A Few Algebraic Problems in the Theory of Quantum Entanglement

M. Smolu howskiInstitute of Physi s

A Few Algebrai Problems

in the Theory of Quantum Entanglement

Thesissubmittedforthefulllmentofthedegreeof Do torofPhilosophy

Šukasz Skowronek

Questions related to the pra ti aluse of quantum me hani s have grown

ex-tremelypopular among physi ists in the pasttwo de ades. The literature on

thesubje tisextensive,but itseemsnotto makeuseofthe advan es of

om-putationalalgebrai geometry,whi hisquiteanaturalframeworkwhendealing

with algebrai varieties likethe set of produ t states. The la k of general

in-terest an be partly attributed to the little popularity of algebrai geometry

among the physi ists working in the eld, and partly to the fa t that

fun -tionsusedasentanglementmeasuresarenotpolynomials. Anotherreasonmay

bethe appearan e of pairs of omplex onjugate variableslike

z

and

z

¯

in the polynomialequationsthatprevailinquantuminformations ien e,in ludingthe

Knill-Laammeequations,equationsforMutuallyUnbiasedBasesand

Symmet-ri InformationallyComplete ve tors,or forexpli itly nding produ t ve tors

in the kernelof an entanglementwitness. This makesthe equationsnot truly

polynomial,butfun tionsofboth

z

and

¯

z

atthesametimeandthusapparently moredi ultto solve. Animportantaim ofthe thesisisto presentanumber

ofspe i questionsthat anneverthelessbesolvedusingresultsfromalgebrai

geometry,and inparti ularthete hniqueof Groebnerbases. Themain result,

ontheother hand,whi h isa hara terization ofPPT bound entangledstates

ofminimal rank,equalfour, makessubstantialuseofBezout'stheorem, whi h

anbedes ribed asabasi theoremin interse tion theory. We alsopresenta

fewproblemssolvedbyelementaryalgebratri ks.

Thestru tureofthethesisisthefollowing. Therstpart, onsistingofthree

hapters,dis ussesthebasi softhetheoryofquantumentanglement,its

pra -ti aluses,andthephenomenonofboundentanglement. InChapter1,thefo us

isonquestionsrelatedtolo al realisti models ofquantum me hani s. I

famil-iarizethereaderwithseparablequantumstatesandseparability riteria. Later,

we onsider developments that go beyond the so- alled separability paradigm.

Inother words, wetakeatrip outsidethereignof quantum entanglement. In

Chapter 2,I brieydes ribetheideas behind several pra ti alappli ationsof

quantumentanglement,su hasquantum ryptography,quantumteleportation

anddense oding,aswellasquantummetrology. InChapter3,Iin ludedbasi

information aboutthe distillationof quantum entanglementand about bound

entangledstates.

makethis partas rigorousaspossible,however,in anumberofpla esIhad to

refertoliteratureforproofs. The hapterbeginsfromthedenitionofanane

varietyand itsideal, andwepro eed to thedenition ofamonomial ordering

and a Groebner basis. I introdu e the S-pair riterion and the Bu hberger

algorithm,whi h anbeusedto ndGroebnerbasesofanideal. Intheendof

Chapter4,itshouldbe ome learwhyGroebnerbasiste hniques anbeuseful

for solvingsystemsof polynomialequations. InChapter 5,thefo usis onthe

basi sofinterse tiontheory. I tryto explainhowthedimensionofananeor

proje tivevarietyrelatetothenumberofmonomialsof ertaintotaldegreenot

inthe orrespondingideal. Ialsointrodu etheimportantnotionofthedegreeof

aproje tivevariety. Atheorem thattwoproje tivevarietiesof omplementary

dimension interse t appears aswell. Finally, I give theBezout'stheorem in a

simpleform,whi hplaysanimportantrolelater,inChapter9.

Part III of the thesis, whi h starts with Chapter 6, mostly ontains the

original resultsobtainedand toyexamplessolvedby theauthor. I rstgivea

hara terizationtheoremfora lassof onvex onesofmapsfrom

n

×n

to

m

×m

matri es, whi h appear as

k

-positive and

k

-superpositive maps in the theory of entanglement. Next, in Chapter 7, wepresentthree algebrai problems in

thetheory ofquantum information, allofwhi h anbesolved by hand. They

on ernthefollowingsubje ts:

ˆ Produ tnumeri alrangeforathree-parameterfamilyof operators,

ˆ Higherordernumeri alranges(HONR)forthree-by-threematri es,

ˆ SeparablestateoflengththreeandS hmidtrankfour.

InChapter8,weapplytheGroebnerbasisapproa htoseveraltypesofequations

that areofinterestforthequantum information ommunity. Theproblemswe

managetosolverelateto:

ˆ Compressionsubspa esforQuantumErrorCorre tion(QEC),

ˆ CompletelyEntangledSubspa es(CES),

ˆ Maximallyentangledstatesin alinearsubspa e,

ˆ MutuallyUnbiasedBases(MUBs),

ˆ Symmetri InformationallyCompleteve tors(SICs).

Itshould bekeptinmindthat thelasttwooftheabovesubje tsarepresented

here in a fully expository manner, be ause better solutions by other authors

wereavailablein theliteraturebefore Istartedmyproje t. Finally,Chapter9,

whi histhe oreelementofthethesis, ontainsaproofoftheabovementioned

theoremrelatingpositive-partial-transpose(PPT)statesofminimalrank,equal

four,toso- alledUnextendibleProdu tBases(UPBs). Ipresentaproofthatthe

theorem isapplied, and somegeneralobservations on erningPPTstatesand

so- alled general Unextendible Produ t Bases are made. I on lude on page

143 and subsequently give a list of papers I published as a part of my PhD

proje t. Most of them have strong relations to the results presented in this

thesis. However,someofthe ontentshasneverbeenpublished.

There areafew people andorganizationswhohelpedme tosu eed in my

resear h proje t. Looking ba k in time, I an ertainly say that my whole

PhD studies were marked with a fair amount of good lu k. Under dierent

ir umstan es, it would have been mu h more di ult, if not impossible, to

omplete the thesis. Hen e, I must rst mention the support I re eived from

theFoundationfor Polish S ien e. Thanksto them, Iwasableto travel,meet

other s ientists, and to live a de ent life for the most of the duration of my

studies. Part of my ontra t with the foundation was a visit to Sto kholm,

where I got to know JanMyrheim andPer Øyvind Sollid. Few months later,

ourintera tionturnedouttobeveryfruitfulandresultedinaproofofTheorem

9.27 of Chapter 9, whi h is the ba kbone of the thesis. This ould probably

haveneverbeenpossible,hadInotre eivedadditionalsupportfromSto kholm

University and the University of Oslo, all thanks to Ingemar Bengtsson and

ErlingStørmer. IwishtothankIngemarformakingagreatdis ussionpartner

during my months in Sweden, and Erling for his grandhospitalityduring my

twovisitstoNorway. ItisOslowheremybestideaswereprovisionallyformed,

in luding theresults of Chapters 6and 9. Forthe rst visitthere, I re eived

additional funding from the S holarship and Training Fund, operated by the

FoundationfortheDevelopmentoftheEdu ationSystem,whi hIamsin erely

thankful for. It is also indisputable that the su ess of my resear h ru ially

depended on the onstant support by my supervisor, Karol ›y zkowski. His

en ouragement,wisejudgmentandgreatamountofunderstandingaredi ult

to overvalue. Besides theabove, I owespe ial personalthanksto PerØyvind

Sollidfor arefulproofreadinganddete tingaawinapreliminaryversionofthe

manus riptonPPTstatesofrankfour, in ludedhereasthe ru ialChapter9.

