2d chaotic channel (holographic channel) Late time: E ≡ E E − E E 3 ( A, B 1 , B 2 ) → − 2 × 3 4 S (1 / 2) = − 2 E A, ¯ A A Time B B 2 Output system 1 B is the whole of output system. t B 2 B 1 and are the halves of output system. B A is subsystem in input system. Input system 0 A 0 , ¯ 0 , ¯ A A
2d chaotic channel (holographic channel) Lower bound Late time: E ≡ E E − E E 3 ( A, B 1 , B 2 ) → − 2 × 3 4 S (1 / 2) = − 2 E A, ¯ A A Time B B 2 Output system 1 B is the whole of output system. t B 2 B 1 and are the halves of output system. B A is subsystem in input system. Input system 0 A
2d chaotic channel (holographic channel) Lower bound Late time: E ≡ E E − E E 3 ( A, B 1 , B 2 ) → − 2 × 3 4 S (1 / 2) = − 2 E A, ¯ A A Time B B 2 Output system 1 B is the whole of output system. t B 2 B 1 and are the halves of output system. B We expect QFT-channels with strong scrambling ability to satisfy A is subsystem in input system. this lower bound, eventually . Input system 0 A
2d chaotic channel (holographic channel) Late time: E ≡ E E − E E 3 ( A, B 1 , B 2 ) → − 2 × 3 4 S (1 / 2) = − 2 E A, ¯ A A Time B B 2 Output system 1 B is the whole of output system. t B 2 B 1 and are the halves of output system. B A is subsystem in input system. This shows all information is scrambled. Input system 0 A
How channels scrambles information Scrambling channel Free fermion channel Compact boson channel ( maximally scramble ) Unscrambling Scrambling
Bipartite operator mutual information in the replica trick What we compute is � � S ( n ) + S ( n ) − S ( n ) � � I ( A, B ) = S A + S B − S A ∪ B = lim − − A B A ∪ B ∪ n → 1 → 1 1 − n [log tr A ( ρ A ) n + log tr B ( ρ B ) n − log tr A ∪ B ( ρ A ∪ B ) n ] = lim n → 1 State: Setup: B A
Bipartite operator mutual information in the replica trick B A A B Setup: B A
B B A A
B A B A : Twist operator in A : Twist operator in B
B A Independent of channels B A : Twist operator in A : Twist operator in B
B A Depend on channels!! B A : Twist operator in A : Twist operator in B
Bipartite operator mutual information in the replica trick − → 1 1 − n [log tr A ( ρ A ) n + log tr B ( ρ B ) n − log tr A ∪ B ( ρ A ∪ B ) n ] I ( A, B ) = lim n → 1
Bipartite operator mutual information in the replica trick E � � n e � ρ T B � E A,B = lim n e → 1 log tr A ∪ B A ∪ B
Bipartite operator mutual information in the replica trick E � � n e � ρ T B � E A,B = lim n e → 1 log tr A ∪ B A ∪ B � T B � B σ n e A
Bipartite operator mutual information in the replica trick E � � n e � ρ T B � E A,B = lim n e → 1 log tr A ∪ B A ∪ B � T B � � B σ n e → σ A σ A σ B n e σ B � � ∼ log n e ¯ n e ¯ n e A
Free fermion channel We consider the following setups to extract properties of free fermion channel: 2. Partially overlapping case 1. Fully overlapping case 3. Disjoint case B B B A A A
1. 1. F Fully o overl rlapping c case Red cu curve : � � Purple cu curve : � Blue cu curve : I ( A, B ) B � � A ) t
� � 2. 2. P Part rtially o overl rlapping c case � I ( A, B ) Red cu curve : B Purple cu curve : Blue cu curve : � � A ) t
Red cu curve : � � Purple cu curve : 3. D 3. Disjoi oint c case � I ( A, B ) Blue cu curve : B � � A ) t
Red cu curve : � � Purple cu curve : 3. 3. D Disjoi oint c case � I ( A, B ) Blue cu curve : B Ti Time evolution of operator logarithmic is quite � � si similar to operator mutual information. A ) t
� � � � � � I ( A, B ) I ( A, B ) � � � � ) t ) t
� � � � Slopes and bumps shows properties of free fermion ch channel are interpreted in terms of the relativistic c propagation of quasi-particl cles . � � I ( A, B ) I ( A, B ) � � � � ) t ) t
Tripartite operator mutual information B B 1 B 2 A
Tripartite operator mutual information B B 1 B 2 A does do esn’ n’t depend o depend on t n the t he time and t e and the c he cho hoice f e for su subsys ystems. s.
