Gravity duals of quantum distances

Published for SISSA by Springer Received: April 6, 2021 Revised: July 2, 2021 Accepted: August 7, 2021 Published: August 27, 2021 Run-Qiu Yang Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Yaguan Road 135, Jinnan District, 300350 Tianjin, P. R. China E-mail: Abstract: This paper provides a holographic approach to compute some most-frequently used quantum distances and quasi-distances in strongly coupling systems and conformal field theories. By choosing modular ground state as the reference state, it finds that the trace distance, Fubini-Study distance, Bures distance and Rényi relative entropy, all have gravity duals. Their gravity duals have two equivalent descriptions: one is given by the integration of the area of a cosmic brane, the other one is given by the Euclidian on-shell action of dual theory and the area of the cosmic brane. It then applies these duals into the 2-dimensional conformal field theory as examples and finds the results match with the computations of field theory exactly. Keywords: AdS-CFT Correspondence, Black Holes, Gauge-gravity correspondence ArXiv ePrint: 2102.01898 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2021)156 JHEP08(2021)156 Gravity duals of quantum distances Contents 1 2 Distances to modular ground state and holographic proposals 2 3 Derivation via holographic replica trick 4 4 Application in CFTs 7 5 Summary 9 1 Introduction In recent years it has been suggested that quantum information theory and gravity theory have deep connection. The gauge/gravity duality, which shows an equivalence between strongly coupling quantum field theories (QFTs) and weakly coupling gravitational theories in one higher dimensions [1–3], offers us a powerful tool towards such connection. As a result, the quantum information theoretic considerations have provided various useful viewpoints in the studies of gauge/gravity duality and quantum gravity. One example is the Ryu-Takayanagi (RT) formula [4–6], which connects the area of a codimension-2 minimal surface in the dual spacetime and the entanglement entropy of the boundary QFT. The RT formula 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 information named “complexity”, which measures the difference of two states according to the size of quantum circuits in converting one state into the other, also has been studied widely in gravity and black hole physics [14–19]. From a general viewpoint, the complexity is a kind of “distance” between quantum states [20]. Except for complexity, there are other several different measures of the distance between states, which are widely used in quantum information [21, 22]. For example, given two density matrices ρ and σ in the same Hilbert space, two families of distance are widely used in quantum information theory. The first one are based on the fidelity Fi(ρ, σ) = Tr q√ √ 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 DF (ρ, σ) = arccos Fi(ρ, σ) and the Bures distance DB (ρ, σ) = p 1 − Fi(ρ, σ). The other family of distances, depending on a positive number n, is provided by the n-distances 1 Dn (ρ, σ) := 1/n (Tr|ρ − σ|n )1/n . (1.2) 2 –1– JHEP08(2021)156 1 Introduction √ P Here Tr|X|n := i λni and λi is the i-th eigenvalue of X † X. When X is hermitian {λi } are just the absolute values of eigenvalues of X. Two special choices are widely applied. One is Hilbert-Schmidt distance, which chooses n = 2. This distance leads to some conveniences in mathematics because the calculation is straightforward by its definition. The other choice is 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 (and so q Fubini-Study distance and Bures distance) are bounded by each others: p D1 (ρ, σ) Tr(OO† ) ≤ 1 − Fi(ρ, σ)2 Tr(OO† ). Thirdly, they supply lower bounds for the p relative entropy S(ρ k σ): [1 − Fi(ρ, σ)] ≤ D1 (ρ, σ) ≤ S(ρ k σ)/2. Despite that trace distance and fidelity have these important properties, their computations are highly challenging in quantum field theory. The first breakthrough towards this issue is achieved by ref. [23], which develops a replica trick to compute the fidelity for 2-dimensional conformal field theory. Refs. [24–26] then also develop replica method to compute the trace distance for a class of special states for single short interval in 1+1 dimensional CFTs. However, there are still huge difficulties in the calculations of trace distance even for 1+1 dimensional CFTs, such as to compute the trace distance between two thermal states or two large intervals in CFTs. There is also no compact method to compute the trace distance in higher dimensional CFTs or general quantum field theories. On the other hand, the holographic descriptions of entanglement, relative entropy [27] and complexity have been found, however, the trace distance does not yet. Refs. [28, 29] propose holographic duality to compute the fidelity between a state and its infinitesimal perturbational state, however, they cannot be used into the case when the difference between two states are not infinitesimal. In this paper, it will develop holographic duals to compute the trace distance, FubiniStudy distance, Bures distance and Rényi relative entropy. By choosing a characteristic reference state which will be called “modular ground state”, this paper will show that they all have gravity duals. Their gravity duals have two equivalent descriptions. The one is given by the integration of the area of a cosmic brane with respective to its tansion. The other one description contains two parts, one of which is the on-shell action of gravity theory and the other one of which is the area of the cosmic brane. We then apply them to the calculations of the trace distance in 1+1 dimensional CFT and show our holographic calculations exactly match 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) ≈ 0 and so their trace distance will almost saturate the upper bound. In this case, it will be more convenient to study a “refined trace distance” DT (ρ, σ) = − ln[1 − D1 (ρ, σ)] . –2– (2.1) JHEP08(2021)156 1 − D1 ≤ Fi ≤ 1 − D12 . Secondly, they offer us dimension-independent bounds on the difference qbetween the expected values qof an operator O in different states: |hOiρ − hOiσ | ≤ Due to the monotonicity, the refined trace distance and trace distance contain same information. Similarly, we also defined a “refined Fubini-Study distance” DF (ρ, σ) and “refined Bures distance” DB (ρ, σ) as follows: DF (...truncated)