East Asian String Webinar Series @May 1st, 2020

Holographic Pseudo Entropy
Tadashi Takayanagi
Yukawa Institute for Theoretical Physics (YITP), Kyoto U.
Based on a paper in preparation arXiv:2005.*****
withYoshifumi Nakata (YITP→U. of Tokyo)
Yusuke Taki (YITP)
Kotaro Tamaoka (YITP)
Zixia Wei (YITP)

① Introduction
Quantum Entanglemenｔ (QE)BA

Two parts (subsystems) A and B in a total system arequantum mechanically correlated.
e.g. Bell state :[1ΨBell =↑ ⊗↓ +↓ ⊗↑BABA2]

Pure States: Non-zero QE ⇔ Ψ AB ≠ Ψ1 A ⊗ Ψ2 B .
Direct ProductMixed States: Non-zero QE ⇔ ρ AB ≠∑ p ρ ⊗ρii(i )
ASeparable(i )B.

The best (or only) measure of quantum entanglement
for pure states is known to be entanglement entropy (EE).
Divide a quantum system into two subsystems A and B:

H tot = H A ⊗ H B .

Define the reduced density matrix by
The entanglement entropy

is defined by(von-Neumann entropy)

Quantum Many-body SystemsεAB

Quantum Field Theories (QFTs)

ContinuumLimit ε→0BA∂A = ∂B

Holographic Entanglement Entropy for Static Spacetimes[Ryu-TT 06; derived by Lewkowycz-Maldacena 13]

For static asymptotically
AdS spacetimes:

 Area(ΓA ) S A = Min ∂ΓA = ∂A4GN
ΓA ≈ AΓA is the minimal area surface(codim.=2) on the time slice
such that∂A = ∂γ A and A ~ γ A .
homologous

CFTd +1(We omit the time direction.)ΓAA

AdSd + 2Bzz >ε(UV cut off)

2−+dz
dtdx
∑i=1 ids 2 = R 2 ⋅z222dCovariant Holographic Entanglement Entropy

A generic Lorentzian asymptotic AdS spacetime is dual toa time dependent state |Ψ(t)〉 in the dual CFT.
The entanglement entropy gets time-dependent:

𝑆𝑆𝐴𝐴 𝑡𝑡 .

This is computed by the holographic formula:𝐴𝐴(Γ𝐴𝐴 )𝑆𝑆𝐴𝐴 𝑡𝑡 = MinΓ𝐴𝐴 Ext Γ𝐴𝐴
4𝐺𝐺𝑁𝑁∂A = ∂γ A and A ~ γ A .

Boundary (CFTd)𝜌𝜌𝐴𝐴 𝑡𝑡 = Tr𝐵𝐵 [|Ψ(𝑡𝑡)⟩⟨Ψ 𝑡𝑡 |]AdSd+1

A

Lorentzian Time

[Hubeny-Rangamani-TT 07, derived by Dong-Lewkowycz-Rangamani-TT 16]

ΓA t

Question: How about the area of minimal surface
in Euclidean time dependent asymptotically AdS spaces ?

Boundary (CFTd)

Euclidean Time τ

τ=∞

τ=-∞

�𝜑𝜑| = ⟨0|𝑂𝑂 𝑥𝑥3 𝑂𝑂 𝑥𝑥4 𝑂𝑂(𝑥𝑥5 )AdSd+1O(x4)O(x3)ΓA AO(x1)O(x5)τ=0O(x2)|𝜓𝜓⟩ = 𝑂𝑂 𝑥𝑥1 𝑂𝑂 𝑥𝑥2 |0⟩

Final
State

〈φ||Ψ〉
InitialStateWe argue that the area of minimal surface calculates
the following EE-like quantity:𝑆𝑆𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴= −Tr

𝜓𝜓|𝜑𝜑𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴 log𝜏𝜏𝐴𝐴.

We call this quantity pseudo (entanglement) entropy.
𝜓𝜓|𝜑𝜑Here , the reduced transition matrix 𝜏𝜏𝐴𝐴 is defined as𝜓𝜓|𝜑𝜑𝜓𝜓|𝜑𝜑,𝜏𝜏𝐴𝐴 = Tr𝐵𝐵 𝜏𝜏

from the transition matrix:𝜏𝜏𝜓𝜓|𝜑𝜑|𝜓𝜓⟩⟨𝜑𝜑|= ⟨ ⟩.
𝜑𝜑|𝜓𝜓

Note that in general this transition matrix is not Hermitian.

