deterministic mediated superdense coding with linear optics.pdf

7/25/2019 Deterministic mediated superdense coding with linear optics.pdf

1/8

Physics Letters A 380 (2016) 848855

Contents lists available at ScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

Deterministic mediated superdense coding with linear optics

Mladen Pavicic a,b,

a DepartmentofPhysicsNanooptics,FacultyofMathematicsandNaturalSciences,HumboldtUniversityofBerlin,Germanyb CenterofExcellenceforAdvancedMaterialsandSensingDevices(CEMS),PhotonicsandQuantumOpticsUnit,RuderBokovicInstitute,Zagreb,Croatia

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received20October2015Receivedinrevisedform22December2015

Accepted

30

December

2015Availableonline6January2016CommunicatedbyP.R.Holland

Keywords:

QuantuminformationQuantumopticsQuantumcommunication

We

present

a

scheme

of

deterministic

mediated

superdense

coding

of

entangled

photon

states

employingonly linear-opticselements. Ideally,weareabletodeterministicallytransfer fourmessages

by

manipulating

just

one

of

the

photons.

Two

degrees

of

freedom,

polarization

and

spatial,

are

used.

A new

kind

of

source

of

heralded

down-converted

photon

pairs

conditioned

on

detection

of

another

pair

with

an

efficiency

of

92%

is

proposed.

Realistic

probabilistic

experimental

verification

of

the

scheme

with

suchasourceofpreselectedpairsisfeasiblewithtodaystechnology.Weobtainthechannelcapacityof1.78

bits

for

a

full-fledged

implementation. 2016ElsevierB.V.All rights reserved.

1. Introduction

Superdensecoding(SC)[1] (sendinguptotwobitsofinforma-tion,i.e.,fourmessages,bymanipulatingjustoneoftwoentangled

subsystems

of

a

quantum

system)

is

considered

to

be

a

protocol

thatcangivequantumcomputationyetanotheredgeoveraclas-sicalone.

So

far

the

attempts

to

implement

photon

SC

concentrated

on

the

Bell

states.

The

idea

was

to

send

four

messages

via

four

Bell

states [see Eq. (1)] and herewith achieve a log2 4= 2 bit trans-fer.Tothisaim,arecognitionofallfourBellstateswasrequired.However,Vaidmans[2]andLtkenhaus[3]groupsprovedthefol-lowing no-go result: Deterministic discrimination of all four Bellstateswithlinearopticselementsandonlyonedegreeoffreedom(DOF)

(e.g.,

polarization)

is

not

possible.

One

can

deterministically

discriminate

only

three

Bell

states

and

they

enable

the

so-called

densecoding(channelcapacitylog2 3= 1.585 bits)[4].Fortunately,theno-goproofdoesallowadeterministicdiscriminationwithtwo

DOFs

in

a

hyperentanglement

setup.

Such

hyperentanglement

ex-periments

have

been

put

forward

and

carried

out

[58].Hyperentanglement

of photon polarization and its orbital an-gular

momentum

recently

served

Barreiro,

Wei,

and

Kwiat

to

beat

the

channel

capacity

of

the

dense

coding

[4] by

a

tight

margin

1.63>1.585[8] inapostselectionexperiment.TheresulthasbeenrecognisedasbreakingthecommunicationbarrierandsuchaSC

* Correspondenceto:CenterofExcellence forAdvancedMaterialsandSensingDevices(CEMS),PhotonicsandQuantumOpticsUnit,RuderBokovicInstitute,Za-greb,Croatia.

E-mailaddress:[email protected].

bymeansofachosenprimaryDOFsupportedbyanotherDOFhasbeenreferredasamediatedSC[9].

Another

kind

of

hyperentanglement

of

photon

polarization

me-diated by a time-spatial DOF has been proposed by Kwiat andWeinfurter[10] andcarriedoutbySchuck,Huber,Kurtsiefer,andWeinfurter

[5].

