A holographic proof of Rényi entropic inequalities

Published for SISSA by Springer Received: July 12, 2016 Revised: November 28, 2016 Accepted: December 20, 2016 Published: December 23, 2016 Yuki Nakaguchia,b and Tatsuma Nishiokab a Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa City, Chiba 277-8568, Japan b Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan E-mail: , Abstract: We prove Rényi entropic inequalities in a holographic setup based on the recent proposal for the holographic formula of Rényi entropies when the bulk is stable against any perturbation. Regarding the Rényi parameter as an inverse temperature, we reformulate the entropies in analogy with statistical mechanics, which provides us a concise interpretation of the inequalities as the positivities of entropy, energy and heat capacity. This analogy also makes clear a thermodynamic structure in deriving the holographic formula. As a by-product of the proof we obtain a holographic formula to calculate the quantum fluctuation of the modular Hamiltonian. A few examples of the capacity of entanglement are examined in detail. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 1606.08443 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP12(2016)129 JHEP12(2016)129 A holographic proof of Rényi entropic inequalities Contents 1 2 Analogy to statistical mechanics 2.1 Partition function Z and the escort density matrix ρn 2.2 Improved Rényi entropy S̃n 2.3 Capacity of entanglement C(n) 2.4 Rényi entropic inequalities from the viewpoint of the analogy 4 4 5 6 6 3 Holographic formula of the Rényi entropy 3.1 The area prescription 3.2 Derivation revisited from the viewpoint of the analogy 7 7 8 4 Proof of the Rényi entropic inequalities 4.1 A holographic proof 4.2 Legendre transformed expression for capacity of entanglement 11 11 12 5 Calculations of the capacity of entanglement 5.1 Conformal field theory 5.2 Free fields 5.3 Gravity duals 5.4 Large and small n limits 14 14 15 17 19 6 Discussion 20 A On holographic calculation of C(1) using graviton propagator 22 B Comments on the strong sub-additivity of Rényi entropies 24 1 Introduction A key concept in modern quantum gravity theory is holography that opened the door to the non-perturbative definition as the dual quantum theory in one lower dimensions. A considerable number of dictionaries have been composed to translate physical quantities in one theory to the other. The holographic duality remains as mysterious as quantum gravity, though, especially on how the bulk spacetime information is encoded in the boundary quantum field theory. There have been a huge amount of attempts to probe the bulk structure via holography, of which one of the most important breakthroughs is the holographic formula of entanglement entropy [1] associating a unit area per four times the Planck length of a codimension-two bulk surface with one bit of information for a given –1– JHEP12(2016)129 1 Introduction region in the boundary field theory. In fact, the formula is a realization of the original idea of the holographic principle [2, 3] that states in quantum gravity theory, the degrees of freedom live not in volumes but in areas. Overviews on the recent developments of the holographic entanglement entropy are available in reviews e.g. [4, 5]. In quantum theories, entanglement entropy SA of a state subspace HA is defined as the von Neumann entropy SvN [ρ] ≡ −Tr[ρ log ρ] of the reduced matrix ρA = TrĀ [ρtotal ] as1 SA ≡ −Tr[ρA log ρA ] . (1.1) SAC + SBC ≥ SC + SABC , (1.2) showing a kind of concavity of the entropy. The sub-additivity SA + SB ≥ SAB (1.3) follows by taking C as ∅. As a field application, the strong sub-additivity is utilized for constructing c-functions, monotonically decreasing functions along RG flows, such as the entropic c-function in two dimensions [7] and the F -function in three dimensions [8]. One of the novel aspects of the holographic entanglement entropy formula is the simplicity of proving the strong sub-additivity (1.2) [9–11]. The proof only relies on the geometric properties of a codimension-two surface in the bulk, and suggests a profound way of the emergence of the bulk spacetime as it translates a quantum mechanical constraint to a purely geometric one. More extensive studies of the inequalities satisfied by the holographic formula were carried out in [12, 13] to classify the characteristics of the geometry which has a field theory dual. Recently, the holographic formula was proposed [14] for the entanglement Rényi entropy Sn [ρ] which is a one-parameter generalization of the von Neumann entropy defined with a non-negative real number n as Sn [ρ] ≡ − 1 log Tr[ρn ] . n−1 (1.4) It reduces to the von Neumann entropy when n = 1, S1 [ρ] = SvN [ρ]. The derivation of the holographic formula by [14] is based on so-called the Lewkowycz-Maldacena prescription [15] employed to derive the holographic entanglement entropy where the replica Zn 1 Throughout this paper, we always normalize a density matrix as Tr[ρ] = 1. –2– JHEP12(2016)129 It measures how much quantum information of the degrees of freedom in HA is entangled with the outer degrees of freedom, namely, how much quantum information will be lost for the subspace HA if the outer subspace is ignored. In quantum field theories, entanglement entropy is defined for a space region A on a time slice, assuming that we can construct a state space HA representing degrees of freedom on the region A by some appropriate procedures. The total state is often taken as the vacuum ρtotal = |0i h0| for simplicity. Entanglement entropy has many mathematical properties, among which the most important one is an inequality called the strong sub-additivity [6] symmetry is assumed in the bulk geometry.2 We will review the derivation in section 3 so as to fix our notations and for later use. Then it is natural to think about how mathematical properties of the Rényi entropy are transcribed to the bulk side in a geometric language. It is known that the Rényi entropy is not strongly sub-additive, but it satisfies inequalities involving the derivative with respect to n [19, 20]3 ∂n Sn ≤ 0 ,  n−1 Sn ≥ 0 , n ∂n ((n − 1)Sn ) ≥ 0 , ∂n2 ((n − 1)Sn ) ≤ 0 . (1.5) (1.6) (1.7) (1.8) 1 P n These inequalities are originally proved for the classical Rényi entropy Sn [pi ] ≡ − n−1 i pi of a probability distribution pi , but are still true for the quantum Rényi entropy (1.4). The proof for a quantum case immediately follows by diagonalizing the density matrix ρ as U ρ U † = diag(p1 , p2 , . . . ) with a unitary matrix U . The first inequality (1.5) implies the positivity of the Rényi entropy Sn ≥ 0 as S∞ = mini (− log pi ) ≥ 0. The aim of this paper is to prove these inequalities by the holographic formula of the Rényi entropy. Before proceeding to the proof, we rewri (...truncated)