In the end, I wish to warmly thank my parents and my younger brother

Mi haª, whowere alwaysthere to helpme whenI needed it,espe iallyduring

Basi s of quantum

Fundamental questions

1.1 Lo al hidden variables

Somestrange onsequen esofquantumme hani shavebotheredphysi istsfrom

theverybeginningof quantum theory. A lassi alexampleof this is the

Ein-stein,PodolskyandRosenpaper[1℄,wheretheauthorsarguethatthequantum

des riptionofrealitymustbein omplete ifwea epttworathernatural

prop-erties everyphysi al theoryshould have. Therst is the prin ipleof physi al

reality, whi h says that properties of physi al systems su h as spin dire tion

orenergy anbepredi ted with ertaintybefore arryingoutthe

orrespond-ing measurement. They areelements of physi al reality. These ond prin iple

onsidered byEinstein, PodolskyandRosen is that oflo ality, whi hrefersto

therequirementthateverysystemhasitsownproperties,independentlyofany

operations arriedoutonother, spatiallyseparatedsystems. Toseewhere the

abovetwoprin iples lashwiththepi tureofrealitygivenbyquantum

me han-i s, letus onsider a quantum system onsisting of twotwo-level 1

subsystems

A

and

B

, initiallypreparedin the so- alledBellstate

∣Φ

+

⟩ = (∣00⟩ + ∣11⟩) /

√

2

. Iftheholderoftherstsubsystemmeasuresitinthebasis

{∣0⟩ , ∣1⟩}

,heorshe obtains theresult

0

or

1

, both with probability

1

/2

. This is not too surpris-ingand maywellhappen in lassi alphysi s,howeverassumingthat thestate

∣Φ

+

⟩

doesnot ontaina ompleteinformation aboutthe degreesof freedomof thesystem. Whatis somewhatmoreinteresting,is thepredi tionofquantum

me hani sthatafter

0

or

1

ismeasuredinthesubsystem

A

usingthe

{∣0⟩ , ∣1⟩}

basis, a orresponding measurement on the

B

side yields identi ally thesame resultasthe aforementionedmeasurementon the

A

side. More generally, the holdersof

A

and

B

neverget twodistin tresultsif theymeasure in thesame basis. Thisispossibletore on ilewiththeprin ipleoflo alityonlyifwea ept

thattheout omesofallpossiblemeasurementsonthe

A

and

B

sidesareknown beforehand, i.e. before any measurements are done. It is possible to ompare

this to ama ros opi situation where a fa tory produ es table tennis bats in

1

two olors,sayredandgreen,putseverysingleoneintoaboxandgroupsthese

boxes into pairs with batsof the same olor inside. It then sells these pairs

without dis losing what olour the batsinside a parti ular pair of boxes are.

Thebuyerofatable tennissetknowsforsurewhatheorshehasaretwobats

in thesame olour, but doesnotknowanythingmore. As soon asoneof the

boxesisopened,the olour ofthebatinside the se ondboxis revealedto the

buyer. Nomatterhowrealisti thewhole situationmightseeminreallife,itis

learlynotex ludedby lassi alphysi s,andit loselyresemblestheexperiment

with twotwo-levelsystemsin thestate

(∣00⟩ + ∣11⟩) /

√

2

,with

0

orresponding to green and

1

orrespondingto red, ortheother wayround. Ouraim in the followingwill be to shortlyexplain why a lassi almodel similar to the table

tennis set fa tory annot nevertheless giveus aproperdes riptionofthe

phe-nomena predi ted by quantum me hani s. In order forthe dis ussion to stay

general,letusintrodu ethefollowingdenition.

Denition1.1. Alo al hiddenvariablemodelofanexperimentona

bipar-tite system( onsistingof parts

A

and

B

) isaprobability spa e

(Ω, Σ, P )

anda setoffun tions

S

x

A

∶ Ω →

Rand

S

y

B

∶ Ω →

R,where

x

and

y

refertothepossible measurement setups on subsystems

A

and

B

, respe tively and

S

x

A

(λ)

,

S

y

B

(λ)

orrespondtothe measurements'out omes. Here

λ

representsthe hidden vari-ables or the true lassi al degreesof freedom of thesystem. Assumingthatthe

measurementsetupisxedto

x

for

A

andto

y

for

B

,the orrelation oe ient between the measurement out omesisgiven bythe following formula

ǫ

_{(x, y) = ∫}

Ω

S

x

A

(λ) S

y

B

(λ) dP (λ) .

(1.1)

Tomakea onne tiontothe tabletennisset fa torymodel, letusmention

that

λ

in formula(1.1) orrespondstoamode of thefa tory, whi h iseither theprodu tionofapairofgreenbatsortheprodu tionaredpair. Themode

is hidden from the buyer atable tennis set, just as the additional degrees of

mathemati alargument,torefutetheideaofalo alhiddenvariablemodelfor

quantumme hani s.

To this aim, let us onsider a system onsisting of twospin-

1

/2

parti les, initially preparedin thestate

∣Φ

+

⟩ = (∣00⟩ + ∣11⟩) /

√

2

, where

∣0⟩

,

∣1⟩

represent the

±1

eigenstatesoftheoperator

σ

z

. Wemeasurethespinoftherstparti le in the dire tion

⃗a

and the spin of the se ond parti le in the dire tion

⃗b

. The orrespondingobservablesare

⃗a ⋅ ⃗σ

A

and

⃗b ⋅ ⃗σ

B

,wherethesubs ripts

A

,

B

refer tooperatorsontherstandthese ondsubsystem,respe tively. The orrelation

oe ientbetweenthetwomeasurements,aspredi tedbyquantumme hani s,

is

˜

ǫ

(⃗a,⃗b) = ⟨Φ

+

∣ (⃗a ⋅ ⃗σ

A

) (⃗b ⋅ ⃗σ

B

) ∣Φ

+

⟩ = a

1

b

1

− a

2

b

2

+ a

3

b

3

,

(1.2) where thenumbers

a

i

,

b

i

for

i

= 1, 2, 3

denote the oordinatesof theve tors

⃗a

and

⃗b

, resp. For the spe i hoi e of the ve tors

⃗a = [sin α, 0, cos α]

and

⃗b =

[sinβ,0,cos β]

, weget

˜

ǫ

(⃗a,⃗b) = ˜ǫ(α,β) = cos(β − α)

. Letusnowsupposethat this form of orrelation fun tion anbereprodu ed byalo al hiddenvariable

model. Thusweneedtohaveaprobabilityspa e

(Ω,Σ,P)

andasetoffun tions

S

_A

α

∶ λ ↦ S

_A

α

(λ)

and

S

β

B

∶ λ ↦ S

β

B

(λ)

givingthemeasurementout omesof the spin measurements for a xed hoi e of the hidden variables

λ

. Sin e a spin measurement anonlygive

±1

asananswer,wehave

S

α

A

(λ) ∈ {−1,+1}

,

S

β

B

(λ) ∈

{−1,+1}

. Let us now onsider the following ombination of the fun tions

S

α

A

and

S

β

B

,

S

α

2

A

(λ) [S

β

1

B

(λ) + S

β

2

B

(λ)] + S

α

1

A

(λ) [S

β

1

B

(λ) − S

β

2

B

(λ)].

(1.3) It is easy to see that for xed

λ

, one of the expressions in squared bra kets equals

0

,while theother oneisequalto

±2

. All inall, thewhole expressionin (1.3) equals

±2

. Thereforewehave, assumingthat thehidden variable model we onsider des ribesthe quantum me hani alworld, thefollowinginequality

forthepreviously onsidered orrelationfun tions,

∣˜ǫ(α

2

, β

1

) + ˜ǫ(α

2

, β

2

) + ˜ǫ(α

1

, β

1

) − ˜ǫ(α

1

, β

2

)∣ ⩽

⩽ ∫

_Ω

∣S

α

2

A

(λ)[S

β

1

B

(λ) + S

β

2

B

(λ)] + S

α

1

A

(λ)[S

β

1

B

(λ) − S

β

2

B

(λ)]∣ ⩽ 2.

(1.4) TheaboveisthefamousCHSHinequality,namedforJ.F.Clauser,M.A.Horne,

A. Shimony and R. A. Holt [2℄. Forthe hoi e

α

1

= 45

○

,

β

1

= 90

○

,

α

2

= 135

○

and

β

2

= 180

○

,one anreadily he kthatthe orrelationfun tionspredi tedby

quantumme hani sdonot obey(1.4),sin e

∣˜ǫ(α

2

, β

1

) + ˜ǫ(α

2

, β

2

) + ˜ǫ(α

1

, β

1

) − ˜ǫ(α

1

, β

2

)∣ = 2

√

2.

(1.5)

Moreover,theaboveviolationoftheCHSHinequalityisthemaximumallowed

byquantum me hani s[3℄. Thevalue

2

√

2

in (1.5), alledtheTsirelson bound, is in lear ontradi tionwith theassumption that quantum me hani s anbe

lusion that there exists no lo al realisti model for quantum me hani s. The

question whether the quantum me hani al orrelations are really observed in

experiments, andhow to lose thepossibleexperimental loopholes,isthe

sub-je tofaseparateeldofresear h, withtherstandmostfamousexperiments

donebytheA. Aspe tgroup[4℄.

