a ainstitutetext: Department of Physics, Kyoto University, Kyoto 606-8502,
Japan b binstitutetext: Kavli Institute for Theoretical Sciences, University
of Chinese Academy of Sciences, Beijing 100190, China c cinstitutetext:
Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

# The Petz (lite) recovery map for scrambling channel

Yasuaki Nakayama b , Akihiro Miyata c , Tomonori Ugajin
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study properties of the Petz recovery map in chaotic systems, such as the
Hayden-Preskill setup for evaporating black holes and the SYK model. Since
these systems exhibit the phenomenon called scrambling, we expect that the
expression of the recovery channel $\mathcal{R}$ gets simplified, given by
just the adjoint $\mathcal{N}^{\dagger}$ of the original channel $\mathcal{N}$
which defines the time evolution of the states in the code subspace embedded
into the physical Hilbert space. We check this phenomenon in two examples. The
first one is the Hayden-Preskill setup described by Haar random unitaries. We
compute the relative entropy
$S(\mathcal{R}\left[\mathcal{N}[\rho]\right]||\rho)$ and show that it vanishes
when the decoupling is archived. We further show that the simplified recovery
map is equivalent to the protocol proposed by Yoshida and Kitaev. The second
example is the SYK model where the two dimensional code subspace is defined by
an insertion of a fermionic operator, and the system is evolved by the SYK
Hamiltonian. We check the recovery phenomenon by relating some matrix elements
of an output density matrix
$\bra{T}\mathcal{R}[\mathcal{N}[\rho]]\ket{T^{\prime}}$ to Rényi-two modular
flowed correlators, and show that they coincide with the elements for the
input density matrix with small error after twice the scrambling time.

###### Keywords:

††preprint: KUNS-2985

##  1 Introduction

Advances in our understanding of the relationship between quantum information
theory and holographic principles have revealed the connection between the
structure of spacetime and quantum entanglement. In particular, the island
formula Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni ;
Penington:2019kki ; Almheiri:2019qdq  for the entropy of Hawking radiation
implies the island region in the interior of an old black hole is
reconstructed from the information of Hawking radiation.

However, it still remains to be understood the precise way to recover black
hole interior region from Hawking radiation. It has been realized that for
this purpose it is convenient to regard the black hole interior as a code
subspace embedded in the Hilbert space of Hawking radiation as a quantum error
correcting code Hayden:2007cs ; Verlinde:2012cy ; Papadodimas:2012aq . For
instance, the decoupling theorem by Hayden and Preskill Hayden:2007cs  implies
that the black hole interior region is protected against the erasure of black
hole degrees of freedom, which assures the recovery. Once we regard an
evaporating black hole as a quantum error correcting (QEC) code, then the
general argument of QEC Barnum2000ReversingQD  tells us that the recovery is
achieved by applying the Petz recovery map Petz:1986tvy ; ohya2004quantum .

In this paper, we study properties of the Petz recovery map in chaotic
systems, such as the Hayden-Preskill (HP) setup for evaporating black holes
and the SYK model. Since these systems exhibit the phenomenon called
scrambling, we expect that the recovery channel $\mathcal{R}$ gets simplified,
given by just the adjoint $\mathcal{N}^{\dagger}$ of the original channel
$\mathcal{N}$ which defines the embedding of the black hole interior into the
Hawking radiation. Therefore schematically we have

$\mathcal{R}\sim a\;\mathcal{N}^{\dagger},$ (1.1)

where $a$ is some numerical factor depending on the dimensions of the Hilbert
spaces of black hole and Hawking radiation.

We will see this phenomenon in two examples. The first one is the Hayden-
Preskill setup where the dynamics of an evaporating black hole and Hawking
radiation is described by Haar random unitaries. We do this by computing the
relative entropy $S(\mathcal{R}\left[\mathcal{N}[\rho]\right]||\rho)$ and show
that it is vanishing when the decoupling is archived. We further show that the
simplified recovery map is equivalent to the Yoshida-Kitaev protocol111This
equivalence has not been directly shown, but such a equivalence is suggested
by Yoshida in Yoshida:2021xyb ; Yoshida:2021haf .. The second example is one
of the SYK model versions of the Hayden-Preskill setup, discussed in
Chandrasekaran:2022qmq 222In Nakata:2023hwg , the authors discuss another
Hayden-Preskill setup in the SYK model, and the setup is different from our
setup.. In this setup, code information is expressed as excitations, and a
system is evolved by the SYK Hamiltonian. We check the recovery phenomenon by
relating some elements of an output density matrix
$\bra{T}\mathcal{R}[\mathcal{N}[\rho]]\ket{T^{\prime}}$ to Rényi-two modular
flowed correlators, and show that they gives an input density matrix
$\bra{T}\rho\ket{T^{\prime}}$ with small error after twice the scrambling
time. However, there are still remaining matrix element, which we need to
check, but it is difficult to evaluate them directly. In upcoming paper WIP ,
we will give their direct evaluations. In this paper, we do not evaluate them
directly, but indirectly guess their expectations for them based on our
obtained results.

Our paper is organized as follows. In section 2, we start with introducing a
quantum channel induced by the Hayden-Preskill setup, and explain how we write
down the simplified recovery map in the original Hayden-Preskill setup, which
is applicable to the SYK case. We also explain a convenient notation to treat
quantum channels induced by the Hayden-Preskill setup, and in the notation,
one can imagine gravitational interpretation simply. In section 3, by using
the convenient notation, we compute some relative entropies to check the
sufficiency that we can use the simplified recovery map as a recovery map.
Also, we show that the Yoshida-Kitaev protocol can be written as the recovery
map. In section 4, we explain one of the Hayden-Preskill setup using the SYK
model, and introduce a corresponding quantum channel. After that, we give the
simplified recovery map, and show that some matrix elements of output results
can be written as “Rényi-two modular flowed correlator”. By evaluating the
“Rényi-two modular flowed correlators” analytically, we show some matrix
elements of output results by the simplified recovery map give desired
results. In section 5, from the previous section result we have computed, we
estimate the remaining matrix elements of output results, which we are
evaluating. The detail of the remaining ones will be reported in upcoming
paper WIP . In section 6, we conclude this paper by the discussion of our
results and future directions. In appendix A, we give another derivation of
the simplified recovery map using a Kraus representation. In appendix B, we
show the relation that holds for an EPR state, which is used in section 3. In
appendix C, conventions used in section 4 are listed. In appendix D, we show
that, in the SYK version of the Hayden-Preskill setup, some recovery results
can be written as “Rényi-two modular flowed correlators” .

##  2 Recovery map for the Hayden-Preskill channel

The Hayden-Preskill setup is a tractable toy model for studying information
flow in evaporating black holes. The setup consists of a black hole $A$ that
has been emitting Hawking radiation $B$. We are particularly interested in the
system after the Page time where the black hole has emitted more than half of
its original entropy 333We follow the notation of Yoshida-Kitaev
Yoshida:2017non ., therefore approximately forming a maximally entangled state
$|\text{EPR}\rangle_{AB}$. Suppose Alice throws a quantum state $\rho_{T}$
(often called a diary) into this old black hole. Then as the black hole
further evaporates $A\rightarrow C+D$ by emitting late Hawking radiation $D$,
information thrown into the black hole will eventually appear in total Hawking
radiation $DB$. Here we denoted by $C$ the remaining black hole after emitting
the late radiation $D$, see the left panel of figure 1. The analysis of Hayden
and Preskill Hayden:2007cs  showed the diary appears in Hawking radiation
almost immediately, namely after the scrambling time.

To see this, it is useful to introduce an additional system called reference
$R$ and form a maximally entangled state $|\text{EPR}\rangle_{RT}$ with the
diary $T$. Then in this setup the initial condition of the process is
$|\text{EPR}\rangle_{RT}\otimes|\text{EPR}\rangle_{AB}$.

Owing to its chaotic dynamics, information of the diary thrown into the black
hole gets scrambled and spread over the entire degrees of freedom. The
resulting state is given by

$\ket{\Psi_{HP}}=(I_{R}\otimes U_{T,A\to C,D}\otimes
I_{B})\ket{\text{EPR}}_{R,T}\otimes\ket{\text{EPR}}_{A,B},$ (2.1)

where $U_{T,A\to C,D}$ is a random unitary matrix from $A,T$ to $C,D$, which
models the chaotic dynamics of the black hole. By finding the Hilbert space
with which $R$ is mostly entangled, one can find where is information of the
original diary in the final time slice. See again the left panel of figure 1.

$U$$R$$R$$T$$C$$A$$D$$B$$B$

$U$$V$$T$$\rho_{T}$$A$$D$$T$$\rho_{T}$$B$$E$

Figure 1:  Left: Hayden-Preskill setup, corresponding to state (2.1). Right:
Its decoder

The surprising results of HP is summarized into the following inequality.

$\overline{\norm{\rho_{RC}-\rho_{R}\otimes\rho_{C}}_{1}^{2}}\leq\left(\frac{d_{T}}{d_{D}}\right)^{2},$
(2.2)

where $\norm{A}_{1}=\tr\sqrt{A^{\dagger}A}$, $\rho_{RC},\rho_{R},\rho_{C}$ are
the reduced density matrices of (2.1) on the indicated subsystems, and in the
left hand side we take average over random unitaries. This inequality
(2.2)implies that if one collects a sufficient number of late Hawking quanta
so that $d_{D}\gg d_{T}$ the system of the remaining black hole and the
reference becomes no longer correlated $\rho_{RC}=\rho_{R}\otimes\rho_{C}$,
and there information of the diary has to be encoded in Hawking radiation
$DB$.

This result is also natural from the view point of the framework of quantum
error correction444 We note that the possible maximum number of late Hawking
radiation $d_{D}$ is given by the input for the Haar random unitary, implying
$d_{D}\leq d_{T}d_{B}$. Due to this bound, the combination $d_{T}/d_{D}$ can
not be $0$, but at most $1/d_{B}$. Thus, the exact equality does not hold
$\rho_{RC}=\rho_{R}\otimes\rho_{C}$ , as long as $d_{B}$ is finite. This means
that strictly speaking the recovery of the diary from Hawking radiation is at
best approximate. However, for a sufficiently large dimension of the early
radiation, $d_{B}\gg 1$, we can almost ignore the deviation from the exact
factorization of $\rho_{RC}$ for late times.. A quantum error correcting code
is a scheme to protect quantum states (logical states) in code subspace
$H_{\rm code}$ against various errors. Such an error is mathematically modeled
by a CPTP map called quantum channel $\mathcal{N}$. The basic idea of quantum
error correction is protecting these quantum states in code subspace $H_{\rm
code}$ by embedding it to the larger Hilbert space often called physical
Hilbert space $H_{phys}$. In the HP protocol, the Hilbert space of the diary
$H_{T}$ corresponds to $H_{\rm code}$ in QEC, and $H_{{\rm phys}}$ is
$H_{DB}$. The quantum channel $\mathcal{N}:T\rightarrow DB$, is obtained by
tracing out the remaining black hole and the reference system degrees of
freedom $C$ and $R$ from $\ket{\Psi_{HP}}$ in (2.1) with replacing the
reference state $\ket{\text{EPR}}_{R,T}$ by
$\sqrt{d_{T}\rho_{T}}\ket{\text{EPR}}_{R,T}$ ($\rho_{T}$ is an input state),

$\displaystyle\mathcal{N}_{T\to D,B}\left[\rho_{T}\right]$
$\displaystyle=\tr_{C}\left[(U_{T,A\to C,D}\otimes
I_{B})(\rho_{T}\otimes\ket{\text{EPR}}_{A,B}\\!\bra{\text{EPR}})(U_{T,A\to
C,D}^{\dagger}\otimes I_{B})\right]$ (2.3)
$\displaystyle=\frac{1}{d_{B}}\sum_{\tilde{D},\tilde{D}^{\prime}=1}^{d_{D}}\,\sum_{\tilde{B},\tilde{B}^{\prime}=1}^{d_{B}}\ket{\tilde{D}}_{D}\\!\bra{\tilde{D}^{\prime}}\otimes\ket{\tilde{B}}_{B}\\!\bra{\tilde{B}^{\prime}}\,\sum_{C=1}^{d_{C}}\,\sum_{\tilde{T},\tilde{T}^{\prime}=1}^{d_{T}}U_{C,\tilde{D};\tilde{T},\tilde{B}}\,(\rho_{T})_{\tilde{T}\tilde{T}^{\prime}}\,U_{C,\tilde{D}^{\prime};\tilde{T}^{\prime},\tilde{B}^{\prime}}^{\dagger}.$

We call this quantum channel the HP channel.

Then a general theorem of QEC555See e.g., Schumacher:1996dy ; Nielsen_2007
for the theorem. tells us that the decoupling condition is equivalent to the
existence of a recovery map $\mathcal{R}:DB\rightarrow T$ which satisfies

$\mathcal{R}\left[\mathcal{N}[\rho_{T}]\right]=\rho_{T}\quad\forall\rho_{T}\in
H_{T}.$ (2.4)

This again implies the information of the diary is recoverable from Hawking
radiation $DB$. See the right panel of figure 1. Moreover, the concrete
expression of the recovery map is known Barnum2000ReversingQD , and is called
the Petz recovery map

$\mathcal{R}_{\sigma,\mathcal{N}}^{\text{Petz}}[\tau]=\sigma^{\frac{1}{2}}\mathcal{N}^{\dagger}[(\mathcal{N}[\sigma])^{-\frac{1}{2}}\tau(\mathcal{N}[\sigma])^{-\frac{1}{2}}]\sigma^{\frac{1}{2}}.$
(2.5)

where $\sigma$ is full rank arbitrary density matrix on the code subspace
$H_{{\rm code}}$. $\mathcal{N}^{-1/2}$ factor of the Petz recovery map is
difficult to compute in general. One way for doing this is, as in
Penington:2019kki  first making the replacement
$\mathcal{N}^{-1/2}\rightarrow\mathcal{N}^{n}$, where $n$ is a positive
integer, computing it for all $n$, then taking analytic continuation
$n\rightarrow-\frac{1}{2}$. Also $\mathcal{N}^{-1/2}$ part is preventing us
from having an operational meaning of the map.

However, in systems exhibiting quantum chaos, we expect that the recovery map
gets simplified, because $\mathcal{N}[\sigma]$ has a flat spectrum, therefore
the approximation $\mathcal{R}\sim\mathcal{N}^{\dagger}$ appears to be
possible666In appendix A, we give another equivalent argument supporting our
expectation of this simplification in terms of the Kraus representation of the
HP channel.. If this is the case, since
$\rho\sim\mathcal{N}^{\dagger}\left[\mathcal{N}[\rho]\right]$ for arbitrary
density matrix $\rho$ in the code subspace, therefore the relative entropy
between them $S(\rho||\mathcal{N}^{\dagger}\left[\mathcal{N}[\rho]\right])$
vanishes.

For the HP channel, the adjoint HP channel $\mathcal{N}^{\dagger}$ is given by

$\displaystyle\mathcal{N}_{D,B\to T}^{\dagger}[\mathcal{O}_{DB}]$
$\displaystyle=\tr_{A,B}\left[\ket{\text{EPR}}_{A,B}\\!\bra{\text{EPR}}(U_{T,A\to
C,D}^{\dagger}\,\mathcal{O}_{DB}\,U_{T,A\to C,D})\right]$ (2.6)
$\displaystyle=_{A,B}\\!\bra{\text{TFD}}(U_{T,A\to C,D}^{\dagger}\otimes
I_{B})\,(\mathcal{O}_{DB}\otimes I_{C})\,(U_{T,A\to C,D}\otimes
I_{B})\ket{\text{TFD}}_{A,B}.$

Here, the adjoint channel is defined by the relation777More generally, for a
quantum channel $\mathcal{N}$, its adjoint channel is defined by the similar
relation
$\tr\left[\mathcal{N}\left[\rho\right]O\right]=\tr\left[\rho\,\mathcal{N}^{\dagger}\left[O\right]\right]$
(2.7)

$\tr_{D,B}\left[\mathcal{N}_{T\to
D,B}\left[\rho_{T}\right]\,O_{DB}\right]=\tr_{T}\left[\rho_{T}\,\mathcal{N}_{D,B\to
T}^{\dagger}[\mathcal{O}_{DB}]\right].$ (2.8)

For later convenience, we introduce a correctly normalized recovery map

$\mathcal{R}^{\text{Lite}}_{D,B\to
T}[\mathcal{O}_{DB}]\coloneqq\frac{1}{N}\cdot\frac{d_{B}d_{D}}{d_{T}}\mathcal{N}_{D,B\to
T}^{\dagger}\left[\mathcal{O}_{DB}\right],$ (2.9)

and define it as the Petz-lite888The terminology “Petz-lite” is introduced in
Penington:2019kki , and we also use this terminology in this paper.. Here $N$
is the normalization constant

$N=\left(\frac{d_{D}}{d_{T}}\right)^{2}+1,$ (2.10)

determined by the condition
$\overline{\tr_{T}\left[\mathcal{R}^{\text{Lite}}_{D,B\to
T}\left[\,\mathcal{N}_{T\to D,B}[\sigma_{T}]\,\right]\right]}=1$, where
$\sigma_{T}$ is some reference state in $T$. In the Haar random case, the
choice of the reference state $\sigma_{T}$ is not important as long as it is
normalized.

With this $N$, the Petz-lite can be expressed as

$\displaystyle\mathcal{R}^{\text{Lite}}_{D,B\to T}[\mathcal{O}_{DB}]$
$\displaystyle=\frac{1}{\left(\dfrac{d_{D}}{d_{T}}\right)^{2}+1}\cdot\frac{d_{B}d_{D}}{d_{T}}\mathcal{N}_{D,B\to
T}^{\dagger}\left[\mathcal{O}_{DB}\right]$ (2.11)
$\displaystyle=\frac{1}{1+\left(\dfrac{d_{T}}{d_{D}}\right)^{2}}\cdot
d_{C}\,\mathcal{N}_{D,B\to T}^{\dagger}\left[\mathcal{O}_{DB}\right]$

where in the second line, we used the relation $d_{B}d_{T}=d_{C}d_{D}$ due to
the unitarity of the Haar random unitary. For the parameter region
$d_{T}/d_{D}\ll 1$, the normalization is just given by $d_{C}$, which
coincides with an expression obtained from another discussion. In appendix A,
we give the discussion.

