JHEP02(2021)017 Springer October 4, 2020 February 2, 2021 : : December 21, 2020 : Received Published Accepted Published for SISSA by https://doi.org/10.1007/JHEP02(2021)017 [email protected] , c,d . 3 2009.13493 The Authors. c Random Systems, Black Holes

and Yuta Kikuchi In the Hayden-Preskill thought experiment, the Hawking radiation emitted , [email protected] a,b Department of Physics, Brookhaven NationalUpton, Laboratory, New York 11973-5000, U.S.A. E-mail: Upton, New York, 11973, U.S.A. Center for Theoretical Physics andBerkeley, Department U.S.A. of Physics, UniversityRIKEN of BNL California, Research center, BrookhavenUpton, National NY, Laboratory, 11973, U.S.A. Computational Science Initiative, Brookhaven National Laboratory, b c d a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: before a quantumcollected state later for is the thrown purpose ofrecoverability into decoding is the the affected quantum if state. black the A storedand/or hole natural early the question is radiation is decoding is how used protocol the damaged isthought or along experiment imperfectly subject in with performed. to the decoherence, the presence We of study radiation decoherence the or recoverability noise in in the the storage of early radiation. Abstract: Ning Bao Hayden-Preskill decoding from noisyradiation Hawking JHEP02(2021)017 19 10 ], some in the slightly modified 6 11 2 – ]. 9 13 5 , 12 7 15 2 14 ] asked the question of how much information 1 – 1 – ˜ Q 10 8 and 3 9 ]. In more recent works, explicit quantum circuit implementations Q 8 ]). This result was critical to the development of black hole quantum – 4 3 , 2 17 1 16 Of particular interest to those without a strong interest in quantum gravity, this pro- 3.3 Decoding fidelity 4.1 Decoherence in4.2 storage Imperfect noisy backward evolution 2.2 Decoding2.3 protocol Decoding fidelity and mutual information 3.1 Decoding3.2 protocol with erasure Mutual errors information 2.1 Hayden-Preskill thought experiment tocol does not require thatindeed, the all quantum state that even is be requiredbling thrown is quantum into specifically system. that a it Such black becomeschaos understandings, hole; associated in from with turn, the a have viewpoint boosted sufficiently rapidly the of scram- study the of quantum quantum information theory and quantum many body channel for the recoverythrown of in (see the also [ quantuminformation information as associated a with subfield, theinformation in paradox state particular [ that being was criticalof to the the thought formulation experiment ofcontext the of have traversable black been wormholes hole proposed in AdS/CFT by [ [ The Hayden-Preskill thought experimentcould [ be recovered regardingthe a assumption that quantum the state blackresult hole that is was that, was a given sufficient thrown fast time scrambling after into quantum the a message system. was black thrown The in, hole surprising there existed under a quantum 1 Introduction 5 Comparison: erasure vs6 decoherence Conclusion A Haar average B Relation between 4 Hayden-Preskill decoding with decoherence in storage 3 Hayden-Preskill decoding with erasure errors Contents 1 Introduction 2 Preliminary: ideal Hayden-Preskill decoding JHEP02(2021)017 ]. , we 14 , we , 5 3 4 ], and have 29 , to end up with 28 , in particular by U 5 , we will review the 2 , which she throws into a A , which can be noisy or imper- ]. The effect of decoherence on U 10 , we compare the recoverabilities with 5 ]. While out-of-time ordered correlation 27 ] could have potential applications for NISQ 1 ]. – 2 – 10 ] proposed a Hayden-Preskill decoding protocol as 10 when Alice throws her diary in, e.g. existing past its Page time [ 0 ] to experimentally assess the amount of chaos in strongly correlated B 10 ]. In this context, the work of [ . In particular, let the black hole have already emitted half of its qubits as 26 B – 14 The organization of this work will be as follows. In section In this work, we study the robustness of the Hayden-Preksill thought experiment to Recently, the effects of certain types of noise and decoherence on the recoverability have 2.1 Hayden-Preskill thoughtSuppose experiment Alice has ablack quantum hole diary represented byHawking a radiation state on The composite system will then experience some unitary evolution the erasures and the decoherenceFinally, based we on the will computations conclude donecontrast in with to the a the previous earlier discussion sections. work of performed these in effects [ in2 section Preliminary: ideal Hayden-Preskill decoding contrast to decoherence, which has a weaker effect. Hayden-Preskill thought experiment in thewill absence discuss of the decohering effect effects.will of discuss In erasure the section errors effects on of the decoherence. Hayden-Preskill protocol. In section In section in the storedrecoverability radiation to collected the earlier mutualin for information general between the setup, the purpose we input carryrecoverability, of out and state explicit the recovery. and computations effects collected of of Afterrandom decoding radiation errors relating fidelity, unitary. provided that the the is We time a find evolution proxy that is of the described by erasure a errors Haar severely affects the decoding fidelity in paradox. Here, however,the we robustness of will the focuscould Hayden-Preskill recovery on occur channel the because against purely erasure ofwith or the circuit information environment. decoherence, imperfections theoretic as In or this questionquality context, inadequate of of we Hayden-Preskill shielding assess decoding the against assuming severity interactions that of those the errors erasure in error terms or of decoherence the occurs fectly implemented. decohering effects different fromblack hole those quantum considered information has inbeen already been [ shown studied to in have for potentially example quite [ interesting effects in the context of the information quantum systems on a(OTOC) noisy functions quantum device are [ signals powerful diagnostic can tools bemental of hard setups. information to scrambling/chaos, extract Yoshidaan their due and alternative to tool Yao decoherence to [ the circumvent and the effect noise difficulty. of effects Accordingly, those their in errors primary on realistic interest lies the experi- in operations of unitaries quantum algorithms and devices,Studying perhaps this as a model subroutine inagainst or decoherence. this a context, kind however, of naturally quantum memory. begs thebeen question studied of in robustness [ physics [ JHEP02(2021)017 , (2.1) (2.2) on his own = i to distill the U 0 B i V . V i ⊗ | , then Bob is capable B i R | that forms a EPR state B is given by =1 d i X R ), Bob’s decoding strategy U B d 1 2.1 . A natural question to ask at ( √ C i . 0 := HP in the absence of error and noise. A 0 0 Ψ = . The dimension of the total Hilbert | 0 R V 0 i BB B ]. We assume that Bob has collected all i d 9 BB i = EPR . The dot in the graphical representation : | 0 EPR B D | d B d EPR C – 3 – , are defined by d ⊗ | 0 and and = after the time evolution R RA BB D 0 i i d i (e.g., that he is capable of implementing ], the answer is affirmative, with Bob needing only to B 1 d = HP U = , denoted by R Ψ A i EPR EPR d | | | d A RA ) i 0 i = and acting on it with some quantum recovery channel. If yes, B I and B D i ⊗ | d ⊗ EPR | R A i RA | d i and the remainder of the black hole AB A =1 U d = i . The state X D A ⊗ A d EPR | shares nearly maximal amount of information with d 1 R I √ E 0 with B := := ( 2 i d from the collected radiation RA 0 i HP R Ψ Prepare a copy of | Decoding Alice’s state is accomplished by applying some operation For the purpose of decoding, we attach a reference system and the late radiation 1. EPR 0 | of the early radiation anddriving the the black late hole’s radiation, dynamics andqubits for has the complete purpose knownledge of ofis the decoding). the unitary following: Given the state of successfully decoding Alice’s state by the2.2 quantum recovery channel Decoding protocol We review how toMuch of construct this a is review decoding of operation work first performed by [ If the state state with Hilbert space dimensions space is stands for the normalization factor of a EPR state. The EPR states with the state on this point would beB if Bob can reconstructa Alice’s further diary question by wouldthe collecting be reconstruction. the how much early According late radiation tocollect time [ a radiation small Bob amount needs of in the order late to time perform radiation to perform the recovery. Hawking radiation JHEP02(2021)017 1 . 0 (2.6) (2.7) (2.3) (2.4) (2.5) DD i .  2 , EPR 1 0 | d A 0  R i O + EPR 2 D ]. It is technically more d 1 2 A ⊗ | 31 , d 0 is a Haar random unitary − 30 BB , U i 2 D 1 14 d , 1 + EPR 2 A 1 ⊗ | d = RA i and it is given by, is ]. ]. In what follows we will use the Haar random ! D 9 i 0 1 assuming that d 32 in , B − 1 EPR 1 DD Ψ | d EPR 17 | i ) 2 A 2 C 0 2 A P EPR – 4 – d d d R P I = be the projection operator, we find the probability − EPR . In other words, Bob repeatedly performs projec- 0 | 0 ⊗ 0 ] = 2 C 0 in d A DD DD 0 DD ρ | 0 i ∗ Π + B until his state is successfully projected onto in U 0 2 B DD EPR ρ d 0 ⊗ EPR P EPR =