1.2 Separable states and separability riteria

Ournexttopi is loselyrelatedtohiddenvariablemodels,andwasrststudied

byR.Wernerinthelate80s[5℄. Heintrodu eda lassofmixedstates,whi hhe

alled lassi ally orrelated,buttheyarenowgenerallyreferredtoasseparable.

Denition1.2. Astaterepresentedbyadensitymatrix

ρ

onabipartitespa e

K⊗H

is alledseparableifandonlyifit anbewrittenasa onvex ombination of proje tionsontoprodu tstates, i.e. asum

ρ

=

n

∑

i=1

λ

i

∣φ

i

⊗ ψ

i

⟩ ⟨φ

i

⊗ ψ

i

∣

(1.6) with

n

nite,

λ

i

⩾ 0

,

∑

n

i=1

λ

i

= 1

and

φ

i

∈ K

,

ψ

i

∈ H

.

A tually,in[5℄,innitesumsofthetype(1.6)were onsidered,butitfollows

from Carathéodory'stheorem ( f. e.g.[6,Chapter 13℄)that anysu h sum an

berewrittenasaniteone. AgeneralizationofDenition1.2toamultipartite

settingisimmediate.

Denition 1.3. A state represented by a density matrix

ρ

on a multipartite spa e

K

1

⊗ . . . ⊗ K

k

is alledseparable ifandonlyifit anbewrittenasasum

ρ

=

n

∑

i=1

λ

i

∣φ

1

i

⊗ . . . ⊗ φ

k

i

⟩ ⟨φ

1

i

⊗ . . . ⊗ φ

k

i

∣

(1.7) with

n

nite,

λ

i

⩾ 0

,

∑

n

i=1

λ

i

= 1

and

φ

l

i

∈ K

l

∀

l=1,2,...,k

.

Itisnowalsogenerallya eptedthatstateswhi harenot oftheformgiven

in Denitions1.2and1.3are alled entangled.

Denition 1.4. A state represented by a density matrix

ρ

on a multipartite spa e

K

1

⊗ . . . ⊗ K

k

is alledentangledif andonlyifit annotbewritteninthe form (1.7).

In aseofpurestates

ρ

,it anbeshown[7℄( f. also[8℄)thatthepropertyof beingentangledimpliesthela kofalo alrealisti modelofthelo al

measure-mentsone anperformon

ρ

. Morepre isely,forapureentangledstate

ρ

,there alwaysexistsaBell-typeinequality

2

thatisnotfullledbythe orrelation

fun -tionsresultingfrom

ρ

. However,ifmixedstates

ρ

aretakeninto onsideration, 2

whi h do admitalo al realisti des ription. Itshouldalsobenotedthatin the

paper[5℄, theauthorneverusedthewordentangled himself. Itmaythusbe

rather surprising to hear that what is now generally a epted as a synonym

of somethingquantum-like, somethingentangled, wasborn for thepurpose to

showthatit ansometimesbedes ribedinafully lassi alway. Fortunately,the

apparentparadoxwaspartially resolvedby[9℄, where theauthor showedthat

sometimeshiddennonlo alityin quantum states anberevealed bysequential

measurements. A step in asimilar dire tion wasalso taken byN. Gisin, who

showedthat lo al intera tion anturn a statethat doesnotviolate any

Bell-type inequality into onethat is nonlo al [10℄. Additional justi ation for the

importan eofthenotionofinseparabilitywasprovidedbyL.Masanes[11,12℄,

whoshowedthatentangledstatesarealwaysusefulfor ertaintasksinquantum

informationpro essing. Finally,the questionaboutnonlo alityof allbipartite

entangledstateswassettledinthepaper[13℄,byMasanes,LiangandDoherty.

They managed to prove that bipartite entangled states

ρ

are pre isely those whi hdo violate someinequalityof CHSH type,possibly aftertheyare tensor

multiplied by some state

σ

that doesnot violate any CHSH inequality itself. Beingmorepre ise,

ρ

isentangled

⇐⇒ ρ ⊗ σ

violatesaCHSHtypeinequality (1.8) where

σ

doesnot violateanyinequalityofCHSHtype,evenafteritundergoes arbitrary sto hasti lo al operations with ommuni ation[13℄. Note that the

tensor multipli ation by

σ

only plays a role of a atalyst in the pro ess of dis overingthenonlo alityof

ρ

. Hen e,itislegitimatetosaythatallbipartite entangledstateshavesomekindofnon-lo allyrealisti properties,andvi eversa.

Be ause of the result by Masanes, Liang and Doherty, we feel it is

well-justiedtoa eptthedenitionofentangledstatesasitis. Hen e we onform

to the separability paradigm. However, we shall go ba k to the question of

separabilityversuslo alrealismwhenwedis ussdistillationofentanglementin

Se tion3.1. Weshouldalsogiveadditional redittotheWerner'spaper[5℄and

mentionthefamous familyof statestheauthorused toprovehisresult. They

arenow alledWernerstates andareofthesimpleform

W

=

1

d

3

_{− d}

[(d − Ξ)

1

+ (d Ξ − 1) V ]

(1.9) where

Ξ

∈ [−1, 1]

,

d

is thedimensionality ofthe Hilbert spa e

K

su h that

W

is dened on

K ⊗ K

, and

V

∶=

∑

d

i,1=1

∣i⟩⟨j∣ ⊗ ∣j⟩ ⟨i∣

. The hoi e of the spe i parametrizationin(1.9)ismotivatedbytheequality

Ξ

=

Tr

(WV )

. Adistin tive futureoftheWerner statesisthattheyareinvariantunderthetransformation

W

↦ (U ⊗ U)W (U

∗

⊗ U

∗

)

for an arbitrary unitary

U

. In [5℄, it was shown that the state

W

is separable for

Ξ

⩾ 0

and entangled otherwise. Moreover, for

Ξ

= −1 + (d + 1) /d

2

it admits a hidden variable des ription. Sin e

−1 +

(d + 1)/d

2

_{⩽ −1/4 ⩽ 0}

, the orresponding

W

isentangledand at thesametime it anbedes ribedinalo alrealisti manner.

widely a epted that the distin tion between entanglement and separability

playsafundamental rolein thetheory ofquantum information. Entanglement

dete tion hasbe ome thesubje tofaseparate resear harea,whi h wewould

verysparselyexplorein therestofthisse tion. Mu hmoreinformation anbe

foundin reviewarti leslike[14,15℄.

Probably the most famous separability test is the PPT riterion by A.

Peres [16℄, where PPT stands for positive partial transpose. The riterion

was qui kly proved by the Horode ki family to be a ne essary and su ient

onditioninthe aseof

2

× 2

and

2

× 3

systems 3

[18℄. The riterionsimplysays

thatadensitymatrix

ρ

onabipartitespa e

K⊗K

,ifseparable,mustbepositive under thefollowingtransformation

ρ

↦ (

id

⊗t) ρ,

(1.10)

where

t

denotesthetransposition mapin

B (K)

. Thus,ifthepartial transpose

ρ

T

2

∶= (

id

⊗t) ρ

ofadensitymatrix

ρ

is foundnot to bepositive,weknowthat

ρ

isentangled. Letusstatethisasaproposition.

Proposition1.5(PPT riterion). Ifastate

ρ

a tingonabipartitespa e

K⊗K

is separable,the partialtransposeof

ρ

,given bythe r.h.s. of (1.10),must bea positive operator.

States whi h do not satisfy the impli ation of Proposition 1.5 are alled

NPT entangled, where NPT stands for negative partial transpose. It was

a natural question to ask whether there exists entangled states with positive

partialtranspose(PPT).For

2

× 2

and

2

× 3

systems,thisisimpossibleby[18℄, but for

3

× 3

systems,aPPT entangled state wasfound byP. Horode ki [19℄. Dierent examples were earlier studied, in a slightly dierent ontext, by E.

Størmer[20℄andM.-D.Choi[21℄. Inordertoprovehisresult,theauthorof[19℄

needed a dierent separability test than the PPT riterion. What he used is

now alled therange riterion forseparability.

Proposition1.6(Range riterion). Foraseparablestate

ρ

onabipartitespa e

K ⊗K

,theremustexistasetof produ tve tors

φ

i

⊗ ψ

i

thatspantherangeof

ρ

,

R

(ρ)

. In additiontothat,thepartially onjugatedve tors

φ

i

⊗ ψ

∗

i

needtospan the rangeof

ρ

T

2

,

R

(ρ

T

2

)

.