###  2.1 West-coast notation and replica-wormhole-like objects

In the following, we are interested in the typical properties of the recovery
map $\mathcal{R}$ for the HP channel $\mathcal{N}$. To investigate these
properties, we will consider replicated quantities, such as ${\rm
tr}(\mathcal{N}[\rho_{T}])^{n}$ involving a product of Haar random unitaries
and its average. Since such averaging involves Wick type contractions between
various pairs of Haar random unitaries in the product, it is convenient to
introduce a graphical notation that manifests which pair of unitaries are
contracted. Therefore, here we introduce a notation similar to the one
employed in Penington:2019kki  for modeling the black hole microstates and
their statistical properties, and call this West-coast notation.

To begin with, let us define the following black hole microstate on $C$,
involving a Haar random unitary

$\ket{\psi_{i}^{T}}_{C}\coloneqq\sqrt{d_{C}\,d_{D}}\sum_{C=1}^{d_{C}}\ket{C}U_{C,T;i}\quad.$
(2.12)

Here $\\{|C\rangle\\}$ is the set of basis states on the Hilbert space $H_{C}$
and the index $i$ collectively denote the indices for both late radiation $D$
and early radiation $B$, $i:(D,B)$ or more concretely
$\ket{i}=\ket{D}\otimes\ket{B}$, so the label $i$ rums from $1$ to
$d_{D}d_{B}\equiv k$.

In the following, we use this type of states $\ket{\psi_{i}^{T}}_{C}$ to write
quantities of our interest, instead of random unitary matrices $U_{C,D;T,B}$.
Under this notation, we can write

$\innerproduct{\psi_{i}^{T}}{\psi_{j}^{T^{\prime}}}=d_{C}\,d_{D}\sum_{C=1}^{d_{C}}U^{\dagger}_{i;C,T}\,U_{C,T^{\prime};j}$
(2.13)

and therefore the HP channel (2.3) is given by

$\displaystyle\mathcal{N}_{T\to D,B}[\rho_{T}]$
$\displaystyle=\frac{1}{kd_{C}}\sum_{i,j=1}^{k}\ket{i}\\!\bra{j}\,\cdot\sum_{\tilde{T},\tilde{T}^{\prime}=1}^{d_{T}}\innerproduct{\psi_{j}^{\tilde{T}^{\prime}}}{\psi_{i}^{\tilde{T}}}\,(\rho_{T})_{\tilde{T}\tilde{T}^{\prime}}\quad.$
(2.14)

In this notation, we call the subscript index $i$ Hawking radiation index, and
the superscript $T$ code index.

West-coast model treats each of these microstate $|\psi_{i}\rangle$ by a
single-sided AdS black hole with insertion of “end of the world brane” (or EoW
brane in short) labeled by the index $i$ behind the horizon. This state has a
Hartle Hawking type preparation, in terms of an Euclidean path integral with
the EoW brane which starts from the Euclidean conformal boundary. In this
model the overlap between two such states $\langle\psi_{i}|\psi_{j}\rangle$ is
computed by an Euclidean gravitational path integral on a region of Euclidean
disc enclosed by the part of the asymptotic boundary (an interval) and the
brane in the bulk.

With this gravitational path integral picture in mind, here we explain the
fact that there is a simple diagrammatic prescription to compute a product of
such overlaps
$\overline{\prod_{m=1}^{n}\langle\psi_{i_{m}}^{a_{m}}|\psi_{j_{m}}^{b_{m}}\rangle}$.
999 In the west-coast paper, this quantity is just called the product of
overlap and denoted without the bar, i.e.,
$\overline{\prod_{m=1}^{n}\langle\psi_{i_{m}}^{a_{m}}|\psi_{j_{m}}^{b_{m}}\rangle}\big{|}_{ours}=\prod_{m=1}^{n}\langle\psi_{i_{m}}^{a_{m}}|\psi_{j_{m}}^{b_{m}}\rangle_{WC}$.
We will use the convention with the bar to keep in mind we do average over
random unitaries in the computation.  without directly applying the formulae
for the Haar random averages, which becomes quite involved when the number of
unitary matrices appearing increases.

Then the prescription is the following:

  1. 1.

For each overlap in the product
$\langle\psi_{i_{m}}^{a_{m}}|\psi_{j_{m}}^{b_{m}}\rangle$ draw an interval
with two end points, and associate the labels $(i_{m},a_{m})$ to one end and
$(j_{m},b_{m})$ to the other. (In the west-coast model this interval with
indices at the end points provides the boundary condition to the gravitational
path integral for the product of the overlaps.)

  2. 2.

The $n$ intervals prepared in this way have 2$n$ endpoints in total. We pick
up two of these end points and connect them by a line, which we call the end
of the world brane. We repeat this until all the endpoints are connected to
the other by a brane. There are many different ways to do this. One
possibility is that the end point of $m$-th interval is always connected to
the other end point of the same interval. Or the other possibility is that the
end point of $m$-th interval is always connected to the point on the next
$(m+1)$-th interval.

  3. 3.

Each diagram $D$ constructed in this way contains $n$ end of the world branes.
We then associate each brane in the diagram with a Kronecker delta factor. If
the brane is connecting two endpoints with the labels$(i_{l},a_{l})$ and
$(j_{m},b_{m})$, then this factor is given by
$\delta_{i_{l}j_{m}}\delta_{a_{l}b_{m}}$. We compute this for all branes in
the diagram and then multiply these factors. Let us denote this factor for the
diagram by $I_{D}$.

  4. 4.

Since each diagram can be regarded as (disjoint union of ) two dimensional
surfaces, we can associate a Euler number $\chi_{D}$ to the diagram. We then
pick up the factor $(d_{C})^{\chi_{D}}$ which corresponds to the gravitational
path integral part in the west-coast model. We then sum the total factor
$I_{D}(d_{C})^{\chi_{D}}$ for all possible diagram $D$.

  5. 5.

The average of the overlaps is equal to the sum of these factors over all
possible diagrams;

$\overline{\prod_{m=1}^{n}\langle\psi_{i_{m}}^{a_{m}}|\psi_{j_{m}}^{b_{m}}\rangle}=\sum_{D\in{\rm
All\;diagrams}}I_{D}\;(d_{C})^{\chi_{D}}.$ (2.15)

Let us provide a few examples. First, for the single overlap
$\overline{\innerproduct{\psi_{i}^{T}}{\psi_{j}^{T^{\prime}}}}$. We can easily
evaluate it

$\displaystyle\overline{\innerproduct{\psi_{i}^{T}}{\psi_{j}^{T^{\prime}}}}$
$\displaystyle=d_{C}\,d_{D}\sum_{C=1}^{d_{C}}\overline{U^{\dagger}_{i;C,T}\,U_{C,T^{\prime};j}}$
(2.16)
$\displaystyle=d_{C}\,\underbrace{\delta_{D_{i}D_{j}}\,\delta_{B_{i}B_{j}}}_{{\delta_{ij}}}\,\delta_{TT^{\prime}}$
$\displaystyle=d_{C}\,\delta_{ij}\,\delta_{TT^{\prime}},$

where in the second line we used the general result for two Haar random
unitaries

$\overline{U_{a,b}U_{c,d}^{\dagger}}=\frac{1}{d}\,\delta_{ad}\delta_{bc}\qquad(a,b,c,d=1,\cdots,d).$
(2.17)

This result is easily reproduced from the west-coast prescription.

$\tilde{T}_{1}$$j_{1}$$T_{1}$$i_{1}$$\tilde{T}_{2}$$j_{2}$$T_{2}$$i_{2}$

$\tilde{T}_{1}$$j_{1}$$T_{1}$$i_{1}$$\tilde{T}_{2}$$j_{2}$$T_{2}$$i_{2}$

Figure 2:  Diagrams for computing the average of overlaps (2.19). A black line
connects two points that appear in the same overlap, and the blue lines
correspond to the end of the world branes in Haar random averaging. Left: The
disconnected diagram Right: The connected diagram

Next, let us evaluate the Haar average of the combination of the overlaps for
later convenience,

$\innerproduct{\psi_{i}^{T_{1}}}{\psi_{j}^{T^{\prime}_{1}}}\cdot\innerproduct{\psi_{j}^{T^{\prime}_{2}}}{\psi_{i}^{T_{2}}}.$
(2.18)

Clearly, by setting $T_{1}=T_{2}=T$ and
$T_{1}^{\prime}=T_{2}^{\prime}=T^{\prime}$, the above combination reduces to
the variance of the overlap
$\overline{\left|\innerproduct{\psi_{i}^{T}}{\psi_{j}^{T^{\prime}}}\right|^{2}}$.
We can evaluate the above quantity by the diagrammatic prescription mentioned
above (See figure 2),

$\displaystyle\overline{\innerproduct{\psi_{i}^{T_{1}}}{\psi_{j}^{T^{\prime}_{1}}}\cdot\innerproduct{\psi_{j}^{T^{\prime}_{2}}}{\psi_{i}^{T_{2}}}}\approx\left(d_{C}\right)^{2}\delta_{ij}\delta_{T_{1}T_{1}^{\prime}}\cdot\delta_{ji}\delta_{T_{2}^{\prime}T_{2}}+d_{C}\,\delta_{ii}\delta_{T_{1}T_{2}}\cdot\delta_{jj}\delta_{T_{2}^{\prime}T_{1}^{\prime}}.$
(2.19)

This coincides with the result obtained by using the Weingarten formula,

$\displaystyle\overline{U_{a_{1},b_{1}}U_{c_{1},d_{1}}^{\dagger}\cdot
U_{a_{2},b_{2}}U_{c_{2},d_{2}}^{\dagger}}$
$\displaystyle=\frac{1}{d^{2}-1}\left(\delta_{a_{1}d_{1}}\delta_{b_{1}c_{1}}\cdot\delta_{a_{2}d_{2}}\delta_{b_{2}c_{2}}+\delta_{a_{1}d_{2}}\delta_{b_{1}c_{2}}\cdot\delta_{a_{2}d_{1}}\delta_{b_{2}c_{1}}\right)$
(2.20)
$\displaystyle\qquad+\frac{1}{d\left(d^{2}-1\right)}\left(\delta_{a_{1}d_{1}}\delta_{a_{2}d_{2}}\delta_{b_{1}c_{2}}\delta_{b_{2}c_{1}}+\delta_{a_{1}d_{2}}\delta_{a_{2}d_{1}}\delta_{b_{1}c_{1}}\delta_{b_{2}c_{2}}\right)$
$\displaystyle\hskip 227.62204pt(a,b,c,d=1,\cdots,d).$

In general, the prescription introduced here correctly computing the average
over Haar random unitaries in the product of overlaps, as long as the rank of
the random unitaries $d=d_{C}\;d_{D}=d_{T}\;d_{A}$ is large.

Furthermore the adjoint channel (2.6), in terms of the west-coast notation is
given by

$\displaystyle\mathcal{N}_{D,B\to T}^{\dagger}[\mathcal{O}_{DB}]$
$\displaystyle=\frac{1}{kd_{C}}\sum_{T,T^{\prime}=1}^{d_{T}}\ket{T^{\prime}}\\!\bra{T}\,\cdot\sum_{i,j=1}^{k}\innerproduct{\psi_{j}^{T^{\prime}}}{\psi_{i}^{T}}\,\bra{j}\mathcal{O}_{DB}\ket{i}\quad.$
(2.21)

Below, using this graphical expression, we evaluate several relative entropies
to check the validity of the approximation
$\mathcal{R}\sim\mathcal{N}^{\dagger}$.

##  3 Relative entropy: Sufficiency

As we have mentioned, the decoupling condition (2.2) implies there is a
recovery map for the Hayden-Preskill channel (2.3). Another characterization
of the existence of the recovery map $\mathcal{R}$ for given $\mathcal{N}$ is
the notion of sufficiency Petz:1986tvy ; Petz:2002eql ; ohya2004quantum . To
state this, let us first recall the fact that relative entropy satisfies the
monotonicity property

$S(\rho||\sigma)\geq S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])$ (3.1)

for any CTPT map $\mathcal{N}$. By repeating this we have

$S(\rho||\sigma)\geq S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])\geq
S(\mathcal{R}\left[\mathcal{N}[\rho]\right]||\mathcal{R}\left[\mathcal{N}[\sigma]\right]),$
(3.2)

therefore if the recovery map exists $\mathcal{R}\circ\mathcal{N}=1_{{\rm
code}}$, then $S(\rho||\sigma)=S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])$, for
any density matrices on the code subspace. This condition is known as
sufficiency, and it was shown that if $\mathcal{N}$ satisfies this condition,
the recovery map is given by (2.5). Here we would like to check the HP channel
(2.3) does satisfy sufficiency, by directory computing the relative entropy
$S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])$ in the presence of the quantum
channel $\mathcal{N}$ 101010See Vardhan:2021mdy ; Kudler-Flam:2022jwd  for
related discussions on original Petz map cases..

Since our interest is a typical result under the Haar random average, we
consider the Haar averaged relative entropy,
$\overline{S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])}$. To evaluate the
relative entropy, we use the replica trick

$\displaystyle\overline{S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])}=\lim_{n\to
1}\frac{1}{n-1}\left(\overline{\log\tr\left[\mathcal{N}[\rho]^{n}\right]}-\overline{\log\tr\left[\mathcal{N}[\rho]\mathcal{N}[\sigma]^{n-1}\right]}\right).$
(3.3)

Generally, since it is difficult to evaluate the Haar average of logarithmic
functional, instead of the expression, we consider

$\overline{S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])}\approx\lim_{n\to
1}\frac{1}{n-1}\left(\log\overline{\tr\left[\mathcal{N}[\rho]^{n}\right]}-\log\overline{\tr\left[\mathcal{N}[\rho]\mathcal{N}[\sigma]^{n-1}\right]}\right).$
(3.4)

It is known that in the large Hilbert dimension limit, this quantity is almost
equal to the original one. For a moment let us focus on the first term of
(3.4). Using the west-coast notation (2.14), the trace
$\tr\left[\mathcal{N}[\rho]^{n}\right]$ can be written in terms of overlaps,

$\displaystyle\tr\left[\mathcal{N}[\rho]^{n}\right]$
$\displaystyle=\frac{1}{\left(k\,d_{C}\right)^{n}}\sum_{\bm{i}=1}^{k}\sum_{\bm{T},\bm{\tilde{T}}=1}^{d_{T}}\,\prod_{m=0}^{n-1}\left(\langle\psi_{i_{m}}^{\tilde{T}_{m}}|\psi_{i_{m+1}}^{T_{m}}\rangle\;\rho_{T_{m}\,\tilde{T}_{m}}\right)$
(3.5)

where the bold fonts $\bm{i},\bm{T}$ in the summation symbol means the sum
with respect to the set of indices;
$\sum_{\bm{i}=1}^{k}=\sum_{i_{1}=1}^{k}\cdots\sum_{i_{n}=1}^{k}$.

$\tilde{T}_{m}$$i_{m+1}$$T_{m}$$i_{m}$$\tilde{T}_{m-1}$$i_{m}$$T_{m-1}$$i_{m-1}$

$\tilde{T}_{m}$$i_{m+1}$$T_{m}$$i_{m}$$\tilde{T}_{m-1}$$i_{m}$$T_{m-1}$$i_{m-1}$

Figure 3:  Left: The dominant diagram for (3.5) when $d_{D}\ll d_{T}$
(disconnected diagram) Right: The connected diagram dominating the sum at
$d_{T}\ll d_{D}$.

In computing the Rényi entropy (3.5) we need to evaluate the product of
overlaps
$\prod_{m=0}^{n-1}\langle\psi_{i_{m}}^{\tilde{T}_{m}}|\psi_{i_{m+1}}^{T_{m}}\rangle$
with $|\psi_{i_{n}}^{T}\rangle\equiv|\psi_{i_{0}}^{T}\rangle$ and its Haar
random average. We do this using the diagrammatic technique introduced in the
previous section.

Among all possible diagrams, we are particularly interested in the ones
dominating the sum, both in for early time regime ($d_{D}\ll d_{T}$) and late
times ($d_{D}\gg d_{T}$). We now argue that fully-disconnected diagram (the
left panel of Figure 3) where for all EoW branes starting point and end point
are on the same interval dominates in the early time regime, and the fully
connected diagram (the right panel of Figure 3)where the indices form a single
loop dominates the late time regime by explicit calculations. The calculation
here is very similar to the ones in Penington:2019kki ; Kudler-Flam:2021alo .

First, let us evaluate the contribution of the fully disconnected diagram.
Since the contribution of this diagram is evaluated as

$\left(\overline{\prod_{m=0}^{n-1}\langle\psi_{i_{m}}^{\tilde{T}_{m}}|\psi_{i_{m+1}}^{T_{m}}\rangle}\right)_{\text{discon}}=d_{C}^{n}\;\prod_{m=0}^{n-1}\left(\delta_{i_{m}i_{m+1}}\delta_{\tilde{T}_{m}T_{m}}\right).$
(3.6)

The contribution of this diagram to the Rényi entropy is

$\displaystyle\left.\overline{\tr\left[\mathcal{N}[\rho]^{n}\right]}\right|_{\text{fully
discon}}=\frac{1}{\left(k\,d_{C}\right)^{n}}\cdot
k\,\left(d_{C}\right)^{n}\,\sum_{\bm{T}=1}^{d_{T}}\rho_{T_{1}T_{1}}\,\rho_{T_{2}T_{2}}\,\cdots\rho_{T_{n}T_{n}}=\frac{1}{\left(k\right)^{n-1}}\left(\tr\left[\rho\right]\right)^{n}.$
(3.7)

Similarly, the value of the fully connected diagram is given by

$\left(\overline{\prod_{m=0}^{n-1}\langle\psi_{i_{m}}^{\tilde{T}_{m}}|\psi_{i_{m+1}}^{T_{m}}\rangle}\right)_{\text{fully
conn}}=d_{C}\prod_{m=0}^{n-1}\left(\delta_{\tilde{T}_{m+1}T_{m}}\right)\;\Rightarrow\left.\overline{\tr\left[\mathcal{N}[\rho]^{n}\right]}\right|_{\text{fully
conn}}=\frac{1}{\left(d_{C}\right)^{n-1}}\tr\left[\rho^{n}\right].$ (3.8)

Combining these two results,
$\overline{\tr\left[\mathcal{N}[\rho]^{n}\right]}$ is given by

$\displaystyle\overline{\tr\left[\mathcal{N}[\rho]^{n}\right]}$
$\displaystyle=\frac{1}{\left(k\right)^{n-1}}\left(\tr\left[\rho\right]\right)^{n}+\frac{1}{\left(d_{C}\right)^{n-1}}\tr\left[\rho^{n}\right]+\cdots,$
(3.9)

where $\cdots$ means contributions coming from partially connected saddles.

