Quantum Information Measures in QFT

Tatsuma Nishioka

Quantum Information

Measures in IFT

O . Introduction

Dh Quantum Information TheoryI AIT I

State conversion

143 → 14 )

A

. What operationsA

are allowed ?

. When 14 ) is transformed

to lots ?

→ Ordering of quantum states

147

Space of quantumStates

④ AFT

RG flow

Space of JETS

Under RG flow

T, → Tz

. When T . flows to % ?

→ Ordering IFT s

( C - theorems )

1. IIT-

14 ) i quantum states

A : LO cc

local opclassical

Communication

→

"i' Eiko.

"

classicalCommunication

Classification of quantum states

-

. pure bipartite system

7- ( = HA ④ 743

It's as Py -- 14741

( mixed statec- , p -

- E Pitti ) Hel )

. Separable state I¥14 ) -

- 14A ) ④ 1413 )in in

HA 743

a Entangled state

= non - separable

Bell pair CEPR )

I It ) = # ( I OA ) 1113 )

t I 1A ) I 013 ) )

=L 14A ) ④ His )

for any 4A,

TB

Schmidt decomposition-

Canonical form

of a bipartite quantum state

In general tuk eHB

(4) =

§.Cij IIA ) 113 )

ACpk

By changing the orthonormal basis

IIA ) → Vik IRA ) U .V

IIB ) → Ye le , ,/ unite ties

Cig i da x dB rectangular

as ..

a.matrix

singnlarvaluedecomposit.io#

Cig -- µ . A . V ) .

y

Oa-

to:#a )

Haz't ,

't"

. azo

f !¥¥"iii. iii. s. ( 4147=1

min IDA ,dB )

c⇒ ¥,Ai = I

4 x Xa R , I 12 I - - 2 Amin HAHN

I O

It = ( IT,

It,

- - -

,

iftminldaidol)

A

Schmidt vector

. We want to Introduce

"

order"

between quantum

States

Partido h"

"n

f÷÷÷a.

For 7L = Gli .kz .- - I

Y = ( Gi,

Yz ,-

- - )

xd is it ¥4

x'is .

÷

yetX is major 2- ed

by yfor any k

Example

Iht,I

,- - It t Int int

.

- - int,

o )

I fat ,ni

.

- .

.nt

,o

. o )

L - -.

L ( I,

o.

- - -

.o )

There are vectors

which cannot be compared

K 43 , Y Xx

e . six .. ht .

E. It

" CE . ¥ .

" " I ' # . ,

74+12 = ¥^

Yet Gz = ¥

OiitproblemWhen can a state 14 )

be transformed

I i:c:c::" '

14 ) 147

10€c

SIE( Separable op )

A- let = I I Am But P I An ④ BaitIAIAm④BniBm=IA④I

L OCC=

A performs op. HE

A sends the results i to B

Then B performs some op

B's"

I Ai BY'

m = I E. j )

Michael y

theorem ( Nielsen's major ization )

For bipartite pure states 147,14 )

147 is transformed to 14 )

under SEP iff It at|147*7,141 itI 's at

. How toquantity the order ?

f- n

n;÷ie ! !'

II:* . ,

" ti¥*:*.

Schur - concave function

x LG ⇒ fix , zflyi

NI.

fix , I fly ) does not

necessarily mean K LY

but.

it means YKX

Quantum information measures

( entanglement )

Al. Monotonicity

H ),

147 ⇒ ftp.ilzfl

A 2. Vanishing for sep . states

14 ) : separable .⇐ I flex ) -

- O

A 3. Convexity

HEAR ) EEP . ftp.i

Relafiveentropyn" distance "

bet quantum States

TP : Trip ) -- I ⇒

SIPHO ) Truth ' I

= tr-pll.ge - logo ) )

. Monotonicity In . SEP

CCPTPStreet 1111011 a

completelyE S ( p 115 ) positive

trace preserving

i ÷±i÷

. PositivitySIP lls ) 20

. SIP 1101=0

⇐ ) p -- o

Example of CPTP

Tarte A B e-

suppose Paup J,

T

T -- IA TRB

¥00Ti PAUB → PA B C

⇒ SLPAHOA) I SLPAUBHOAUB )

Entanglement measures

-

. Entanglement entropyf ,

=-

TRIPI

SACP ) ± - Era EPA log PA )A

= log da - S l Pall IAI datT

maximally

entangledstate

. From the monotonicity of rel.

ent .

1140 Not

seronf¥t"

Save + S BUC

Z SA t SB AUDI

/SSAc⇒M0NI

E. g. 21 Relative ent . of entanglement

Exp Ip ) = inf 'S IP Ilo )

or EX

In AFT .

[ Hollands - Sunders 117 )

X : separablestate

GIT Resource thy-

separable states free states

CPTP free operationsentangled

states resource

entanglement resource measure

measures

Ref )

. M.

Nielsen"

An introduction to major 't talion

and its application to quantum mechanics"

• Plenio - Vir mani quaint - ph 10504163

. Chiam bar - Gour 1806 - 06107

2.Quantumentanglement.no. Algebraic formulation

(Path - integral formulation

¢useful for proving inequalities

) practical computation

Algebraic formulation- OA

H -- HA ④ HE

toA

. Reduced density matrix

PA = Tra )At -

- da ④ IF

( OA ) = Tr [ P A ④ IAD= TraCPA OA ] for any AA

( OA ) invariant under transformation

On.→ U AA Ut U' - HA → HA

Pa → U Pa.Ut

SA IP ) is also inv .

""

. .

.If DEE ) -

. DEA )a

11/11€7;Us .

t.

DEA )

PE -

- UPA Ut domain ofdependence

U

A.FI?IeordgetraAuassociated with

a spacetime region U n

Postulatesaca

⇒ Aachen a-

→ Partial order In a

2. If U .

I are spacelikeex

[ ten.ten ] -

- 0 / ④ G)

3.

Covariance ( Haag 's textbook )

Classification of A-

.

Type I ( IMI . Type It I QFTIA

. Type I Iti

Relatiueentropy.no(

Pg,

!HI HI

For a spacetime region lol ) Coll

N -- DEA )

the rel . Ent. between 14 ) , 14 ) E 7C

SA I Pull Poll = - HI log 0414143

Out = G. a ④ Ria?C Araki

963( relative modular operator )

SA I Px It Poll = S ( G. A H Poli At

ProofTA I kill Pol I = - tr C Px log Oxio )

= - tr [ Py ( log Pol, A ④ HE

- 1A ④ log Putt ) ]= - tr A [ Pa

, A log Pol, A ) Schmidt

de comp -

+ tr # [ Rett log PHIL,

CE → At

= - tr A [ Px. A log Pol

. A )

t tht [ Px . A log Px. A )

-- tht [ Px

. A I log Px. A

- log Pol. At )

= S I Px .All Pol

.A )

yThis is the case for Type I

The new definition of rel . eat

is valid for Type II (wt Tomeihteo;Take saki)

Ref ). Witten 1803.04993