Tripartite operator logarithmic negativity � � B ) = E 3 ( A, B 1 , B 2 ) � ≡ E ( A, B 1 ) + E ( A, B 2 ) B 1 B 2 ) − E ( A, B 1 ∪ B 2 ) A � Re Relativistic propagation t E 3 ( A, B 1 , B 2 ) = 0 of of quasi si-par particle. e. I ( A, B 1 , B 2 ) = 0
Toy model The time evolution of operator mutual information (logarithmic negativity) and tripartite operator mutual information (logarithmic negativity) for free fermion channel can be interpreted in terms of the relativistic propagation of local objects as follows: 1. Each point in the input subsystem A has two particles. - One of them propagates in the right direction ( ) at speed of light. - The other ( ). - particle size . A -# of particles in A is proportional to the input subsystem size .
Toy model 2. The particles in the output subsystem B contribute to . - # of particles in B . B
2. 2. P Part rtially o overl rlapping c case B A Red cu curve : Purple cu curve : B A B A
Red cu curve : Purple cu curve : 3. 3. D Disjoi oint c case Blue cu curve : B A
Red cu curve : Purple cu curve : 3. 3. D Disjoi oint c case Blue cu curve : B A
Red cu curve : Purple cu curve : 3. 3. D Disjoi oint c case Blue cu curve : B A B A
Red cu curve : Purple cu curve : 3. D 3. Disjoi oint c case Blue cu curve : B A B A B A
Red cu curve : Purple cu curve : 3. D 3. Disjoi oint c case Blue cu curve : B A B A B A B A
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. B B 2 1 B A
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. @time =0 B B 2 1 A B
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. @time =0 B B 2 1 A B
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. @time =0 B B 2 1 A B
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. @time =0 B B 2 1 A B
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. @time = t B B 2 1 B
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. @time = t B B 2 1 B
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic Op Oper erator mutual information for free ee propagation of local objects, too. fe fermion channel can be interpreted in @time = t te terms of the relativistic propagation of B B 2 1 B lo local o al obj bject s suc uch as h as quas quasi-part partic icle les. s.
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. Fo For free fermion channel, quantum correlation @time = t between input be n input and o and out utput put sub subsystems is s is B B 2 explained by lo ex local o al obj bject (quas (quasi-part 1 partic icle les)! s)!! B
Tri-partite information • Tri-partite information can be interpreted in terms of the relativistic propagation of local objects, too. Quantum information for free fermion Qu @time = t ch channel el is s carried ed by lo local o al obj bject (quas (quasi- B B 2 part partic icle les)! s)!! 1 B
Comparison We consider the following setups to extract properties of compact boson and holographic channels by comparing them to free fermion channel: 2. Partially overlapping case 1. Fully overlapping case 3. Disjoint case B B B A A A
Comparison We consider the following setups to extract properties of compact boson and holographic channels by comparing them to free fermion channel: 2. Partially overlapping case 1. Fully overlapping case 3. Disjoint case B B B A A A
B 2. 2. P Part rtially o overl rlapping c case A Holographic channel: Free fermion channel: Compact boson channel: :Purple :Blue Red Purple :Red :Black dash Black Blue
B 2. 2. P Part rtially o overl rlapping c case A Qu Quasi-par particle pi pictur ure w works w well. Holographic channel: Free fermion channel: Compact boson channel: :Purple :Blue Red Purple :Red :Black dash Black Blue
B 2. P 2. Part rtially o overl rlapping c case A Ho Hologra raphic channel el does esn’t show w a platea eau. Holographic channel: CB channel: FF channel: :Purple :Blue Red Purple :Red :Black dash Black Blue
B 2. P 2. Part rtially o overl rlapping c case A This pr Thi prope perty i is be beyond t nd the he par particle i interpr pretat ation. n. Holographic channel: CB channel: FF channel: :Purple :Blue Red Purple :Red :Black dash Black Blue
Ho Holo lographic aphic channel hannel B Red Purple A Black Blue
Ho Holo lographic aphic channel hannel B Red Purple A Black Blue Monotonically y decreasing Once On ce quantum information le leak aks from B, , it keeps to leak k be before al all i inf nformat ation l n leak aks .
Ho Holo lographic aphic channel hannel B Red Purple A Black Blue Monotonically y decreasing If quantum inform If rmation is carri rried ed by y local objects, this th is does esn’t t hap appen en!! !! On Once ce quantum information leak le aks from B, , it keeps to leak k be before al all i inf nformat ation l n leak aks .
Ho Holo lographic aphic channel hannel B Red Purple A Black Blue Th The slo lope chan anges at .
Heur Heuris istic tic explana planatio tion n B A
Heur Heuris istic tic explana planatio tion n B A Quantum information keeps s to go out from the left boundary y wi with . At At t=0, right-mo moving ng signal gnal appe appear ars at at the he righ ght bo boundar undary of A. . It Its speed eed is is the e lig light’s.
Heur Heuris istic tic explana planatio tion B A B A
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