Since there are upper and lower bounds on $\tr\left[\rho^{n}\right]$, that is,
$1/(d_{T})^{n-1}\leq\tr\left[\rho^{n}\right]\leq 1$, we can see that

$\displaystyle\overline{\tr\left[\mathcal{N}[\rho]^{n}\right]}$
$\displaystyle=\frac{1}{\left(k\right)^{n-1}}\left(\tr\left[\rho\right]\right)^{n}+\frac{1}{\left(d_{C}\right)^{n-1}}\tr\left[\rho^{n}\right]+\cdots,$
(3.10) $\displaystyle\approx\begin{dcases}\frac{1}{\left(k\right)^{n-1}}&k\ll
d_{C}\Leftrightarrow d_{T}\ll\left(\frac{d_{T}}{d_{D}}\right)^{2}\\\
\frac{1}{\left(d_{C}\right)^{n-1}}\tr\left[\rho^{n}\right]&d_{C}\,d_{T}\ll
k\Leftrightarrow\left(\frac{d_{T}}{d_{D}}\right)^{2}\ll 1\end{dcases}.$

Thus, when the necessary condition for the decoupling condition,
$d_{T}/d_{D}\ll 1$, holds, the dominant contribution is given by the fully
connected saddle.

We have to carefully evaluate the precise range of $m$ where the value of the
connected saddle gets larger than that of the disconnected saddle. This value
of $m$ depends on the density matrix $\rho$ on the code subspace, and gets
maximized when it is the maximally mixed state $\rho=I_{T}/d_{T}$. Therefore
after $k>d_{C}d_{T}$ the connected saddle become the dominant one for all
density matrices in $H_{{\rm code}}$.

Next, let us evaluate the second term of (3.4). This computation is completely
parallel to the above computation. In terms of the overlaps, it is given by

$\displaystyle\tr\left[\mathcal{N}[\rho]\mathcal{N}[\sigma]^{n-1}\right]$
$\displaystyle=\frac{1}{\left(k\,d_{C}\right)^{n}}\sum_{\bm{i}=1}^{k}\sum_{\bm{T},\bm{\tilde{T}}=1}^{d_{T}}\,\left(\prod_{m=0}^{n-1}\langle\psi_{i_{m}}^{\tilde{T}_{m}}|\psi_{i_{m+1}}^{T_{m}}\rangle\right)\;\rho_{T_{0}\,\tilde{T}_{0}}\left(\prod_{m=1}^{n-1}\sigma_{T_{m}\,\tilde{T}_{m}}\right).$
(3.11)

The contribution of the fully disconnected diagram and the connected diagram
to the second term of (3.4) can be evaluated, again by substituting the result
(3.6) and (3.8)

$\begin{aligned}
\left.\tr\left[\mathcal{N}[\rho]\mathcal{N}[\sigma]^{n-1}\right]\right|_{\text{discon}}=\frac{1}{\left(k\right)^{n-1}}\,\tr\left[\rho\right]\,\left(\tr\left[\sigma\right]\right)^{n-1},\end{aligned}\quad\begin{aligned}
\overline{\left.\tr\left[\mathcal{N}[\rho]\mathcal{N}[\sigma]^{n-1}\right]\right|}_{\text{conn}}=\frac{1}{\left(d_{C}\right)^{n-1}}\tr\left[\rho\,\sigma^{n-1}\right].\end{aligned}$
(3.12)

Thus using these results, we obtain

$\displaystyle\overline{\tr\left[\mathcal{N}[\rho]\mathcal{N}[\sigma]^{n-1}\right]}$
$\displaystyle=\frac{1}{\left(k\right)^{n-1}}\tr\left[\rho\right]\left(\tr\left[\sigma\right]\right)^{n-1}+\frac{1}{\left(d_{C}\right)^{n-1}}\tr\left[\rho\sigma^{n-1}\right]+\cdots,$
(3.13) $\displaystyle\approx\begin{dcases}\frac{1}{\left(k\right)^{n-1}}&k\ll
d_{C}\Leftrightarrow d_{T}\ll\left(\frac{d_{T}}{d_{D}}\right)^{2}\\\
\frac{1}{\left(d_{C}\right)^{n-1}}\tr\left[\rho\sigma^{n-1}\right]&k\gg
d_{C}d_{T}\Leftrightarrow\left(\frac{d_{T}}{d_{D}}\right)^{2}\ll
1\end{dcases},$

where $\cdots$ again means contributions coming from partially connected
saddles, and also, in the second approximate equality we assumed that
$1/(d_{T})^{n-1}\lesssim\tr\left[\rho\sigma^{n-1}\right]\leq 1$ in order to
obtain the conditions111111If the support of the density matrix $\rho$ is not
contained in that of $\sigma$, then $\tr\left[\rho\sigma^{n-1}\right]=0$,
implying the divergent relative entropy $S(\rho||\sigma)=\infty$. In that
case, we would need another treatment, so we do not consider such a case in
this paper..

Now that we have evaluated the two terms appeared in the relative entropy, we
can obtain the resulting relative entropy

$\displaystyle\overline{S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])}$
$\displaystyle\approx\lim_{n\to
1}\frac{1}{n-1}\left(\log\overline{\tr\left[\mathcal{N}[\rho]^{n}\right]}-\log\overline{\tr\left[\mathcal{N}[\rho]\mathcal{N}[\sigma]^{n-1}\right]}\right),$
(3.14) $\displaystyle\approx\begin{dcases}0&k\ll d_{C}\Leftrightarrow
d_{T}\ll\left(\frac{d_{T}}{d_{D}}\right)^{2}\\\ \lim_{n\to
1}\frac{1}{n-1}\left(\log\tr\left[\rho^{n}\right]-\log\tr\left[\rho\sigma^{n-1}\right]\right)&k\gg
d_{C}d_{T}\Leftrightarrow\left(\frac{d_{T}}{d_{D}}\right)^{2}\ll
1\end{dcases}$ $\displaystyle=\begin{dcases}0&k\ll d_{C}\Leftrightarrow
d_{T}\ll\left(\frac{d_{T}}{d_{D}}\right)^{2}\\\ S(\rho||\sigma)&k\gg
d_{C}d_{T}\Leftrightarrow\left(\frac{d_{T}}{d_{D}}\right)^{2}\ll
1.\end{dcases}$

Thus we can conclude that, when the condition $d_{T}/d_{D}\ll 1$ is satisfied,
the relative entropies obeys the relation

$\overline{S(\mathcal{N}[\rho]||\mathcal{N}[\sigma])}\approx S(\rho||\sigma).$
(3.15)

This result implies that the condition of sufficiency holds for the Hayden-
Preskill channel when $\left(\frac{d_{T}}{d_{D}}\right)^{2}\ll 1$.

###  3.1 Check the recovery map

We argued that in chaotic systems (2.5) the Petz recovery map gets simplified
and is reduced to so called Petz-lite map $\mathcal{R}^{\text{Lite}}$defined
in (2.11). In this section, we show this by checking

$\overline{S(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]||\rho_{T})}=0,\quad{\rm
when}\;\left(\frac{d_{T}}{d_{D}}\right)^{2}\ll 1.$ (3.16)

for any density matrix $\rho_{T}$ on the code subspace. This means that at
sufficiently late time one can recover $\rho_{T}$ from the state of the
Hawking radiation $\mathcal{N}[\rho_{T}]$ by applying the recovery map
$\mathcal{R}^{\text{Lite}}$.

One can show this by computing the relative entropy by the replica trick
similar to (3.3),

$\overline{S(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]||\rho_{T})}=\lim_{n\rightarrow
1}\frac{1}{n-1}\left(\log\overline{{\rm
tr}(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho]\right])^{n}}-\log\overline{{\rm
tr}(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho]\right]\rho^{n-1})}\right).$
(3.17)

In terms of Haar random unitaries,
$\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]$ is given by

$\displaystyle\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]$
$\displaystyle=\frac{1}{N}\cdot\frac{d_{B}d_{D}}{d_{T}}\cdot\mathcal{N}^{\dagger}_{D,B\to
T}\left[\mathcal{N}_{T\to D,B}\left[\rho\right]\right]$ (3.18)
$\displaystyle=\frac{1}{N}\sum_{\tilde{T},\tilde{T^{\prime}}=1}^{d_{T}}\ket{\tilde{T}}_{T}\\!\bra{\tilde{T}^{\prime}}\cdot\frac{1}{k(d_{C})^{2}d_{T}}\sum_{T,T^{\prime}=1}^{d_{T}}\sum_{i,j=1}^{k}\innerproduct{\psi_{i}^{\tilde{T}}}{\psi_{j}^{\tilde{T}^{\prime}}}\innerproduct{\psi_{j}^{T^{\prime}}}{\psi_{i}^{T}}(\rho)_{TT^{\prime}}.$

Therefore the first term in (3.17) is given by

${\rm
tr}(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho]\right])^{n}=\frac{1}{(Nkd_{C}^{2}d_{T})^{n}}\sum_{\bm{T},\bm{T}^{\prime}=1}^{d_{T}}\sum_{\bm{\tilde{T}},\bm{\tilde{T}}^{\prime}=1}^{d_{T}}\sum_{\bm{i},\bm{j}=1}^{k}\;\prod_{m=1}^{n}\;\left(\innerproduct{\psi_{i_{m}}^{T_{m}}}{\psi_{j_{m}}^{T_{m+1}}}\innerproduct{\psi_{j_{m}}^{\tilde{T}_{m}}}{\psi_{i_{m}}^{\tilde{T}^{\prime}_{m}}}\rho_{\tilde{T}_{m}\tilde{T}^{\prime}_{m}}\right).$
(3.19)

We compute this by following the procedure explained in section 2.1, namely by
preparing an interval for each overlap, and connecting the endpoints of the
intervals by EoW branes, then evaluating each diagram generated in this way.
As shown in the figure, $m$ th replica consists of two intervals with indices
for Hawking radiation $i_{m},j_{m}$. Therefore it is clear that the dominant
diagram when $k=d_{D}d_{B}$ is sufficiently large is the one connecting the
endpoint with the index $i_{m}$ in the first interval of to the endpoint of
the second replica with the same index in the same replica (the right panel of
Figure 4). Similarly, we connect the endpoints with $j_{m}$ in this replica.
This is because, if there is an EoW brane connecting endpoints with distinct
Hawking indices (say $i,j$), then the value of the diagram is significantly
reduced in the large $k$ limit because of the Kronecker delta factor
$\delta_{ij}$ coming from the brane.

$T_{m}$$i_{m}$$T_{m+1}$$j_{m}$$\tilde{T}_{m}$$j_{m}$$\tilde{T}_{m}^{\prime}$$i_{m}$

$T_{m}$$i_{m}$$T_{m+1}$$j_{m}$$\tilde{T}_{m}$$j_{m}$$\tilde{T}_{m}^{\prime}$$i_{m}$

Figure 4: Diagrams for the product of overlaps appearing in the calculation of
(3.19) . Left: disconnected diagram. Right: The connected diagram.

This means that in the dominant saddle two different replicas are not
connected by any EoW brane, because they start and end at the same replica.
This means that the Rényi entropy is a self averaging quantity

$\overline{{\rm
tr}(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho]\right])^{n}}={\rm
tr}\left(\overline{\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho]\right]}\right)^{n}$
(3.20)

A similar statement holds for the second term of (3.17), therefore we conclude
that the relative entropy of our interest is also self averaging,

$\overline{S(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]\;||\rho_{T})}=S(\overline{\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]}\;||\rho_{T}).$
(3.21)

when $k$ is sufficiently large. This implies that in the relative entropy one
can replace $\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]$ to
its average
$\overline{\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]}$. The
average of the density matrix is given by

$\overline{\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]}=\frac{1}{1+\left(\dfrac{d_{T}}{d_{D}}\right)^{2}}\left(\rho+\left(\dfrac{d_{T}}{d_{D}}\right)^{2}\cdot\dfrac{I_{T}}{d_{T}}\right).$
(3.22)

A more precise way to argue this is the following: Let us compute

$\displaystyle\overline{{\rm
tr}\left[\left(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]-\overline{\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]}\right)^{2}\right]}=\overline{{\rm
tr}\left[\left(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]\right)^{2}\right]}-{\rm
tr}\left[\left(\overline{\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]}\right)^{2}\right].$
(3.23)

Then, the right hand side of the above equation is given by

$\displaystyle\frac{(k\,d_{C})^{4}}{\left(Nk(d_{C})^{2}d_{T}\right)^{2}}\Bigg{\\{}\frac{1}{k\,d_{C}}\Bigg{[}\frac{1}{k^{2}}\left(2+d_{T}\tr\left[\rho^{2}\right]+(d_{T})^{2}\right)$
(3.24)
$\displaystyle+\frac{1}{k\,d_{C}}\left((d_{T})^{2}\tr\left[\rho^{2}\right]+2d_{T}+2\tr\left[\rho^{2}\right]\right)+\frac{1}{(d_{C})^{2}}d_{T}\tr\left[\rho^{2}\right]\Bigg{]}\Bigg{\\}}$

which becomes small when $k\gg d_{C}d_{T}$. By plugging this expression, we
have

$\displaystyle\overline{S(\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]||\rho_{T})}$
$\displaystyle\approx
S(\overline{\mathcal{R}^{\text{Lite}}\left[\mathcal{N}[\rho_{T}]\right]}\;||\rho_{T})$
(3.25)
$\displaystyle=\begin{dcases}S\bigg{(}\rho\,\bigg{|}\\!\bigg{|}\dfrac{I_{T}}{d_{T}}\bigg{)}&k\ll
d_{C}\Leftrightarrow d_{T}\ll\left(\frac{d_{T}}{d_{D}}\right)^{2}\\\ 0&k\gg
d_{C}d_{T}\Leftrightarrow\left(\frac{d_{T}}{d_{D}}\right)^{2}\ll
1.\end{dcases}$

Thus, for early times $k\ll d_{C}$, the relative entropy is non-vanishing
unless $\rho=I_{T}/d_{T}$, but for late times $d_{C}\,d_{T}\ll k$, the
relative entropy is vanishing. This result implies that when $k\gg
d_{C}d_{T}$, $\mathcal{R}^{\text{Lite}}$ indeed works as a recovery map.

###  3.2 Relation to the Yoshida-Kitaev protocol

So far we have shown that when $k\gg d_{C}d_{T}$, the Petz-lite
$\mathcal{R}^{\text{Lite}}\sim\mathcal{N}^{\dagger}$ indeed works as a
recovery map. However, we have not discussed the physical interpretation of
the Petz-lite. So, in this subsection, we explain the interpretation by
showing the equivalence between the Petz-lite and the well-known Yoshida-
Kitaev (YK) protocol. The relation between the Yoshida-Kitaev protocol and the
Petz map has been suggested by Yoshida Yoshida:2021xyb ; Yoshida:2021haf .

In Yoshida:2017non , Yoshida and Kitaev proposed an interesting recovery
protocol for the object thrown into the black hole $T$ from late and early
radiation $DB$. Brief summary of their protocol is as follows:

  1. 1.

In addition to the original Hayden-Preskill setup, introduce a copy of the
diary and the reference, denoted by $R^{\prime}T^{\prime}$. We choose the
state on $R^{\prime}T^{\prime}$ to be an EPR state. Bob can manipulate Hawking
radiation $DB$ and $R^{\prime}T^{\prime}$. Before applying the decoding
protocol the state of the total system is

$|\Psi_{HP}\rangle\otimes|\text{EPR}\rangle_{R^{\prime}T^{\prime}}$ (3.26)

where $|\Psi_{HP}\rangle$ is the state on $RCDB$ given by (2.1).

  2. 2.

We then use the early Hawking radiation $B$ and the copy of the diary
$T^{\prime}$ to simulate the black hole dynamic by applying $U^{*}$ which is
the complex conjugate of $U$ for the time evolution of the original system.
After the simulation total system consists of
$RCDR^{\prime}C^{\prime}D^{\prime}$, and the state is

$|\Psi_{YK}\rangle_{RCDD^{\prime}C^{\prime}R^{\prime}}=\left(I_{RC}\otimes
I_{D}\otimes U^{*}_{D^{\prime}C^{\prime}\rightarrow BT^{\prime}}\otimes
I_{R}\right)|\Psi_{HP}\rangle\otimes|\text{EPR}\rangle_{R^{\prime}T^{\prime}}$
(3.27)

  3. 3.

Post select to the EPR pair on $DD^{\prime}$. If it succeeds, the state on
$RR^{\prime}$ is the EPR state with high fidelity, meaning the success of
information recovery.