A whole family of separability riteria an be derived from the following

resultbytheHorode kifamily[18℄,whi hgeneralisesthePPT riterion.

Proposition 1.7 (Positivemaps riterion). A separablestate

ρ

on abipartite spa e

K ⊗ K

isseparable ifandonly if

(

id

⊗Λ) ρ ⩾ 0

(1.11)

for alllinearmaps

Λ

∶ B (K) → B (K)

that preserve the positivity of opera-tors.

3

thename of the riterion. Oneof themain resultsof this thesis,presentedin

Chapter6,isabroadgeneralizationofthepositivemaps riterionfordierent

sub lassesof theset of all densitymatri es, in luding statesof S hmidt rank

k

[22℄.

The problem with ondition (1.11) is that it needs to be he ked for all

positivemaps,whi hisimpossibleaslongaswedonotknowtheirfullstru ture.

However, for a xed hoi e of the map

Λ

, the positive maps riterion always givesane essary onditionforseparability. Anexampleofthisiswhen

Λ

(ρ) =

1Tr

ρ

− ρ

, so- alled redu tion map. For su h hoi e of the positive map, we get[23℄

Proposition 1.8 (Redu tion riterion). Aseparable stateon abipartite spa e

has tofulllthe following ondition

(

Tr

B

ρ

) ⊗

1

− ρ ⩾ 0,

(1.12)

where Tr

B

denotes the partial tra e of

ρ

with respe t to the se ondsubsystem,

(

Tr

A

ρ

)

ij

=

∑

k

ρ

ik,jk

.

Anotherpossible hoi eof

Λ

is

Λ

∶ ρ ↦

1Tr

ρ

− ρ − V ρ

t

_V

∗

,so- alled

Breuer-Hallmap[24,25℄. Here

ρ

t

standsforthetranspositionof

ρ

and

V

isan antisym-metri ,unitary matrix,

V

t

_{= −V}

. Su h matri es

V

onlyexist ifthedimension ofthespa e

K

iseven.

Yet another, experimentallyfeasible approa h to the dis rimination of the

set of separable states is by the use of so- alled entanglement witnesses. An

entanglementwitness 4

isanoperator

W

onamultipartitespa e

K

1

⊗K

2

⊗. . .⊗K

k

withtheproperty

⟨φ

1

⊗ φ

2

⊗ . . . ⊗ φ

k

∣ W ∣φ

1

⊗ φ

2

⊗ . . . ⊗ φ

k

⟩ ⩾ 0

(1.13) forall

φ

1

,...,

φ

k

in

K

1

,...,

K

2

, resp. Intermsofsu hoperators,wehavethe followingseparability riterion

Proposition 1.9 (Entanglement witness riterion). A density matrix

ρ

on a multipartite spa e

K

1

⊗ K

2

⊗ . . . ⊗ K

k

is separable if and only if the following inequality,

Tr

(Wρ) ⩾ 0

(1.14)

holdsfor allwitnesses

W

on

K

1

⊗ K

2

⊗ . . . ⊗ K

k

.

A big advantage of witnesses over positive maps is that the tra e on the

l.h.s. of (1.14) an be measured in an experiment asan expe tation valueof

an observable. Moreover,one an oftennd an optimalde omposition of the

witnessinto lo ally measurable quantities [26,27℄, i.e. ade omposition of the

form

W

=

r

∑

l=1

γ

l

X

1

l

⊗ X

l

2

⊗ . . . ⊗ X

l

k

(1.15) 4

with

r

minimal. One analsoaskwhetherawitness

W

is optimalin thesense that forno otherwitness

W

′

theinequalityTr

(Wρ) < 0

impliesTr

(W

′

_ρ

_{) < 0}

[28℄.

Nevertheless, we should note that by the Jamioªkowski-Choiisomorphism

[29,30℄( f. also[31℄),everywitnesshasa orrespondingpositivemap

Λ

W

,and the orrespondingpositivemap riterion

(

id

⊗Λ

W

)ρ ⩾ 0

is mu h strongerthan the riterionTr

(Wρ) ⩾ 0

. However,therst riterionismu hmoredi ultto measure inanexperiment[32℄.

To nish, let us explain a relation of the CHSH inequality, introdu ed in

Se tion 1.1, to entanglement witnesses. It was rst pointed out in [33℄ that

Bell-typeinequalities an beper eivedasve torsintheFarkaslemma[6℄,

dis- riminatingbetweenthesetof orrelationswithalo alrealisti des riptionand

thequantum orrelations. TheFarkasve tors aninturnberelatedto

observ-ables,whi h havetheinterpretation ofwitnesses. Intheparti ular aseofthe

CHSH inequality, the Terhal's theory boils down to the observation that the

expression

˜

ǫ

(α

2

, β

1

) + ˜ǫ(α

2

, β

2

) + ˜ǫ(α

1

, β

1

) − ˜ǫ(α

1

, β

2

)

inequation(1.4) anbe written intheformTr

(Bρ)

,where

B ∶= ⃗a

1

⋅ ⃗σ ⊗ (⃗b

1

+ ⃗b

2

) ⋅ ⃗σ − ⃗a

2

⋅ ⃗σ ⊗ (⃗b

2

− ⃗b

1

) ⋅ ⃗σ

(1.16) is the CHSH operator, rst introdu ed in [34℄,and

⃗a

1

,

⃗a

2

,

⃗b

1

and

⃗b

2

are the spindire tionve tors, orrespondingtothepreviouslyuseddete torangles

α

1

,

α

2

,

β

1

and

β

2

, respe tively. Using

B

, one an easily onstru t the operator

W

_{= 2}

1

− B

,whi hisawitnessa ordingto theCHSH inequalityandthefa t that all separable states admit a hidden variable des ription. Moreover, the

inequality

Tr

(Wρ) = 2 −

Tr

(Bρ) < 0

(1.17) observedfor somestate

ρ

, doesnotonlyindi ate that

ρ

is entangled, but also that itisnonlo al. Thus

W

playsadoubleroleofanentanglementand nonlo- alitywitness.

1.3 Beyond quantum entanglement

Questionsbeyondtheseparabilityparadigm,orevenbeyondtheframesof

quan-tum me hani s, have been onsidered in the quantum information literature

sin etheearlydaysofthesubje t. Awell-knownexampleofthisisthefamous

paper[35℄byPopes uandRohrli h, wherenonlo alityis onsideredasa

possi-bleaxiomforquantumme hani s. Morepre isely,theauthors onsidernonlo al

theories that do obeyrelativisti ausality. It turns out that there anexist,

at least in prin iple,theories of this type whi h are not identi al to quantum

me hani s. Toexplain this in more detail, letus briey repeat the simplied

versionoftheargumentin[35℄,asitwaspresentedin alaterpaper[36℄.

We onsider a theory of a pair of

spin-1

2

parti les whi h yields, for some reason,identi alprobabilitiesforthemeasurementout omes

↑↑

and

↓↓

,aswellas identi alprobabilitiesfortheout omes

↓↑

,

↑↓

,nomatterwhatthemeasurement

possibilityofsupraluminal ommuni ationusingthetwoparti les. Wesaythat

thereareonlynon-signalling orrelations ( f. e.g.[37℄)betweenthem. Another

onsequen eisthattherespe tive orrelationfun tion

ǫ

mustdependonlyonthe relativeangle

θ

betweentherst andthe se ondmeasuringdevi e. Moreover, ithastofulll

ǫ

(π − θ) = −ǫ (θ)

. Onepossible hoi eofsu h afun tionis [36℄,

ǫ

(θ) =

⎧⎪⎪⎪⎪

⎨⎪⎪⎪

⎪⎩

1

for

θ

∈ [0,

π

4

]

2

(1 −

2x

_π

)

for

θ

∈ (

π

4

,

3π

4

)

−1

for

θ

∈ [

3π

4

, π

]

(1.18)

By hoosingthesu essiveangles

α

1

= 0

,

β

1

=

π

4

,

α

2

=

π

2

and

β

2

=

3π

4

inanEPR experimentofthetypedis ussedinSe tion 1.1,weget

∣ǫ (α

2

− β

1

) + ǫ (α

2

− β

2

) + ǫ (α

1

− β

1

) − ǫ (α

1

− β

2

)∣ = 4

(1.19) asananalogueofequation(1.5). However,thistimetheviolationofthe lassi al

bound(1.4)isbiggerthanpossibleinquantumme hani s. Thus,atheorywith

a orrelationfun tionoftheform(1.18)obeysrelativisti ausality,yetitisnot

onsistentwith thequantum-me hani aldes riptionoftheworld.

