Properties of entropy in holographic theories Matthew Headrick randeis University Contents 0 Definitions 1 Properties of entropy Entanglement entropy in QFT 3 Ryu-Takayanagi formula 6 Monogamy 8 5 SS of covariant holographic EE? 10 1
0 Definitions Suppose we have a quantum system in a state ρ (could be pure or mixed) Let,,... be disjoint subsystems (H = H H ) = etc. Reduced density matrix: ρ = tr c ρ Entanglement entropy: S() = tr ρ ln ρ Three examples ( is first bit): 1. ρ = 1 ( 00 + 11 ) ( 00 + 11 ). ρ = 1 ( 00 00 + 11 11 ) 3. ρ = 1 ( 00 00 + 01 01 + 10 10 + 11 11 ) = 1 ( 0 + 1 ) ( 0 + 1 ) 1 ( 0 + 1 ) ( 0 + 1 ) In all three cases, ρ = 1 ( 0 0 + 1 1 ), S() = ln So entanglement entropy can be due to: (1) entanglement; () classical correlation; (3) being in a mixed state without entanglement or correlation; or a combination thereof 1 Properties of entropy ased purely on properties of Hilbert spaces & density matrices; no other physics input
One subsystem: S() 0, S() = 0 iff ρ is pure Hence S is a measure of mixedness Two subsystems: (1) raki-lieb: S() S() S() In particular if ρ is pure then S() = S() () Subadditivity: S() S() + S(), S() = S() + S() iff ρ = ρ ρ Hence mutual information I( : ) = S() + S() S() is a measure of correlation (classical + quantum) between,. In examples above: 1. ρ = 1 ( 00 + 11 ) ( 00 + 11 ) I( : ) = ln. ρ = 1 ( 00 00 + 11 11 ) I( : ) = ln 3. ρ = 1 ( 00 00 + 01 01 + 10 10 + 11 11 ) = 1 ( 0 + 1 ) ( 0 + 1 ) 1 ( 0 + 1 ) ( 0 + 1 ) I( : ) = 0 Mutual information is a key quantity in classical & quantum information theory. Numerous operational senses in which it measures total amount of correlation. Examples: I( : ) = S() S( ) = how much your ignorance about decreases if you know state of, where S( ) = S() S() = conditional entropy of given (N.. in quantum setting S( ) can be negative, so interpretation is less clear) I( : ) gives the rate at which information can reliably be sent over a noisy channel (Shannon) ound on correlators between normalized operators: (Wolf, Verstraete, Hastings, Cirac 07): ( O O O O ) I( : ) 3
Three subsystems: If mutual information measures total amount of correlation, then it should increase under adjoining another subsystem to or : I( : C) I( : ) i.e. S() + S(C) S(C) + S() Strong subadditivity. Easy to show at classical level; conjectured in quantum systems by Lanford, Robinson 68, proved by Lieb, Ruskai 73 Four subsystems: Constrained inequality (Linden, Winter 0): If I( : C) = I( : ) = I( : C), I( : CD) = I( : D) then I(C : D) I(C : ) Five or more subsystems: Infinite hierarchy of constrained inequalities (Cadney, Linden, Winter 11) Entanglement entropy in QFT In a QFT, we can take subsystems to be spatial regions:
In vacuum, or any low-energy state, EE is UV-divergent along boundary of, S() = ɛ D area( ) + (with subleading divergences in D ) or, in two dimensions, S() = c UV 6 ln ɛ #( ) + (Note that a generic state will have S() = ɛ 1 D vol(), so this is very little entanglement: ground states and low-energy states are special in that they are highly unentangled. This provides a massive simplification e.g. in simulations of lattice systems.) Mutual information is divergent along common boundary, I( : ) = ɛ D area( ) + rea-law divergence manifestly obeys SS, since area ( (C)) = area( ) + area( C) area( ) C lso obeys Linden-Winter constrained inequality, since if I( : C) = I( : ), I( : CD) = I( : D) 5
then C is separated from,, hence I(C : ) = 0, hence I(C : D) I(C : ) If, are separated then I( : ) is finite & universal. However, it is very difficult to compute in practice. E.g. for free compact scalar in 1 + 1, mutual information between two intervals is not known (Calabrese, Cardy, Tonni 09, 10) 3 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 ) in a state described in the bulk by a static, classical field configuration ( distinguished constant-time surfaces): S() = 1 ( ) min area(m) G N m where area(m) is computed w.r.t. spatial, Einstein-frame metric m means bulk region r s.t. r = m call minimizer m(), r() r() m() bulk Notes: 6 black hole m() = horizon
Is r() the holographic dual (in some sense) of ρ? Raamsdonk 1) (See also Czech, Karczmarek, Nogueira, Van Corrections believed to take general form (α = classical higher-derivative; G N = quantum = 1/N ): S() = 1 ( ) min area(m) + O(α ) + O(G 0 G N) N m See Hung, Myers, Smolkin 10, de oer, Kulaxizi, Parnachev 10 Ryu-Takayanagi formula obeys all the general properties listed above. Obviously S() 0. Triangle inequality is triangle inequality: m() m() m() Strong subadditivity (MH, Takayanagi 07): C S() + S(C) = m() m(c) = m(c) m() = S(C) + S() General, formal proof: Take r() = r() r(c), r(c) = r() r(c) Notes: 7