[cf. “conditional entropy of post-selected states” Salek-Schbert-Wiesner 2013]

In this talk, we would like to study properties ofpseudo entropy in quantum many-body systems,
CFTs and AdS/CFT.
Note: In quantum information theory, the transition
matrices arise when we consider post-selection.

⟨𝜑𝜑|𝑂𝑂𝐴𝐴 |𝜓𝜓⟩𝜓𝜓|𝜑𝜑
= T𝑟𝑟[𝑂𝑂𝐴𝐴 𝜏𝜏𝐴𝐴 ]⟨𝜑𝜑|𝜓𝜓⟩Initial StateFinal stateafterpost-selection

This quantity can becomplex valued ingeneral and is called
weak value.

[Aharanov-Albert-Vaidman 1988,…]

Contents① Introduction② Basics of Pseudo (Renyi) Entropy③ Pseudo Entropy in Qubit Systems④ Holographic Pseudo Entropy⑤ Pseudo Entropy for Locally Excited States⑥ Mixed State Generalization⑦ Conclusions

② Basics of Pseudo (Renyi) Entropy(2-1) Definition of Pseudo (Renyi) Entropy
Consider two quantum states |𝜓𝜓⟩ and |𝜑𝜑⟩ , and define
the transition matrix: 𝜓𝜓|𝜑𝜑 |𝜓𝜓⟩⟨𝜑𝜑|

𝜏𝜏=⟨

𝜑𝜑|𝜓𝜓

.
⟩We decompose the Hilbert space as H tot = H A ⊗ H B .
and introduce the reduced transition matrix:
𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴 = Tr𝐵𝐵𝜏𝜏𝜓𝜓|𝜑𝜑The pseudo n-th Renyi entropy is defined by𝜓𝜓|𝜑𝜑(𝑛𝑛)𝑆𝑆𝜏𝜏𝐴𝐴

1=logTr
1 − 𝑛𝑛𝜓𝜓|𝜑𝜑 𝑛𝑛𝜏𝜏𝐴𝐴.

The n=1 limit defined the (von-Neumann) pseudo entropy:𝑆𝑆𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

=𝑆𝑆(𝑛𝑛=1)

𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

𝜓𝜓|𝜑𝜑Note 1: Since 𝜏𝜏𝐴𝐴is not a quantum state i.e. Hermitianand positive semi-definite, the pseudo (Renyi) entropy iscomplex valued in general.
Note 2: When A=total system (B=empty), we obtain𝜏𝜏𝜓𝜓|𝜑𝜑 𝑛𝑛= 𝜏𝜏𝜓𝜓|𝜑𝜑TrThus, 𝑆𝑆 (𝑛𝑛) 𝜏𝜏 𝜓𝜓|𝜑𝜑 = 0.

𝜏𝜏𝜓𝜓|𝜑𝜑 𝑛𝑛=1.

(2-2) Basic Properties of Pseudo Entropy[1] If either |𝜓𝜓⟩ or |𝜑𝜑⟩ has no entanglement (i.e. directproduct state) , then𝑆𝑆 (𝑛𝑛)[2] 𝑆𝑆[3](𝑛𝑛)

𝑆𝑆 (𝑛𝑛)𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

= 𝑆𝑆(𝑛𝑛)= 𝑆𝑆 (𝑛𝑛)

= 0.

𝜑𝜑|𝜓𝜓
𝜏𝜏𝐴𝐴

𝜓𝜓|𝜑𝜑𝜏𝜏𝐵𝐵.

→Connection toquantum entanglement !

†.

→“SA=SB”[4] If |𝜓𝜓⟩=|𝜑𝜑⟩ , then 𝑆𝑆 (𝑛𝑛) 𝜏𝜏𝐴𝐴𝜓𝜓|𝜑𝜑 = Renyi entropy.