They

make

use

of

the

spatial

DOF

in

order

to

achieveatimedelay.ThemainfeatureofmediatedSCsisthatphotonsstatesarede-

fined

by

one

main

DOF

(e.g.,

polarization)

and

one

ancillary

DOF

(e.g.,

a

time-spatial,

spatial,

or

photon

angular

momentum).

The

latteroneenablesadiscriminationofthestatesoftheformerone.Theyrequireasophisticatedlevelofcontrollingqubitstates,butatthe

same

time

in

the

existing

designs

we

actually

loose

more

in-formationthaninthedensecoding.Forinstance,intheaforemen-tionedhyperentanglementeachhyperentangledstateisauniquesuperposition

of

four

of

the

sixteen

possible

combinations

of

two-photon

spinorbit

Bell

states

[8].On the other hand, it was shown that more entanglement

doesnotnecessarily implymorecomputationalpower[11] and

therefore

we

considered

it

viable

examining

whether

SC

with

me-diated photons might be less entangled. We make use of theso-calledmixedbasisstates,twoofwhicharemediatedbyaspatialDOF,

to

implement

an

ideally

deterministic

2

bit

transfer.We proposed another mixed basis SC protocol previously in

Ref.[12] butthatonecouldnottransfermorethan1.43bits.Thepresent

protocol

enables

Alice

to

transfer

log2 4= 2 bitsof infor-mation,

via

sending

4

messages

to

Bob,

by

manipulating

only

one

photoncalled a travel photonfrom a pair of entangled pho-tons inaBellstate | generatedbyBob.Bobkeeps theotherphotoncalled

a

home

photondelayed

in

a

fibre spool.

Alice

encodes2of4messagesbymanipulatingthetravelphotonsoas

http://dx.doi.org/10.1016/j.physleta.2015.12.0370375-9601/ 2016ElsevierB.V.All rights reserved.
http://dx.doi.org/10.1016/j.physleta.2015.12.037http://www.sciencedirect.com/http://www.elsevier.com/locate/plamailto:[email protected]://dx.doi.org/10.1016/j.physleta.2015.12.037http://crossmark.crossref.org/dialog/?doi=10.1016/j.physleta.2015.12.037&domain=pdfhttp://dx.doi.org/10.1016/j.physleta.2015.12.037mailto:[email protected]://www.elsevier.com/locate/plahttp://www.sciencedirect.com/http://dx.doi.org/10.1016/j.physleta.2015.12.037


2/8

M. Pavicic / Physic s Letters A 380 (201 6) 848855 849

Fig. 1. Schematicoftheprotocol;Alicesendsmessages,3,4;Sisasourceofphotonsinstate|seeSubsec. 3.1;r1,r2arerouters(seetext)whicheitherletthephotonsthrough(off mode,r1,r2)ordeflectthem(on mode,r1+,r2+)intodetectorsdV,dH,respectively;D14 arephotonnumberdissolvingdetectors;BSisastandardbeamsplitter;PBSsarepolarizingbeamsplitters;gisaglassplatewhichpreservespolarizationandmakes-pathlengthidenticaltotheothers;Alicesends

byturningtheroutersoffand+ bykeepingthemoffandslidinginHWP(0);shesends3 (4)byslidinginHWP(45 )andturningr1(r2)onandr2(r1)off,indicatedbyr1+ (r2+)andr2 (r1),respectively;photonsrandomlychoosetoexitPBS1eitherintheH orV stateindicatedas3a vs. 3b an d4a vs. 4b options;in3band4b vacuum(vac)issenttoBob;dV (dH)istriggered[3b ( 4b)]ornot[3a ( 4a)];Bobreceives3-message( 4-message)as|H1|H2 (|V1|V2)3a ( 4a),oras|V1|vac2 (|H1|vac2)3b (4b).

togenerate | statesandsendsthetravelphotonstoBobwhocombines

them

with

his

home

photons

at

a

beam

splitter

(BS)

andmeasures

them.

