BOUNDING NEGATIVE ENERGY WITH QUANTUM INFORMATION, CAUSALITY AND A LITTLE BIT OF CHAOS

Transcription

1 BOUNDING NEGATIVE ENERGY WITH QUANTUM INFORMATION, CAUSALITY AND A LITTLE BIT OF CHAOS based on arxiv: with: Srivatsan Balakrishnan, Zuhair Khandker, Huajia Wang Tom Faulkner University of Illinois at Urbana-Champaign

2 NEGATIVE ENERGY DENSITY Globally energy in relativistic QFT is positive (above vacuum), but local energy density can be negative due to quantum fluctuations How negative? Why is negative bad? Answer 1 (gravity) - you can use it to build time machines when coupling the QFT to gravity some local negative energy density Answer 2 (QFT) - violates micro-causality - violates quantum information bounds

3 THE QUANTUM NULL ENERGY CONDITION (QNEC) Classical field theory, Null Energy Condition (NEC): T uu (y) (@ u ) 2 0 u = light-like coordinate Quantum mechanics: violated by quantum fluctuations, but: ht uu (y)i ~ 2 a d 2 S EE (A u ) du 2 Bousso, Fisher, (Koeller), Leichenauer, Wall `15 u t Proof: free fields A u y

4 A GENERAL PROOF OF THE QNEC? Links many different areas of study: Quantum information Entanglement Hamiltonians chaos bound bulk reconstruction ANEC defect CFTs Entanglement entropy micro causality and the CFT bootstrap CHAOS emergence of gravity spacetime from QFT via AdS/CFT??

5 THINGS TO KEEP IN MIND The QNEC is a conjectured property of all QFTs (no gravity) We will work in flat space, although should extend to QFTs in curved space We will work with general QFTs with an interacting UV fixed point and d>2 We start our story with the ANEC

6 TWO PATHS TO THE ANEC

7 THE ANEC The averaged NEC: Z 1 h b A u (y) i 0 ba u du 0 T uu (u 0,v 0 =0,y) 1 Z Null momentum: P u = dya b u (y) y u v T uu Predicted from GR (e.g. wormholes not traversable) Non trivial in Minkowski space ~ Hofman-Maldacena bounds In d=3+1 CFTs: a c 1 3

8 METHOD I: ENTANGLEMENT HAMILTONIANS TF, Leigh, Parrikar, Wang, `16 H tot = H A H Ā A = e 2 H A A =Tr Ā ih A Full Entanglement Hamiltonian - better behaved in QFT: K A = H A 1 Ā 1 A H Ā Inclusion property: B B A A K A K B 0 Proof: relative entropy monotonicity (Also called modular Hamiltonian) D( A A) D( B B)

9 MODULAR HAMILTONIANS Relativistic QFT - inclusion at level of causal domains u D(B) D(A) x u (y) vacuum state: i! 0i A: half space (Rindler) cut KA 0 = Boost operator v B: small null deformation thereof Z KA 0 KB 0 = dy x u Au b (y) Uniform cuts = null momentum op. Non-uniform cuts = ANEC ba u (y) 0

10 METHOD II: CAUSALITY Hartman, Kundu, Tajdini `16 Space like separated operators: [O, O] =0 t u t Ex v True for any state: h [O, O] i =0 Operator quenches: i (t = i ) 0i 4 point function! (Bootstrap) f = Z 1 h0 OO 0i

11 COMPUTABLE IN LIGHTCONE LIMIT O O f /h0 OO 0i Operator Product Expansion: X = i O i Dominated by lowest dimension operator Light-cone Operator Product Expansion: u O v O = X i Z dug(u) u O i (u) v Dominated by lowest twist operator = dimension - spin O i = T uu

12 COMPUTABLE IN LIGHTCONE LIMIT f /h0 OO 0i Indeed the ANEC operator dominates in this limit: =2 Small correction in light-cone limit: apple / v d 2 v! 0 Same as chaos bound story for OTOC What is this time, s? f(s) =1 applee 2 s/ ha u i +... Hartman, Kundu, Tajdini `16 Maldacena, Shenker, Stanford (MSS)