③ Pseudo Entropy in Qubit Systems(3-1) Classification in 2 Qubit Systems
The two pure states in 2 qubit systems are parameterizedby 7 parameters:|𝜓𝜓⟩ ∝ |00⟩ + 𝑎𝑎|11⟩|𝜑𝜑⟩ ∝ |00⟩ + 𝑏𝑏𝑒𝑒 −𝑖𝑖𝜃𝜃 |11⟩+𝑐𝑐𝑒𝑒 −𝑖𝑖𝜉𝜉 |01⟩+d𝑒𝑒 −𝑖𝑖𝜂𝜂 |10⟩1
𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴 = 1+𝑎𝑎𝑎𝑎𝑒𝑒 𝑖𝑖𝜃𝜃 �|0⟩⟨0| + 𝑎𝑎𝑎𝑎𝑒𝑒 𝑖𝑖𝜉𝜉 |1⟩⟨0| + 𝑑𝑑𝑒𝑒 𝑖𝑖𝜂𝜂 |0⟩⟨1| +=General 2 Qubitsystem(Random states)
Exotic Transition
Matrices
HolographicTransitionMatrix ?
Nice Transition
Matriceswith operationalinterpretation

An Example of Exotic Transition Matrix|𝜓𝜓⟩ =|𝜑𝜑⟩ =1⟩ + 𝑒𝑒 𝑖𝑖𝜃𝜃 |11⟩)(|0021(|00⟩ + |11⟩)21
𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴 = 1+𝑒𝑒 𝑖𝑖𝜃𝜃𝑆𝑆 (𝑛𝑛)𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

→ Complex conjugate|0⟩⟨0| + 𝑒𝑒 𝑖𝑖𝜃𝜃 |1⟩⟨1| . pair of Eigenvalues
𝑛𝑛𝜃𝜃
cos 2

1=log
𝜃𝜃
1 − 𝑛𝑛
𝑛𝑛−1
𝑛𝑛
2 cos
2

→Only special values of θ can give
positive values pseudo entropy.

(3-2) Pseudo Entropy as Entanglement Distillation𝜓𝜓|𝜑𝜑

Let us focus on the class E i.e. 𝜏𝜏𝐴𝐴 And 𝜏𝜏𝐵𝐵𝜓𝜓|𝜑𝜑 are
Hermitian and semi-positive definite.

Remarkably, in this case we can show a quantum
information theoretical interpretation of pseudo entropy:
ClaimPseudo Entropy 𝑆𝑆𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

= # of Distillable Bell Pairs
as an intermediate statesof post-selection |𝜓𝜓⟩ → |𝜑𝜑⟩.

More precisely, we take asymptotic limit M→∞.

Operational Interpretation of EE from LOCC
ABSetup⇒ H tot = H A ⊗ H BA

B

LO (=Local Operations)
Projection measurements and unitary trfs. which act
either A or B only.
CC (=Classical Communications between A and B)⇒These operations are combined and called LOCC.
A basic example of LOCC: quantum teleportation

(|Ψ⟩𝐴𝐴𝐴𝐴 ⟨Ψ|)⊗𝑀𝑀AB⊗𝑀𝑀Entangled in a very
complicated way

LOCCDistillation

AB

N Bell pairs(|Ψ⟩𝐴𝐴𝐴𝐴 ⟨Ψ|)⊗𝑀𝑀 ⇒ (|Bell⟩⟨Bell|)⊗𝑁𝑁

Well-known fact in QI:𝑆𝑆 𝜌𝜌𝐴𝐴

𝑁𝑁
= lim
𝑀𝑀→∞ 𝑀𝑀𝜌𝜌𝐴𝐴 ≡ Tr𝐵𝐵 [|Ψ⟩𝐴𝐴𝐴𝐴 ⟨Ψ|]