Theabovedis ussionshowsthatitis notpossibleto reprodu ethelawsof

quantumme hani sjustbyusingtheprin ipleofnon-signalling. The

Popes u-Rohrli h orrelations onstituteatoymodel,usefulfordemonstratingthisfa t.

However,after theseminal paper[35℄,afair amountof work [3744℄ hasbeen

devoted to understanding thepropertiesof Popes u-Rohrli h orrelationsand

how they would ae t ommuni ation omplexity, had they been present in

reality. Usually, su h questions are formulated in the language of so- alled

nonlo al boxes. In order to demystify this new notion, let us explain that a

nonlo al box orresponding to the pre ise Popes u-Rohrli h setup dis ussed

above,looksasinFigure1.2. Itisanimaginarydevi ewithtwoinputs

a

,

b

and two(random)outputs

x

,

y

thatsatisfy ertainrelation. Theinputs,whi htake values

0

or

1

, orrespondtothemeasurementsetupsfortherstandthese ond parti le,respe tively. Forexample,

a

= 0

meansthatthespinoftherstparti le ismeasuredin abasisrotatedby

α

1

. Similarly,

b

= 1

indi ates ameasurement basisforthese ondparti leisrotatedby

β

2

. Theoutputs

x

and

y

,ontheother hand, orrespondtothemeasurementresults

↑

or

↓

. Forexample,

y

= 1

indi ates that spin

↑

was measured for the se ond parti le. A qui k thought reveals that the abovebox, alled mod2NLB in [45℄,is just amoreabstra t wayto

expressthepropertiesofanimaginaryEPRexperimentwith orrelationsgiven

by the fun tion (1.18). The only mathemati al ontent of any su h box, not

ne essarilyrelatedtothe orrelationfun tion(1.18),isa onditionalprobability

fun tion

p

(xy ab)

thatfulllsso- allednon-signalling onditionsthatguarantee theimpossibilityofsupraluminal ommuni ation, f. [37℄. Generalizationstoa

multipartite s enarioareimmediate.

Notably, mod2NLBwaspostulatedasaunit of nonlo ality [46℄,somewhat

similarto theroleplayedbytheBellsinglet

1

/

√

experiment

theory. However,itwasimmediatelyrealized[46℄thatnotallmultipartiteboxes

anbesimulatedusing anumberof opiesofmod2NLB.Moreover,in [45℄the

authors showedthat in thebipartites enario, there doesnot exist anite set

of nonlo al boxesthat ouldbeused to simulate all bipartitenonlo al boxes.

Interestingly,intheproofpresentedin[45℄theHilbertbasistheoremwasused,

whi halsoappearsin Se tion4.2ofthisthesisasTheorem4.18.

Asintelle tuallyappealingastheyare,generalnonlo alboxesdonotseemto

havea ounterpartintherealworld. Still,mostofthedis ussionbythequantum

information ommunitydoesstaywithintheframeworkofquantumme hani s,

but notne essarily on entratesonentanglement. Inparti ular, itwasqui kly

re ognized that there exist nonlo al phenomenain quantum me hani s whi h

annotbeexplainedbythepresen eofentanglement. Inthewell-knownpaper

[47℄, the authors show an example of a family of nine mutually orthogonal

bipartiteprodu tstatesthat annotbedistinguishedusinglo almeasurements

and lassi al ommuni ation by the two parties. They all this phenomenon

nonlo alitywithoutentanglement,hen epointingouttothedieren ebetween

the twonotionsthat tended to be takenasequivalent. However, itshould be

keptinmindthatnonlo alityintermsoftheviolationofBellinequalitiesisvery

loselyrelated,ifnotequivalent,tothepropertyofbeingentangled. Webriey

explainedthisinSe tion1.1,wherewereferredtoapaperbyL.Masanes,Y.-C.

Liangand A.C. Doherty[13℄. Therefore,thenotionofnonlo alityin[47℄ and

in theresear hwedes ribein therest ofthis se tion,signi antlydiersfrom

what was traditionally per eived asthe equivalentof being nonlo al, i.e. the

violationofBellinequalitiesandthela kofalo alrealisti des ription.

In more re ent days, the study of nonlo ality largely revolves around its

twoquantitativemeasures,whi harethequantumdis ord,introdu edby›urek

Thebasi ideabehindthequantumdis ordisthattwoexpressionsforso- alled

mutualinformation that areequivalentin the aseof lassi alprobability

dis-tributions, do not ne essarily give the same answer when generalized to the

quantums enario. Indeed,letusdene theentropyofa lassi alrandom

vari-able

A

as

H

(A) = −∑

a

p

(A = a)log p (A = a)

(1.20) andthe onditionalentropyof

A

withrespe ttoanother lassi alvariable

B

as

H

(A B) = ∑

b

p

(B = b)H (A B = b),

(1.21) where

H

(A B = b)

istheentropyofthevariable

A

onditionedonaparti ular value

b

ofthevariable

B

. Wedene themutualinformation ofthevariables

A

and

B

as

J

(A ∶ B) = H (A) − H (A B)

(1.22)

A littleinspe tionshowsthat in the aseof lassi alprobabilitydistributions,

theaboveexpressionisequivalentto

I

(A ∶ B) = H (A) + H (B) − H (A,B),

(1.23) where

H

(A,B)

staysfortheentropyofthe olle tivevariable

(A,B)

. Thuswe have

I

(A ∶ B) = J (A ∶ B)

for arbitrary lassi alvariables

A

and

B

. However, as pointed out in [48℄, the equality between the two expressions for mutual

informationdoesnotgenerallyholdinaquantumworld.

To show this, let us onsider a bipartite quantum system des ribed by a

density matrix

ρ

AB

. The states of the subsystems are given by the partial tra es of

ρ

AB

,

ρ

A

=

Tr

B

ρ

AB

and

ρ

B

=

Tr

A

ρ

AB

. We immediately see that a quantumanalogueof (1.23) is

I

(ρ

AB

) = H (ρ

A

) + H (ρ

B

) − H (ρ

AB

)

(1.24) where

H

(ρ) ∶= −

Tr

(ρ log ρ)

. However, it is not obvious how to generalize

J

(A ∶ B)

to the quantum ase. The reason behind this is that the quantum subsystem

B

an be measured in various bases, and one of them has to be sele ted before asum similar to the

∑

b

in formula (1.21) is al ulated. Thus we haveawhole familyof onditional entropies

H

(ρ

AB

{Π

b

})

, where

{Π

b

}

is anarbitrary ompleteset of one-dimensionalproje tionsonthe subsystem

B

, satisfying

∑

b

Π

b

=

1. Expli itly,

H

(ρ

AB

{Π

b

})

isgivenby

H

(ρ

AB

{Π

b

}) = ∑

b

p

b

H

((

1

⊗ Π

b

)ρ

AB

(

1

⊗ Π

b

)

p

b

)

(1.25)

where

p

b

=

Tr

((

1

⊗ Π

b

) ρ

AB

)

is theprobability to obtainaresult

b

in a mea-surement orresponding to

{Π

b

}

. Simply be ause the

H

(ρ

AB

{Π

b

})

are not

allequal,there isnosinglequantum analogueof

J

(A ∶ B)

. Instead,wehavea familyofmutualinformationanalogues,givenby

J

(ρ

AB

{Π

b

}) = H (ρ

A

) − H (ρ

AB

{Π

b

})

(1.26) Thesupremum

C

B

(ρ

AB

) = sup

{Π

b

}

J

(ρ

AB

{Π

b

})

(1.27)

anbe onsideredasameasureof lassi al orrelations[48,51℄. Notethatthere

also existsarelated quantity

C

A

(ρ

AB

)

where theroles of

A

and

B

havebeen inter hanged. The quantum dis ord is now dened as the dieren e between

I

(ρ

AB

)

and

C

B

(ρ

AB

)

,

D

B

(ρ

AB

) = I (ρ

AB

) − C

B

(ρ

AB

)

(1.28) Alternatively,thenamedis ord mayreferto

D

A

(ρ

AB

) = I (ρ

AB

) − C

A

(ρ

AB

)

(1.29) althoughthetwoquantities

D

A

and

D

B

donotgenerally oin ide.

Duetotheequality

I

(A ∶ B) = J (A ∶ B)

validinthe lassi alworld,the non-vanishing ofthe dis ordfor

ρ

AB

isasignof quantumnessof thestate. Unlike separability,thevanishingofthedis ordonlyo urs forameasurezerosubset

oftheset ofallstates[52℄. Inparti ular,

D

A

and

D

B

vanish simultaneouslyif andonlyif

ρ

AB

hasaneigenbasis onsistingofprodu tve tors,i.e.

