Privatecapacity

andSubfactors

Garo lina Dias Alexiou
The

University of Tokyo
Virtual LPO
2020/06/19
work

Yasuyuki Kawahigashi 's group

inprogressOutline.

••

Wiretap channels

Beyond type I
Private Quantum

subalgebras

2/18

wiretapchannels-s.senddasa.cat information using guardschannels( finite dimension )gchro-dingerpicture.JP⑧④ HAAPB✓

g-V : HA

HBe:

->

-

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HE⑤

statesComplementary channels
PB IOBCPA)⇐=Cfa);i.IO/oI--oIB@

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wiretapchannel.sisend.

oatmeal information( finite dimension )(classicalShannon:¥i④ HAPAPB§,HB BO-He⑤"

formal quantum analog
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7/18

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thiretafchannels

send:dashed information@efinition_it fsePanjaakastdapssfhdb.a.at:18?afh?uawerateusing guards channels→

average amount of information that
canA send to B

amount Einacan recoverwayisthat

thenegligibleamoa.ntaigtainbtgmpgtieri-kmoan.tat.iantggm.cat"pie)ETheorem (Dentate 03',Cai-

WinterYeung 047'-

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eedpa.pa.9ie.z.n.edpa.IEYpa.cm

918

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:

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using guards channels@efinition_it fsePanjaateastdapssfhdb.a.at:718?aff?wawenafe→

average amount of information that
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amount Eina

can recoverway
isthatthenegligible.m÷::÷÷÷i÷ii÷*aEet Pa ,Problems-•cmpan t;e.* hpa to:'( pain'll,;

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How to compute ROI ) ?
How todeal

withinfinite dimension ?

a••

Not

everything

Entropyetc.

.

canbeisBAD (type I]

infinite.

10/18

fwe:④
ultraweak

Beyondtypei

strongly&

Quick review

todefine

notation✓

Concrete von Neumann

algebras Deft

( insert favorite:

7

von(Neumanndoubletheorem

Factorsf:£ ( M)=algebraMAM*-

algebra of BCH) with ht✓

com mutant

Abstract von Neumann

topology here) closed

Def 2( Sakai':71)

Mc BCH)M

closed under involution'taC( Von Neumann'=algebra with-49)'"feveryavon

apreduabt

Neumann isdirect"such that M = M

isomorphic tointegral of factors

schri-dingerpicturefaassisw.nl?g;I:-:Ye" """* ''e' " "µ.

Beyondtypein
Normal unitalcompletely Positive Trace Preserving

Completely Positive

map

map

-:÷::÷::::÷::O-N:

N*M → K'',

has a universal

enveloping

vonAnyK* → M *p esc-tfm ( Sherman so Takeda 54 ):Ct-poralgebra

Neumannalgebra

12/18

Beyondtypei

Band Emutually commutingvon

Neumanalgebras"BCHB)④H,H④BCfT"

Schrodinger

sites:::: :c:p::*: :c: anatomy:::missing .
such thatSince:•X ( E) IIi pas ( Pa p)✓where,T Holen X(ECB sub factor

-p =p
weN

ohave a notion,'p=Naaijkens( followingTg papa')

rtof relative size

andScaTquantity.

,)

EB::

E]

relativeis theJones index

entropy!

canbe tooachannel :PGIT-nhjmatnmaxlxcepsx-xce.cm.

Sps , Gpa? 3pm⇐'=XK pal spat),-

XK pal {,pal↳ restriction iste:

B → E

j : E ↳B

is theadjoint of( inclusion map)

13/18

BeyondzpeI-i.iiiiiiiii.IE?xiIiiiiiaii-xap.ne.
thin ( Pimsner('-Popa 86

Hiai',)

91

If

ECB

isirreduciblet(Longo 18g)

.ee/eoEotIB:!#E=
the conditionalE"'-

spam palm } ;,CaiWinterYeung 047'-t.FI#.,.III'E-

hpa pan t

expectation.

mutinies",then.

with N : E → B"*n--Theorem (Dentate 03EH)
,p f.

--=]
p.mn?psaePfSpoieCEY--log-B:E# E-hpmp.CL[Bon: Eon] [B : E)'( ie E' AB

y"-Eino a Enes',tN④n= N ⑤ N ⑤thin above→".ie' e' s.

.

-

*N :

E*n→B$"a¥ity !
44/18.

Beyondtypeic

notgeneralMany assumptions definitely.

,••suggests that thpiFEEI.no?'9a'm7aii jiY.

.

Highlights the difficulty of trans bake the wiretap model tomore•

cannot be extended

general von Neumann algebras

Suggests thataretheremight bea.

classof channels

whichadditivegqayyeadifferentappmaO
15/18

Privatesabalgebras
M Mosca et al.

.

.

Goodenozde④'
classicalPiva••encoder EMehas( El 9*7)

an=p.

-TPB

quantum

state

③Mev

inversetraa

ME ( El label ④ Pa) =p.

ten⑤E cannot•

private spaces

distinguish the inputs from A?

16/18

PrivateSabalgebras④-

5. D. Barlett

et .ae.

→SPOT-/6004)
ft

⑤AMev

f
E : ft

pg

④:

enoH

ancillaCA'Hspace,

Me ( proddimHs=dim

If forthen→a-,isT

Etta Me ( p,subsystem FIA istheMoreoverHaIt

fixedifsaidME ( p ① )Ttobea④ T)

said to=Hattdecoherence--pobe

Fp Eff'andT

distinguish the inputs from A

fixed pot ft

acompletely private with respect

④ oPo

⑤

#A

E cannotME : ft-'full subsystem

independent of ptoMEthen,.

17/18

Privatesabalgebras
J.

Cran net.

al.

( 20157GivenM

Hilbert

S awethatsay

ifavonNeumannalgebra,Paprojection in PCs)

Nc B CPS)

isprivate with respect topoh → BCS)

:M

space ,,Cpe MCM)

andaN'.

( where Cpc ) =P.

.

#)#TLeferenc.es#

Devetak 63

mainCai-.

Mosca-Probe

Naaij kens 17

arXiv(2000)

arXiv( 2004)

arXiv(2015)

arXiv ! 1510 06672 [quantum'-et.

al

5. D. Barlett et al.

J Crown et.

quant ph 10304127

Yeung 04

Winter'M

arXiv :-.

al.

.

.

Inf Trans.

:'-m

40 , 318-336

1704.05562 [ math-ph]quant ph 10003101-:

quant ph 10403161
-.

-ph)

18/18