Distillation from Post-selection
In class D, we can write𝝍𝝍|𝝋𝝋𝑺𝑺 𝝉𝝉𝑨𝑨|𝜓𝜓⟩ = cos𝜃𝜃1 |00⟩ + sin𝜃𝜃1 |11⟩|𝜑𝜑⟩ = cos𝜃𝜃2 |00⟩ + sin𝜃𝜃2 |11⟩cos𝜃𝜃1 cos𝜃𝜃2 |0⟩⟨0| + sin𝜃𝜃1 sin𝜃𝜃2 |1⟩⟨1|
𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴 =cos(𝜃𝜃1 − 𝜃𝜃2 )𝜓𝜓|𝜑𝜑𝜃𝜃1𝑆𝑆 𝜏𝜏𝐴𝐴=cos𝜃𝜃1 cos𝜃𝜃2cos𝜃𝜃1 cos𝜃𝜃2 sin𝜃𝜃1 sin𝜃𝜃2sin𝜃𝜃1 sin𝜃𝜃2−
� log−
� logcos(𝜃𝜃1 −𝜃𝜃2 )cos(𝜃𝜃1 −𝜃𝜃2 ) sin(𝜃𝜃1 −𝜃𝜃2 )sin(𝜃𝜃1 −𝜃𝜃2 )𝜃𝜃2(|𝜓𝜓⟩)⊗𝑀𝑀 = (cos𝜃𝜃1 |00⟩ + sin𝜃𝜃1 |11⟩)⊗𝑀𝑀MCk
𝑀𝑀𝑀𝑀−𝑘𝑘𝑘𝑘= ∑𝑘𝑘=0(cos𝜃𝜃1 )(sin𝜃𝜃1 ) ∑𝑎𝑎=1|𝑃𝑃 𝑘𝑘 , 𝑎𝑎⟩|𝑃𝑃 𝑘𝑘 , 𝑎𝑎⟩
𝑘𝑘 = 0:𝑘𝑘 = 1:

|𝑃𝑃 0 , 1⟩ = |00 ⋯ 0⟩
|𝑃𝑃 1 , 1⟩ = |10 ⋯ 0⟩, |𝑃𝑃 1 , 2⟩ = |01 ⋯ 0⟩, ⋯Projection to maximally entangled stateswith Log[MCk] entropy:
MCk=M!/(M-k)!k!
MCkΠ𝑘𝑘 = � |𝑃𝑃 𝑘𝑘 , 𝑎𝑎⟩⟨𝑃𝑃 𝑘𝑘 , 𝑎𝑎|
𝑎𝑎=1(𝑐𝑐1 𝑐𝑐2 )𝑀𝑀−𝑘𝑘 (𝑠𝑠1 𝑠𝑠2 )𝑘𝑘
probability: 𝑝𝑝𝑘𝑘 = ⟨𝜑𝜑|Π𝑘𝑘 |𝜓𝜓⟩/⟨𝜑𝜑|𝜓𝜓⟩ =� MCk
𝑀𝑀(𝑐𝑐1 𝑐𝑐2 + 𝑠𝑠1 𝑠𝑠2 )# 𝐨𝐨𝐨𝐨 𝐃𝐃𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢 𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩: 𝐍𝐍 = ∑𝑴𝑴
𝒌𝒌=𝟎𝟎 𝒑𝒑𝒌𝒌・Log[MCk]𝝍𝝍|𝝋𝝋≈ 𝑴𝑴 � 𝑺𝑺 𝝉𝝉𝑨𝑨!

(3-3) Monotonicity in 2 Qubit systems
We can prove the following monotonicity under unitarytransformation:
Claim Consider two states related by local unitary trf.
|𝜓𝜓⟩ = (𝑈𝑈𝐴𝐴 ⊗ 𝑉𝑉𝐵𝐵 )|𝜑𝜑⟩.

𝜓𝜓|𝜑𝜑If 𝜏𝜏𝐴𝐴 has non-negative eigenvalues (i.e. class B=C), then𝑆𝑆 (𝑛𝑛)𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

≥ 𝑆𝑆 𝑛𝑛 Tr𝐵𝐵 |𝜓𝜓⟩⟨𝜓𝜓|= 𝑆𝑆 (𝑛𝑛) Tr𝐵𝐵 |𝜑𝜑⟩⟨𝜑𝜑| .

Note: However, this claim is limited to 2 qubit systems.