ρ

AB

=

∑

i,j

λ

i,j

∣φ

i

⟩⟨φ

i

∣ ⊗ ∣ψ

j

⟩⟨ψ

j

∣

(1.30) where

λ

ij

⩾ 0

, while

φ

i

and

ψ

j

onstitute bases for the rst and the se ond subsystem, respe tively. Su hstatesare alled lassi ally orrelated [53℄. They

alsoplayanimportantroleinthealternativeframeworkfor orrelationstudies,

developedbyOppenheimandtheHorode kifamily[49,53℄.

Itis ingeneralnoteasy toevaluatethequantum dis ord,butsomeresults

have been obtainede.g. for

2

× 2

systems [54,55℄. Several onditions forzero and non-zero quantum dis ord are known as well [52,56,57℄, and a missing

operationalinterpretationofthequantityhasbeenprovidedin [58℄intermsof

aquantum statemergingproto ol.

Quantumde it,ontheotherhand,hashadarelatively learphysi al

inter-pretationfromtheverybeginningwhenitwasintrodu edin[49℄. Thequantity

is believed to beequal to theamount of work whi h anbeextra ted from a

multipartite quantum state

ρ

globally, minus the amount of work the parties andrawlo ally,possiblyafter transformingthestatebyanallowedfamilyof

transformations. This des riptionmay seemalittlevague, but on the

mathe-mati al side,the dis ussion aneasilybemademorerigorous. Foraquantum

state

ρ

ina

d

-dimensionalspa e, wedene

asthe information ontainedin

ρ

. For theallowedfamily of transformations, we take so- alled losed lo al operations and lassi al ommuni ation family,

CLOCCforshort[53℄. They anbede omposed intotwobasi typesof

opera-tions

i) Lo alunitarytransformations

ii) Sending subsystemsdowna ompletely dephasing hannel (i.e. a hannel

thatdestroysallnon-diagonalelementsofthetransformeddensitymatrix

insomebasis)

Letusdenotethisfamilyby

CL

. Inthebipartites enario,thequantum de it ofaquantum state

ρ

AB

isdened as

∆

(ρ

AB

) = I (ρ

AB

) − sup

Φ∈CL

(I (

Tr

A

(Φ (ρ

AB

))) + I (

Tr

B

(Φ (ρ

AB

))))

(1.32) orequivalently

∆

(ρ

AB

) = inf

Φ∈CL

(H (

Tr

A

(Φ (ρ

AB

))) + H (

Tr

B

(Φ (ρ

AB

)))) − H (ρ

AB

)

(1.33) Generalizationsto multipartite asesare immediate. Similarlyto the dis ord,

thede itvanishesfor lassi ally orrelatedstates,i.e. statesoftheform(1.30).

Moreover,asexplainedin[53℄,reversibleCLOCC transformsof lassi ally

or-relatedstatesplayanimportantrolein evaluationof

∆

foragivenstate

ρ

. On the physi sside, thetheoreti al possibility to drawamaximal amount

kT

⋅ I (ρ)

of work from a heat bath in temperature

T

using a state

ρ

is a widely believed onje ture. It has been partly onrmed by papers like [59℄

and [60℄. Hen e, it seems plausible that the quantum de it really has the

physi al interpretation wementioned earlier, but oneshould remain autions.

Themathemati al stru tureof the quantity, however,remains inta tin either

ase.

Beforewe losethis hapter,weshoulddenitelymentionthattheprin iple

of non-signalling, whi h appeared in the dis ussion by Popes u and Rohrli h,

anberepla ed by so- alledinformation ausality prin iple,whi h isstronger

thanno-signallingandpre ludes orrelationsthatarenotallowedbyquantum

me hani s [61℄. Hen e, information ausality may possibly be onsidered as

an axiomfor quantum theory [61,62℄, unlike thenon-signalling prin iple[35℄.

Pra ti al appli ations

2.1 Quantum ryptography

The ideaof quantum ryptography orquantum key distribution, rstput

forwardinthefamous1984paper[63℄byBennettandBrassard,hasitsorigins

in anearly workby S.Wiesner [64℄. Themainobservationbehinditwasthat

twophotonpolarizationbases,say

R

and

D

forre tilinearanddiagonal, anbe sele tedinsu hawaythatphotonsfullypolarizedwithrespe ttooneofthem

givetotallyrandom resultswhen measuredin the other basis,and vi e versa.

Equallyimportantwasthefa tthatquantummeasurementsae tthemeasured

systemsingeneral. BennettandBrassardusedthesequantum-me hani al

fea-turesto onstru t aproto ol, now alled BB84,whi h allowstwoparties that

do not initially share any se rets, to generate arandom string of bits that is

knowntobothofthem,but notto anyoneelse. Su hbits ansubsequentlybe

usedasasharedse retkeyforperfe tlyse ure lassi aldatatransmission. Let

us all the two parties

A

and

B

, orAli eand Bob. Theproto ol designedby BennettandBrassard onsistsinthefollowingsteps:

1. Ali e and Bob agree on two polarization bases, say

R

and

D

, whi h are rotated by

45

○

with respe t to ea h other. Let us denote the

orrespond-ing pure polarization photon states by

∣↔⟩

,

∣↕⟩

for the

R

basis and

∣⤡⟩ =

1

/

√

2

(∣↔⟩ + ∣↕⟩)

,

∣⤢⟩ = 1/

√

2

(∣↔⟩ − ∣↕⟩)

forthe

D

basis. 2. Ali e generatesrandomsequen esof bits,

{a

i

}

n

i=1

and

{b

j

}

n

j=1

,using a las-si al randomnumbergenerator.

3. Bob generatesarandomsequen eofbits

{c

k

}

n

k=1

,alsousing a lassi al gen-erator.

4. Ali ethenbeginstosendphotonstoBob. Thepolarizationstateofthe

i

-th photonis hosena ordingtothevaluesof therandombits

a

i

and

b

i

. The bit

a

i

determineswhi h polarizationbasis is used, with

a

i

= 0

standingfor the

R

and

a

i

= 1

forthe

D

basis. Thebit

b

i

determineswhether therstor

Randombits

(0,0) (0,1) (1,0) (1,1)

Photonsent

∣↔⟩

∣↕⟩

∣⤡⟩

∣⤢⟩

Table2.1: Photonpolarizationstates hoi es orrespondingtoAli e's random

bits

(a

i

, b

i

)

.

these ondpure polarizationstatewith respe ttothegivenbasisis hosen.

Table2.1summarizesonAli e's hoi eofphoton,depending on

(a

i

, b

i

)

. 5. Bobmeasuresthere eived

i

-thphotoninthe

R

or

D

basis,dependingonthe

valueof

c

i

. When

c

i

= 0

,Bobuses

R

. Otherwise,heuses

D

. Therstve tor inthesele tedbasis(

∣↔⟩

or

∣⤡⟩

)isassignedthemeasurementresult

0

,while theremainingve tor(

∣↕⟩

or

∣⤢⟩

)isassigned

1

. IfBobhappensto hoosethe samebasisasAli edid(i.e.

a

i

= c

i

),hismeasurementresultexa tlymat hes

b

i

, assumingthephotontransmissionwasnotdisruptednorinterfered with byaneavesdropper.

6. After measuringall the

n

photons, Bob publi ly dis loses the bits

c

i

, and Ali e does the same with

a

i

. Thus done, they know whi h measurement bases they used for individual photonsand ansingle out the aseswhere

their basis hoi es wereidenti al. Onaverage,they would have hosenthe

samebasisin

n

/2

ases.

7. Astheirse retkey,Ali eandBob hoosethebits

b

i

forwhi h

a

i

= c

i

. They bothknowthese bits,asaresultofusingidenti al measurementbases.

Thepoweroftheaboveproto ol omesfromthefa tthatanyinterferen ebyan

eavesdropperwould verylikelyhavebeendete tedbyAli eand Bob,provided

thattheyperformanadditional orre tness he kbeforetheyagreeonthekey.