To

send

the

other

2

messages

Alice

first

generates

a | Bellstateand thencollapses it to2computationalstatesmediatedbyaspatialDOF:twophotonpaths;oneleadstoBobsBSandhemeasuresthetravelandhomephotons;theotherleadsto

Alices

detector

and

Bob

combines

his

home

photon

with

the

vacuum

state

at

his

BS.As in the aforementioned experiments [8,5], we consider the

SCprotocolprimarilyasacomputationalresource.Thus,weonlyelaborate

on

the

information

transferred

from

Alice

to

Bob

without

Eve

(eavesdropping)

being

involved

although

we

briefly

discuss

a

possiblecryptographicimplementationinSec.4.ThespatialDOF,Bobmakesuseof,whenmeasuringthepho-

tons

encoded

by

Alice,

does

not

contain

any

information

about

the

polarization

states

Alice

imposes

on

photons

taking

different

paths

and

therefore

there

is

no

classical

information

transfer

involved

in

Alicesencoding.Theclassical informationcarriedbyphotonspa-tial

DOF

is

tantamount

to

the

mediation

of

messages

via

these

modes

as

in

[5].For our protocol to be feasible, a source of entangled photon

pairsondemandoraveryefficientsourceofheraldedpairsarere-quired

because,

for

an

equal

efficiency

of

measuring

both

vs. only

one

of

two

photons,

Bob

cannot

rely

on

a

postselection

as

in

a

cryptographyapplicationwhereonlydetectionofbothphotonsarekept and those of single ones are discarded. None of the so farexperimentally implemented candidates for such a source, eventhe

most

developed

quantum

dots,

is

sufficiently

reliable

and

ef-ficient.

Therefore inthispaperwecomeforwardwithaproposalforaveryefficientsourceofheraldedpreselectedentangledpho-

ton

pair

in

a

Bell

state

conditioned

on

a

detection

of

another

pair.

The

source

can

be

implemented

with

todays

technology

so

as

to

havearealisticefficiencyof92%.An experiment in a postselection mode, similar to postselec-

tion experiments carried out in [5,6,8], can be carried out withtodays

technology

as

proposed

in

detail

in

Sec. 2.

Actually,

also

afull-fledgedexperimentoftheproposalcanbecarriedoutwithtodaystechnologywithevenhigherefficiency,however,withhighendversionsofallcomponents.

The

paper

is

organised

as

follows.

In

Sec.2we

give

physical

and

technical

details

of

our

protocol

and

all

the

definitions

of

states,

messages,andopticalelementsusedinthepaper.InSec.3 wede-scribethenewsourceofpreselectedentangledphotonpairs(Sub-sec.3.1)

and

propose

a

postselection

proof-of-principle

experiment

(Subsec.3.3).

At

the

end

of

the

section

we

compare

channel

capac-

ityofourproposalwithpreviousexperimentallyobtainedones.In

Sec.4we

summarise

and

discuss

the

obtained

results.

At

the

endof the section we discuss (in)applicability of our SC protocol to

quantum

cryptography.

2.

Protocol

The

superdense

coding

(SC)

is

an

encoding

of

four

messages

intothestatesofentangledpairsofqubitsbymeansofan inter-action

with

one

of

the

qubits

only.WemakeuseofthefollowingthreeBellstates

| = 12

(|H1|V2 |V1|H2),

| =

1

2(|H1|

H2 |

V1|

V2

), (1)

and

the

following

two

states

from

the

computational

basis

|H1|H2, |V1|V2. (2)The

Bell

states

|givenbyEq.(1) togetherwiththestatesgivenbyEq.(2)formabasiscalledmixedstatebasisorsimplymixedbasis.

Bob

prepares

| photon pairs ideally by using a source ofentangledphotonpairsondemandbutrealisticallybymakinguseof

the

source

we

propose

in

Subsec.3.1which

can

be

realised

with

todaystechnologysoastohavetheefficiencyofpreselectingpairsof

92%.