Decreasing Pseudo Entropy Examples
Ex1 |𝜓𝜓⟩ = 1 (|0000⟩ + |0110⟩)21|𝜑𝜑⟩ = 2 (|0000⟩ + |1001⟩)
𝜓𝜓|𝜑𝜑(𝑛𝑛)𝑆𝑆𝜏𝜏𝐴𝐴
=0𝑆𝑆 (𝑛𝑛) Tr𝐵𝐵 |𝜓𝜓⟩⟨𝜓𝜓|

A B= 𝑆𝑆 (𝑛𝑛) Tr𝐵𝐵 |𝜑𝜑⟩⟨𝜑𝜑|

Ex2 Thermofield States in CFTs|𝜓𝜓𝑖𝑖 ⟩ =1� 𝑒𝑒

𝑍𝑍(𝛽𝛽1 ) 𝑛𝑛

𝛽𝛽− 2𝑖𝑖 𝐸𝐸𝑛𝑛

Entanglement
Swapping|𝑛𝑛⟩|𝑛𝑛⟩= log2.

(𝑖𝑖 = 1,2)

𝛽𝛽1 + 𝛽𝛽21
≤ [𝑆𝑆 𝛽𝛽1 + 𝑆𝑆 𝛽𝛽2 ]𝑆𝑆221𝜓𝜓 |𝜓𝜓
𝑆𝑆 𝜏𝜏𝐴𝐴 1 2 ≤ 2 [𝑆𝑆 Tr𝐵𝐵 |𝜓𝜓1 ⟩⟨𝜓𝜓1 | +𝑆𝑆 Tr𝐵𝐵 |𝜓𝜓2 ⟩⟨𝜓𝜓2 | ].

④ Holographic Pseudo Entropy
Holographic Pseudo Entropy (HPE) Formula

Basic Properties of HPE𝑖𝑖 If 𝜌𝜌𝐴𝐴 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝, 𝑆𝑆𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

=0.

𝑖𝑖𝑖𝑖 If 𝜓𝜓 𝑜𝑜𝑜𝑜 𝜑𝜑 𝑖𝑖𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒,𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴𝑆𝑆=0.
→This follows from AdS/BCFT [TT 2011]𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑣𝑣

𝜓𝜓|𝜑𝜑𝑆𝑆 𝜏𝜏𝐴𝐴

𝜓𝜓|𝜑𝜑𝑆𝑆 𝜏𝜏𝐴𝐴

𝜓𝜓|𝜑𝜑= 𝑆𝑆 𝜏𝜏𝐵𝐵

𝜓𝜓|𝜑𝜑+ 𝑆𝑆 𝜏𝜏𝐵𝐵

. “SA=SB”

𝜓𝜓|𝜑𝜑≥ 𝑆𝑆 𝜏𝜏𝐴𝐴𝐵𝐵

Boundary (CFTd)

𝑆𝑆𝐴𝐴(Γ𝐴𝐴 )= MinΓ𝐴𝐴≥04𝐺𝐺𝑁𝑁𝜓𝜓|𝜑𝜑𝜏𝜏𝐴𝐴

AdSd+1A

Euclidean Time(4-1) Holographic Pseudo Entropy and General Properties

ΓA t. “Subadditivity”

• However, it is not clear whether the strong subadditivityholds for the holographic pseudo entropy even for a single classicalgeometry. (cf. Proof of SSA in Lorentzian AdS [Wall 2012] )•Since holographic computations in general leads to𝜓𝜓|𝜑𝜑𝜓𝜓|𝜑𝜑Tr[ (𝜏𝜏𝐴𝐴 )2 ] ≠ Tr[ 𝜏𝜏𝐴𝐴
𝜓𝜓|𝜑𝜑we find that 𝜏𝜏𝐴𝐴𝜓𝜓|𝜑𝜑

� (𝜏𝜏𝐴𝐴 )† ]in holography belongs to class C.

• So far we focus on a single classical geometry. In general,
we can consider a quantum superposition of several classicalgravity states, where we find the linearity⇒We can find examples in this class where SSA is violated.

(4-2) Simple Example: Janus AdS3/CFT2Consider a gravity dual of 2d CFT perturbed
by the exactly marginal operator O(x) such that𝜏𝜏 < 0: 𝜙𝜙− � 𝑑𝑑 2 𝑥𝑥 𝑂𝑂 𝑥𝑥 ,|𝜓𝜓⟩𝜏𝜏 > 0: 𝜙𝜙+ � 𝑑𝑑 2 𝑥𝑥 𝑂𝑂 𝑥𝑥 .