Therequiredadditionalpro edure anbesummarizedasfollows:

7.' AfterperformingStep6.,Ali eandBobsele tarandomsubsetoftheindi es

i

forwhi h

a

i

= c

i

. Assume thesele tedindi es are

{i

k

}

m

k=1

. Ali epubli ly dis losesthebits

{b

i

k

}

m

k=1

,andBobdis losesthe orrespondingmeasurement resultsheobtained. Ifbothmat h,thetransmissionisassumedtobeperfe t

andtheremainingbitsforwhi h

a

i

= c

i

areusedasase retkey. Otherwise, itisassumedthatsomeonewaseavesdropping,andtheresultsofthewhole

se retkeygenerationpro edurearedis arded.

An exemplaryrunofthepro edure onsistingofsteps1.-7.,with 7.' in luded,

ispresentedin Table2.2. Note thatin reallife appli ations,itisimpossibleto

avoidtransmissionerrors,evenifthereisnooneeavesdropping. Hen e,ageneral

strategyhastobedevelopedtodealwithtransmission/eavesdroppingerrors,a

strategythatwouldallowtoprodu ease retkey,evenifthetransmissiondoes

notwork perfe tly. Suitabletools,borrowedfrom lassi al odingtheory,were

dis overedsomeyears after the adventof BB84 [65℄. They areverygenerally

des ribed asinformation re on iliation and priva y ampli ation. For

{a

i

}

1

0

1

1

0

1

0

1

0

{b

i

}

0

1

1

0

1

0

1

1

0

{c

i

}

1

1

1

0

0

0

1

0

1

Ali e's hoi eofbasis

D

R

D

D

R

D

R

D

R

Ali e'sphotonstate

⤡

↕

⤢ ⤡ ↕ ⤡

↕

⤢ ↔

Bob's hoi eofbasis

D

D

D

R

R

R

D

R

D

Bob'sresult

0

∗

1

∗

1

∗

∗

∗

∗

Thesamebasis?

Y

N

Y

N

Y

N

N

N

N

Randomlysele tedbits

1

Dotheymat h?

Y

Se urekey

0

1

Table2.2: AnexemplaryrunoftheBB84 proto ol. Thesymbol

∗

denotesthe fa tthateither

0

or

1

ouldhavebeenobtained. Theletters

Y

and

N

standfor Yes andNo.

We need to point out that in the above pro edures, no use of

entangle-ment was made. However, in the early nineties, A. Ekert proposed the rst

entanglement-basedquantumkeydistributionproto ol,knownasE91[66℄.

Al-thoughthegeneralideabehindE91is thesameasforBB84,thereare several

keydieren es:

1) Instead of leaving the photon state preparation to Ali e, both parties are

assignedtheidenti altaskofmeasuringasubsysteminatwo-partite

maxi-mally entangled photonstate

(∣00⟩ + ∣11⟩)/

√

2

. Thestate isassumed to be externallygiven. Ali emeasurestherstandBobthese ondsubsystem.

2) ThreeinsteadoftwophotonpolarizationbasesareusedatrandombyAli e

andBob.In aseofAli e,thepolarizerangles

φ

A

1

= 0

○

,

φ

A

2

= 45

○

and

φ

A

3

= 90

○

areused. ForBob,itis

φ

B

1

= 45

○

,

φ

B

2

= 90

○

and

φ

B

3

= 135

○

.

3) Bob and Ali e publi ly dis lose whi h bases they used in whi h

measure-mentround. Then, theyrevealthemeasurementresultsfor whi h dierent

measurementsetupswere used. This permits them to al ulate theCHSH

quantity

E

(φ

A

3

, φ

B

3

) + E (φ

A

3

, φ

B

1

) + E (φ

A

1

, φ

B

3

) − E (φ

A

1

, φ

B

1

),

(2.1) where

E

(φ,ψ)

isthe orrelation oe ientbetweenthemeasurementresults forAli eandBob whentheirpolarizeranglesare

φ

and

ψ

,respe tively. As in the exampledis ussed in Se tion 1.1, thevalue of thefun tion (2.1)for

a truly maximally entangled sour e state is

2

√

2

. By testing whether the equalitybetween

2

√

2

and(2.1)reallyo urs,BobandAli emakesurethat noeavesdroppingtakespla e,northatthesour eis orrupted.

4) Ifthereis(anapproximate)equalitybetween(2.1)anditstheoreti alvalue,

far,sothey anbeused asase retkey.

Shortly after Ekert published his paper, Bennett, Brassard and Mermin [67℄

suggested another entanglement-based proto ol, now alled BBM92, whi h is

basi ally aversionof BB84that exploits the properties ofentangledquantum

states. Thus, thedieren efrom BB84des ribedbyitem

1

)

abovestillexists, buttheotheronesdonot.

It isnaturaltoaskhowtheabovetwo-qubitkeydistribution methods

gen-eralize to higher dimensional quantum systems. The question wasaddressed

by the authors of the paper [68℄, who used so- alled mutually unbiased bases

(MUBs) as a higherdimensional analogueof the pair of bases

{∣↔⟩ ,∣↕⟩}

and

{∣⤡⟩ ,∣⤢⟩}

. Letusexplainthat twoorthonormalbases

{φ

i

}

d

i=1

and

{ψ

j

}

d

j=1

of C

d

are alled unbiased ifandonlyifthefollowingequality

∣⟨φ

i

, ψ

j

⟩∣

2

=

1

d

(2.2)

holdsforall

i

and

j

. Theunbiasedness onditionguaranteesthedesirable prop-ertythat anelementofoneof thebases givesfullyrandom resultswhen

mea-suredintheotherbasis.

There anexist atmost

d

+ 1

mutuallyunbiasedbases inC

d

[69℄. Weshall

dis uss someof theirfurther aspe ts in Se tion 8.4. Either apairof them, or

more an be used to design quantum key distribution proto ols based on

d

-dimensional quantum systems[68℄. These proto ols do notdiersigni antly

from the qubit ones. Let us also remark that in the qubit setting, there are

threeMUBsavailable,sothatthereexists analternativetoBB84thatusessix

quantumstatesinsteadoffour. Thispossibilitywasrststudiedin apaperby

Bruss[70℄.

2.2 Quantum teleportation and dense oding

As our next example of how the lawsof quantum me hani s an be used for

pra ti alpurposes, weshalldis uss thetwointer onne ted on eptsof dense

oding[71℄ andquantum state teleportation[72℄.

In its most basi form, dense oding permits two parties, say Ali e and

Bob, to ex hange two lassi al bits of information by just transmitting one

qubit. The fundamental tri kbehindthis feature is theuse ofone-sided Pauli

transformations,a tingonamaximallyentangledstate. Wehave

(

1

⊗

1

)∣Φ

+

⟩ = ∣Φ

+

⟩ ,

(σ

x

⊗

1

)∣Φ

+

⟩ = ∣Ψ

+

⟩ ,

(2.3)

(σ

y

⊗

1

) ∣Φ

+

⟩ = −i ∣Ψ

−

⟩,

(σ

z

⊗

1

) ∣Φ

+

⟩ = ∣Φ

−

⟩ ,

sothatthefourstatesresultingfromone-sidedPaulia tionon

∣Φ

+

⟩

areperfe tly distinguishable. Hen e,they an arrytwobitsof lassi alinformation. Inthe

entangledstate

∣Φ

+

⟩

ofatwo-partitesystem,andea hofthemhasa esstoonly oneofthesubsystems. Ali ethenperformsoneofthefourPaulitransformations

on her subsystem, and sends the subsystem to Bob. After this step, Bob is

in possession of one of the two-partite maximally entangled states from the

list (2.3). Be ause these states an be perfe tly distinguished by a quantum

measurement,Bob aninprin ipletellwhi hofthefourPaulioperationsAli e

used. Consequently, two bits of lassi al information havebeen transmitted,

eventhoughonlyonequbit wasex hangedbetweenAli eandBob.

Theaim of quantum state teleportation is, ontheother hand,to transmit

an unknown quantum state

∣ψ⟩

between the two parties. In the basi qubit teleportationmodel[72℄,therequiredresour esareamaximallyentangledstate,

i.e.