| DD ih | DD 1 to be given by AB in Π 0 U − 1 Ψ = Tr[Π . We call this state 0 EPR 2 = ih | ⊗ DD d A i in 0 R I out := EPR Ψ = B | ρ 0 P EPR = = ( on | to be a 2-design as a true Haar random unitary operator takes an exponential number DD EPR i 2 ∗ in Π in U ρ U Ψ UP | d Z For later convenience, we calculate operator: The state that is projected onto tive measurements on Letting of getting Project the state onto Apply This protocol is calledThis the assumption probabilistic decoding is in justified [ by the fast scrambling conjecture [ 3. 2. 1 2 correct to take of 2 qubit gatesentanglement to entropy prepare, this but assumption forapproximation, the is as sake sufficient it of [ is quantities appropriate that for depend the only situation on being the considered. second moment such as JHEP02(2021)017 is R

D

(2.8) (2.9) 0

A

. 0 N because . RB 5 0 2 A . 0 2 A d . 5 5 d . 0 1 . 7 10 EPR P 0 . = log , implying nearly = 10 2 A is the Haar random R , d B 5 ) . /d U d 7 over the Haar measure 1 D 3 A D 0 d d U in the ideal case. The total . The left panel shows that 0 . ≈

D C 5 D RB

N d ( N ) = log EPR is a pure state and 5 EPR . R P 2 EPR (2) F ( P S and = 0 (2) . CD 0 A 0 − S ). ) N 0 2 4 6 8 0 ...... 1 0 0 0 0 D RB 0 1 0

PEPR B ( (2) quantifies the quality of decoding (see sec- S 0 ) reduces to . We find ] because 2 (see figure RR – 5 – ρ 2.7 C i ) + . The base of logarithm is taken to be 2 through- 1 d

and decoding fidelity

A R

A (

≈ 0 . N . 5 ). For computational convenience we consider Rényi- 0 ). Here, we make a connection between the mutual . EPR (2) 0 | D 2 EPR . log Tr[ 5 S 5 . P N 0 2.8 2.1 EPR . 7 − ( ) = log F . 10 and D := ) := D 0 A 0 0 . is satisfied, ( N D out B 0 D 10 ] (2) ρ RB d D S ( 5 d B out . 7 d ρ leads to high fidelity. In particular, provided (2) R,B 0 A 3 ( 1  S 0 d .