13 SECRETLY AN OUT OF TIME ORDER CORRELATION f /h0 OO 0i u s A Vacuum entangled like a thermofield double state: s v Light cone limit achieved with boost after sending OO together Defines: But reduced density matrix for A looks thermal w.r.t. boost operator! 0i = X f(s) =h0 O(s)O(s) 0i e E /2 i A iā A = e 2 H A =2

14 SECRETLY AN OUT OF TIME ORDER CORRELATION f /h0 OO 0i u s v Reduce to thermal correlator (Schwinger-Keldysh contour) s A Boosts in imaginary time become rotations around `thermal circle s Can be used to extract Commutator squared: f(s) 3 Tr s! s + i /4 e MSS H A /2 [, O(s)] 2

15 BOUND ON CHAOS: Properties of chaos function: f(s) /2 Here: Im (s) Computable L apple 2 / f(s) =1 e Ls f(s) =1 applee 2 s/ ha u i +... Re (s) Ims = ± /4 f(s) apple 1 (Cauchy-Schwarz, exact bound here!) Hartman, Kundu, Tajdini `16 Saturates the Chaos bound, but ANEC: ha u i 0

16 BOUND ON CHAOS: Note that we did not need a large N limit - small parameter determined by kinematics of light-cone limit All (interacting) theories have same Lyapunov exponent in this limit - so this kind of Chaos not very discriminating Chaotic behavior: ( i ) 0i Disrupts local correlations/entanglement between A and Ā that exist in vacuum. Results in exponential decay of boosted correlator: h O A (s)o Ā (s) i

17 HOW ARE THESE RELATED?? Quick answer: I have no idea

18 COMBINING THEM D(B) D(A) Inclusion property: D(B) D(A) Causally disconnected: D(B), D( Ā) =0

19 COMBINING THEM Bob B = e 2 H B Alice Time evolution with entanglement Hamiltonian! [M B, M A ]=0 Now allow for non-local operators Interesting ops: M B (s) = is BO B is B M A (s) = is Ā O Ā is Ā Note B reduced density matrix for: ih

20 SETUP u, x v, x + D(B) Our new Chaos function is: f(s) /h M B (s)m A (s) i D(Ā) s>0 Using results from Algebraic QFT, one can show that f(s) has all the desired properties Challenge: computing it!! New operators are non-local and messy objects. light cone limit!

21 RESULT working in same ANEC light-cone limit for the operator insertions: f(s) =1 applee 2 s/ Q u +... apple 1 Q u du 0 T uu (u 0,v 0 =0,y)+ SEE ( A ) X u (y) Computed using a defect OPE in the replica trick S EE ( B ) X u (y) Evolving with entanglement Hamiltonian ~ to leading order is the vacuum boost plus computable corrections

22 PROPERTIES: Causality: entanglement time evolution ~ thermal time for non-equilibrium states ~ analog KMS condition ~ analyticity in the thermal strip: Borchers (+others) Im (s) =2 Ims = ± /4 Re (s) Cauchy-Schwarz bound also applies, so by same logic arrive at QNEC: Q u 0 Balakrishnan,TF, Khandker, Wang

23 SOME INTUITION - CAUSALITY u v Vacuum flow: h0 M B 0 (s)m A 0 (s) 0i x u x u (1 e s ) Composition of two null separated boosts results in a mild null translation

24 SOME INTUITION - CAUSALITY u v Vacuum flow: h0 M B 0 (s)m A 0 (s) 0i x u Q u > 0 Composition of two null separated boosts results in a mild null translation Q u < 0 In excited state boosts slightly misalign and error amplified bigly.

25 NO TIME FOR AdS/CFT interpretation (this was the main motivation for this calculation - and also gave us the idea to use entanglement time evolution) Relation to bulk reconstruction (see above) The actual calculation (we used several new techniques for dealing with entanglement in QFT) Higher spin version of the QNEC

26 QUESTIONS FOR AUDIENCE Are there useful bounds one can derive using similar methods in CM context? (i.e. no relativistic symmetry) Entanglement spectrum/hamiltonians studied in topological phases. What happens if you time evolve with such Hamiltonians? What are the implications for a bound on the acceleration of entanglement entropy? (i.e. the QNEC.)