M )] . Then the summation (A.3) can be computed analytically D n ( ρ, Ω) n = e − (2 n − I n (cid:88) k =0 C k n ( − k e − ka − e − na + e B n = e − (2 n − I (cid:104)(cid:0) − e − a (cid:1) n − e − na + e B n (cid:105) . (A.4)Analytically continue it into n = 1 / and we find D ( ρ, Ω) = 1 − e − a . (A.5)Thus, we find ˆ D T ( ρ ) = I bulk ( M ∞ ) − I bulk ( M ) − I ( ∞ ) brane . (A.6)This gives us Eq. (3.7). B Calculations from CFT
To calculate the intrinsic refined trace distance from CFT, we still start from the Eqs. (3.1)and (3.2). The difference is that now we will use the replica trick of field theory. In 2D CFT,the trace of ρ m can be obtained by using twist operators. Follows the usual computationsin CFT (e.g. see Ref. [6]), one can find that the result in large c -limit reads ln Tr ( ρ m ) = − c m − /m ) ln(2 R/(cid:15) ) . (B.1)Taking this into Eq. (3.2) and considering the large m limit, we find F m,n,k = − (2 n − c R/(cid:15) ) + kc R/(cid:15) ) + O (1 /m ) . (B.2)if k (cid:54) = 2 n and F m,n, n = c n − / (2 n )] ln(2 R/(cid:15) ) . (B.3)Then we take Eqs. (B.2) and (B.3) into Eq. (3.1). Following the same steps of appendix A,the summation can be computed analytically and we finally find that the trace distancereads D ( ρ, Ω) = 1 − exp (cid:104) − c R/(cid:15) ) (cid:105) . (B.4)Thus, we see ˆ D T ( ρ ) = c ln(2 R/(cid:15) ) , which matches with the our holographic result exactly.– 9 – eferences [1] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity , Int. J.Theor. Phys. (1999) 1113–1133, [ hep-th/9711200 ].[2] S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncriticalstring theory , Phys. Lett. B (1998) 105–114, [ hep-th/9802109 ].[3] E. Witten,
Anti-de Sitter space and holography , Adv. Theor. Math. Phys. (1998) 253–291,[ hep-th/9802150 ].[4] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT , Phys. Rev. Lett. (2006) 181602, [ hep-th/0603001 ].[5] V. E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglemententropy proposal , JHEP (2007) 062, [ ].[6] B. Chen, Holographic Entanglement Entropy: A Topical Review , Commun. Theor. Phys. (2019) 837.[7] L.-Y. Hung, R. C. Myers, M. Smolkin and A. Yale, Holographic Calculations of RenyiEntropy , JHEP (2011) 047, [ ].[8] X. Dong, The Gravity Dual of Renyi Entropy , Nature Commun. (2016) 12472,[ ].[9] X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity , JHEP (2014) 044, [ ].[10] J. Camps, Generalized entropy and higher derivative Gravity , JHEP (2014) 070,[ ].[11] R.-X. Miao and W.-Z. Guo, Holographic Entanglement Entropy for the Most General HigherDerivative Gravity , JHEP (2015) 031, [ ].[12] T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographicentanglement entropy , JHEP (2013) 074, [ ].[13] N. Engelhardt and A. C. Wall, Quantum Extremal Surfaces: Holographic EntanglementEntropy beyond the Classical Regime , JHEP (2015) 073, [ ].[14] D. Harlow and P. Hayden, Quantum Computation vs. Firewalls , JHEP (2013) 085,[ ].[15] D. Stanford and L. Susskind, Complexity and Shock Wave Geometries , Phys. Rev. D (2014) 126007, [ ].[16] L. Susskind, Computational Complexity and Black Hole Horizons , Fortsch. Phys. (2016)24–43, [ ].[17] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic ComplexityEquals Bulk Action? , Phys. Rev. Lett. (2016) 191301, [ ].[18] B. Czech,
Einstein Equations from Varying Complexity , Phys. Rev. Lett. (2018) 031601,[ ].[19] P. Caputa and J. M. Magan,
Quantum Computation as Gravity , Phys. Rev. Lett. (2019)231302, [ ].[20] L. Susskind and Y. Zhao,
Switchbacks and the Bridge to Nowhere , . – 10 –
21] M. A. Nielsen and I. L. Chuang,
Quantum Computation and Quantum Information: 10thAnniversary Edition . Cambridge University Press, Cambridge, UK, 10th ed., 2011.[22] J. Watrous,
The Theory of Quantum Information . Cambridge University Press, Cambridge,UK, 1st ed., 2018.[23] N. Lashkari,
Relative entropies in conformal field theory , Phys. Rev. Lett. (Jul, 2014)051602.[24] J. Zhang, P. Ruggiero and P. Calabrese,
Subsystem Trace Distance in Quantum FieldTheory , Phys. Rev. Lett. (2019) 141602, [ ].[25] J. Zhang, P. Ruggiero and P. Calabrese,
Subsystem trace distance in low-lying states of (1 + 1) -dimensional conformal field theories , JHEP (2019) 181, [ ].[26] J. Zhang and P. Calabrese, Subsystem distance after a local operator quench , JHEP (2020) 056, [ ].[27] D. D. Blanco, H. Casini, L.-Y. Hung and R. C. Myers, Relative Entropy and Holography , JHEP (2013) 060, [ ].[28] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance betweenquantum states and gauge-gravity duality , Phys. Rev. Lett. (Dec, 2015) 261602.[29] M. Alishahiha and A. Faraji Astaneh,
Holographic fidelity susceptibility , Phys. Rev. D (Oct, 2017) 086004.[30] E. J. König, A. Levchenko and N. Sedlmayr, Universal fidelity near quantum and topologicalphase transitions in finite one-dimensional systems , Phys. Rev. B (Jun, 2016) 235160.[31] P. Zanardi, M. Cozzini and P. Giorda, Ground state fidelity and quantum phase transitions infree fermi systems , Journal of Statistical Mechanics: Theory and Experiment (Feb.,2007) L02002–L02002.[32] R. Haag,
Local quantum physics: Fields, particles, algebras . Springer, Berlin, Germany, 05,1992.[33] H. Li and F. D. M. Haldane,
Entanglement spectrum as a generalization of entanglemententropy: Identification of topological order in non-abelian fractional quantum hall effectstates , Phys. Rev. Lett. (Jul, 2008) 010504.[34] H. Yao and X.-L. Qi,
Entanglement entropy and entanglement spectrum of the kitaev model , Phys. Rev. Lett. (Aug, 2010) 080501.[35] A. Lewkowycz and J. Maldacena,
Generalized gravitational entropy , Journal of High EnergyPhysics (Aug., 2013) .[36] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr and M. Tomamichel,
On quantum rényientropies: A new generalization and some properties , Journal of Mathematical Physics (2013) 122203, [ https://doi.org/10.1063/1.4838856 ].[37] M. M. Wilde, A. Winter and D. Yang, Strong converse for the classical capacity ofentanglement-breaking and hadamard channels via a sandwiched rényi relative entropy , Communications in Mathematical Physics (July, 2014) 593–622.[38] Y. Nakaguchi and T. Nishioka,
A holographic proof of Rényi entropic inequalities , JHEP (2016) 129, [ ].[39] R. M. Wald, Black hole entropy is the Noether charge , Phys. Rev. D (1993) 3427–3431,[ gr-qc/9307038 ].].