D RR 5 (2) A , and the subsystem sizes are given by N EPR I d P 5 EPR . stands for the integration of the unitary operator 2 P = 10 ) for details of computation). U 0 = Tr[Π = d ). It is computed as, . N 0 is maximally entangled with R A.4 0 2 4 6 8 0 ...... 2.3 0 R 0 0 0 EPR 0 1 . The projection probability F FEPR Thus, small unitary and maximal decoding quality, tion Here, (see ( The fidelity between 4. with the Rényi-2 entropy the state out the article. Also, 2.3 Decoding fidelityThe and decoding mutual quality information is quantifiedand by the mutual collected information between radiation information the and reference decoding system fidelity ( 2 mutual information, the fidelity is close to unity for Figure 1 system size is JHEP02(2021)017 is D 0 (2.12) (2.13) (2.10) (2.11) B ρ . EPR ) P ] as originally D 0 B D 10 d d R,B ( = . The state ] for details). There- ) (2) I D 0 10 − B ( = 2 (2) , ) S ) − D D 0 0 B ( (2) ] = 2 R,B , S ( D ) 0 − ) D 2 B (2) 0 R ρ I ( A 2 A 1 R,B (2) d Tr[ d ( S − (2) ) I ] = ]. D 2 0 1 – 6 – CD = 2 B 0 directly measures the Rényi-2 mutual information. RB d ( 1 results in high decoding fidelity [ ), 2 A RB (2) d ) EPR ρ S EPR [ 2.5 D F ( P 0 = RC ] = 2 EPR D until he applies the decoding protocol upon gathering the R,B 0 . Let us compute ( P = Tr 0 2 B C ρ (2) B D we find 0 I B Tr[ is equal to the “averaged out-of-time-ordered correlator (OTOC)”, D ρ A C d d d C d EPR = 2 B P = d 1 is related to 2 A B ] d EPR d D 0 P A 2 B d ] = ρ D = 0 Tr[ d B 2 ρ Tr[ 3 Hayden-Preskill decoding with erasureIn errors real world physicalerrors. systems, however, the In decodingkeeps the protocol early present can radiation work, be subject on we to are various interested in storage errors that occur while Bob which shows thatargued large based on the decoupling principle [ namely, the projection probability We remark that and thus, gives another perspectivefore, of the information decoding scrambling (see fidelity [ is expressed in terms of Using Then, given by, maximally entangled with JHEP02(2021)017 - 2 B (3.2) (3.3) (3.1) N is traced out as 0 2 B , to fill in the erased 2 B after attaching the EPR ∗ /d 2 U B I , and the red wire represents the 0 . The state on 0 1 RR in order to separate them into B 0 B is graphically represented by, . The dimension of the lost Hilbert space in ρ 1) and ≤ B – 7 – p ≤ . The entire state before the application of recovery is the decoding operation that Bob performs on the p . The erasure is modeled by randomly choosing qubits ) (0 4 B V p 2 d 2 B 1 -surviving qubits on d 1 -qubit maximally mixed state, = ( 2 B = B to distill the EPR pair on B N in N pN D ρ 0 B = 2 from 2 0 2 B becomes inaccessible because they are lost or damaged. We will examine B N and the 0 0 B R 0 = 2 A 2 i B d is EPR | 0 2 Following the protocol in the previous section, Bob applies B qubits. Thus, the associated density operator We graphically doubled theerased wire qubits between on Bob is unable to access the lost qubits. state the effect of decoherence into section be discarded withon probability protocol takes the form, 3.1 Decoding protocol withWe erasure model errors the noisethe in storage storage on by erasure errors, i.e. the situation where some qubits in In this graphical representation, collected radiation on noisy storage. late radiation. JHEP02(2021)017 ] ), )+ out 3.2 ρ R (3.7) (3.6) (3.4) (3.5) 0 ( RR (2) S . ) 0 2 ) = EPR B ) ( P = Tr[Π D D 0 1 0 1 (2) on the state ( S . ) EPR RB ) ( )+ EPR D F R,B 0 1 D C F ( (2) ( ( S δ. RB (2) (2) ( − (2) 2 ) I S C B S (2) D − d d ) 0 1 S . Compared with the fidelity B − D D 0 ( 0 1 = ) + d B 2 C (2) ( ( 2 B S RR ] = 2 d (2) 1 2 1 (2) )+ S ) B R S ( )+ D d 0 1 R to obtain, 2 A (2) ( d S 0 D ) = RB 0 (2) 2 2 d ρ S 2 B DD 2 2 B ] = ( i d 1 in Tr[( 1 = (2) ρ B – 8 – ) S 2 B 0 d 2 EPR , d | 1 B A ) DD’ ( 2 A 0 1 d quantifies the effect of error in ) + we find the decoding fidelity d B 0 1 (2) ( 0 δ S B := (2) ( − ] = ) RR S ) onto i 2 ) = Tr[Π C (2) − ) ( ) D S 3.3 D 0 1 D (2) 0 1 0 1 S EPR EPR , δ B | ( R,B ) + RB )+ ( 2 A δ ρ R d (2) D ( (2) , we firstly compute 0 1 S I ) = 2 onto (2) Tr[( )+ RB D ,P ( ), we note that 0 S 1 D 0 ( (2) out EPR 2.8 (2) S ρ DD RB P S − ( Π 2 in (2) EPR EPR ρ = 2 = S 0 P F δ − ) DD D Π 0 1 B = ( (2) out ρ where we have used This equation is recast into the following form, 3.2 Mutual information In order to make a connectionS to the Rényi-2 mutual information Successfully projected state isin the the distilled ideal EPR case pair ( on Upon projecting to be Then, Bob projects the state ( JHEP02(2021)017 − (3.8) (3.9) (3.13) (3.10) (3.11) (3.12) = 1 . p ]. 2  − B 2 10 1 d d . )[ 2  d 1 c O , 3.7 ) ( δ 2 + b δ 0 1 b , using ! ( . 2 p ), the decoding fidelity p ∗ a 2 B 1 d U EPR 2 and the collected radiation 2.12 d P − . 2 . 1 ) c 1 R 0 ) 1 D B 0 2 ! d B