Proof is far simpler than proof of SS in quantum mechanics. General relativity knows some sophisticated quantum information theory! This proof (along with other tests of RT formula) automatically still holds in presence of higher-derivative (α ) corrections, but not quantum (G N ) corrections We will discuss Linden-Winter etc. below Monogamy We saw that mutual information is monotonic: (adjoining a system cannot decrease correlations). I( : C) I( : ) ut what do we expect for I( : C) vs. I( : ) + I( : C)? How are correlations correlated with each other? In principle three possibilities: 1. correlations are uncorrelated: I( : C) = I( : ) + I( : C). correlations are shared: I( : C) < I( : ) + I( : C) 3. correlations are distributed: I( : C) > I( : ) + I( : C) ll three behaviors are possible in general. Example of shared correlations: ρ C = 1 ( 000 000 + 111 111 ) I( : C) = I( : ) = I( : C) = ln Example of distributed correlations: ρ C = 1 ( 000 000 + 110 110 + 101 101 + 011 011 ) I( : C) = ln, I( : ) = I( : C) = 0. Define tripartite information : I 3 ( : : C) = I( : ) + I( : C) I( : C) = S() + S() + S(C) S() S(C) S(C) + S(C) 8
What happens in QFTs? rea-law divergence in QFT has I 3 = 0; reflects purely pairwise correlations across entangling surfaces C In gapped + 1 theories (for simply connected), S() = ɛ 1 area( ) γ In states with topological order, γ > 0, topological entanglement entropy (Kitaev & Preskill, Levin & Wen 05). Hence I 3 ( : : C) = γ < 0: correlations are distributed non-locally Free 1 + 1 examples (Casini, Huerta 08): Massive fermion or boson: I 3 can be positive or negative depending on,, C Fermion in massless limit: I 3 0 for all,, C oson in massless limit: I 3 + for all,, C; after tracing over complement of C, longwavelength modes lead to shared correlations In holographic theories, RT formula implies I 3 ( : : C) 0 for any,, C (Hayden, MH, Maloney 11). Proof is more complicated version of holographic SS proof Inequalities of the form f( : C) f( : ) + f( : C), where f is a measure of entanglement, are called monogamy relations, because they reflect the fact that entanglement is never shared. In general, mutual information does not obey this inequality because it measures classical correlations in addition to entanglement Holds in presence of higher-curvature corrections, not quantum corrections 9
Implies Linden-Winter and Cadney-Linden-Winter inequalities. RT formula obeys every known applicable property of entropy Interpretation is not clear, but it suggests that correlations are distribute non-locally, as in topological order. May also suggest that correlations are dominantly in the form of entanglement 5 SS of covariant holographic EE? RT formula applies only to static bulk spacetimes, and region lying in constant-time slice 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 Two key changes compared to static case: Surface m() is co-dimension two in bulk spacetime; region r() is co-dimension one m() is not minimum but only extremum of area For both reasons, holographic proof of SS does not go through anymore: C S() + S(C) = m() m(c) = m(c) m() = S(C) + S() Question: does covariant HEE formula obey SS? 10
Tests (Callan, He, MH 1; see also llais, Tonni 11, Caceres, Kundu, Pedraza, Tangarife 13) We consider planar ds 3, TZ, ds 3 -Vaidya (preserve spatial translation symmetry) Regions,, C are adjacent single intervals (so, C, C are also single intervals). Two possibilities: constant-time: C non-constant-time: Constant-time intervals: Translation invariance: S() = s(l ), S() = s(l + l ), etc S() + S(C) S() + S(C) s (l) 0 S() + S(C) S(C) + S() s (l) 0 ds 3 : ) s(l) = c 3 ln ( l ɛ TZ: s(l) = c ( ) 1 3 ln sinh(πt L) πt ɛ 1 3 5 11
Vaidya (intervals at different times after the injection of energy): TZ 5 horizon ds shell 3 1 6 8 10 x 1 UV thermalizes first, so short intervals show TZ behavior while long intervals show ds behavior Negative-energy Vaidya: 10 8 6 6 8 10 1 x Transition from TZ to ds behavior leads to loss of concavity. Violation of SS is correlated with violations of null energy condition and second law (area theorem) Non-constant-time intervals: ds, TZ: S() = 1 (s( x + t) + s( x t)) Since s(l) obeys SS, S obeys SS Vaidya: SS was tested for many configurations. Trapezoid with, C null provides most stringent test, since SS saturated in ds, TZ 1
SS violated precisely when NEC violated ppearance of NEC is not surprising; known to be required for area theorem ousso covariant entropy bound holographic c-theorems... Proof? local proof for -dimensional theories can be obtained using geodesic deviation equation. Two assumptions required: Null energy condition bsence of conjugate points along geodesic (plausible but not proven for minimal geodesic) Does not cover cases where minimal surface jumps (phase transitions) Wall 1: Global proof; however, does not apply to spacetimes containing horizons complete proof would Provide a crucial check on HRT formula e an important new theorem in GR, at the level of the area theorem for black holes, and deepen our still-sketchy understanding of the entropy-area connection in GR 13