The quantum circuit for the protocol is shown in the left panel of figure 5.
Combining these steps, the quantum channel $\mathcal{R}_{D,B\to
R^{\prime}}^{KY}$ for the Yoshida-Kitaev (YK) recovery map is given by

$\displaystyle\mathcal{R}_{D,B\to
R^{\prime}}^{\text{YK}}\left[\mathcal{O}_{DB}\right]$ (3.28)
$\displaystyle=\frac{1}{\Delta}\tr_{C^{\prime}}\left[\,{}_{D,D^{\prime}}\\!\bra{\text{EPR}}U^{*}_{B,T^{\prime}\to
C^{\prime},D^{\prime}}\left(\mathcal{O}_{DB}\otimes\ket{\text{EPR}}_{T^{\prime},R^{\prime}}\\!\bra{\text{EPR}}\right)U^{T}_{B,T^{\prime}\to
C^{\prime},D^{\prime}}\ket{\text{EPR}}_{D,D^{\prime}}\right],$

where $\Delta$ is a normalization factor given by

$\Delta=\overline{\left|{}_{D,D^{\prime}}\innerproduct{\text{EPR}}{\Psi_{YK}}\right|^{2}}\approx\frac{1}{(d_{T})^{2}}+\frac{1}{(d_{D})^{2}}.$
(3.29)

$U$$U^{\ast}$$R$$R$$T$$C$$A$$D$$B$$D^{\prime}$$T^{\prime}$$C^{\prime}$$R^{\prime}$$R^{\prime}$

$=$$U^{T}$$U^{\ast}$$U$$U^{\dagger}$$\rho_{DB}$$\rho_{DB}$$T$$T^{\prime}$$C$$C^{\prime}$$D$$B$$B$$D^{\prime}$$D$$D$$C$$T$$T^{\prime}$$B$$B$$A$$D$$D$$A$$T$$C$$C$$T$

Figure 5:  Left: Yoshida-Kitaev decoding protocol. Right: operator transpose
that providing the key equivalence (3.33).

For the above YK recovery map, we show the equivalence between the YK recovery
map $\mathcal{R}_{D,B\to R^{\prime}}^{\text{YK}}$ and the Petz-lite (2.9),
$\mathcal{R}_{D,B\to T}^{\text{Lite}}$ up to the isomorphism $V_{T\to
R^{\prime}}$ between systems $T$ and $R^{\prime}$,

$\mathcal{R}_{D,B\to R^{\prime}}^{KY}\left[\mathcal{O}_{DB}\right]=V_{T\to
R^{\prime}}\,\mathcal{R}_{D,B\to
T}^{\text{Lite}}\left[\mathcal{O}_{DB}\right]\,V_{T\to
R^{\prime}}^{\dagger}\quad,$ (3.30)

where $V_{T\to R^{\prime}}$ is explicitly given by

$V_{T\to R^{\prime}}\coloneqq
d_{T}\,{}_{T,T^{\prime}}\\!\innerproduct{\text{EPR}}{\text{EPR}}_{T^{\prime},R^{\prime}}=\sum_{\tilde{T}=1}^{d_{T}}\ket{\tilde{T}}_{R^{\prime}\,T}\\!\\!\bra{\tilde{T}}.$
(3.31)

The argument for the equivalence is summarized in the right panel of figure 5.
We start with the YK recovery map (3.28). First, we rewrite the trace of
subsystem $C^{\prime}$ in the YK recovery map as

$\tr_{C^{\prime}}\left[\mathcal{O}\right]=d_{C}\,{}_{C,C^{\prime}}\\!\bra{\text{EPR}}\left(I_{C}\otimes\mathcal{O}\right)\ket{\text{EPR}}_{C,C^{\prime}},$
(3.32)

and introduce two EPR states $\ket{\text{EPR}}_{D,D^{\prime}}$ and
$\ket{\text{EPR}}_{C,C^{\prime}}$.

Next, by using (3.32) and the relation (see appendix B for the derivation)

$\displaystyle U^{T}_{C^{\prime},D^{\prime}\to
B,T^{\prime}}\ket{\text{EPR}}_{C,C^{\prime}}\otimes\ket{\text{EPR}}_{D,D^{\prime}}$
$\displaystyle=U_{A,T\to
C,D}\ket{\text{EPR}}_{A,B}\otimes\ket{\text{EPR}}_{T,T^{\prime}},$ (3.33)

the KY recovery map (3.30) can be rewritten as

$\displaystyle\mathcal{R}_{D,B\to
R^{\prime}}^{KY}\left[\mathcal{O}_{DB}\right]$ (3.34)
$\displaystyle=\frac{d_{C}}{\Delta}\left(\,{}_{A,B}\\!\bra{\text{EPR}}\otimes\,_{T,T^{\prime}}\\!\bra{\text{EPR}}\right)\left[U_{A,T\to
C,D}^{\dagger}\left(\mathcal{O}_{DB}\otimes\ket{\text{EPR}}_{T^{\prime},R^{\prime}}\\!\bra{\text{EPR}}\right)U_{A,T\to
C,D}\right]$ $\displaystyle\hskip
312.9803pt\times\left(\ket{\text{EPR}}_{A,B}\otimes\ket{\text{EPR}}_{T,T^{\prime}}\right)$
$\displaystyle=\frac{d_{C}}{\Delta}\left(\,{}_{T,T^{\prime}}\\!\innerproduct{\text{EPR}}{\text{EPR}}_{T^{\prime},R^{\prime}}\right)$
$\displaystyle\hskip
85.35826pt\times\,_{A,B}\\!\bra{\text{EPR}}\left[U_{T,A\to
C,D}^{\dagger}\,\mathcal{O}_{DB}\,U_{T,A\to C,D}\right]\ket{\text{EPR}}_{A,B}$
$\displaystyle\hskip
284.52756pt\times\left(\,{}_{T^{\prime},R^{\prime}}\\!\innerproduct{\text{EPR}}{\text{EPR}}_{T,T^{\prime}}\right)$
$\displaystyle=\frac{d_{C}}{(d_{T})^{2}\,\Delta}\,V_{T\to
R^{\prime}}\,\mathcal{N}_{D,B\to
T,A}^{\dagger}\left[\mathcal{O}_{D,B}\right]\,V_{T\to R^{\prime}}^{\dagger},$

where, in the final line, we used the definition of the isomorphism (3.31) and
the adjoint HP channel (2.6). Additionally, the above overall constant
$\frac{d_{C}}{(d_{T})^{2}\,\Delta}$ coincides with that of the Petz-lite
(2.11), since

$\frac{d_{C}}{(d_{T})^{2}\,\Delta}=\frac{d_{C}}{1+\left(\dfrac{d_{T}}{d_{D}}\right)^{2}},$
(3.35)

where we used the definition of $\Delta$, (3.29). Therefore the above
expression implies the desired relation (3.30).

##  4 Recovery map for the Hayden-Preskill channel in SYK

So far we have given the evidence that the Petz-lite works as a recovery map,
under the Haar random unitary which is highly chaotic. In this section, we
argue that this continues to hold for a more realistic but tractable model of
chaotic dynamics; the Sachdev-Ye-Kitaev (SYK) model Sachdev:1992fk ;
KitaevTalks ; Sachdev:2015efa . In this paper, we briefly explain the relevant
calculations, leaving details in the upcoming paper WIP .

###  4.1 Setup of SYK Hayden-Preskill protocol

In this section, we explain the setup to study the Hayden Preskill like
protocol (what we call SYK HP channel) in the SYK model. This was first
introduced in Chandrasekaran:2022qmq ; Chandrasekaran:2021tkb .

The SYK model is a theory of $N$ Majorana fermions $\psi_{i}$, and its
Hamiltonian is given by

$H=\left(i\right)^{q/2}\sum_{1\leq i_{1}<i_{2}<\cdots<i_{q}\leq
N}j_{i_{1}i_{2}\cdots i_{q}}\psi_{i_{1}}\psi_{i_{2}}\cdots\psi_{i_{q}},$ (4.1)

where $q\in 2\mathbb{N}\,(q>2)$, $j_{i_{1}i_{2}\cdots i_{q}}$ is a random
coefficient drawn from a Gaussian random distribution with zero mean and the
variance $\left\langle j_{i_{1}i_{2}\cdots
i_{q}}^{2}\right\rangle=J^{2}(q-1)!/N^{q-1}$.

Following Chandrasekaran:2022qmq , we consider two copies of the Hilbert space
of the SYK model, say left SYK system $L$ and right one $R$. Hereafter we
denote the Majorana fermions on the left system by $\psi_{i,L}$ and
$\psi_{i,R}$ for the right. For notational simplicity, we use the convention

$\left\\{\psi_{i},\psi_{j}\right\\}=2\delta_{i,j},$ (4.2)

for the anti commutation relation for the fermions on the same side. In this
set up, the right SYK system corresponds to early radiation degrees of freedom
of the original Hayden-Preskill setup, and the left SYK system corresponds to
the rest; the union of the diary system and the initial black hole before the
action of the random unitary, or equivalently the remaining black hole plus
late radiation degrees of freedom after the unitary evolution. In particular,
the left system $L$ is divided into two subsystems, say $\tilde{L}$ and $K$,
the former correspond to the remaining black hole and the latter to the late
radiation part of the original HP setup.

On the union of the above SYK systems $L$ and $R$, we consider the following
thermo-field double (TFD) state ;

$\ket{\text{TFD}}_{L,R}=Z^{-1/2}(\beta)\,e^{-\beta(H_{L}+H_{R})/4}\ket{0}_{L,R},$
(4.3)

where $Z(\beta)$ is a normalization factor of the state, and $\ket{0}_{L,R}$
is given by Qi:2018bje

$\left[\,\psi_{j,L}(0)+i\psi_{j,R}(0)\,\right]\ket{0}_{L,R}=0\quad\text{for
}\forall j.$ (4.4)

Note that the thermo-field double state (4.3) satisfies the relation
$(H_{L}-H_{R})\ket{\text{TFD}}=0$. This TFD state corresponds to an entangled
state between the initial black hole and the early radiation.

The code subspace (a diary system) of our interest is two dimensional, and
let’s denote two basis vectors by $\ket{0}$ and $\ket{1}$. This code subspace
is embedded into the physical Hilbert space $LR$ by an isometry. The image of
the code subspace is spanned by the TFD state $\ket{\text{TFD}}_{L,R}$ and the
excited state $\psi_{i,L}(0)\ket{\text{TFD}}_{L,R}$. Here we assume that the
Majorana fermion $\psi_{i,L}(0)$ acting on the TFD state lives in the
subsystem $\tilde{L}$, $i\in\tilde{L}$. More explicitly, by the isometry, the
states in the code subspace $\ket{T}\,(T=0,1)$ are mapped to

$\left(V_{T,L\to L}\otimes
I_{R}\right)\left(\ket{T}_{T}\otimes\ket{\text{TFD}}_{L,R}\right)\coloneqq\begin{dcases}\ket{\text{TFD}}_{L,R}&\text{
for }T=0\\\
\dfrac{1}{\left(Z_{\delta}\right)^{\frac{1}{2}}}\psi_{i,L}(i\delta)\ket{\text{TFD}}_{L,R}&\text{
for }T=1,\end{dcases}$ (4.5)

where $\psi_{i,L}(i\delta)$ is the regulated Majorana fermion operator

$\psi_{i,L}(i\delta)=e^{-\delta H_{L}}\psi_{i,L}(0)e^{\delta H_{L}},$ (4.6)

and $\delta$ is an infinitesimal cutoff parameter to normalize the state with
the operator insertion even in the conformal limit, where the SYK model has an
effective description in terms of the reparametrization modes
Maldacena:2016hyu . $Z_{\delta}$ is its normalization factor given by the two
point function

$\displaystyle Z_{\delta}$
$\displaystyle=\frac{1}{N-K}\sum_{i=1}^{N-K}\frac{1}{Z(\beta)}\tr\left[e^{-\beta
H_{L}}\psi_{i,L}(-i\delta)\psi_{i,L}(i\delta)\right]$ (4.7)
$\displaystyle=\frac{1}{N-K}\sum_{i=1}^{N-K}\frac{1}{Z(\beta)}\tr\left[e^{-\beta
H_{L}}e^{2\delta H_{L}}\psi_{i,L}(0)e^{-2\delta
H_{L}}\psi_{i,L}(0)\right]=G_{\beta}(2\delta).$

This normalization factor is not for the specific Majorana fermion “$i$”, but
averaged over the region $\tilde{L}$ with $N-K$ sites. We expect the
difference between the two only appears in sub-leading terms with respect to
$K/N$ because of typicality. Therefore, we use this normalization factor (4.7)
for later convenience.

Using the above embedding, we can holographically prepare an initial entangle
state between the early radiation and an initial black hole containing a diary
in the SYK model. For this system, we consider an unitary time evolution on
the left system $L$ by the SYK Hamiltonian $H_{L}$,

$U_{L}(t)=\exp\left(itH_{L}\right).$ (4.8)

By this time evolution, information in the diary gets scrambled and uniformly
distributed over the left SYK system after the scrambling time. The resulting
state is

$\ket{\Psi_{\text{SYK HP}}}=(I_{\text{Ref}}\otimes U_{L}(t)\otimes
I_{R})\left(I_{\text{Ref}}\otimes V_{T,L\to L}\otimes
I_{R}\right)\left(\ket{\text{EPR}}_{\text{Ref},T}\otimes\ket{\text{TFD}}_{L,R}\right),$
(4.9)

which corresponds to the state (2.1). In figure 6, we give the circuit diagram
corresponding to the state (4.9).

We are interested in recovering the diary information from the early and late
radiations $R$ and $\tilde{L}$ by using the Petz-lite for the SYK HP protocol.

Figure 6: Circuit diagram corresponding to state (4.9).

As in (2.3), the SYK HP channel $\mathcal{N}^{\text{SYK}}_{T\to K,R}$
representing error is obtained by tracing out the remaining black hole part
$\tilde{L}$ in the final state (4.9),

$\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\rho_{T}]\coloneqq\tr_{\tilde{L}}\left[U_{L}V_{T,L\to
L}\left(\rho_{T}\otimes\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right)V_{T,L\to
L}^{\dagger}U_{L}^{\dagger}\right].$ (4.10)

This channel maps a density matrix on the diary $T$ to the one on the late and
early radiation system $K,R$. Also, the adjoint
$\mathcal{N}^{\text{SYK}\dagger}_{K,R\to T}$ of the SYK HP channel is given by

$\displaystyle\mathcal{N}^{\text{SYK}\dagger}_{K,R\to T}[\mathcal{O}_{KR}]$
$\displaystyle\coloneqq\tr_{L,R}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\left(V_{L\to
T,L}^{\dagger}U_{L}^{\dagger}\,\mathcal{O}_{KR}\,U_{L}V_{L\to
T,L}\right)\right]$ (4.11)
$\displaystyle=\,_{L,R}\expectationvalue{\left(V_{L\to
T,L}^{\dagger}U_{L}^{\dagger}\,\mathcal{O}_{KR}\,U_{L}V_{L\to
T,L}\right)}{\text{TFD}}_{L,R}.$

The above quantum channels are analogous to the original HP channel and its
adjoint for the Haar random unitary. However, we note that the difference that
the SYK HP channel and its adjoint includes the embedding map $V$, which
induces (fermionic) excitations.

###  4.2 Some matrix elements of the Petz-lite and Rényi-two correlators

Now that we have prepared the SYK HP channel and its adjoint, we can construct
the Petz-lite map for this channel. As in the Petz-lite for the Haar random
case (2.9), we consider the Petz-lite for the SYK case,

$\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{O}_{KR}\right]=\frac{1}{N_{\text{SYK}}}\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}[\mathcal{O}_{KR}],$ (4.12)

where $N_{\text{SYK}}$ is the normalization factor, which is determined by the
condition

$\tr_{T}\left[\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to K,R}[\sigma_{T}]\right]\right]=1.$
(4.13)

Here $\sigma_{T}$ is some reference state in $T$ for the normalization. We
take it to be $\sigma_{T}=\ket{0}_{T}\\!\bra{0}$. For this choice, the
normalization factor is given by

$\displaystyle N_{\text{SYK}}$
$\displaystyle=\sum_{T=0,1}\matrixelement{T}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{T}.$ (4.14)

We note that due to this normalization, we can see that the Petz-lite (4.12)
for the SYK HP protocol have a similar overall constant with the Petz-lite for
the original HP protocol (2.11). To see the similarity, we first rewrite the
Petz-lite (4.12) with the normalization factor (4.14) as follows

$\displaystyle\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{O}_{KR}\right]=\dfrac{\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}}{1+\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}}\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}[\mathcal{O}_{KR}],$ (4.15)

where $\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}$ is an effective
dimension of subsystem $\tilde{L}$ defined by the purity
$\tr_{\tilde{L}}\left[\left(\rho_{\tilde{L}}\right)^{2}\right]$ of the TFD
state with respect to the subsystem121212We note that in our setting,
subsystem $\tilde{L}$ is smaller than the complement system $KR$.,

$\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{0}=\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]=\tr_{\tilde{L}}\left[\left(\rho_{\tilde{L}}\right)^{2}\right]\eqqcolon\frac{1}{\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}}.$
(4.16)

The effective dimension is analogous to the dimension of the remaining black
hole in the original HP setup. Indeed in the infinite temperature limit
$\beta\to 0$, the effective dimension is almost reduced to the actual
dimension of subsystem $\tilde{L}$, $d_{\tilde{L}}=2^{N-K}$. However in
general the effective dimension is smaller than the actual dimension due to
the property of the purity and thermal effects;

$1\,\leq\,\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\,\leq\,d_{\tilde{L}},$
(4.17)

where this effective dimension becomes closed to 1 in $\beta\to\infty$ and
$d_{\tilde{L}}$ in $\beta\to 0$. With this effective dimension, we can compare
the Petz-lite (4.15) for the SYK model to that for the original one (2.11) in
the HP setup

$\displaystyle\mathcal{R}^{\text{Lite},HP}_{D,B\to
T}[\mathcal{O}_{DB}]=\frac{d_{C}}{1+\left(\dfrac{d_{T}}{d_{D}}\right)^{2}}\,\mathcal{N}_{D,B\to
T}^{\dagger}\left[\mathcal{O}_{DB}\right].$

The similarities between the quantities in the HP and the SYK are summarized
in the following identifications;

$\displaystyle d_{C}$
$\displaystyle\longleftrightarrow\quad\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta},$
(4.18) $\displaystyle\left(\dfrac{d_{T}}{d_{D}}\right)^{2}$
$\displaystyle\longleftrightarrow\quad\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}.$

Also, we have the unitarity constraint on the dimensions of the Hilbert
spaces; $d_{T}\,d_{B}=d_{C}\,d_{D}$. By using the relation, we can rewrite the
dimension as

$\left(\dfrac{d_{T}}{d_{D}}\right)^{2}=\frac{d_{C}\,d_{T}}{d_{B}\,d_{D}},$
(4.19)

from which we have the following identification

$\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}\quad\longleftrightarrow\quad\frac{1}{d_{C}}\cdot\left(\dfrac{d_{T}}{d_{D}}\right)^{2}=\frac{d_{T}}{d_{B}\,d_{D}}=\frac{d_{T}}{k}.$
(4.20)

This might be a good ratio to understand current physics; if we have a
sufficiently large amount of Hawking radiation compared with the diary,
$d_{T}\ll d_{B}\,d_{D}=k$, the ratio becomes almost 0. As we see soon after,
the left quantity also becomes almost 0 around and/or after a critical time.