d b ( 0 1 2 D 1 b 2 D ( 1 d (2) = 2 d S a . − ) U )+ + . For small 1 1 D D p 0 d 1 0 1

2 B 2 1 c B ) d ) ( 2 A = 0 0 2 , R,B B d ( b ) (2) = d 1 p D b S (2) ( 0 1 # I 1 − 1 B ∗ ) 2 a ( – 9 – D U − ( (2) 1 = 2 B d S p (2) 2 C 1 2 B d − c 1 S d d ) 2 A (2 ln 2 log − 2 EPR d p b − 1 F b − ( ] = 2 ) we find the Haar random average of = 2 2 C 1 2 ] = 1 d a ) ) we find is given by a combination of Rényi-2 entropies instead of 2 ) U D A.5 ≈ + 0 1 D EPR 3.9 D ¯ 0 δ p B 1 2 P d 2 B ρ B EPR d 2 d ρ P 2 B " we get, d 1 ) provided that the time evolution is governed by a Haar random Tr[( 1 1 ) Tr[( B 2 ), and ( − d p 1 3.5 takes a different form from the ideal case ( ( ( 2 A 3.7 d d O δ EPR = = ), ( P ) + δ B 3.5 U δ d d Z , to which Bob has an access. The effect of erasure is translated to the reduction of Using ( We next compute := D ¯ δ 0 1 (2 ln 2 log It is reducedp to 1 without erasure errors, Executing the Haar integral ( 3.3 Decoding fidelity Let us compute unitary, Even though rightly reflects the mutual informationB between the state on the mutual information and vice versa. Rényi-2 mutual information, the latterThe of ratio which of encodes these the quantities recoverability of is Alice’s precisely state. captured by the error factor This can be written as With the erasure error, JHEP02(2021)017 = 2 (4.1) (4.2) B (3.14) (3.15) (3.16) N ) I | ⊗ . B 2 j . d ih 1  # c B ) 1 i 2 2 | 1 b d ]( . 0 1 − | . b ) 0  ( 0 p B 1  2 B j ∗ O a 2 2 C d BB ih 1 U | d d 0 2 A + 2 B d d  i | 2 2 D c ) d O − EPR 0 2 p Tr[ b 2 B 1 2 C + 0 1 ih I d b d ρ. i,j ( ) 1 2 B X 2 A | 2 ⊗ d 0 1 + d a Tr 2 B I − B p ) d EPR U j I d p p − | p − 1 ( ih p 2 B − d p + + . 0 0 2 d ) 2 B 0 0 2 D B c 0 1 + i d ) 2 A , the quantum channel may be understood 2(1 B d | 0 BB 0 2 ( d d ρ BB BB b BB | | + ) | 1 B " Q + b p + ( 1 2 D p 2 2 B d ∗ 2 D | ⊗ Q − a EPR EPR – 10 – EPR − ) = d 1 d B 1 0 U ih ih j ih 2 2 A 1 d ih d d = 1 BB B | c i ) EPR EPR | = = EPR ) = (1 | | 2 | ) ) b ( ρ p p 0 1 i,j ( EPR ¯ X b δ ¯ ( EPR P − − Q BB 1 EPR B 1 2 A a ih ¯ d P Q d U = (1 = (1 = U D = ) = d 3 d 0 EPR , | 0 2 ( Z 2 B 0 BB | EPR d BB B 1 | 1 ) results in, := F Q B d EPR A.6 2 A ih EPR EPR d ¯ P ih , and imperfect implementation of the backward unitary evolution combined 0 = EPR | B ( 0 EPR B is similarly computed: | EPR Q P . B EPR d P This identity is checked as follows: 3 log Although the decoherence affects theto state on act on with decoherence. 4.1 Decoherence in storage We model the decoherence by the depolarizing channel, 4 Hayden-Preskill decoding with decoherenceWe consider in the storage effects ofradiation two on types of errors: decoherence in storage of the collected early Therefore, the decoding fidelity is Its Haar average ( p The error factor is approximately proportional to the number of erased qubits JHEP02(2021)017 ∗ U (4.5) (4.6) (4.3) (4.4)
0 A 0 R | EPR , ih EPR leads to the decoding 0 ⊗ | 0 RR i BB | EPR | followed by an application of EPR 0 ih A 0 R EPR ⊗ | D d 1 and then on RA A | 0 d – 11 – DD := i EPR D ih , that are initially maximally entangled, resulting in d 1 2 A B EPR | d EPR , δ | = on 2 A 0 δ d in BB ρ ) = EPR 0 Q P A ) 0 0 T B A EPR 0 U ∗ P B U ⊗ EPR ⊗ † AB F and the black hole U is given by, AB ( 0 U B × EPR = = ( P in ρ Below, we will discuss itsexplicitly. relation to the mutual information, and calculate these quantity where Sequential projections of fidelity, To start decoding, Bob introducesto an obtain EPR the state, pair on radiation the state, The decoherence in storage is responsible for the decay of entanglement between the early JHEP02(2021)017 = B ) . 