Quantum Information and Entanglement in Holographic Theories

Transcription

1 Quantum Information and Entanglement in Holographic Theories Matthew Headrick randeis University Contents 1 asic notions Entanglement entropy & mutual information EE in QFT Ryu-Takayanagi formula 3 3 few checks 5 4 Two intervals (MH 10) Prediction from RT formula First-principles calculations

2 5 Important open questions 9 1 asic notions 1.1 Entanglement entropy & mutual information Entropy is extensive: if two systems, are uncorrelated, then On the other hand, if they are correlated then ρ() = ρ() ρ(), and S() = S() + S() ρ() ρ() ρ() (where ρ() tr ρ(), ρ() tr ρ()) and S() < S() + S(), (subadditivity of entropy) S(), S() are called entanglement entropies, because they can be non-zero even if ρ() is pure. Example: In general, if ρ() is pure, then S() = S(). entanglement, or even correlation!) ψ = 1 ( ) 21/2 ρ() = ψ ψ = 1 ( ) ( ) 2 ρ() = ρ() = 1 ( ) 2 S() = 0, S() = S() = ln 2 (Note that there can be entanglement entropy without Mutual information: I( : ) = S() + S() S() measures total amount of correlation (both classical correlation & entanglement). In above example, I( : ) = 2 ln 2 Mutual information bounds correlators between normalized operators (Wolf, Verstraete, Hastings, Cirac 07): ( O O O O ) 2 2I( : ) Mutual information increases under adjoining another system to or : I( : C) I( : ) i.e. (strong subadditivity of entropy) S() + S(C) S() + S(C) 2

3 1.2 EE in QFT In a QFT, we can take subsystems,,... to be spatial regions. The EE is UV-divergent: S() = ɛ 2 D area( ) + or, in two dimensions, S() = c UV ln ɛ #( ) + 6 Mutual information for separated regions is finite. Subleading terms are difficult to compute in practice even in free QFTs. (For example, for compact free scalar in 1 + 1, in vacuum, mutual information between two intervals on the line is not known see Calabrese, Cardy, Tonni 09, 10) EE & mutual information are very useful non-local observables: gain, I( : ) bounds all possible two-point functions between &. (EE knows a lot about the system!) EE can be used as an order parameter (e.g. to detect topological phase transitions) EE is the most efficient method for numerically extracting central charges of fixed points in two dimensions... 2 Ryu-Takayanagi formula Ryu-Takayanagi formula for EE of a spatial region in a holographic theory dual to classical Einstein gravity ( large N, strong coupling ) 3

4 in a state described in the bulk by a static, classical field configuration ( distinguished constant-time surfaces): S() = 1 ( ) min area(m) 4G N m where area(m) is computed wrt spatial, Einstein-frame metric m means bulk region r s.t. r = m call minimizer m(), r() r() m() bulk Notes: Simple, elegant, easy to compute black Deep yet simple statement about quantum gravity (inhole particular: area-entropy relation does not just apply to horizons) m() = horizon Many checks, but no proof or derivation (purported proof by Fursaev = 06 is incorrect, see MH 10) entire boundary Example to illustrate homology condition m() : r() r() black hole or black hole m() m() r() r() Is r() the holographic dual (in some sense) of ρ()? (See Czech, Karczmarek, Nogueira, Van Raamsdonk 12) 4

5 Corrections believed to take general form (α = classical higher-derivative; G N = quantum): S() = 1 ( ) min area(m) + O(α ) + O(G 0 4G N) N m See Hung, Myers, Smolkin 10, de oer, Kulaxizi, Parnachev 10 Conjecture for covariant generalization by Hubeny, Rangamani, Takayanagi 07: replace minimal surface with minimal extremal surface. Has been applied to various systems, but subjected to fewer tests than static RT formula. 3 few checks Ryu-Takayanagi formula: reproduces EE of interval of CFT in vacuum or thermal state (Holzhey, Larsen, Wilczek 94), e.g. in vacuum on R: S([u, v]) = c ( ) v u 3 ln ɛ u 1 v 1 u 2 v 2 reproduces structure of UV divergences part of EE in arbitrary dimensions reproduces ekenstein-hawking entropy: black hole S() = S H m() = horizon = entire boundary (hence also works for regions that can be mapped to black holes: Casini, Huerta, Myers 10) obeys S() = S() when is in a pure state: m() = m() 5