With this discussion of the normalization factor in mind, we consider a matrix
element of $\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to K,R}[\rho_{T}]\right]$ for a general
density matrix $\rho_{T}$ in the Hilbert space of the diary,

$\matrixelement{T}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to K,R}[\rho_{T}]\right]}{T^{\prime}}.$
(4.21)

To check whether the Petz-lite works as the recovery map, it is sufficient to
see whether the following relation holds (approximately) or not,

$\matrixelement{T}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\rho_{T}]\right]}{T^{\prime}}\overset{?}{\approx}\matrixelement{T}{\rho_{T}}{T^{\prime}}\quad\text{for
}\forall\rho_{T}.$ (4.22)

Checking the above relation is equivalent to focus on the matrix elements

$\matrixelement{T}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}\left[\ket{\tilde{T}}_{T}\\!\bra{\tilde{T}^{\prime}}\right]\right]}{T^{\prime}}\overset{?}{\approx}\innerproduct{T}{\tilde{T}}\innerproduct{\tilde{T}^{\prime}}{T^{\prime}},\qquad\forall
T,T^{\prime},\tilde{T},\tilde{T}^{\prime}$ (4.23)

Generally, we have 16 components of the above matrix, but half of them,
including odd Majorana fermions, are trivially vanishing due to the fermionic
parity of the SYK model. In other words, matrix elements which satisfies
$(T+T^{\prime}+\tilde{T}+\tilde{T}^{\prime})\equiv 1\mod 2$ are vanishing.

Now, we focus on three of non-zero matrix elements, and briefly explain how we
can evaluate them 131313The details of the calculation will be discussed in
upcoming paper WIP .  . First, we consider the
$T,T^{\prime},\tilde{T},\tilde{T}^{\prime}=0$ case. If (4.23) holds then since
its right-hand side is $1$, and therefore the following identity holds,

$\displaystyle
1\overset{?}{\approx}\matrixelement{0}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}\left[\ket{0}_{T}\\!\bra{0}\right]\right]}{0}$
$\displaystyle=\left(1+\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}\right)^{-1}.$ (4.24)

The second one is for the $T,T^{\prime}=1$, $\tilde{T},\tilde{T}^{\prime}=0$
case, where the matrix element is expected to become 0. In this case, we can
see that this matrix element has the same ratios as above,

$\displaystyle
0\overset{?}{\approx}\matrixelement{1}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}\left[\ket{0}_{T}\\!\bra{0}\right]\right]}{1}=\dfrac{\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}}{1+\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}}.$ (4.25)

The final one is for $T,\tilde{T}=0$, $T^{\prime},\tilde{T}^{\prime}=1$, the
matrix elements (4.23), which is expected to be 1, becomes

$\displaystyle
1\overset{?}{\approx}\matrixelement{0}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}\left[\ket{0}_{T}\\!\bra{1}\right]\right]}{1}=\dfrac{\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}}{1+\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}}.$ (4.26)

The rest of matrix elements

$\displaystyle\matrixelement{0}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}\left[\ket{1}_{T}\\!\bra{0}\right]\right]}{1},\quad\matrixelement{1}{\mathcal{R}^{\text{Lite,SYK}}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}\left[\ket{1}_{T}\\!\bra{1}\right]\right]}{1}.$

are difficult to directly evaluate as we will mention in footnote 16. In the
next section, we evaluate these matrix elements indirectly from the results of
this section.

Thus, to see the recovery (4.23), we need to study the behaviours of the
matrix elements of $\mathcal{N}^{\dagger}\mathcal{N}$ which appear in the
right hand side of (4.24), (4.25), (4.26). In order for the recovery to
happen, these have to satisfy

$\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}\overset{?}{\approx}0,$ (4.27)
$\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}\overset{?}{\approx}1,$ (4.28)

We study the behavior of the left hand side of(4.27) (4.28) below. To this
end, it is convenient to rewrite the quantities as correlators. From the
definitions of the channels (2.3) and (4.11), we obtain the left-left
correlators

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}=\frac{1}{Z_{\delta}}\cdot\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(I_{\tilde{L}}\otimes\rho_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]},$
(4.29)
$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}$
$\displaystyle=\frac{1}{Z_{\delta}}\cdot\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(\rho_{\tilde{L}}\otimes
I_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]},$
(4.30)

where the two fermions are put on the left system, and $\rho_{KR}$ and
$\rho_{\tilde{L}}$ are defined by

$\rho_{\tilde{L}}=\tr_{KR}\left[\ket{\text{TFD}}_{LR}\\!\bra{\text{TFD}}\right],\qquad\rho_{KR}=\tr_{\tilde{L}}\left[\ket{\text{TFD}}_{LR}\\!\bra{\text{TFD}}\right].$
(4.31)

We give the derivation of the correlators in appendix D.

We also note that the numerators in the above correlators can be written as

$\displaystyle\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(I_{\tilde{L}}\otimes\rho_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}$
(4.32)
$\displaystyle=\tr_{KR}\left[\tr_{\tilde{L}}\left[\psi_{i,L}(t+i\delta)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\psi_{i,L}(t-i\delta)^{\dagger}\right]\rho_{KR}\right]$

and

$\displaystyle\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(\rho_{\tilde{L}}\otimes
I_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}$ (4.33)
$\displaystyle=\tr_{\tilde{L}}\left[\tr_{KR}\left[\psi_{i,L}(t+i\delta)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\psi_{i,L}(t-i\delta)^{\dagger}\right]\rho_{\tilde{L}}\right].$

These expressions are also useful to see that these quantities are related to
“Renyi-2” quantities as explained below.

Below we would like to evaluate these correlators analytically, but the
expressions (4.29) and (4.30) are not suitable for an analytic treatment as
they are“specific site” correlators, so we can not apply the large-$N$
techniques to evaluate them. However, since we are basically interested in
typical behaviors under highly chaotic dynamics in our setup, the specific
choice of the embedding would not be essential. Therefore, below we consider
the “typical” embedding of the code information into the whole $\tilde{L}$
system uniformly. Therefore we replace these correlators with their averages
on $\tilde{L}$,

$\displaystyle\frac{1}{Z_{\delta}}\cdot\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(I_{\tilde{L}}\otimes\rho_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]}$
(4.34)
$\displaystyle\qquad\to\frac{1}{N-K}\sum_{i=1}^{N-K}\frac{1}{Z_{\delta}}\cdot\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(I_{\tilde{L}}\otimes\rho_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]}$

and

$\displaystyle\frac{1}{Z_{\delta}}\cdot\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(\rho_{\tilde{L}}\otimes
I_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]}$
(4.35)
$\displaystyle\qquad\to\frac{1}{N-K}\sum_{i=1}^{N-K}\frac{1}{Z_{\delta}}\cdot\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(\rho_{\tilde{L}}\otimes
I_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]}.$

These replacements would change the correlators in sub-leading orders of $N$,
but the essential physics would not be changed, because of typicality.

These averaged two point functions are special cases of the (right-left)
modular-flowed correlators of the form

$\frac{1}{N-K}\sum_{i=1}^{N-K}\frac{\matrixelement{\text{TFD}}{\psi_{i,R}(\tau)\left(\rho_{\tilde{L}}^{n-1-k}\otimes\rho_{KR}^{k}\right)\psi_{i,L}(\tau^{\prime})}{\text{TFD}}}{\tr\left[\rho_{KR}^{n}\right]},$
(4.36)

where one of the fermions is put on the left system and the other one is on
the right system. In the Euclidean regime, they are computed by using the
replica trick in Chandrasekaran:2022qmq  when $K\ll N$.

We use the result to compute “Rényi-2” (left-left) modular-flowed correlators
(4.34) and (4.35) from the Euclidean (right-left) correlator (4.36), by taking
the limits $k\to n-1$ (and $k\to 0$), and $n\to 2$, then analytically
continuing to the Lorentzian regime. We note there is a difference between the
above correlator (4.36) computed in Chandrasekaran:2022qmq  and our
correlators (4.34) and (4.35), namely that in(4.36) two fermions are living on
opposite sides but in our correlators they live on the same side. In our
setup, one can relate the correlator to the following diagrams (figure 7).

Figure 7: Diagrams for the path integral calculation of the correlator (4.29)
with using the relation (4.32) (Top), and the other correlator (4.30) with
(4.33) (Bottom). The red regions in the figure correspond to subsystem $RK$,
and the blue regions correspond to subsystem $\tilde{L}$. The semicircles
correspond to the Euclidean segments that prepares the TFD states. Orange dots
represent the insertions of the SYK Majorana fermion with the regularization,
$\psi_{i,L}(t+i\delta)$. The combination of the upper two semicircles with the
operator insertions corresponds to the density matrix
$\tr_{\tilde{L}}[\psi_{i,L}\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\psi_{i,L}^{\dagger}]$
(and
$\tr_{KR}[\psi_{i,L}\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\psi_{i,L}^{\dagger}]$),
and the remaining combination represents the other one $\rho_{KR}$ (and
$\rho_{\tilde{L}}$). Solid green arrows in the figure correspond to $\beta/2$
Euclidean evolutions. The two insertions are separated by Euclidean time
$2\beta$ (Top), and $\beta$ (Bottom). These separations are directly related
to $\tau+2\beta$ and $\tau+\beta$ appearing in (4.37)and (4.38) respectively.

We study the correlators in the large $\beta J$ limit because their analytic
expressions of are available in the limit. One can instead work in the large
$q$ limit while keeping the value of $\beta J$ finite. We will not do this
here because it is the former limit where the generalization to two-
dimensional CFT is straightforward WIP . The right hand side of (4.34) and
(4.35) in the Euclidean regime are evaluated in the large $\beta J$ and $K\ll
N$ limit as

$\displaystyle\frac{1}{N-K}\sum_{i=1}^{N-K}\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(\tau)\,\left(I_{\tilde{L}}\otimes\rho_{KR}\right)\,\psi_{i,L}(\tau^{\prime})}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]}$
(4.37)
$\displaystyle=G_{2\beta}(\tau+2\beta-\tau^{\prime})+2\frac{K}{N}\left(\mathcal{F}(\tau+2\beta,\tau^{\prime};\beta,0)-\mathcal{F}_{0}(\tau+2\beta,\tau^{\prime};\beta,0)\right)+\mathcal{O}\left(\left(\frac{K}{N}\right)^{2}\right),$
$\displaystyle\frac{1}{N-K}\sum_{i=1}^{N-K}\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(\tau)\,\left(\rho_{\tilde{L}}\otimes
I_{KR}\right)\,\psi_{i,L}(\tau^{\prime})}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]}$
(4.38)
$\displaystyle=G_{2\beta}(\tau+\beta-\tau^{\prime})+2\frac{K}{N}\left(\mathcal{F}(\tau+\beta,\tau^{\prime};\beta,0)-\mathcal{F}_{0}(\tau+\beta,\tau^{\prime};\beta,0)\right)+\mathcal{O}\left(\left(\frac{K}{N}\right)^{2}\right),$

Here, $G_{2\beta}(\tau)$ is an Euclidean thermal SYK two point function for
subsystem $\tilde{L}$ with periodicity $2\beta$,
$\mathcal{F}_{0}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$ is the connected SYK
four point function, which is related to the bare one
$\mathcal{F}_{0}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$ by the so-called ladder
kernel $K_{c}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$,

$\displaystyle\mathcal{F}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$
$\displaystyle=\int d\tau\,\int
d\tau^{\prime}\,\frac{1}{1-K_{c}(\tau_{1},\tau_{2};\tau,\tau^{\prime})}\mathcal{F}_{0}(\tau,\tau^{\prime};\tau_{3},\tau_{4}),$
(4.39) $\displaystyle\mathcal{F}_{0}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$
$\displaystyle=G_{2\beta}(\tau_{13})G_{2\beta}(\tau_{42})-G_{2\beta}(\tau_{14})G_{2\beta}(\tau_{32}),\qquad\tau_{ij}=\tau_{i}-\tau_{j},$
$\displaystyle K_{c}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$
$\displaystyle=-J^{2}(q-1)G_{2\beta}(\tau_{13})G_{2\beta}(\tau_{24})\left(G_{2\beta}(\tau_{34})\right)^{q-2}.$

In the SYK model, these two-point and four point functions are well-studied in
many papers e.g., Maldacena:2016hyu ; Polchinski:2016xgd ; Bagrets:2017pwq ;
Gross:2017aos ; Kitaev:2017awl ; Romero-Bermudez:2019vej . See also
Trunin:2020vwy ; Sarosi:2017ykf  for the review and references therein.

The Euclidean times $\tau,\tau^{\prime}$ (4.37) and (4.38) are continued to
the Lorentzian time with a regularization parameter $0<\delta\ll 1$; $\tau\to-
it-\delta,\tau^{\prime}\to-it+\delta$. In this way, the correlator (4.37) is
continued to Lorentzian time as an out-of-time ordering correlator (OTOC),
$\tau_{1}>\tau_{3}>\tau_{2}>\tau_{4}$, under the condition $1\ll\beta J\ll
N/K$. This correlator with the ordering is given by Maldacena:2016hyu ;
Trunin:2020vwy ,

$\mathcal{F}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})=G_{2\beta}(\tau_{12})G_{2\beta}(\tau_{34})\frac{2\beta
J}{q^{2}\pi
C}\left[1-\frac{\pi}{2}\dfrac{\sin\left(\dfrac{\pi}{\beta}\Delta\tau\right)}{\sin\left(\dfrac{\pi}{\beta}\cdot\dfrac{\tau_{12}}{2}\right)\sin\left(\dfrac{\pi}{\beta}\cdot\dfrac{\tau_{34}}{2}\right)}\right],$
(4.40)

where $\Delta\tau=(\tau_{1}+\tau_{2})/2-(\tau_{3}+\tau_{4})/2$, and $C$ is a
constant related to an overall constant of the Schwarzian action derived from
the Schwinger-Dyson equation of the SYK model Maldacena:2016hyu ;
Trunin:2020vwy . Thus, we have the following continuation

$\displaystyle\mathcal{F}(\tau+2\beta,\tau^{\prime};\beta,0)\to$
$\displaystyle\mathcal{F}(-it-\delta+2\beta,-it+\delta;\beta,0)$ (4.41)
$\displaystyle=2G_{2\beta}(2\beta-2\delta)G_{2\beta}(\beta)\cdot\frac{2\beta
J}{q^{2}\pi
C}\left[1-\frac{\pi}{2}\frac{\cosh\left(\frac{\pi}{\beta}t\right)}{\sin\left(\frac{\pi\delta}{\beta}\right)}\right]$
$\displaystyle\approx-2G_{2\beta}(2\beta-2\delta)G_{2\beta}(\beta)\cdot\frac{\beta
J}{2q^{2}C}\cdot\frac{\exp\left(\dfrac{\pi}{\beta}t\right)}{\sin\left(\dfrac{\pi\delta}{\beta}\right)}.$

In particular, the correlator is exponentially growing in time. On the other
hand, the other correlator (4.38) is continued to Lorentzian time with the
ordering $\tau_{3}>\tau_{1}>\tau_{2}>\tau_{4}$ under the condition $1\ll\beta
J\ll N/K$, therefore it is not OTOC. The correlator with the ordering
$\tau_{3}>\tau_{1}>\tau_{2}>\tau_{4}$ is given by

$\displaystyle\mathcal{F}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$ (4.42)
$\displaystyle=-G_{2\beta}(\tau_{12})G_{2\beta}(\tau_{34})\frac{2\beta
J}{q^{2}\pi
C}\left[\left(\dfrac{\pi\tau_{12}}{2\beta\tan\left(\dfrac{\pi}{\beta}\cdot\dfrac{\tau_{12}}{2}\right)}+\dfrac{\pi}{\tan\left(\dfrac{\pi}{\beta}\cdot\dfrac{\tau_{12}}{2}\right)}-1\right)\left(\dfrac{\pi\tau_{34}}{2\beta\tan\left(\dfrac{\pi}{\beta}\cdot\dfrac{\tau_{34}}{2}\right)}-1\right)\right],$

and its analytic continuation is

$\displaystyle\mathcal{F}(\tau+\beta,\tau^{\prime};\beta,0)$ (4.43)
$\displaystyle\to\mathcal{F}(-it-\delta+\beta,-it+\delta;\beta,0)=-2G_{2\beta}(\beta-2\delta)G_{2\beta}(\beta)\cdot\frac{2\beta
J}{q^{2}\pi
C}\left[1-\left(\dfrac{\pi}{2}-\dfrac{\pi\delta}{\beta}\right)\tan\left(\dfrac{\pi\delta}{\beta}\right)\right].$

Clearly, this is time-independent unlike the previous case.