0 ) (4.9) (4.7) (4.8) B 0 (4.10) (4.11) ( is em- B ( (2) ˜ . Q . S (2) C S δ )+ ) d − EPR R ) D ( P 0 D = 0 (2) B D B S d d ( R,B ( (2) = (2) = 2 S ˜ I 2 d )+ D ( and (2) satisfying (see appendix S ) ˜ p 2 D 0 = ) RB D ( ) 0 C ) is converted to, ( (2) RB S ( (2) 4.3 S − p. (2) − ) S = ) , ) = − D D ) 0 0 D 2 0 D ˜ B p B 0 ( B ( ( 2 A B 1 – 12 – ( − (2) d (2) (2) S p (2) S S 2˜ = S )+ − 2 A 1 on the state ( R d )+ ( )+ ) R is to emphasize that the depolarizing channel ( R D (2) = 0 ( ] = 2 S (2) 2 2 ) (2) S (2) RB I ( 2 D S 0 (2) B = S ρ Let us relate the decoding fidelity to the Rényi-2 mutual infor- ) ) = − C ( D 0 Tr[( (2) ] = 2 2 A S 1 2 d ) )+ R,B 2 A ( 1 D D ) implies the following, 0 0 d ] = (2) . The tilde on 2 ˜ 4.7 ) I RB RB ) ( ρ D ] = D ( 2 0 (2) ) (2) S describes the depolarizing channel with probability D Tr[( RB S 0 − ρ ˜ B Q )+ ρ C = 2 ( We next compute Tr[( δ (2) Tr[( S We have used 2 ployed for the computation. The relation ( where for derivation), In the last equality we used, mation. Mutual information. JHEP02(2021)017 (4.14) (4.15) (4.16) (4.12) (4.13) dependence in and discussed : p 3 . . 2 D EPR  p  d 2 P 2 1 1 d d + ). Its   # # 1 O 4.5 O 1 − + + − ) channel is applied. 0 channel involves participation 2 A 2 C # ! B 2 A ˜ . 2 D d d Q ( ˜ 2 2 D d 2 D d 1 Q d d d (2) 1 − 1 c − S 2 B 1 − . 2 C d ) is shown in figure b 2 C ) )+ 2 d 1 ∗ d a D D D + " 0 0 d + U 3.16 + B 2 2 A 2 B 2 D ( p p 1 2 B d 2 A d d d 2 A R,B 2 d d ( c (2) d A " 2 S − d " − b (2) 1 2 ˜ − I p # 1 ) a 2 B 2 + 1 − D 2 D U d − – 13 – ( − p 1 d 2 = 1 d 1 2 d : 2 A (2) − 2 d c δ S d 1 2 = − b +

1 EPR − ∗ a p p F U 2 D = 2 1 1 EPR − − d d ¯ P 1 c channel which does not involve the ancillas at all. In this + U 1 EPR b = 1 = 1 d 1 2 A P 1 Q a d ). The intuition here is that the Z U " Uδ d D 4.9 := = Let us compute d ( Z 2 B p p d EPR := A ≤ ¯ as ancilla qubits, which can in general change the mutual information of d P ¯ δ ˜ p 3 D B ) + p d p . B with ) is a consistency condition that must be satisfied to ensure that the ancilla − d 5 − and ˜ 1 A Q 4.9 2 d ) and we find the error rate, B = (1 = Similarly, we carry out the Haar integral to find Haar average of A.7 δ The decoding fidelity iscomparison now with computed the by one pluggingin with these section the into erasure ( errors ( described by Haar random unitary.in Haar ( random average of the second term is computed In order to further proceed with the computation, we assume that the time evolution is when contrasted with the sense, ( qubits have no affect on the mutualDecoding information fidelity. when the Therefore, the larger thebecomes. Rényi-2 Note, mutual however, information thatchannel the is mutual the information better isfrom the computed by decoding using quality thestates quantum after application of a quantum channel even of systems non involving the ancilla which, in turn, implies This is equivalently written as JHEP02(2021)017 (4.19) (4.20) (4.21) (4.17) (4.18) by using † AB U U
0 , , A 0 2 D p R d | + η ) EPR p ih − because of either imperfect EPR U = (1 ⊗ | ρ. 0 2 D p d Tr BB I d | + p , 0 + † EPR DD ˜ i ih U D d ˜ B Uρ 1 ) d EPR EPR p | A – 14 – d − ⊗ | D d RA | B ∆ := 1 d ) = (1 2 A ρ ( d EPR Q , ih = is given by, 2 A ∆ projected onto d in EPR D EPR = | ρ EPR p d P 0 F B − B d 0 EPR 1 T A A P d Q AB EPR that is not precisely the complex conjugate of F U ∆ = ∗ = = ˜ U is computed as in ρ ∆ Then, with the probability of The decoding fidelity where the quantum channel used here is Suppose Bob is only capablea of matrix implementing noisy backward timeimplementation, evolution decoherence, of or both. We model the situation by the following initial state: 4.2 Imperfect noisy backward evolution JHEP02(2021)017 . .