Bob

then

sends

one

photon

from

each

pair

to

Alice

who

manipulates it soas tosend fourdifferent messages to Bob. Wecall

her

photon

a travel photon.Theother (Bobs)photon froma

pair

we

call

a

home photon.

Alice

ideally

deterministically

encodesthe

following

four

messages

and

sends

them

to

Bob:

+-message, -message, 3-message, 4-message, (3)

as

shown

in

Fig. 1.

We

will

discuss

non-ideal

realistic

implementa-tionoftheprotocolandtakelossesintoaccountinSec.3.Wealsodiscuss

a

particular

aspect

of

a

realistic

implementation

at

the

end

ofthissection.To

send

a -message Alice keeps both routers (r1, r2) off,

meaning that they let photons through without affecting theirstates,

indicated

as

r1 and

r2 in

Fig. 1.

The

routers

make

use

of

electro-optical modulators based on rubidium titanite phosphate[13].

When

they

are

turned

on,

they

can

deflect

incoming

pho-tonsindependentlyoftheirpolarizationunlikethestandardoptical

switches

like,

e.g.,

Pockels

cells,

based

on

polarization

selection.


3/8


4/8


amountstoapreselectionofanentangledpairconditionedonde-tection

of

another

pair

[20].

The

spontaneous

emission

occurs

only

once

and

with

them

we

have

a

genuine

preselection:

by

detecting

two photons at a beam splitter we entangle theother two pho-tonsintothestate| althoughtheirtrajectoriesnevermixorcross

[20].

With

photon

pairs

down-converted

in

two

nonlinear

crystals,

on

the

other

hand,

the

probabilities

of

their

conversion

in

eachofthecrystalsorinjustoneofthemareaboutthesameand

the

latter

ones

will

ruin

the

preselection

if

we

stick

to

the

stan-dard

way

of

combining

photons

from

two

different

crystals

at

a

BS

in

an

attempt

to

entangle

the

other

two

photons.Having such standard way of combining photons at a BS in

mind, SliwaandBanaszekwrote:generationofadoublepair inone

crystal

and

none

in

the

second

crystal

[23][is]

a

fundamental

obstacle in theconditionalpreparationofmaximalentanglementfromfourdown-convertedphotons:ithasbeenshown[byKokandBraunstein[24]]thatamaximallyentangledstatecannotbegen-erated

[conditioned]

on

detection

of

two

auxiliary

photons.

This

rules

out

the

possibility

[making

use]

of

overall

four

photons,

to

producemaximallyentangledpairsbymeansofconditionaldetec-tion[25].

Kok

and

Braunstein

obtained

their

result

under

the

following

assumption:

every

detector

needs

to

detect

at

most

one

photon[24]andifwefoundawaytopreselectapairofentangledphotons

conditionedondetectingmorethanonephotonbysomedetectorswe

would

be

able

to

go

around

Sliwa

and

Banaszeks

conclusion.

This

is

exactly

what

we

did

in

the

design

of

our

source

below.However,beforewedwellonit,letusfirstdealwiththeprob-

lem of low collection efficiency. To overcome it, we turn to ex-periments

with

heralded

generations

of

entangled

signal

and

idler

colinearphotonsinapairofoppositepolarizationdown-convertedin a type II poled potassium titanyl phosphate (PPKTP) crystalplaced inacavitycarriedoutbyBensonsgroups[2629].Signaland

idler

are

filtered

from

the

central

peak

of

a

comb-like

distribu-tionofphotonfrequenciesinacavity[30]soastohavethesamefrequency what makes them indistinguishable up to their polar-ization.Avery importantfeatureofsofilteredsignaland idler isthat

the

visibility

of

their

coincidence

rate

is

over

96.5%,

without

backgroundsubtraction[28,31].Closelyrelatedistheefficiencyofgeneratinganentangledpairincontrasttogeneratingjustoneofthe

photons

(i.e.,

loosing

one

of

them)

which

can

be

extrapolated

from

the

recently

obtained

experimental

results

[31] for

which

data show that with the current setup an efficiency of 80% canbeachieved.