AdS3 gravity coupled to a massless scalar[Bak-Gutperle-Hirano 2007]

𝜙𝜙+Boundary

Janus Solution

ρ=∞|𝜑𝜑⟩𝜙𝜙−ρ = −∞

𝜏𝜏ρ

𝑦𝑦|𝜑𝜑⟩|𝜓𝜓⟩𝑦𝑦

Holographic Pseudo Entropy
We take A=[0,l]. The minimal surface is given by𝜏𝜏 = 0,𝑥𝑥 2 + 𝑦𝑦 2 = 𝑙𝑙2 .

Thus the holographic pseud entropy reads𝑆𝑆𝜓𝜓 |𝜓𝜓
𝜏𝜏𝐴𝐴 1 2𝑐𝑐
1 + 1 − 2𝛾𝛾 2𝑙𝑙
� log .
= �32𝛿𝛿

When γ is small, we obtain𝑆𝑆𝜓𝜓 |𝜓𝜓
𝜏𝜏𝐴𝐴 1 2𝑔𝑔𝑥𝑥𝑥𝑥 = 𝜖𝜖 −2 → 𝛿𝛿 =1 + 1 − 2𝛾𝛾 2� 𝜖𝜖2𝑐𝑐𝛾𝛾 2𝑙𝑙𝑐𝑐 2
� log + 𝛾𝛾 .
≈ � 1−34𝜖𝜖 12Agreeing with the CFT perturbation (universal in any 2d CFTs):

⑤ Pseudo Entropy for Locally Excited States(5-1) Pseudo Renyi Entropy in Free CFTConsider the following 2 locally excited states:|𝜓𝜓⟩ = 𝑁𝑁1 � 𝑂𝑂(𝑥𝑥1 , 𝜏𝜏1 )|0⟩,|𝜑𝜑⟩ = 𝑁𝑁2 � 𝑂𝑂(𝑥𝑥2 , 𝜏𝜏2 )|0⟩

We would like to calculate their pseudo Renyi entropy.
We take the replica ofcomplex plane 𝑤𝑤 = 𝑥𝑥 + 𝑖𝑖𝜏𝜏.
⇒Σ n𝜓𝜓|𝜑𝜑𝑆𝑆 (𝑛𝑛) 𝜏𝜏𝐴𝐴

1= 1−𝑛𝑛 logTr(𝑛𝑛)𝜓𝜓|𝜑𝜑𝜓𝜓|𝜑𝜑 𝑛𝑛𝜏𝜏𝐴𝐴

We compute the difference:∆𝑆𝑆𝐴𝐴= 𝑆𝑆 (𝑛𝑛) 𝜏𝜏𝐴𝐴.

-𝑆𝑆 (𝑛𝑛) Tr𝐵𝐵 |0⟩⟨0| .

For simplicity, we will focus on 2nd Renyi pseudo entropy
in the 2d massless free scalar CFT.
[EE in the same setup: Nozaki-Numasawa-TT, He-Numasawa-Watanabe-TT 2014]† (𝑤𝑤 )𝑂𝑂(𝑤𝑤 )𝑂𝑂 † (𝑤𝑤 )𝑂𝑂(𝑤𝑤)𝑂𝑂
1234 Σ2(2)∆𝑆𝑆𝐴𝐴 = −log
𝑂𝑂(𝑤𝑤1 )𝑂𝑂† (𝑤𝑤2 ) Σ21

We choose 𝑂𝑂 𝑥𝑥, 𝜏𝜏 = 𝑒𝑒 𝑖𝑖𝜙𝜙(𝑥𝑥,𝜏𝜏)/2 + 𝑒𝑒 −𝑖𝑖𝜙𝜙(𝑥𝑥,𝜏𝜏)/2 ~Bell pair .
After we perform the conformal map
𝑤𝑤 − 𝑎𝑎= 𝑧𝑧 2𝑤𝑤 − 𝑏𝑏2(2)we obtain ∆𝑆𝑆𝐴𝐴 = log 1+ 𝜂𝜂 + 1−𝜂𝜂

≤ 0.