2 δ EPR (4.22) P . Total 0 8 6 4 2 0 ...... 1 0 0 0 = 1 0 0 EPR

0 D δ F

2 N 4 6 . 4 . p 8 is fair. Nevertheless, and the decoherence , the error factor p p , in contrast to the p 2 p 10 . D of the radiation that N = 0 D 0 p N . The erasure error induces  D U 2 (b) † N ˜ . U 4 A  0 and

A N 6 proportional to

N A ). For general DD δ N 8 Π 3.13 ⊗ due to the erasure errors (red surface) and 10 δ C I d . The decoding fidelity declines rapidly as is small. This is understood from the fact  4 A † ) given sufficiently large ˜ N U , and becomes precisely their two-norm overlap

– 15 –

and U ˜ δ U quantifies the deviation from unity of is smaller ( 0 3.12 3 δ ( A DD 0 8 6 4 2 p 0 ...... N and 1 0 0 Π 0 0 0

0 D

U

⊗ 2 N I also displays a sharp difference between the erasure and the 4  6 8 EPR ), that induces the decay of ¯ 1 10 F . = Tr increases, which significantly limits the recoverability of Hayden- the Haar random average of unitary evolutions is taken. Both η p = 0 4.16 δ 0 p , and the subsystem sizes are given by than the decoherence. 2 (a) δ ]. , the effect of erasure error is proportional to the number of erased qubits . It is, however, not clear whether this comparison with the same = 10 p 4 p 10 [ N A 6 N = 1 . The Haar random average of error factor 8 , that becomes larger when C d B 10 The decoding fidelity Here, we made a comparison between the erasure errors with error probability pN 4 is well protected against theBob decoherence collects effect later as long is as large the relative size to that ofwith Alice’s probability state we believe our analysis is still useful to see how these effects depend on the parameter decays exponentially with respect to effect of decoherence ( decoherence effects, as shown inthe erasure figures probability Preskill thought experiment in the presence of erasure errors. In contrast, the recoverability For computations of plots clearly show thatparticularly the when erasure the error have sizethat, severe of for impacts Alice’s small relative state to∼ the decoherence, 5 Comparison: erasure vsAs decoherence we have discussed, theIndeed, error the factor error factor isthe identically erasure unity errors in and the the absence of decoherence errors. in terms Comparisons between of the error factor are shown in figure quantifies the difference between when the decoherence (blue surface) insystem the size storage. is Without errorsbigger the depletion error in factor satisfies where Figure 2 JHEP02(2021)017 0 . 10 1 = 10 2 4 . . = 0 = 0 = 0 N p p p 8 . 8 0 . A N 6 . 6 0 D p N 4 . 4 0 ) as a consequence of the (b) Decoherence (b) Decoherence 3.10 2 . = 2 = 4 = 6 2 is large relative to 0 D D D N N N D N 0 . 0 0 0 8 6 4 2 0 0 8 6 4 2 0 ......

1 0 0 0 0 0

1 0 0 0 0 0 EPR EPR F F – 16 – 0 . 10 1 2 4 . . = 2 = 4 = 6 = 0 = 0 = 0 D D D p p p N N N 8 . 8 0 6 . 6 0 D p N 4 . 4 0 (black vertical line), respectively. The erasure errors have bigger impacts on (a) Erasure (a) Erasure = 2 (the size of late radiation) dependence of the decoding fidelity in the presence of 2 A . 2 0 D . The erasure errors severely affect the decoding fidelity compared while the it is well N , respectively. The black solid horizontal line indicates the lower bound of decoding N 2 A . Error-probability dependence of the decoding fidelity in the presence of (a) erasure . /d = 2 and 1 0 . 0 A 0 2 0 8 6 4 0 0 0 8 6 4 2 ......