Now, in Fig. 2 we present the design of our source in whichwe

make

use

of

a

MachZehnder

interferometer

(MZI)

to

single

out photons entangled in a Bell state while discriminating be-tween them and the photons coming fromjust one crystalthelatter

photons

will

bunch

at

either

exit

of

the

MZI

obeying

its

ba-sic

symmetry:

what

comes

in

from

one

of

its

sides,

will

exit

from

theother.Asfortheformerphotons,thedesignenablesthemto

interfere

after

taking

either

of

two

indistinguishable

paths,

like

in

Fransons interferometer [3234]onlyherewedonothave longvs. short

paths

but

equally

balancedi vs. ii pathsaswellascorre-

latediiivs. iv paths.The intensity of the pump beam is lowered down so as to

down-convert

predominantly

two

pairs

of

entangled

photons,

each

in

state

|H V within a chosen time window. Each pair can bedown-convertedineachcrystalorbothoftheminoneofthecrys-tals,withthesameprobability.

Photons

from

a

cavity

can

take

four

different

paths

after

being

split

at

the

first

polarizing

beam

splitter

(PBS).

When

we

look

at

optionstodetectexactlytwophotonsbyourdetectorsbehindMZI,weseethatthisisrealisedinexactlytwoways.

The

first

way

is

when

the

signal

(H)comingfromthetopcavity

takes

path

i and

idler

(V)

from

the

bottom

one

takes

path

ii.

The

Fig. 2. Proposalofaheraldedgenerationofanentangledphotonpairinthe|stateconditionedondetectinganotherpairofphotonsinthe|+ state.Allpathsareequalinlengthtoenableinterference(notshown).Routers(r)whichblockexitphotonsthatarenotinstate| aretriggeredbydetectorsdi ,i= 1,. . . ,4.

top

idler

(V)isthengoingto iii andthebottomsignal(H)to iv.This combination is indistinguishable (with respect to BS1) fromthe

top

idler

(V)

going

to

i and

bottom

signal

(H)

to

ii while

top

signal(H)isgoingtoiiiandbottomidler(V)toiv.Becauseoftheaforementionedindistinguishability,theinputstatetoBS1 is

|+ = 12

(|H1|V2 + |V1|H2). (4)

At

BS1 ittransformsto[35,Eqs. (1.160,1),pp. 72,73],[36,Eq. (4.28),p. 68]

|+ BS1 12

(|H1|V1 |H2|V2)= |12, (5)

andatBS2 itgivesthefollowingoutput

|

12

BS2

1

2(|H

1|V

2+ |

V

1|H

2)= |

+

. (6)

ThephotonswillbedetectedatoppositesidesofBS2 andthede-tectors

will

trigger

two

rs

so

as

to

let

theiii,ivphotonsthrough.

The

companion

photons

go

toiiiandivintwoaforementioned

indistinguishablecombinationsandformthefollowingstate

|+ = 12

(|H1|V2 + |V1|H2), (7)

whichHWPtransformsintothefinaloutputstate:

| = 12

(|H1|V2 |V1|H2). (8)

Whenrouters(r)doletthephotonsthroughinstate| theymustrelayphotonssoasnottoaffecttheirpolarization.Thereforewe

make

use

of

routers

with

electro-optical

modulators

based

on

rubidium

titanite

phosphate

so

as

to

preserve

the

entangled

state

[13].Theonlyotherwayofhavingonlytwophotonswithdifferent

polarizationgoingtoi andiiiswheneitherboththetopidlerandsignalaregoingto i (whilethebottomsignalandidleraregoingto

iv)or(indistinguishably)thebottomidlerandsignalaregoingto

ii (whilethetopsignalandidleraregoingtoiii)inwhichcasewehave

|+12 =1

2(|H1|V1 + |H2|V2) (9)

asaninputtoBS1 .MZIdoesnotchangethestate,i.e.,thephotonswillbedetectedbunchedatonesideofBS2 andthedetectorswill

trigger

two

routers

(r)

so

as

not

to

let

the

iii,

ivphotons

through.


