quantum information measures in qft tatsuma...
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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