Cross ratio:(𝑧𝑧1 −𝑧𝑧2 )(𝑧𝑧3 −𝑧𝑧4 )𝜂𝜂 =.
(𝑧𝑧1 −𝑧𝑧3 )(𝑧𝑧2 −𝑧𝑧4 )τ=5

τ=4Plotsτ=2

τ=2

τ=0.1

τ=0.1

Pseudo Entropy is reduced when ∂A gets closer to the excitation.
⇒ This is due to the decreasing property under entanglement swap.

O210-τ
τ
O1

O2AXm

20

Aτ
O1

τ

20

Xm
10

(5-2) Holographic CFT Calculations
We can calculate the pseudo entropy for the same
setup in holographic CFTs:
Positive contribution is new !
ΔS>0

τ=0.1

τ=0.1

We have ΔS<0 when ∂A get closer to Os.
(in the same way as in free scalar.)

O210-τ
τ
O1AXm

10

τ
O1

AO2

τ

10
Xm 20

⑥ Mixed State Generalization
Can we define pseudo entropy when the total system AB is mixed ?
𝜓𝜓|𝜑𝜑Assume 𝜏𝜏𝐴𝐴𝐵𝐵 is given.
One possibility is to extend the reflected entropy. [Dutta-Faulkner 2019,
Kusuki-Tamaoka 2019]

𝑋𝑋𝐴𝐴𝐴𝐴 = � 𝑋𝑋𝑖𝑖𝑖𝑖 |𝑖𝑖 � ⟨𝑗𝑗|𝑖𝑖,𝑗𝑗⇒ |𝑋𝑋⟩𝐴𝐴𝐴𝐴𝐴𝐴�𝐵𝐵� = � 𝑋𝑋𝑖𝑖𝑖𝑖 |𝑖𝑖 ⟩𝐴𝐴𝐴𝐴𝐴𝐴�𝐵𝐵� |𝑗𝑗 ∗ ⟩𝐴𝐴𝐴𝐴𝐴𝐴�𝐵𝐵�
𝑖𝑖.𝑗𝑗

𝜓𝜓|𝜑𝜑
Define pseudo reflected entropy by 𝑆𝑆𝑅𝑅 (𝜏𝜏𝐴𝐴𝐵𝐵) = 𝑆𝑆 Tr𝐵𝐵𝐵𝐵�𝜓𝜓|𝜑𝜑𝜓𝜓|𝜑𝜑|𝜏𝜏𝐴𝐴𝐴𝐴 ��(𝜏𝜏𝐴𝐴𝐴𝐴 )† |𝜓𝜓|𝜑𝜑𝜓𝜓|𝜑𝜑�(𝜏𝜏𝐴𝐴𝐴𝐴 )† |𝜏𝜏𝐴𝐴𝐴𝐴 �

This coincides with the twice of entanglement wedge cross section:𝑨𝑨(𝜮𝜮𝑨𝑨𝑨𝑨 )𝝍𝝍|𝝋𝝋
𝑺𝑺𝑹𝑹 (𝝉𝝉𝑨𝑨𝑩𝑩 ) = 𝟐𝟐 � 𝟒𝟒𝑮𝑮 .
𝑵𝑵𝜮𝜮𝑨𝑨𝑨𝑨[cf. EW=EoP: Umemoto-TT 2017
EW=Odd entropy: Tamaoka 2018
EW∝Negativity: Kudler-Flam-Ryu 2019]AB.

⑦ Conclusions
Main claim of this talk
Pseudo Entropy = Area of Minimal surface in Euclidean
asymptotically AdS with time-dependence• Pseudo Entropy is in general complex valued.
⇒ Why is holographic pseudo entropy non-negative ?
[ Euclidean path-integrals are positive valued !]• Pseudo Entropy for `non-exotic states’ measures the amountof quantum entanglement in the intermediate states.
[How about general states ? What is the meaning of complex values?]• Pseudo entropy tends to decrease in CFT computations due toentanglement swapping. However, we find also a positive
contribution only for holographic CFTs, as opposed to free scalar CFT.

Thank you very much !