We do not evaluate bare four point functions
$\mathcal{F}_{0}(\tau_{1},\tau_{2};\tau_{3},\tau_{4})$ for (4.37) and (4.38),
because they are particular combinations of the thermal SYK two point
functions with the power low behavior with respect to time, therefore they do
not give dominant contributions to the correlators (4.37) and (4.38).

Combining the above results, we can obtain the analytic expressions of the
quantities (4.37) and (4.38),

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}$ (4.44)
$\displaystyle\approx\frac{1}{Z_{\delta}}\left[G_{2\beta}(2\beta-2\delta)-G_{2\beta}(2\beta-2\delta)G_{2\beta}(\beta)\cdot\frac{2\beta
J}{q^{2}C}\cdot\frac{K}{N}\frac{\exp\left(\dfrac{\pi}{\beta}t\right)}{\sin\left(\dfrac{\pi\delta}{\beta}\right)}+\mathcal{O}\left(\left(\frac{K}{N}\right)^{2}\right)\right]$
$\displaystyle\approx\frac{G_{2\beta}(2\beta-2\delta)}{G_{\beta}(2\delta)}\left[1-\frac{G_{2\beta}(\beta)}{\sin\left(\dfrac{\pi\delta}{\beta}\right)}\cdot\frac{2\beta
J}{q^{2}C}\cdot\frac{K}{N}\exp\left(\dfrac{\pi}{\beta}t\right)\right],$

and

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}$ (4.45)
$\displaystyle\approx\frac{1}{Z_{\delta}}\left[G_{2\beta}(\beta-2\delta)-G_{2\beta}(\beta-2\delta)G_{2\beta}(\beta)\cdot\frac{8\beta
J}{q^{2}\pi
C}\cdot\frac{K}{N}\left[1-\left(\dfrac{\pi}{2}-\dfrac{\pi\delta}{\beta}\right)\tan\left(\dfrac{\pi\delta}{\beta}\right)\right]+\mathcal{O}\left(\left(\frac{K}{N}\right)^{2}\right)\right]$
$\displaystyle\approx\frac{G_{2\beta}(\beta-2\delta)}{G_{\beta}(2\delta)}\left[1-G_{2\beta}(\beta)\cdot\frac{8\beta
J}{q^{2}\pi
C}\cdot\frac{K}{N}\left[1-\left(\dfrac{\pi}{2}-\dfrac{\pi\delta}{\beta}\right)\tan\left(\dfrac{\pi\delta}{\beta}\right)\right]\right],$

where we ignored would-be sub-leading terms, coming from the replacements
(4.34) and (4.35) in (4.29) and (4.30), and the sub-sub-leading terms of the
averaged correlators.

Let us consider the consequences of the above results. First, we focus on the
ratios $G_{2\beta}(2\beta-2\delta)/G_{\beta}(2\delta)$ and
$G_{2\beta}(\beta-2\delta)/G_{\beta}(2\delta)$ appearing in the above results.
Since the SYK two point function under the conformal limit $\beta J\gg 1$ is
given by Maldacena:2016hyu ,

$G_{\beta}(\tau)=b\left[\frac{\pi}{\beta\sin\frac{\pi\tau}{\beta}}\right]^{2\Delta},\qquad\Delta=\frac{1}{q}\quad
J^{2}b^{q}\pi=\left(\frac{1}{2}-\Delta\right)\tan\pi\Delta,$ (4.46)

we can evaluate the ratios as follows

$\displaystyle\frac{G_{2\beta}(2\beta-2\delta)}{G_{\beta}(2\delta)}=\cos^{2\Delta}\left(\frac{\pi\delta}{\beta}\right),$
(4.47)

and

$\displaystyle\frac{G_{2\beta}(\beta-2\delta)}{G_{\beta}(2\delta)}=\sin^{2\Delta}\left(\frac{\pi\delta}{\beta}\right).$
(4.48)

Thus, these ratios can not be $1$ simultaneously for general $\delta$ and
$\beta$. However since $\Delta=1/q$ when $q$ is large these ratios are close
to $1$. We give plots of the above two functions for several $q$ in figure 8.
As we can see from plots 8 or directly from (4.47) and (4.48), we need to
consider a (relatively) large-$q$ regime, which implies that the SYK Majorana
fermion has a small conformal dimension, $\Delta=1/q\ll 1$, in order to
achieve recovery.

One may wonder why here we take the large $q$ limit, because the (SYK$)_{q}$
is chaotic for all $q\geq 4$ so the identities (4.27), (4.28) are expected to
hold for any value of $q$ in this range. Nevertheless here we have to take the
large $q$ limit because we define the code subspace using the SYK Majorana
fermion operator $\psi_{i,L}$ and the calculations of the relevant correlation
functions can be possible only in the large $\beta J$ limit where the
entanglement between $L$ and $R$ is weak. Because of the weakness of the
entanglement, the recovery is only possible when the dimension of the operator
that defines the code subspace is small, implying the necessity of taking the
large $q$ limit.

Figure 8: Plots of ratios (4.47) and (4.48) as a function of $\beta J$ for
smaller $q$ (Top), and for larger $q$ (Bottom). Here, we set $\delta J=0.1$.
For large $q$ regions, all the ratios become close to $1$.

Next, we consider the two point function $G_{2\beta}(\beta)$ appearing in the
sub-leading terms. The two point function $G_{2\beta}(\beta)$ can be written
as

$G_{2\beta}(\beta)=b\left[\frac{\pi}{2\beta\sin\frac{\pi}{2}}\right]^{2\Delta}=\left[\left(\frac{1}{2}-\Delta\right)\frac{\pi\tan\pi\Delta}{(2\beta
J)^{2}}\right]^{\Delta}.$ (4.49)

The above expression includes $(1/\beta J)^{\Delta}$, so in $\beta J\to\infty$
limit, the SYK two point function $G_{2\beta}(\beta)$ vanishes. We also note
the $q$-dependence of the SYK two point function. Plots of the above function
and $\beta JG_{2\beta}(\beta)$ for several $q=\Delta^{-1}$ are given in figure
9 and 10 respectively. The plots show that as $q$ increases, the two point
function $G_{2\beta}(\beta)$ and $\beta JG_{2\beta}(\beta)$ take larger
values.

Figure 9:  Plots of the SYK two point function $G_{2\beta}(\beta)$, (4.49) as
a function of $\beta J$ for several $q=\Delta^{-1}$. Figure 10:  Plots of
$\beta JG_{2\beta}(\beta)$ as a function of $\beta J$ for several
$q=\Delta^{-1}$. The dotted line is just $\beta J$, which is equivalent to
$\beta JG_{2\beta}(\beta)$ under the $q\to\infty$ limit.

Thus, from the above discussion, in the strict $\beta J\to\infty$
limit141414We note that to consider the perturbative expansion, we have
assumed $\beta J\ll N/K$, and also implicitly assumed $q\ll N/K$ for large
$q$. Thus, we can not take the $\beta J\to\infty$ or $q\to\infty$ limits,
unless we take the $N/K\to\infty$ limit. However, the limit $N/K\to\infty$
implies that there is almost no Hawking radiation compared to the entire
Hawking radiation $K/N\to 0$. Intuitively, in such a situation, we would not
be able to recover the diary information from the Hawking radiation. , we have
$G_{2\beta}(\beta)\to 0$, hence the second terms including $G_{2\beta}(\beta)$
in (4.44) and (4.45) vanish if we keep the exponential factor $\exp\left(\pi
t/\beta\right)$ in (4.45) fixed. Therefore, in this strict $\beta J\to\infty$
limit, we can not have contributions from the second terms including
$G_{2\beta}(\beta)$ in (4.44) and (4.45). These terms are of order $K/N$ and
crucial for the following discussion.

Finally, let us focus on the time dependence of the results (4.44) and (4.45).
First, we focus on the second case (4.45). This result is time-independent at
least up to the $K/N$-order, and the second term is always suppressed by the
time-independent factor at the $K/N$-order, so the second terms is very small
compared with the first term. This implies that the quantity (4.45) is almost
given by the ratio $G_{2\beta}(\beta-2\delta)/G_{\beta}(2\delta)$, which
becomes close to $1$ when $q$ is large.

Next, we focus on (4.44). Because of the exponential time dependent factor,
this correlator has crucially different behavior as a function of time from
(4.45). For early times $t\ll 1$, the exponential in the second term can be
approximated by $1$, they are similar. However, because of the exponentially
growing factor, the perturbative expansion with respect to $K/N$ breaks down,
similar to the fact that the perturbative calculations of OTOCs in $1/N$
become invalid. The time scale of this break down can be estimated by equating
the second term with the first term in (4.44). From the condition, we can find
a critical time $t_{\text{*}}$151515 In defining the critical time, we might
have the ambiguity that which factors should be included into the critical
time (or correspondingly the scrambling time), e.g., $\beta J$ and also
$G_{2\beta}(\beta)$. However, as we saw before, the two point function is
typically order one $G_{2\beta}(\beta)=\mathcal{O}(1)$, so we might need not
to include the factor to the scrambling time. Another factor
$1/\sin\left(\frac{\pi\delta}{\beta}\right)$ can be set to be $\mathcal{O}(1)$
by setting the cutoff $\delta$ suitably. For the other factor $\beta J$, since
we have the condition $\beta J\ll N/K$, the factor can not give a significant
contribution compared to the leading factor $N/K$, thus including the factor
would be redundant. Therefore, the critical time here would be the simplest
choice.,

$\frac{K}{N}\exp\left(\frac{\pi}{\beta}t_{*}\right)\sim
1\qquad\Longrightarrow\quad
t_{*}=\frac{\beta}{\pi}\log\left(\frac{N}{K}\right)=2t_{\text{Scram}},$ (4.50)

where we introduce the usual scrambling time $t_{\text{Scram}}$, Sekino:2008he
given by

$t_{\text{Scram}}=\frac{\beta}{2\pi}\log\left(\frac{N}{K}\right).$ (4.51)

Using this time scale, we can rewrite the correlator (4.44) as

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}$ (4.52)
$\displaystyle\approx\frac{G_{2\beta}(2\beta-2\delta)}{G_{\beta}(2\delta)}\left[1-\frac{G_{2\beta}(\beta)}{\sin\left(\dfrac{\pi\delta}{\beta}\right)}\cdot\frac{2\beta
J}{q^{2}C}\cdot\exp\left(\frac{\lambda_{L}}{2}\left(t-2t_{\text{Scram}}\right)\right)\right],$

where we introduce the Lyapunov exponent $\lambda_{L}$ for a black hole with
temperature $\beta$,

$\lambda_{L}=\frac{2\pi}{\beta}.$ (4.53)

Thus around the critical time, which is twice the scrambling time, we can see
that the overall coefficient of
$G_{2\beta}(2\beta-2\delta)/G_{2\beta}(2\delta)$ becomes very small as usual
OTOC correlators. This reproduces the expected result (4.27) under the
condition $\beta J\gg 1$.

From the discussion so far, we have confirmed that the matrix elements (4.24),
(4.25) and (4.26) do behave as we expect them to under the condition
$1\ll\beta J\ll N/K$.

##  5 Expected properties of the Petz-lite under the SYK dynamics

So far, we have confirmed that the matrix elements we computed (4.27)and
(4.28) reproduce our expected results under the conditions of relatively
large-$q$ interaction, after the critical time $t_{*}=2t_{{\rm Scram}}$.
Additionally, of course, the following trivial matrix element is equal to $1$
by the definition,

$\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{0}=1.$ (5.1)

Also, we can obtain the same consequences for two related matrix elements. Let
us explain them. First, the matrix element (4.27), which becomes close to $0$,
is directly related to

$\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{1}]]\right]}{0}$ (5.2)

via the definition of the adjoint channel (2.7). Thus, this matrix element
also becomes close to $0$ after the critical time, and the behavior is
consistent with our expectation.

Next, for the matrix element (4.28), being almost equal to $1$, we have the
following relation through the definition of the adjoint channel (2.7) again,

$\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{0}]]\right]}{0}.$ (5.3)

Thus, although we have eight non-trivial matrix elements (4.23) that should be
checked, we already know the behavior of the above five matrix elements, and
there are still three matrix elements. However, since two of them are related
by the complex conjugation, essentially we need to investigate following two
matrix elements

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{0}]]\right]}{1},$ (5.4)

and

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{1}]]\right]}{1}.$ (5.5)

Here, the first matrix element is related to the following one

$\displaystyle\left(\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{0}]]\right]}{1}\right)^{*}=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{0}.$ (5.6)

In evaluating these matrix elements, we can not directly use the technique of
Chandrasekaran:2022qmq  unlike the cases for the matrix elements (4.29) and
(4.30)161616We briefly explain the reason why the evaluations of the matrix
elements (5.5) and (5.6) are difficult. The reason is that they do not have
simple expressions like (4.32) and (4.33) naively. Of course, for matrix
element (5.5), we can consider the similar expression like (4.32) with
replacing the TFD state with the excited state
$\psi_{i,L}\ket{\text{TFD}}_{L,R}$, but in that case, we can no longer use the
techniques in Chandrasekaran:2022qmq , and we need to consider the modular
operator for the excited state. For the other matrix element (5.6), we naively
need to introduce transition matrices, not density matrices, to write it in
terms of a correlator.. In the upcoming paper WIP , we will report their
results, but here we explain their expected behaviors from our obtained
results. To this end, it would be useful to introduce the Kraus representation
of the quantum channel (4.10),

$\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\rho_{T}]=\sum_{m=1}^{d_{\tilde{L}}}E_{m}^{\text{SYK}}\rho_{T}E_{m}^{\text{SYK}\dagger}$
(5.7)

given by

$E_{m}^{\text{SYK}}=\,_{\tilde{L}}\\!\bra{m}U_{L}V_{T,L\to
L}\ket{\text{TFD}}_{LR}.$ (5.8)

We can obtain this Kraus representation by introducing an orthonormal basis of
the subsystem $\tilde{L}$ as
$\left\\{\ket{m}_{\tilde{L}}\right\\}_{m=1}^{d_{\tilde{L}}}$. We also note
that the adjoint channel (4.11) can be written as

$\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}[\mathcal{O}_{KR}]=\sum_{m=1}^{d_{\tilde{L}}}E_{m}^{\text{SYK}\dagger}\mathcal{O}_{KR}E_{m}^{\text{SYK}}.$
(5.9)

Using this Kraus representation, it is possible to extract the very important
“typical” relation from our results. Here, the “typical” means that the
relation almost does not depend on the detail of a specific state
$\ket{m}_{\tilde{L}}$ in the subsystem $\tilde{L}$, corresponding to a black
hole microstate. First, the matrix elements (5.1) is equal to $1$ and can be
expressed as

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{0}$
$\displaystyle=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\sum_{m,n=1}^{d_{\tilde{L}}}\,{}_{T}\matrixelement{0}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}\ket{0}_{T}\\!\bra{0}E_{n}^{\text{SYK}}E_{m}^{\text{SYK}\dagger}}{0}_{T}$
(5.10)
$\displaystyle=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\sum_{m,n=1}^{d_{\tilde{L}}}\left|\,{}_{T}\matrixelement{0}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}}{0}_{T}\right|^{2},$

and we expect the typical relation

$\,{}_{T}\matrixelement{0}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}}{0}_{T}\sim\frac{1}{\sqrt{d_{\tilde{L}}\cdot\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}}}\delta_{mn}.$
(5.11)

Next, we focus on the matrix element (4.28). This matrix element is also equal
to $1$, and we can express the matrix element in terms of the Kraus operators,

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}$
$\displaystyle=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\sum_{m,n=1}^{d_{\tilde{L}}}\,{}_{T}\matrixelement{0}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}\ket{0}_{T}\\!\bra{1}E_{n}^{\text{SYK}}E_{m}^{\text{SYK}\dagger}}{1}_{T}.$
(5.12)

By using the relation (5.11), we extract a similar relation,

$\,{}_{T}\matrixelement{1}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}}{1}_{T}\sim\frac{1}{\sqrt{d_{\tilde{L}}\cdot\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}}}\delta_{mn}.$
(5.13)

Finally, the time-dependent matrix element (4.27), which almost vanishes
around the critical time $t_{*}$, can be written as

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}$
$\displaystyle=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\sum_{m,n=1}^{d_{\tilde{L}}}\,{}_{T}\matrixelement{1}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}\ket{0}_{T}\\!\bra{0}E_{n}^{\text{SYK}}E_{m}^{\text{SYK}\dagger}}{1}_{T}$
(5.14)
$\displaystyle=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\sum_{m,n=1}^{d_{\tilde{L}}}\left|\,{}_{T}\matrixelement{1}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}}{0}_{T}\right|^{2}.$

From this expression, we expect the following relation and its complex
conjugation,

$\,{}_{T}\matrixelement{1}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}}{0}_{T}\sim
0,$ (5.15)

around and/or after the critical time.

Combing the above expectations, we obtain the typically expected
relation171717Here, we check the Knill-Laflamme condition from our obtained
results. However, in principle, it would be possible to investigate the Knill-
Laflamme condition directly by introducing a basis Qi:2018bje . It would be
interesting to investigate this topic.

$\,{}_{T}\matrixelement{T}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}}{T^{\prime}}_{T}\sim\frac{1}{\sqrt{d_{\tilde{L}}\cdot\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}}}\delta_{mn}\delta_{TT^{\prime}}\qquad\text{
for }t\gtrsim t_{*},$ (5.16)

which corresponds to the Knill-Laflamme condition Knill:1996ny .

Using this relation, the remaining matrix elements (5.4), (5.5) are expected
to behave as follows

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{0}]]\right]}{1}$
$\displaystyle=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\sum_{m,n=1}^{d_{\tilde{L}}}\,{}_{T}\matrixelement{0}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}\ket{1}_{T}\\!\bra{0}E_{n}^{\text{SYK}}E_{m}^{\text{SYK}\dagger}}{1}_{T}$
(5.17) $\displaystyle\sim 0\qquad\text{ for }t\gtrsim t_{*},$

and

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{1}]]\right]}{1}$
$\displaystyle=\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\sum_{m,n=1}^{d_{\tilde{L}}}\,{}_{T}\matrixelement{1}{E_{m}^{\text{SYK}\dagger}E_{n}^{\text{SYK}}\ket{1}_{T}\\!\bra{1}E_{n}^{\text{SYK}}E_{m}^{\text{SYK}\dagger}}{1}_{T}$
(5.18) $\displaystyle\sim 1.$

These results are, of course, consistent with our original expectation (4.23),
but the discussion so far using the typical relation is indirect (5.16).
Nevertheless, since this typicality is strong enough for a highly chaotic
theory, we expect that nearly identical results can be obtained by direct
calculations of the matrix elements (5.4) and (5.5).