6 m() = horizon = entire boundary m() = m() obeys strong subadditivity: S() + S(C) S() + S(C) (MH, Takayanagi 07) C S() + S(C) = m() m(c) = m(c) m() = S(C) + S() (formal proof: take r() = r() r(c), r(c) = r() r(c)) This proof fails for covariant holographic EE formula, and it s not known whether it obeys SS. llais, Tonni, 11, Callan, He, MH 12 studied ds 3 -Vaidya, found no violations of SS as long as matter in bulk satisfied null energy condition. obeys monogamy property for mutual information (Hayden, MH, Maloney 11): I( : C) I( : ) + I( : C) i.e. S() + S(C) + S(C) S() + S() + S(C) + S(C) (llais, Tonni 11 found that monogamy also holds in time-dependent examples.) 6

7 Notes: Unlike SS, monogamy is not a general property of EEs; counterexample: ρ(c) = 1 2 ( ); in general QFTs it can go either way see Casini, Huerta 08. Rather it is a special property of holographic theories. Physical interpretation is still obscure, but suggests that entanglement dominates over classical correlations. However, monogamy implies an infinite list of general inequalities (Linden, Winter 04; Cadney, Linden, Winter 11). In fact, RT formula obeys every known applicable general property of EE. general relativity knows a lot of sophisticated quantum information theory! most of these tests automatically still hold with higher derivative corrections, but not quantum (1/N) corrections. 4 Two intervals (MH 10) 4.1 Prediction from RT formula Ryu-Takayanagi formula makes many new predictions. n interesting one is a novel phase transition in the mutual information between separated regions as a function of their separation. Simplest case is 2 intervals on R in CFT: u 1 v 1 u 2 v 2 Mutual information is UV-finite & conformally invariant: u 1 v 1 u 2 v 2 I(x) I( : ) = S() + S() S() x = (v 1 u 1 )(v 2 u 2 ) (u 2 u 1 )(v 2 v 1 ) u 1 v 1 u 2 v 2 x 1/2 x 1/2 m() = m() m() m() m() m() 7

8 { } 0 x 1/2 I(x) = c 3 ln + O(c 0 ) (N: G x N c 1 ) 1 x x 1/2 I x c Qualitative features: phase transition at x = 1/2 I(x) = 0 for x 1/2 (Expect both features to be preserved by higher-derivative classical corrections, but not by quantum corrections) 4.2 First-principles calculations Unlike for 1 interval, no general formula for EE of 2 intervals in CFT much harder to compute from first principles than for 1 interval (unknown even for free scalar (Calabrese, Cardy, Tonni 09, 10)). We will apply replica trick (Holzhey, Larsen, Wilczek 94): 1. For all integer n > 1, consider orbifold theory C n /Z n, and compute I (n) (x) = 1 n 1 ln σ 1(0)σ 1 (x)σ 1 (1)σ 1 ( ) σ 1 (0)σ 1 (x) σ 1 (1)σ 1 ( ) (mutual Rényi information) where σ 1, 1 are twist operators 2. nalytically continue to non-integer n and take limit n 1: I(x) = lim n 1 I (n) (x) Two approaches: Holographic: 4-point function of twist operators in C n /Z n is related to partition function of C on genus- (n 1) Riemann surface whose complex structure depends on x 8

9 In holographic theories, this has a first-order phase transition (à la Hawking-Page) at x = 1/2 (since x = 1/2 is fixed point of mapping-class group) for all n Confirms phase transition in I(x) at x = 1/2 CFT: First expand I (n) (x) in x, then take limit n 1. Using OPE, (from identity & stress tensor) σ 1 (0)σ 1 (x)σ 1 (1)σ 1 ( ) σ 1 (0)σ 1 (x) σ 1 (1)σ 1 ( ) = O m C n /Z n c σ σmc m σσx 2dm = 1 + (n2 1) 2 c 144n 3 x 2 + O(x 3 ) I (n) (x) = (n + 1)2 (n 1)c 144n 3 x 2 + O(x 3 ) I(x) = lim n 1 I (n) (x) = O(x 3 ) To go to higher order in x, use conformal blocks of C n /Z n. t each order expand in 1/c: I (n) (x) = (n 1)(n + 1) 2 144n 3 ( x 2 + x n4 2n n 4 x n4 2n 2 ) n 4 x 5 + O(x 6 ) c Confirms predicted vanishing of order-c part of I(x) + O(c 0 ) 5 Important open questions Can RT formula (or covariant generalization) be proved from first principles? What is the general form for the corrections to the RT formula in the presence of higher-derivative classical corrections to the Einstein-Hilbert action (or even for a general classical bulk theory: Chern-Simons, TMG, Vasiliev,...)? Is there a general formalism for computing bulk quantum corrections to RT formula? Does r() represent (in some sense) the holographic dual of ρ()? Does the covariant holographic EE formula obey SS in general? What does monogamy of mutual information imply about entanglement structure of holographic states? How do EEs behave in large-n theories in general? 9