∣Ψ

−

⟩ = (∣01⟩ − ∣10⟩)/

√

2

, whi h is shared between Ali e an Bob, and the statetobeteleported,initiallyheldbyAli e. Altogether,theyhaveatripartite

system, initially in thestate

∣ψ⟩ ∣Ψ

−

⟩

. The rsttwosubsystems are ontrolled by Ali e, and the third one by Bob. In order to teleport

∣ψ⟩

to Bob, Ali e performsa measurementon thersttwoqubits, using the measurementbasis

{∣Φ

+

⟩,∣Φ

−

⟩ ,∣Ψ

+

⟩ ,∣Ψ

−

⟩}

. She then ommuni ates theresult to Bob. Provided thisinformation,Bob anre over

∣ψ⟩

byperformingasuitableunitaryrotation onhis subsystem. Toseethat thisisa tuallythe ase,itsu esto noti ethe

followingidentity

∣ψ⟩ ∣Ψ

−

⟩ =

1

2

(−∣Ψ

−

⟩∣ψ⟩ − ∣Ψ

+

⟩ σ

z

∣ψ⟩ + ∣Φ

−

⟩σ

x

∣ψ⟩ − i ∣Φ

+

⟩ σ

y

∣ψ⟩)

(2.4) After the Ali e's measurement on the rst two qubits, Bob's subsystem is in

oneof thestates

− ∣ψ⟩

,

−σ

z

∣ψ⟩

,

σ

x

∣ψ⟩

,

−iσ

y

∣ψ⟩

. Moreover,Ali e anperfe tly dierentiatebetweenthese four ases, as she knows whi h of the states

∣Ψ

−

⟩

,

∣Ψ

+

⟩

,

∣Φ

−

⟩

and

∣Φ

+

⟩

shegotinhermeasurement. Ifshe issokindtoshare this knowledgewith Bob,he an thenre overthestate

∣ψ⟩

by simplyundoing the suitablerotation

σ

x

,

σ

y

or

σ

z

,ifhisstateisnotalreadyamultipleof

∣ψ⟩

.

Naturally,theabovedense oding andteleportations hemesforqubits are

expe ted to have generalizations to higher dimensional systems. Su h

gen-eralizations do indeed exist and for the so- alled tight type, they have been

ompletely hara terized byWerner [73℄. Moreover,he showedthat there isa

one-to-one orresponden ebetween tight dense oding and tight teleportation

s hemes. Inorder tofully understandhis result,werstneed toexplain what

ageneraldense odingandteleportations hemeis.

Denition2.1.Let

X

beasetof

d

2

elements. Atightquantumteleportation

s heme onsistsof

ˆ Adensityoperator

ω

on C

d

_⊗

C

d

ˆ A olle tion of ompletely positive and tra e preserving maps

T

x

,

x

∈ X

, a tingon operatorsonC

d

ˆ A olle tionofobservables

F

x

onC

d

_⊗

C

d

,

x

∈ X

,su hthatforalldensity operators

ρ

onC

d

andalloperators

A

onC

d

,the following equalityholds

∑

x∈

X

Denition2.2. Let

X

beasetof

d

2

elements. Atightdense oding s heme

onsistsofthesameelementsasatightquantumteleportations heme, however

the ondition (2.5)isrepla edby

Tr

(ω (T

x

⊗

1

)(F

y

)) = δ

xy

(2.6) for all

x, y

∈ X

Note that in the above mentioned example of a dense oding s heme for

qubits, wehad

{F

x

}

x∈

X

= {∣Φ

+

⟩ ⟨Φ

+

∣ ,∣Ψ

+

⟩⟨Ψ

+

∣ ,∣Ψ

−

⟩⟨Ψ

−

∣ ,∣Φ

−

⟩ ⟨Φ

−

∣}

. Weused themaximallyentangledstate

ω

= ∣Φ

+

⟩ ⟨Φ

+

∣

andthetransformations

{T

x

}

x∈

X

=

{

1

,

Ad

σ

x

,

Ad

σ

y

,

Ad

σ

z

}

, where Ad

σ

x

∶ ρ ↦ σ

∗

x

ρσ

x

, and similarly for

σ

y

and

σ

z

. In the qubit teleportations heme, on the other hand,wehad

{F

x

}

x∈

X

=

{∣Ψ

−

⟩⟨Ψ

−

∣ ,∣Ψ

+

⟩⟨Ψ

+

∣ ,∣Φ

−

⟩ ⟨Φ

−

∣ ,∣Φ

+

⟩⟨Φ

+

∣}

,

ω

= ∣Ψ

−

⟩⟨Ψ

−

∣

, aswell as

{T

x

}

x∈

X

=

{

1

,

Ad

σ

z

,

Ad

σ

x

,

Ad

σ

y

}

Werner provesthefollowinggeneralresult[73℄.

Theorem 2.3. All tight teleportation or dense oding s hemes in C

d

are

ob-tainedby hoosing

ω

= ∣Ω⟩ ⟨Ω∣

foramaximallyentangledstate

∣Ω⟩ ∈

C

d

_⊗

C

d

,

F

x

=

∣Φ

x

⟩ ⟨Φ

x

∣

for an orthonormal basis of maximally entangled states

{∣Φ

x

⟩}

x∈

X

⊂

C

d

_⊗

C

d

and

T

x

=

Ad

U

x

,where

U

x

is hosen su hthat

∣Φ

x

⟩ = (U

x

⊗

1

)∣Ω⟩

. InParti ular,Theorem2.3appliesthatthereisaone-to-one orresponden e

betweentightteleportationanddense odings hemes. Everysu hs hemeneeds

abasisof maximallyentangled states. Let usremark that Werner proposed a

onstru tion of su h bases, based on Latin squares and omplex Hadamard

matri es, whi h also appear in the ontext of mutually unbiased bases, to be

dis ussedinmoredetailin Se tion8.4.

2.3 Quantum metrology

Inthelastse tion on erningpra ti alappli ations ofquantumentanglement,

we shall givean example of how entanglement an beused to in rease phase

sensitivity in aphoton interferometryexperiment. Our dis ussion isbased on

the paper [74℄ by Gerry and Benmoussa, but we make a few remarks about

related work byother authors. The verysimple experimental setupwewould

liketodis ussisdepi tedinFigure2.1. It onsistsoftwophotodete tors,abeam

splitter, andaphaseshifter. Together,theymakeupasimpleinterferometer.

Animportantpartoftheexperimentisalsothephotoni quantumstatewhi h

isfedintothearmsoftheinterferometer,aswellastheobservableone al ulates

usingthemeasurementresultsfromthephotodete tors. Theaimistoestimate

thephase

φ

, indu edbythe phaseshifter onsingle photons. Su h phasemay resulte.g. frompropagationthroughathinlayerofamediumthathasanindex

ofrefra tiongreaterthantheenvironment. Inthefollowing,wearguethat the

estimation of

φ

an be made more pre ise if one does exploit entanglement between

N

photons impinging onthe beamsplitter, instead of just repeating single-photonmeasurements

N

times.

theexperimentistoestimatethephase

φ

usinganappropriateinputstateand measurement

Weshallusethequantum-me hani aldes riptionoftheopti alexperiment

inFigure2.1,thebasi sforwhi h anbefoundinthetextbook[75,Chapter6.℄.

Inthisformalism,thequantumstateofthephotonsleavingthebeamsplitteris

des ribedasanelementofatwo-parti leFo kspa e,with reation/annihilation

operators

a

/a

∗

and

b

/b

∗

orresponding totheupperand theloweroutputarm

oftheinterferometer,respe tively. Itshould leadto no onfusionifwe allthe

upperand the lowerarm itself

a

and

b

for onvenien e ( f. Figure 2.1). The orresponding reation/annihilationoperatorssatisfythe ommutationrelations

[a,a

∗

_{] = [b,b}

∗

_{] =}

1

[a,b] = [a

∗

, b

] = [a,b

∗

] = [a

∗

, b

∗

] = 0

(2.7) The va uum state

∣0,0⟩

orresponds to no photons in arms

a

and

b

, and it satises

a

∣0,0⟩ = b ∣0,0⟩ = 0

. We assume that

∣0,0⟩

is normalized. Photon numberstatesaresubsequentlydenedas

∣n,m⟩ = (

a

∗

√

)

n

(b

∗

)

m

n! m!

∣0,0⟩

(2.8)

They have the lear interpretation of states with

n

photons in arm

a

and

m

photons in arm

b

of the interferometer. An analogous onstru tion works for theupperandlowerinputarmoftheinterferometer,whi hwe all

b

′

and

a

′

,the

sameasthe orrespondingannihilation operators. Note thattheupperarmis

denoted with

b

′

and not with

a

′

,thesameasinFigure2.1. The orresponding

photonnumberstatesaredenotedwith

∣n,m⟩

′

.

Ina ordan ewith[75℄,ifwehaveaninput state

∣Φ⟩ = f (a

′∗

_{, b}

′∗

_)∣0,0⟩

′

for

somefun tion

f

ofthe reationoperators

a

′∗

and

b

′∗

, thentheoutputstateof

theinterferometerequals