5/8


6/8


Fig. 3. Proposalofaproof-of-principlepostselectionexperiment.PathsthatleadtoBSarebalanced.

3.3. Postselectionexperiment

We make use of the source described in Subsec. 3.1. As forthe other components needed for the postselection experimentwewillkeeptothestandardoff-the-shelfelements,exceptforat

least

three

of

the

detectors

that

should

have

a

rather

low

dark

countrate.ThesuperconductingTESdetectorshavepracticallynodarkcountsatall,butInGaAsavalanchephotodiodesoperatingat60 Cwithca. 0.1%darkcountprobability[40] wouldserveourpurpose as D4 and D

4 as shown in Fig. 3. A review of 16 de-

tectorswithrespect to theirefficiencies,darkcountrates,etc., isgivenin[41].Forourpostselectionexperimentitisactuallybettertochoosedetectorswith lowerefficiencybecausethedarkcountprobability always rises with the efficiency. We shall assume tohavedetectorswithanefficiencyof50%andthedarkcountprob-ability of 1%. Alice needs to have reliable confirmations on sentmessageswithonlyhomephotonsarrivingatD4 orD

4 sothather

dH shouldalsobeanInGaAsone.We need not have a full fledged setup but only its essential

part

as

for

example

in

the

SC

experiment

carried

out

by

Wein-furtersgroup[5].Thismeansthatweshouldbeabletocarryoutcalibratedcomparisonofdatacollectedforallfourmessages.ForthatBobneedsonlyhistwoInGaAsdetectorsD4 andD

4 andthe

fourotherones (D1D3)canbe thestandardoff-the-shelfdetec-

tors operating at the room temperature, because he can alwayscalibratetheobtaineddatawithrespecttotheformertwodetec-tors.Tosimplifythecalculationweshall,however,assumethatallsixBobsdetectorsD1D

4 areofthesamekind.

WhentwophotonswithparallelpolarizationarriveatBS1 andBS2 showninFig. 3theyhave50%probabilityofsplittingatthemand

we

should

calibrate

all

other

measurements

with

respect

to

the measurements by detectors aftersuch apossible splitting asfollows.

Bob

detects

both

photons

of messages. -message

triggers

either

[D1 and

(either

D3 or

D3)]

or

[D2 and

(either

D4 or

D4)](seeTable 1)withthedetectionprobabilityof100%(ideally).Similarly,

+-messagetriggerseither[D1 andD2]or[(eitherD3 orD3)and (eitherD4 orD

4)]alsowith thedetectionprobabilityof

100%

(ideally).As

for 3,4-messages with both home and travel photons at

BS,

they

will,

according

to

the

HongOuMandel

effect

[14],

ei-ther

go

to

the

left

or

to

the

right

from

BS

in

Fig. 3.

Those

going

to

the

left

(50%

of

them)

will

end

up

either

in

D1 (bothphotons)orD2 (bothphotons).SincetheywillgiveBobasingleclickinei-ther

D1 orD2,Bobwilldiscardtheserecordings.Theotherhalfof3,4-messages withbothhomeand travelphotonswillgo to theright

and

half

of

them

will

bunch

together

and

the

other

half

will

split

at

either

BS1 (4) or BS2 (3). Here the HongOuMandel

effect

does

not

apply

because

both

photons

come

from

the

same

side.ThusD3-and-D3 simultaneousclicksaswellasD4-and-D

4 si-

multaneousclickswillaltogethercollect25%ofall3,4 messagesof

home

and

travel

photons

and

the

statistics

of

postselected

mea-surements

of

all

four

kinds

of

messages

should

be

then

calibrated

because

when

a

correlated

detection

is

carried

out

and

two

detec-torsareusedandtheefficiencyofeachofthemis0.5,thentheir

joineddetection iscarriedoutwiththeefficiencyof0.52 = 0.25.However,whensingledetectionsarecarriedoutfortheotherhalf