0 0 0 0 0 1

0 1 0 0 0 0 N EPR EPR F F = 10 quantum recovery channel fromrious the viewpoint breakdown of of its decodingwhile decoding fidelity the fidelity. occurs protocol We when found isbe the that relatively attributed erasure se- resilient to error against thedecreased exists the reduction entanglement in decoherence. due of the to the storage the This mutual lost may information qubits. equivalently ( 6 Conclusion Both erasure errors andthat he decoherence collected are at likelysequently, an affect to earlier the happen time quality of in in decoding the Bob’s algorithm. Hayden-Preskill storage thought We of experiment, have assessed radiation and their con- impact on the (a) erasure errors andN (b) decoherence. Thethe sizes decoding of total fidelity/recoverability compared systemsmall with and influence the Alice’s on decoherence. state the are Indeed, recoverability fixed the of to the decohrence be protocol. has a Figure 4 and fidelity, protected against the decoherence effect particularly when Figure 3 errors and (b) decoherence. The sizes of total system and Alice’s state are fixed to be JHEP02(2021)017 (A.1) (A.2) (A.3) , 4 j 2 3 j j δ 2 3 j j δ 1 4 . j j ) δ 1 3 j i δ 2 3 i VU i δ ( 2 4 i i f δ 1 d 1) 4 i i δ 1 i − Z 1 δ + 2 3 − d + j ( ) = 2 2 4 d j d j δ ]. In the latter context, projective 2 acts on. , the integral over the Haar measure 4 j j UV δ 40 V 1 ( U 3 j – j f δ 1 4 j d 33 , i δ 2 1 4 i Z j i δ 2 2 3 j i i δ δ 1 d 2 3 i ) = i i – 17 – δ 1 1 i i U ( δ δ − f d = = 2 4 j j Z 2 4 ∗ ∗ i i U U 1 3 j j 1 3 , ∗ i i and a unitary operator U f 2 = 1 j 2 UU i d U U d 1 Z j 1 Z i ]. They explored these effects on the unitary evolution, and proposed an UU 10 d Z is the dimension of the Hilbert space that d It is also interesting to see our recovery analysis in the presence of the erasure in In the present work, we primarily focused on the noise and decoherence occurring where Here are the Haar integral formulas useful for our computation: For an arbitrary function satisfies the following properties: ported by RIKEN-BNL Research Center. A Haar average We would like to thank Elizabethsions. Wildenhain and N.B. Aidan is Chatwin-Davies supported for13067-44-PHPXH, useful by by discus- the the Department National ofby Science Energy the Foundation under Computational under grant Science number grant Initiative DE-SC0019380, number at and 82248- Brookhaven National Laboratory. Y.K. is sup- that randomly chosen qubitsmay are be discarded used in to the retain erasure a while certain the amount measurements of results entanglement. Acknowledgments relation to the entanglementmeasurements and phase random transition unitaries [ ment are rates repeatedly the applied. latechanged time Depending if entanglement on the structure the rate of measure- the at a rate which quantum of the system random system circuit can entanglement is be generation. disentangled drastically A by crucial the difference measurements here, overcomes however, is during the storage phasehowever, a motivated chance by that theaddressed these setup in can of [ affect the otherexperimental parts thought implementation of experiment. to circuit, diagnose somethinga information There noisy which scrambling is, device. has in been quantum systems on JHEP02(2021)017 2 2 d d 2 2 2 2 d d d d δ δ 1 1 2 (A.5) (A.6) (A.7) (A.4) 1 1 d d d c c δ δ 2 2 2 2 2 2 d d c c c c 1 δ 1 δ δ 2 2 d 1 d 0 2 0 2 c c 2 1 c b b δ δ δ d d 0 2 0 2 2 0 2 0 2 2 1 b b c δ b b c δ δ d 0 2 0 2 1 δ 2 0 1 0 1 b b c δ 2 c b b δ δ 2 b 2 δ 1 1 c 0 1 0 1 c 1 b b 2 b b b 2 δ δ δ b 1 1 c 2 δ 2 2 b b 1 b δ 2 b a a δ δ 2 2 2 1 a 2 2 δ b b a a 2 a a 2 δ 2 δ δ 2 1 a a b 2 a a δ 2 1 1 δ a δ δ a d d 2 1 1 1 1 δ 2 2 a a d d d d d 1 δ 1 δ δ 2 1 1 a d 2 2 d d d 1 δ c c δ δ δ 1 d 1 1 1 1 2 d ), c c 2 1 c c c δ δ δ d d 1 1 1 1 2 2 c c 1 c δ c b b δ δ d 2 δ 1 2 2 2 0 2 2 D c δ 4.15 b b 2 c b b 1 b 1 d δ δ δ 2 0 2 c c 1 0 1 0 1 b b 2 C b 1 δ b b δ δ b c 1 1 1 δ d 0 1 0 1 1 b b b δ b b 2 D 1 b δ δ B 1 1 1 1 2 2 2 D 2 D a δ d 1 2 b b b b d d 1 d d d 1 a a δ δ δ 1 C 1 1 1) a a 1 1 1 2 C C b 2 A 1) c c 2 1) 1) δ d a a 1 a a ) ) d d δ a d δ δ 1 1 a 2 2 − 2 1 1 2 B − − B B a a − b b δ 1 1 1) + a a + + 2 2 2 δ δ d d d 0 1 0 1 + 1 2 δ b b d d 1 1 1 − − − d 2 A A ), A a − ( ( 1) ( ( + + + d d d 1 d ( ), 2 2 d d D 1 2 2 2 δ 2 + d d 2 2 2 d ( 2 2 ∗ ∗ 1) 1) 2 d d d d d a a − d d d d 1 d d d − + + d d δ δ 2 2 – 18 – + + 2 1 d ( − − δ 2 U U 2 2 2 C 2 2 d d c 3.12 D D 2 2 2 2 d 2 c c d d 2 2 d δ δ d 3.15 2 d D d d d d c 2 2 ( 1 d d d d δ 1 1 b 2 c c 2 δ ( ( c d c c 2 2 2 C 2 C 2 B 1 d 1 δ δ c d 1 2 2 2 c c d d ∗ c d d a d 2 C 2 2 c c b ) ) c δ δ ) ) 2 . b b δ δ 2 2 2 0 0 2 d 2 1 c p p U A 0 2 0 2 ∗ b b c 2 0 2 a b c . δ # b b 2 − − b b d 1 1 0 0 B 2 2 δ δ δ 2 d 0 2 2 1 b b U  b c 1 d b 0 1 0 1 b b ( ( 2 2 b 2(1 B 2(1 B δ δ 1 b b 2 c δ δ 2 2 − 2 A d 2 0 1 0 1 . 1 d d 1 b − 2 0 1 0 1 a a b b b 2 b a b b b d − δ A 2 A c # δ δ δ 2 0 1 0 1 2 2 U U D 1 d d 2 2 A 2 C . b 1 2 b b 1 2 2 A a a 1 1 a b − d a a δ δ δ d d a d  δ d d 2 2 2 C 2 2 1 2 U 2 − − 2 − p 2 C 2 2 a a 1 a a d a a D 1 2 B a a c c 2 δ δ δ d 2 2 − ) ) D D d 2 d d 2 A δ d U 2 D a a − 2 2 0 0 2 a d d 1 B 2 2 1 1 δ δ d d b b 2 C c 2 C p 2 d d δ d d d − 2 C 2 C d 1 1 2 C 1 2 B d 1 1 d 1 1 2 δ d 1 b b b d d d d d − d 1 1 d d 2 A c ( ( 2 C 1 d ), 2 2 B 1 δ δ 1 1 d B B d 2 2 1 c ∗ d δ d a 2 C + 1 1 d d b ∗ ∗ d 1 1 d d δ a a − 2 c c δ δ 1 c d d 1 U − c 1 1 2 A A A ∗ + 2.7 2 2 a U U + 2 B c δ δ 2 C c c 1 1 1 c c d d ) d 1 1 2 1 + 1 2 c δ δ d d d 1 1 U p c d d b c 1 1 0 2 0 2 c c 2 D 1 d δ − + + 1 " 2 A 1 1 + 1 b b 1 δ c − δ δ 1 + d 2 d c c b c − − 2 2 2 D 2 D 2 1 d 0 2 0 2 b ) ) 1 b b 2 D b 2 2 δ d b p b b δ d d C D 1 2 c " 2 2 δ δ 2 B 1 2 2 2(1 B 1 1 2 d d 2 b d b b 1 d d d C C 1 1 b b b b d a a δ d b 1 1 b b δ δ d d C D δ 1 1 b b 2 B B 2 1 1 1 −   ) ) 1 1 b 1 ( ( 1 b b a d d a a b b p p d d δ 1 1 a δ δ 2 δ 1 1 1 D D − − B 2 B 1 a a 1 − 2 1 b b A 2 A a d 1 1 UU d d a a a δ δ a d d 2 d d δ 1 1 1 d 2 1 − 2(1 B B 2(1 B B δ − − d 2 A A a a UU a a a d d d d 2 2 δ δ 1 1 d d δ d Finally, we carry out the following computation for ( We next carry out the integral for ( The second computation is used in ( We present the computations that are skipped in the main text. The first one is the d d = = = − UU UU A 2 A 2 A A Z a a d d δ d d δ d d − − = = = Z ======Z Z Haar integral for ( JHEP02(2021)017 (B.2) (B.1) 14 4 B 1 ρ, B (2009) 2537 I Phys. Rev. D Tr 2 . , ˜ p I d 2 .
465 ˜ 3 p p 3 B 2 − B d B 2 + ˜ (1993) 1291 − 1 p ]. B 2˜ ) B ρ p ]. i | ˜ p ) 71 ˜ 2 p + 4 B ˜ 2 B p − d 4 = 2˜ = d − B EPR EPR p
(1 | 1 SPIRE The mother of all protocols: 4 ˜ +
p ih B IN 2 | B 1 B ][ 1 B + | ) = (1 A decoupling approach to the quantum B EPR | ρ | EPR Proc. Roy. Soc. A ( , ˜ 2 ih Q EPR B quant-ph/0702005 1 EPR Phys. Rev. Lett. ih [ B , ih EPR ˜ Q | – 19 – ) are proved as follows: 3 ) ρ, 2 B EPR | 2 ˜ ) p EPR Tr | ˜ B p 4.11 4 | (2008) 7 I d − ˜ 2 arXiv:0708.4025 B Q 2 B 2 − B p 1 p B [ ) d d 1 15 B ˜ p are respectively defined by + (1 (2˜ B EPR ˜ Q ), which permits any use, distribution and reproduction in p h ˜ − ρ ˜ Q − Q ) ) and ( and B B 1 p Black holes as mirrors: Quantum information in random 1 d d ]. + ⊗ 4.7 Q − = = = = (1 and (2007) 120 ]. Q SPIRE ]. 09 IN CC-BY 4.0 ) = (1 ][ ρ Breakdown of Predictability in Gravitational Collapse ( SPIRE This article is distributed under the terms of the Creative Commons Q IN . JHEP 2 Average entropy of a subsystem [ ˜ , p Open Syst. Info. Dyn. − , p = 2˜ gr-qc/9305007 quant-ph/0606225 p capacity (1976) 2460 [ subsystems restructuring quantum information’s family tree [ The identities used in ( S.W. 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