Out of Time Ordered. Beni Yoshida (Perimeter) [1] Chaos in quantum channel, Hosur, Qi, Roberts, BY

Transcription

1 Out of Time Ordered Beni Yoshida (Perimeter) [1] Chaos in quantum channel, Hosur, Qi, Roberts, BY [2] Complexity by design, (in prep, with Daniel Roberts)

2 Scrambling

3 Out-of-time ordered correlation function Scrambling: delocalization of quantum information, hidden into non-local DOF. Fast scrambling is the defining feature of black holes. Out-of-time ordered (OTO) correlators can detect scrambling. Definition OTO = ha(0)b(t)c(0)d(t)i B C local operators B(t) =e iht B(0)e iht A D D(t) =e iht D(0)e iht system of qubits [Larkin-Ovchinnikov, Hayden-Preskill, Kitaev, Shenker-Stanford-Roberts-Susskind]

4 Scrambling implies decay of OTO Local perturbation to an initial state cannot be detected by any local measurement on an output state. Consider OTO = A(0)B(t)A (0)B (t) (group commutator) [A, B] = 0 then OTO = 1 {A, B} =0 then OTO= 1 Non-commutativity between A(0) and B(t) B(0) 0 i 1 i 0 (t)i B(t) 1 (t)i Expand B(t): B(t) =e iht Be iht = X j P j j scrambling high-weight Pauli operators OTO ' 0 scrambling/chaos (butterfly effect) [Roberts-Susskind-Stanford]

5 Key Questions How do we define scrambling? Quantum information theoretic meaning of OTO? Is the converse true? scrambling? OTO ' 0 Relation to entanglement entropy (and geometric quantities)?

6 State-Channel duality Quantum channel on n qubits can be viewed as a state on 2n qubits. unitary operator as a state (T=infty) cf) perfect tensor U = X i,j U i,j iihj Ui = X i,j U i,j ii ji in out in out in U out in U state out in out in unitary out out Thermofield double state (finite T) length=o(t) out in = e H Tr e H out out code out = e H Tr e H CFT 1 CFT 2 i = X j e E j /2 e ie jt j i j i ER bridge = quantum channel [Choi, Jamilkowski, Hayden-Preskill]

7 Average value of OTO Average of OTO over local operators A and D at T=infty A(0)D(t)A (0)D (t) average over A, D input output A C U If then, is large OTO ' 0 This implies the mutual information B X D S (2) BD For finite T, we consider TFD state. = 1 X A(0)D(t)A (0)D (t) 4 a+d A A,D =2 n a d S(2) BD I (2) BD = S(2) B + S(2) D S (2) BD Pauli operators Renyi-2 entropy D is small B and D are not correlated, so the system is scrambling. (unitary 1-design) [Hosur, Qi, Roberts, BY]

8 [Hosur, Qi, Roberts, BY]

9 Scrambling in AdS black hole RT surface TFD(0) A B 0 0 mutual information I(A, B) is large t TFD(T) A B 0 ~ T mutual information I(A, B) is (almost) zero

10 Toy model of the ER bridge Operators grow ballistically, leading to decay of OTO correlators. Left CFT Right CFT network of perfect (or random) tensors input op AdS ERB AdS [Hosur, Qi, Roberts, BY]

11 OTO average for Alice and Bob Alice and Bob are playing a catch ball. Alice may apply some perturbation X Bob asks Charlie to throw the same ball, and compare it with Alice s ball. Alice applies Ai with equal probability. Bob performs a joint measurement on two systems. The outcome : Alice U Alice Bob Ai X superoperator A i ( )A i i X B j B j U j SWAP operator X ha i (0)B j (t)a i (0)B j (t)i gravitational shockwave! i,j Bob measuring horizon shift! Bj U Ai EPR EPR U U Bj U T Charlie [Kitaev, Roberts-BY]

12 Complexity

13 Can we detect complexity growth? The complexity of the TFD state still keeps growing? Reference state Target state How many quantum gates do we need? X X t i X i i (0)i = X j e E j /2 j i j i (t)i = X j e E j /2 e ie jt j i j i X [Hayden-Harlow, Susskind, Brown et al]

14 Entanglement can detect complexity? Entropy grows as the complexity grows. entropy perturbation quench dynamics of local perturbation entanglement propagation After the scrambling time, entanglement entropies get saturated. complexity entanglement Entanglement is not enough! scrambling time time [Susskind]