3,4-messageswith

only

home

photons

which

did

not

split

at

ei-ther BS1 or BS2 (altogether 75% of

3,4-messages) then they arecarried

out

with

the

efficiency

of

0.5,

which

means

that

Bob

will

have

to

divide

the

number

of

these

detections

by

2

what

corre-spondstotheefficiencyof0.52 = 0.25.forcorrelateddetectionsoftwodetectorsforalltheothermeasurements.Intheend,hemul-tiplies all the obtained 3,4 data by two to compensate for thediscardedD1 ,D2 data.

Thenextstepistoseewhichlossesandefficienciescanaffectour

final

result.

The

efficiency

of

the

source

is

0.92

and

the

ef-ficiencies

of

the

optical

elements

we

obtain

as

follows.

BS

losses

can

be

lower

than

1%

but

only

the

differences

between

the

right

side with additional two BS and the left side without them arerelevantforourpostselectionexperiment.Misalignmentsandpath

differences

might

cause

up

to

3%

of

losses.

Router

losses

can

beaslowas1%.Thenwetakethedarkcountsintoaccountbymul-

tiplying the50%ofsplit 3,4-messages by0.99,corresponding tothe

detector

efficiency

(1

0.01= 0.99, where 0.01 is the darkcount

probability).

We

should

add

ca. 4%

for

miscalibration.

So,

the element losses, detector dark count gains, and miscalibra-tionamount(bymultiplication)toanefficiencyofca. 0.91,whatislowerthanforafullfledgedexperimentobtainedinSubsec.3.2but this isdueto themiscalibrationanddarkcounterrors.Alto-gether,anoverallefficiencyof0.92 0.91 0.84 coverslossesinthe

source,

in

the

optical

elements,

by

dark

counts,

and

miscalibra-tion.

To

obtain

the

number

of

successfully

transferred

messages

we

assume tohave10m low-lossfibre andfollowing theprocedure

from

Subsec. 3.2we

get

3.36

messages.

From

this

we

obtain

that

the

mutual

information

between

Alice

and

Bob,

i.e.,

the

channel

capacityislog2 3.36 1.75 bits.Thiscanbecomparedwith thebestpostselectionchannelca-

pacities

achieved

so

far:

1.63>log2 3 1.585[8] and1.18[5].

4. Resultsanddiscussion

WehaveshownthatAlicecancarryoutafullmediateddeter-ministicsuperdensecoding(SC)bymanipulatingonlyonephotonfromapairofentangledphotonstogeneratefourmessageswhichBob can unambiguously discriminate by beam splitters (BS) andtwo

polarizing

beam

splitters

(PBS)

as

shown

in

Fig. 1.For

this

to

work,

ideally

a

would-be

source

of

such

pairs

on

demand

would

be

required

but

for

an

immediate

implementation

with

todays

technology,

in

Subsec.3.1,

we

propose

a

new

source

of

heraldedpairsinaBellstateconditionedondetectionofanotherpairofancillaryphotons(seeFig. 2)whoserealisticefficiencycanbeashighas92%.

Alice

makes

use

of

two

Bell

states

| tosend-messagesand

one

|tosend3,4-messages[seeEqs.(1) and(3)].Tosendthe

latter

messages

she

first

collapses

| into twostates fromthecomputationalbasis,|H1|H2 and|V1|V2 ,atherPBS1withthe50:50probability, i.e.,completely randomly. Thatmeans thatAlice is not able to obtain |H1|H2 or |V1|V2 at will (such apossibilitywouldbetantamounttohersendingsuperluminalmes-sagestoBob)butacleverdesignwithopticalroutersandherowndetectors

dV anddH shown inFig. 1 enablesher toneverthelessideally

deterministically

send3-message (4-message)mediated

by

the

spatial

degrees

of

freedom

(DOF)two

paths:

one

leading


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