##  6 Discussion

In this paper, we studied a recovery map for the Hayden-Preskill type
scrambling channel $\mathcal{N}$. We showed that one can use a simplified
recovery map, called Petz-lite, consisting of the adjoint channel
$\mathcal{N}^{\dagger}$ with a suitable normalization factor for this purpose.
We considered two examples, the Hayden-Preskill setup and the SYK model, and
show that in both cases the Petz-lite indeed works as a recovery map. Also, we
find that if the Petz-lite for the SYK case is used to recover information of
given code subspace, it takes twice the scrambling time for the recovery.
However, the SYK model case includes we did not evaluate all of the matrix
elements necessary to show the recovery because of technical difficulties.
Instead, we evaluate them in an indirect way in section 5. In the upcoming
paper WIP , we will explain their results, and also some generalizations of
our results.

Let us discuss our results. First, we focus on the physical interpretation of
the critical time given by twice the scrambling time,
$t_{*}=2t_{\text{Scram}}$, when the matrix elements gives the input
information, $\mathcal{R}[\mathcal{N}[\rho]]\sim\rho$. It was argued in
Sekino:2008he  that information of a diary thrown into a black hole appears
after the scrambling time. This means that, after the scrambling time, the HP
scrambling channel $\mathcal{N}$ maps the diary information to Hawking
radiation completely. However, even if the diary information appears in the
Hawking radiation, it is difficult to get it directly since the information is
uniformly embedded into the Hawking radiation. To extract the information, we
need a recovery operation given by the Petz-lite
$\mathcal{R}\sim\mathcal{N}^{\dagger}$. Since it is the adjoint of the HP
channel $\mathcal{N}$, it again takes the scrambling time to apply the
recovery map. Thus, in total, we need to wait for twice the scrambling time
for the identity (4.23) to get satisfied.

Next, let us explain the bulk interpretation of our results181818We note that
since currently there is no clear understanding of a dual gravitational theory
for a subset of the SYK Majorana fermions (or Majorana spin chain), we can not
check the interpretation using the gravity side explicitly at least in the
context of NAdS2/NCFT1 context. However, there are several proposals for such
a gravitational treatment, e.g., in Chandrasekaran:2022qmq . One would be able
to use them to check the bulk interpretation.. The bulk interpretation comes
from the island prescription Penington:2019npb ; Almheiri:2019psf . First, the
Hayden-Preskill setup concerns post-Page time regimes. In these regimes, there
is an island, which is a non-trivial entanglement wedge of Hawking radiation
in the black hole interior. Thus, if one throws a diary into a black hole and
waits for the scrambling time, then the diary enters the island region,
implying that the diary is encoded into the Hawking radiation in a very
complicated way. The mechanism that the thrown diary is encoded into the
Hawking radiation corresponds to our quantum channel $\mathcal{N}$. To recover
the diary information from the Hawking radiation, we need to consider the
recovery operation corresponding to the map
$\mathcal{R}\sim\mathcal{N}^{\dagger}$. The recovery map is given by the
adjoint channel of the quantum channel $\mathcal{N}$. In the bulk side, the
action of the adjoint channel $\mathcal{N}^{\dagger}$ means that the “reverse”
process of the original quantum channel $\mathcal{N}$191919Here, we note that
in these two processes, we need to use two different (remaining) black holes
since, in defining the quantum channel, (remaining) black holes are treated as
internal degrees of freedom of the quantum channel.. More precisely, the
“reverse” process is given as follows: First, we start from the output state
provided by the action of the quantum channel $\mathcal{N}$, implying the
diary is located on the island at some time slice $\Sigma$. The application of
the adjoint channel $\mathcal{N}^{\dagger}$ then is interpreted as replacing
the future of this time slice $\Sigma$ by a white hole. Because of the
replacement, the diary on the island region of the original black hole is
coming out from the horizon of the white hole. Here, the reason why the white
hole appears is that the adjoint channel includes the Hermit conjugation of
unitaries $U$ (and $U^{\dagger}$) compared to the quantum channel
$\mathcal{N}$. Thus, the diary thrown into the black hole reappears from the
white hole induced by $\mathcal{N}^{\dagger}$. This bulk interpretation is
consistent with the critical time. This is because, after throwing the diary,
it takes the scrambling time for the diary to enter the island region, and in
the “reverse” process, it would also take the scrambling time for the diary to
go outside the island region and the horizon.

Finally, we end with discussing some of our in-progress works and future
directions:

#### Analysis in high temperature regime, $\beta J\ll 1$

In this paper, we have focused on the large $\beta J$ limit (low-temperature
limit) in the SYK model to make the calculation analytic and for the purpose
of the generalization to the CFT 2 case. In the limit, we can use emergent
conformal symmetry of the SYK model and also we would be able to use semi-
classical intuition of the dual Jackiw-Teitelboim gravity, but we have a
relatively weak initial entangle state $\ket{\text{TFD}}_{L,R}$ between the
left and right SYK systems. Due to this weak entangle state, we would require
some conditions to consider successful recovery protocol, e.g., large-$q$
regime. Thus, analysis without taking the large $\beta J$ limit would be
interesting. In that case, we would need to consider numerical approaches.

#### Direct bulk analysis and relation to other protocols

In this paper, we studied the recovery protocol from the boundary CFT
perspective. One would be able to consider corresponding bulk computations.
Also, it would be interesting to figure out the relation between other
proposed protocols e.g., Gao:2019nyj ; Brown:2019hmk ; Schuster:2021uvg ;
Nezami:2021yaq  and ours202020For such protocols, one can characterize
protocol by computing “price”, “distance”, etc. Pastawski:2016qrs ;
Chandrasekaran:2022qmq ; Bentsen:2023xlu . One would be able to find the
relation between our results and such quantities..

#### Generalization to (Holographic) CFT2 and other systems

While this paper focuses on the SYK model, which is a $0+1$-dimensional
quantum system, it can also be interpreted as a spin chain with $q$-body SYK
interactions. Thus, we can interpret that the SYK model has a spatial
direction effectively. As a result, we expect that a similar analysis can be
applied to a two-dimensional CFT exhibiting chaos, e.g., holographic CFT2.
Indeed, one of the Hayden-Preskill setups in a two-dimensional holographic CFT
is introduced in Chandrasekaran:2021tkb .

Also, there are other possibilities for generalizations to other systems
exhibiting chaos. For example, studying the Petz-lite in a chaotic spin chain
would be interesting.

#### Chaotic-Integrable transition

In this paper, the chaotic nature is important for the simplification of the
Petz map to the Petz-lite. Thus, if a system do not exhibit the chaotic
nature, in other words, the system is integrable, then the Petz-lite (also the
original Petz map) is not expected to works correctly. This is because, in an
integrable system, the decoupling condition is not expected to hold. In the
framework of the SYK model, we can prepare integrable and non-integrable
(chaotic) situation by adding two-body interaction Garcia-Garcia:2017bkg .
Using the setup, we would be able to study Petz-lite.

#### Higher dimensional code sub-space?

The SYK version of the HP setup studied in this paper treats the two-
dimensional code sub-space spanned by the vacuum and the excited state.
However, in a more realistic situation, one needs to deal with code sub-spaces
with dimensions greater than two. For example, the interior of a black hole,
when it is viewed as a code subspace embedded into the Hawking radiation, the
dimension of its Hilbert space has to be large enough to accommodate a part of
the semi-classical QFT degrees of freedom to have a geometric interpretation
of the black hole interior212121Of course, the interior degrees of freedom may
appear to be infinite, but almost all of them can not contribute due to post-
selection Akers:2022qdl . Even in that case, there can be degrees of freedom
with Bekenstein-Hawking entropy.. To this end, one would need to consider a
more complicated embedding involving for example states like,
$\psi_{i,L}\psi_{j\neq i,L}\ket{\text{TFD}}_{L,R}$. In that case, we can
evaluate corresponding matrix elements in principle, but it would be difficult
to them analytically since we encounter higher-point functions.

Another possibility for higher dimensional code sub-space is to consider a
random embedding and the double-scaling limit. For example, we might be able
to use the state $\kappa_{ij}\psi_{i,L}\psi_{j,L}\ket{\text{TFD}}_{L,R}$,
where $\kappa_{ij}$ is random like observables in the double-scaled SYK model
Berkooz:2018jqr . In this case, by taking the double-scaling limit and using
chord diagram techniques, we might be able to evaluate the resulting matrix
element analytically. Also, this might open up an interesting connection
between QEC in the SYK model and recent discussions of the von Neumann algebra
of quantum gravity, in particular, Lin:2023trc .

## Acknowledgments

We thank Yoshifumi Nakata for discussions. AM thanks Norihiro Iizuka, Tomoki
Nosaka, Masahiro Nozaki and Jia Tian for comments. AM also thanks Chen Bai for
related discussions. AM thanks the workshop “Beijing Osaka
String/Gravity/Black Hole Workshop” at KITS, where this work was presented. AM
also thanks the long-term work shop “Quantum Information, Quantum Matter and
Quantum Gravity” YITP-T-23-01 at YITP, where this work was also presented. YN
was supported by JST, the establishment of university fellowships towards the
creation of science technology innovation, Grant Number JPMJFS2123. TU was
supported in part by JSPS Grant-in-Aid for Young Scientists 19K14716 and in
part by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A “Extreme
Universe” No.21H05184.

##  Appendix A Derivation of the Petz lite using Kraus representation

In this appendix, we derive the Petz-lite with a different normalization
factor based on paper barnum2000reversing . See e.g., §10.3 of
nielsen_chuang_2010  for related reviews.

We start with the Kraus representation of the HP channel (2.3). The Kraus
representation can be introduced by expressing the trace as

$\displaystyle\mathcal{N}_{T\to D,B}\left[\rho_{T}\right]$
$\displaystyle=\tr_{C}\left[(U_{T,A\to C,D}\otimes
I_{B})(\rho_{T}\otimes\ket{\text{EPR}}_{A,B}\\!\bra{\text{EPR}})(U_{T,A\to
C,D}^{\dagger}\otimes I_{B})\right]$ (A.1)
$\displaystyle=\sum_{m=1}^{d_{C}}\,{}_{C}\matrixelement{m}{(U_{T,A\to
C,D}\otimes
I_{B})\ket{\text{EPR}}_{A,B}\,\rho_{T}\,{}_{A,B}\bra{\text{EPR}}(U_{T,A\to
C,D}^{\dagger}\otimes I_{B})}{m}_{C}$
$\displaystyle=\sum_{m=1}^{d_{C}}E_{m}\rho_{T}E_{m}^{\dagger},$

where $\ket{m}_{C}$ is an orthonormal basis of subsystem $C$, and $E_{m}$ is
the Kraus operator defined by

$E_{m}=\,_{C}\bra{m}(U_{T,A\to C,D}\otimes I_{B})\ket{\text{EPR}}_{A,B}.$
(A.2)

Here, we note that since the state $\ket{m}_{C}$ is a basis state of the
remaining black hole $C$. We also note that the adjoint HP channel is
expressed in terms of the Kraus operators,

$\mathcal{N}\left[\mathcal{O}\right]=\sum_{m=1}^{d_{C}}E_{m}^{\dagger}\mathcal{O}E_{m}.$
(A.3)

Using this Kraus operator, let us investigate the Knill-Laflamme condition
Knill:1996ny ,

$P_{code}E_{m}^{\dagger}E_{n}P_{code}=\alpha_{mn}P_{code}\quad\left(\alpha_{mn}=\alpha_{nm}^{*}\in\mathbb{C}\,\text{with
}\sum_{m=1}^{d_{C}}\alpha_{mm}=1\right),\,\text{for }\forall
m,n=1,\cdots,d_{C}.$ (A.4)

where $P_{code}$ is a projection operator onto a code subspace in general, but
in our setup, $P_{code}$ is assumed to be just given by the identity operator
$P_{code}=I_{T}$, since all input states should be recoverable under the
Hayden-Preskill setup. If this condition holds, we can construct a recovery
map222222See e.g., §10.3, in particular, theorem 10.1, of nielsen_chuang_2010
for the review..

Under Haar random averaging, we can easily evaluate the Knill-Laflamme
condition from the expression (A.2) and Haar average (2.17),

$\displaystyle\overline{E_{m}^{\dagger}E_{n}}=\frac{1}{d_{C}}\delta_{mn}I_{T}$
(A.5)

This result appears to imply that the Knill-Laflamme condition holds always
under the averaging, but this is not correct. This is because, even if the
Knill-Laflamme condition is satisfied, higher moments of the Knill-Laflamme
condition, e.g.,
$\overline{\left|P_{code}E_{m}^{\dagger}E_{n}P_{code}\right|^{2}}$, might not
hold due to contributions coming from Weingarten calculus. We can see their
contributions by directly evaluating the second moment232323See also
Liu:2020sqb  for related discussions.,

$\displaystyle\overline{\left|P_{code}E_{m}^{\dagger}E_{n}P_{code}\right|^{2}}\approx\frac{1}{(d_{C})^{2}}\cdot
I_{T}\left[\delta_{mn}+\frac{d_{T}}{d_{D}d_{B}}\right]$ (A.6)

where we used the know result (2.20) with large-$d$ approximation. Thus, when
we do not have enough Hawking radiation $D,B$ compared to the diary $T$, that
is, $d_{D}d_{B}\gtrsim d_{T}$, we can not ignore the second term, implying the
break down of the Knill-Laflamme condition. On the other hand, in the opposite
limit $d_{D}d_{B}\gg d_{T}$, where we have enough Hawking radiation, we can
ignore the second term, and we get the Knill-Laflamme condition. We note that
this is consistent with the decoupling condition (2.2), since the unitarity
means the relation

$\frac{d_{T}}{d_{D}d_{B}}=\frac{1}{d_{C}}\cdot\left(\frac{d_{T}}{d_{D}}\right)^{2},$
(A.7)

and the factor $(d_{T}/d_{D})^{2}$ gives an upper bound of the decoupling
condition (2.2).

Next, we construct a recovery map for the HP quantum channel. With the Knill-
Laflamme condition in mind, we consider the following map, which is equal to
the adjoint HP channel up to the overall factor $d_{C}$,

$\mathcal{R}[\mathcal{O}]\coloneqq
d_{C}\sum_{m=1}^{d_{C}}E^{\dagger}_{m}\mathcal{O}E_{m}=d_{C}\mathcal{N}^{\dagger}\left[\mathcal{O}\right].$
(A.8)

Under the Haar random average, this map gives

$\displaystyle\overline{\mathcal{R}[\mathcal{N}\left[\rho_{T}\right]]}$
$\displaystyle=d_{C}\sum_{m,n=1}^{d_{C}}\overline{E^{\dagger}_{m}E_{n}\rho_{T}E^{\dagger}_{n}E_{m}}$
(A.9) $\displaystyle\approx
d_{C}\sum_{m,n=1}^{d_{C}}\left[\overline{E^{\dagger}_{m}E_{n}}\rho_{T}\overline{E^{\dagger}_{n}E_{m}}+\overline{E^{\dagger}_{m}\overline{E_{n}\rho_{T}E^{\dagger}_{n}}E_{m}}\right]$
$\displaystyle=d_{C}\sum_{m,n=1}^{d_{C}}\frac{1}{(d_{C})^{2}}\left[\delta_{mn}\,\rho_{T}+\frac{\tr\left[\rho_{T}\right]}{d_{D}d_{B}}I_{T}\right]$
$\displaystyle=\rho_{T}+\left(\frac{d_{T}}{d_{D}}\right)^{2}\cdot\frac{1}{d_{T}}I_{T},$

where in the second line we used the fact that in the large-Hilbert space
dimension limit, Weingarten calculus reduces to Wick calculus, and in the
final line, we used $\tr\rho_{T}=1$ and the relation $d_{T}d_{B}=d_{C}d_{D}$.
In the third line, we encountered the Knill-Laflamme condition for the first
term (A.5), and the second terms disturb the Knill-Laflamme condition. These
two terms in the third line correspond to the first and second terms in (A.6).
Thus, under the situation $d_{B}d_{D}\gg d_{T}$ where the Knill-Laflamme
condition holds (approximately), we can ignore the second term of the above
result, implying that the map(A.8) works as a recovery map. This is a quantum
information theoretic derivation of the Petz-lite. However, we note that the
recovery map here is little bit different from the one (2.11) up to the
overall factor, but the difference almost vanishes when the condition
$d_{B}d_{D}\gg d_{T}$ is satisfied.