15 Unitary k-design (complexity of randomness) Imagine an ensemble of Haar (uniformly distributed) random unitary (on n qubits) A typical operator in the Haar ensemble has exp(n) complexity. Consider k copies of the system (kn qubits in total) and consider k-fold twirl :! K( ) = Z Z du (U U) (U U ) k copies kn qubits n qubits We think of approximating the Haar random ensemble by some ensembles which are easier to generate. {p j,u j }! ( ) = X j p j (U j U j ) (U j U j ) If = K, then {p j,u j } is said to form a unitary k-design, i.e. it is as good as Haar up to k-th moment. [For an easy introduction, see a recent paper by Zak Webb]

16 Examples of k-design A group of Pauli operators is 1-design. I, X, Y, Z, ZZ, YYZY,. A group of Clifford operators is 2-design. Clifford operators can prepare arbitrary stabilizer states (eg perfect tensors). A toy model of the wormhole (Hosur, Qi, Roberts, BY) random unitary Left CFT Right CFT this object forms an approximate k-design. [Brandao-Harlow-Horodecki, Hosur-Qi-Roberts-BY, Roberts-BY]

17 Lower bound on complexity Imagine a system of n qubits. (d=2^n states in the Hilbert space). If an ensemble of unitary operators formed a k-design, the ensemble must contain at least 0 12 d + k 1 A. (due to the Schur-Weyl duality) k At each step, the number of implementable quantum gates is ' gn 2 (g: number of different 2-qubit gates) In T step, the number of implementable quantum gates is (gn 2 ) T A typical operator in k-design has a complexity of at least 2kn log(2) log(gn 2 ) roughly linear in k and n! [Roberts-BY]

18 How do we detect the design? Answer : Out-of-time ordered correlation functions 2k-point OTO correlators can detect k-design. h A 1 (0)B 1 (t)a 2 (0)B 2 (t) A k (0)B k (t) i time B 1 B 2 B 3 B k A 1 A 2 A 3 A k

19 OTO Xdetermines k-fold channel E { } Consider a k-fold twirl over an arbitrary ensemble E = {p j,u j } E( ) = X j p j (U j U j ) (U j U j ). (quantum channel) k copies The density matrix can be expanded X by Pauli operators, so we are interested in E(B 1 B k )= X Pauli op C 1,...,C k C 1,...,C k (C 1 C k ). knowing these numbers give complete characterization of the channel. Assume that we know averages of 2k-point OTO correlators for Pauli operators A1,...,A k = A 1 (0)B 1 (t) A k (0)B k (t) T =1 E B i (t) =UB j U. We know these numbers. Question Can we determine C 1,...,C n from A1,...,A k? Yes

20 X Theorem X : OTO and k-fold twirl Goal : A1,...,A k! C1,...,C k A1,...,A k = A 1 (0)B 1 (t) A k (0)B k (t) T =1 E E(B 1 B k )= X C 1,...,C k C 1,...,C k (C 1 C k ). Define : M C 1,...,C k A 1,...,A k =Tr[A 1 C 1 A k C k ]. M A 1,...,A k C 1,...,C k =Tr[C k A k C 1A 1], C 1,...,C k / M A 1,...,A k C 1,...,C k A1,...,A k

21 Effective design of an ensemble We need to know OTO values for Haar random (or k-design) in advance. This is possible by using some heavy math machineries. 4m-point OTO correlation functions that are related to shockwave geometries. 4m-point Haar 1 d 2m 4m-point Cli ord 1 d 2 4m-point 1 d D effective design of the ensemble 2-design shockwave geometries catch ball setup

22 Growth of design in an ER bridge? How do we define design in an ER bridge? Unitary t-design considers an ensemble of unitary operators. Time-evolution of an ER bridge is given by a single Hamiltonian H. Maybe, we can consider an ensemble of Hamiltonians? eg) Sachdev-Ye-Kitaev model H = 1 4! N J ijkl i j k l i,j,k,l=1 random variables we can compute disorder average analytically Or, we can imagine { very } high-energy = DOF, which can be integrated out. ( ) H1 H H2 H3 H4

23 Conclusion / Speculation toy model of AdS/CFT out-of-time ordered correlator very entangled tensor (eg random tensor) probe of space-time OTO = ha(0)b(t)c(0)d(t)i scrambling / chaos complexity / design