Finally, we end this appendix by giving the connection between the Petz map
and the Petz-lite in terms of the Kraus operator and the Knill-Laflamme
condition. Generally, since the coefficients $(\alpha_{mn})$ is Hermitian, we
can diagonalize the Knill-Laflamme condition by some unitary $(U_{mn})$ as
follows nielsen_chuang_2010 ,

$P_{code}F_{m}^{\dagger}F_{n}P_{code}=\lambda_{m}\delta_{mn}P_{code}\quad\left(\lambda_{m}\in\mathbb{R},\text{with
}\sum_{m=1}^{d_{C}}\lambda_{m}=1,\,\lambda_{m}>0\right),\,\text{for }\forall
m,n=1,\cdots,d_{C},$ (A.10)

where $F_{m}=\sum_{n}U_{mn}E_{n}$ is the newly defined Kraus operator. Using
this Kraus operator, one can define the following map

$\mathcal{R}[\mathcal{O}]\coloneqq\sum_{m=1}^{d_{C}}\frac{1}{\lambda_{m}}P_{code}F^{\dagger}_{m}\mathcal{O}F_{m}P_{code}.$
(A.11)

This map can be also expressed in terms of the original quantum channel with
introducing some full rank reference state $\sigma$ as follows
barnum2000reversing

$\mathcal{R}[\mathcal{O}]=\sigma^{1/2}\mathcal{N}^{\dagger}\left[\left(\mathcal{N}[\sigma]\right)^{-1/2}\mathcal{O}\left(\mathcal{N}[\sigma]\right)^{-1/2}\right]\sigma^{1/2},$
(A.12)

and this is exactly the Petz map. In the recovery map (A.11), the factor
$\lambda_{m}$ prevents us from directly giving the adjoint channel
$\mathcal{N}^{\dagger}$, and we need to introduce the curios factors
$\left(\mathcal{N}[\sigma]\right)^{-1/2}$ and $\sigma^{1/2}$. However, for the
case where $\lambda_{m}=1/d_{C}\,(m=1,\cdots,d_{C})$, one can consider the map
(A.8) instead of the above map. As we have seen, the Haar random case with the
Knill-Laflamme condition (A.5) is certainly this case.

##  Appendix B Operator Transpose for the EPR state

In this appendix, we derive the relation (3.33) algebraically. We can show the
relation directly as follows;

$\displaystyle U^{T}_{C^{\prime},D^{\prime}\to
B,T^{\prime}}\ket{\text{EPR}}_{C,C^{\prime}}\otimes\ket{\text{EPR}}_{D,D^{\prime}}$
(B.1) $\displaystyle=\left(I_{C}\otimes I_{D}\otimes
U^{T}_{C^{\prime},D^{\prime}\to
B,T^{\prime}}\right)\ket{\text{EPR}}_{C,C^{\prime}}\otimes\ket{\text{EPR}}_{D,D^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{d_{C}d_{D}}}\sum_{\tilde{C}=1}^{d_{C}}\sum_{\tilde{D}=1}^{d_{D}}\ket{\tilde{C},\,\tilde{D}}_{C,D}\otimes\left(U^{T}_{C^{\prime},D^{\prime}\to
B,T^{\prime}}\ket{\tilde{C},\tilde{D}}_{C^{\prime},D^{\prime}}\right)$
$\displaystyle=\frac{1}{\sqrt{d_{C}d_{D}}}\sum_{\tilde{C}=1}^{d_{C}}\sum_{\tilde{D}=1}^{d_{D}}\sum_{\tilde{B}=1}^{d_{B}}\sum_{\tilde{T}=1}^{d_{T}}\ket{\tilde{C},\,\tilde{D}}_{C,D}\otimes\ket{\tilde{B},\tilde{T}}_{B,T^{\prime}}\cdot\Big{.}_{B,T^{\prime}}\\!\bra{\tilde{B},\tilde{T}}U^{T}_{C^{\prime},D^{\prime}\to
B,T^{\prime}}\ket{\tilde{C},\tilde{D}}_{C^{\prime},D^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{d_{C}d_{D}}}\sum_{\tilde{C}=1}^{d_{C}}\sum_{\tilde{D}=1}^{d_{D}}\sum_{\tilde{B}=1}^{d_{B}}\sum_{\tilde{T}=1}^{d_{T}}\ket{\tilde{C},\,\tilde{D}}_{C,D}\otimes\ket{\tilde{B},\tilde{T}}_{B,T^{\prime}}\cdot\Big{.}_{C,D}\\!\bra{\tilde{C},\tilde{D}}U_{A,T\to
C,D}\ket{\tilde{B},\tilde{T}}_{A,T}$
$\displaystyle=\frac{1}{\sqrt{d_{B}d_{T}}}\sum_{\tilde{B}=1}^{d_{B}}\sum_{\tilde{T}=1}^{d_{T}}\left(U_{A,T\to
C,D}\ket{\tilde{B},\tilde{T}}_{A,T}\right)\otimes\ket{\tilde{B},\tilde{T}}_{B,T^{\prime}}$
$\displaystyle=\left(U_{A,T\to C,D}\otimes I_{B}\otimes
I_{T^{\prime}}\right)\ket{\text{EPR}}_{A,B}\otimes\ket{\text{EPR}}_{T,T^{\prime}}$
$\displaystyle=U_{A,T\to
C,D}\ket{\text{EPR}}_{A,B}\otimes\ket{\text{EPR}}_{T,T^{\prime}},$

where in the fifth equality, we used the unitarity condition of the Hilbert
space dimensions $d_{T}\,d_{B}=d_{C}\,d_{D}$. ∎

The above relation implies that the left and right diagrams in figure 11 are
equivalent.

Figure 11: Diagrams representing left and right hand sides of the relation
(3.33). Left: The left hand side of the relation. Right: The right hand side
of the relation. The left and right diagrams are equivalent.

##  Appendix C Convention in the SYK Hayden-Preskill protocol

In this appendix, we gather some important definitions and conventions which
we use in section 4.

#### Majorana SYK fermions

  * •

Anti-commutation relation

$\left\\{\psi_{i},\psi_{j}\right\\}=2\delta_{ij}$

  * •

The unitary time evolution operator

$U_{\alpha}=U_{\alpha}(t)=\exp\left(itH_{\alpha}\right)\quad\left(\alpha=L,R\right)$

  * •

Positive direction of time evolutions in left and right SYK systems (in
Lorentzian signature)

$\psi_{i,L}(t)\equiv
U_{L}\psi_{i,L}(0)U_{L}^{\dagger}=e^{itH_{L}}\psi_{i,L}(0)e^{-itH_{L}},$
$\psi_{i,R}(t)\equiv
U_{R}^{\dagger}\psi_{i,R}(0)U_{R}=e^{-itH_{R}}\psi_{i,R}(0)e^{itH_{R}},$

which can be written as

$\psi_{i,\alpha}(t)=\Delta_{L}^{-i\frac{t}{\beta}}\psi_{i,\alpha}(0)\Delta_{L}^{i\frac{t}{\beta}}=\Delta_{R}^{i\frac{t}{\beta}}\psi_{i,\alpha}(0)\Delta_{R}^{-i\frac{t}{\beta}}\quad(\alpha=L,R),$
(C.1)

where $\Delta_{L}=\Delta_{R}^{-1}$ is the modular operator defined by

$\Delta_{L}=\rho_{L}\otimes\rho_{R}^{-1}=e^{-K_{L}}\otimes
e^{K_{R}}=e^{-(K_{L}-K_{R})},\qquad K_{\alpha}\equiv\beta
H_{\alpha}\quad(\alpha=L,R).$ (C.2)

Here $\rho_{\alpha}\,(\alpha=L,R)$ is defined by

$\rho_{L}=\tr_{R}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right],\quad\rho_{R}=\tr_{L}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right]$
(C.3)

In the Euclidean signature, one can rewrite the above formal formula as

$\psi_{i,\alpha}(\tau)=\Delta_{L}^{\frac{\tau}{\beta}}\psi_{i,\alpha}(0)\Delta_{L}^{-\frac{\tau}{\beta}}\quad(\alpha=L,R),$
(C.4)

and recover the Lorentzian operator by the analytic continuation $\tau\to-it$.

  * •

Euclidean regularization parametrized by the cutoff $\delta$

$\psi_{i,L}(t+i\delta)\equiv
e^{i(t+i\delta)H_{L}}\psi_{i,L}(0)e^{-i(t+i\delta)H_{L}}=e^{(-\delta+it)H_{L}}\psi_{i,L}(0)e^{(\delta-
it)H_{L}}$ (C.5)

This regularized operator is related to the Euclidean evolved operator (C.4)
by continuation $\tau\to-it+\delta$.

#### SYK Hayden-Preskill channel

  * •

SYK Hayden-Preskill channel (4.10)

$\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\rho_{T}]\coloneqq\tr_{\tilde{L}}\left[U_{L}V_{T,L\to
L}\left(\rho_{T}\otimes\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right)V_{T,L\to
L}^{\dagger}U_{L}^{\dagger}\right].$

  * •

Adjoint SYK Hayden-Preskill channel (4.11)

$\displaystyle\mathcal{N}^{\text{SYK}\dagger}_{K,R\to T}[\mathcal{O}_{KR}]$
$\displaystyle\coloneqq\tr_{L,R}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\left(V_{L\to
T,L}^{\dagger}U_{L}^{\dagger}\,\mathcal{O}_{KR}\,U_{L}V_{L\to
T,L}\right)\right]$ $\displaystyle=\,_{L,R}\expectationvalue{\left(V_{L\to
T,L}^{\dagger}U_{L}^{\dagger}\,\mathcal{O}_{KR}\,U_{L}V_{L\to
T,L}\right)}{\text{TFD}}_{L,R}$

##  Appendix D Derivation of correlator from quantum channels

In this appendix, we give the derivation of the relation (4.29) and (4.30). We
can derive the relation graphically, but below we give an algebraic derivation
of the relation.

We start with the derivation of the relation (4.29), which can be obtained
straightforwardly from the definition of the quantum channels (4.10) and
(4.11). We first note that, from the definition of the quantum channel (4.10),
the state $\ket{0}_{T}\\!\bra{0}$ is mapped to

$\displaystyle\mathcal{N}^{\text{SYK}}_{T\to K,R}[\ket{0}_{T}\\!\bra{0}]]$
$\displaystyle=\tr_{\tilde{L}}\left[U_{L}\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}U_{L}^{\dagger}\right]$
(D.1) $\displaystyle=U_{R}\,\rho_{KR}\,U_{R}^{\dagger},$

where we used the fact that $(H_{L}-H_{R})\ket{\text{TFD}}_{L,R}$ leading to
$U_{L}\ket{\text{TFD}}_{L,R}=U_{R}\ket{\text{TFD}}_{L,R}$, and $\rho_{KR}$ is
defined by (4.31). For this density matrix, we consider the action of the
adjoint channel (4.11), and take the following matrix element;

$\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}=\frac{\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\,U_{R}\,\rho_{KR}\,U_{R}^{\dagger}\,\right]}{1}}{\tr\left[\left(\rho_{KR}\right)^{2}\right]},$
(D.2)

where we used the definition (4.16). Using the definition (4.11), we can
evaluate the denominator as

$\displaystyle\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\,U_{R}\,\rho_{KR}\,U_{R}^{\dagger}\,\right]}{1}$ (D.3)
$\displaystyle=\big{(}\,_{L,R}\bra{\text{TFD}}\otimes\,_{T}\bra{1}\big{)}{\left(V_{L\to
T,L}^{\dagger}U_{L}^{\dagger}\,U_{R}\,\rho_{KR}\,U_{R}^{\dagger}\,U_{L}V_{L\to
T,L}\right)}\big{(}\ket{\text{TFD}}_{L,R}\otimes\ket{1}_{T}\big{)}$
$\displaystyle=\frac{1}{Z_{\delta}}\cdot\,_{L,R}\bra{\text{TFD}}{\left(\psi_{i,L}^{\dagger}(-i\delta)\,U_{L}^{\dagger}\,U_{R}\,\rho_{KR}\,U_{R}^{\dagger}\,U_{L}\,\psi_{i,L}(i\delta)\right)}\ket{\text{TFD}}_{L,R}$
$\displaystyle=\frac{1}{Z_{\delta}}\cdot\,_{L,R}\bra{\text{TFD}}{\left(U_{R}\psi_{i,L}^{\dagger}(-i\delta)\,U_{L}^{\dagger}\,\rho_{KR}\,U_{L}\,\psi_{i,L}(i\delta)\,U_{R}^{\dagger}\right)}\ket{\text{TFD}}_{L,R}$
$\displaystyle=\frac{1}{Z_{\delta}}\cdot\,_{L,R}\bra{\text{TFD}}{\left(U_{L}\psi_{i,L}^{\dagger}(-i\delta)\,U_{L}^{\dagger}\,\rho_{KR}\,U_{L}\,\psi_{i,L}(i\delta)\,U_{R}^{\dagger}\right)}\ket{\text{TFD}}_{L,R}$
$\displaystyle=\frac{1}{Z_{\delta}}\cdot\,_{L,R}\bra{\text{TFD}}{\left(\psi_{i,L}^{\dagger}(t-i\delta)\,\rho_{KR}\,\psi_{i,L}(t+i\delta)\right)}\ket{\text{TFD}}_{L,R},$

where in the 4-th equality, we used the relation
$U_{L}\ket{\text{TFD}}_{L,R}=U_{R}\ket{\text{TFD}}_{L,R}$. Thus, by combining
the above expressions, we obtain the relation (4.29),

$\displaystyle\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}\cdot\matrixelement{1}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{0}]]\right]}{1}=\frac{1}{Z_{\delta}}\cdot\dfrac{\matrixelement{\text{TFD}}{\psi_{i,L}(t-i\delta)\,\left(I_{\tilde{L}}\otimes\rho_{KR}\right)\,\psi_{i,L}(t+i\delta)}{\text{TFD}}}{\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]}.$

Next, we derive the relation (4.30). Since
$\expectationvalue{\hat{d}_{\tilde{L}}}_{\beta}^{-1}=\tr_{KR}\left[\left(\rho_{KR}\right)^{2}\right]$
by the definition (4.16), we focus on the remaining factor
$\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}$. To evaluate the factor, we use the
definition of the adjoint channel (2.7),

$\displaystyle\matrixelement{0}{\mathcal{N}^{\text{SYK}\dagger}_{K,R\to
T}\left[\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]]\right]}{1}$ (D.4)
$\displaystyle=\tr_{K,R}\left[\,\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{0}_{T}\\!\bra{1}]\,\mathcal{N}^{\text{SYK}}_{T\to
K,R}[\ket{1}_{T}\\!\bra{0}]\,\right]$
$\displaystyle=\tr_{K,R}\left[\,\tr_{\tilde{L}}\left[U_{L}\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\psi_{i,L}^{\dagger}(-i\delta)U_{L}^{\dagger}\right]\,\tr_{\tilde{L}}\left[U_{L}\psi_{i,L}(i\delta)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}U_{L}^{\dagger}\right]\,\right]$
$\displaystyle=\tr_{K,R}\left[\,\tr_{\tilde{L}}\left[U_{R}\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\psi_{i,L}^{\dagger}(-i\delta)U_{L}^{\dagger}\right]\,U_{R}U_{R}^{\dagger}\,\tr_{\tilde{L}}\left[U_{L}\psi_{i,L}(i\delta)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}U_{R}^{\dagger}\right]\,\right]$
$\displaystyle=\tr_{K,R}\left[\,U_{R}\tr_{\tilde{L}}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}U_{R}\,\psi_{i,L}^{\dagger}(-i\delta)U_{L}^{\dagger}\right]\,\,\tr_{\tilde{L}}\left[U_{L}\psi_{i,L}(i\delta)U_{R}^{\dagger}\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right]\,U_{R}^{\dagger}\right]$
$\displaystyle=\tr_{K,R}\left[\,\tr_{\tilde{L}}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}U_{L}\,\psi_{i,L}^{\dagger}(-i\delta)U_{L}^{\dagger}\right]\,\,\tr_{\tilde{L}}\left[U_{L}\psi_{i,L}(i\delta)U_{L}^{\dagger}\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right]\,\right]$
$\displaystyle=\tr_{K,R}\left[\,\tr_{\tilde{L}}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\,\psi_{i,L}^{\dagger}(t-i\delta)\right]\,\,\tr_{\tilde{L}}\left[\psi_{i,L}(t+i\delta)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right]\,\right].$

By explicitly introducing bases for the traces, we can rewrite the last
expression as follows,

$\displaystyle\tr_{K,R}\left[\,\tr_{\tilde{L}}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\,\psi_{i,L}^{\dagger}(t-i\delta)\right]\,\,\tr_{\tilde{L}}\left[\psi_{i,L}(t+i\delta)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right]\,\right]$
(D.5)
$\displaystyle=\sum_{\alpha,\alpha^{\prime}=1}^{d_{K}d_{R}}\sum_{a,a^{\prime}=1}^{d_{\tilde{L}}}\left(\,{}_{KR}\bra{\alpha}\otimes\,_{\tilde{L}}\bra{a}\right)\left(\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\,\psi_{i,L}^{\dagger}(t-i\delta)\right)\left(\ket{\alpha^{\prime}}_{KR}\otimes\ket{a}_{\tilde{L}}\right)$
$\displaystyle\hskip
113.81102pt\times\left(\,{}_{KR}\bra{\alpha^{\prime}}\otimes\,_{\tilde{L}}\bra{a^{\prime}}\right)\left(\psi_{i,L}(t+i\delta)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right)\left(\ket{\alpha}_{KR}\otimes\ket{a^{\prime}}_{\tilde{L}}\right)$
$\displaystyle=\sum_{\alpha,\alpha^{\prime}=1}^{d_{K}d_{R}}\sum_{a,a^{\prime}=1}^{d_{\tilde{L}}}\,{}_{L,R}\bra{\text{TFD}}\psi_{i,L}^{\dagger}(t-i\delta)\left(\ket{\alpha^{\prime}}_{KR}\otimes\ket{a}_{\tilde{L}}\right)$
$\displaystyle\hskip
113.81102pt\times\left(\,{}_{KR}\bra{\alpha}\otimes\,_{\tilde{L}}\bra{a}\right)\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\left(\ket{\alpha}_{KR}\otimes\ket{a^{\prime}}_{\tilde{L}}\right)$
$\displaystyle\hskip
227.62204pt\times\left(\,{}_{KR}\bra{\alpha^{\prime}}\otimes\,_{\tilde{L}}\bra{a^{\prime}}\right)\psi_{i,L}(t+i\delta)\ket{\text{TFD}}_{L,R}$
$\displaystyle=\,_{L,R}\bra{\text{TFD}}\psi_{i,L}^{\dagger}(t-i\delta)\left(\tr_{KR}\left[\ket{\text{TFD}}_{L,R}\\!\bra{\text{TFD}}\right]\otimes
I_{KR}\right)\psi_{i,L}(t+i\delta)\ket{\text{TFD}}_{L,R}$