Efficient Computational Information Geometric Analysis of Physically Allowed Quantum Cloning Attacks for Quantum Key Distribution Protocols

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1 Laszlo Gongos Sando Ime Effcent Computatonal Infomaton Geometc Analss of Phscall Allowed Quantum Clonng Attacks fo Quantum Ke Dstuton Potocols LASZLO GYONGYOSI SANDOR IMRE Depatment of Telecommuncatons Budapest Unvest of Technolog Maga tudosok kt.. HUNGARY {gongos Astact: - In secet quantum communcatons the est eavesdoppng attacks on quantum cptogaph ae ased on mpefect clonng machnes. The ncoheent attack ased on quantum clonng s the most common eavesdoppng stateg. Usng a poe the eavesdoppe mpefectl clones the sende s quantum state whch keeps one cop and sends the othe. The phscall allowed tansfomatons of Eve s quantum clone on Bo s qut can e desced n tems of Completel Postve (CP) tace pesevng maps. The map of the quantum clone compesses the Bloch-all as an affne map. Ths affne map has to e a complete postve tace pesevng map whch shnks the Bloch all. The effects of a quantum clone can e gven n tetahedon epesentaton. In ths pape we show a new quantum nfomaton theoetcal epesentaton of eavesdoppng detecton focused on the Fou-state (BB84) and Sx-state quantum cptogaph potocols. We use a fundamentall new computatonal geometcal method to analze the nfomatonal theoetcal mpacts of clonng actvt on the quantum channel. The poposed algothm uses Delauna tessellaton and convex hull calculaton on the Bloch sphee wth espect to quantum elatve entop as dstance measue. The mpoved coe-set appoach can e used to analze effcentl the nfomatonal theoetcal mpacts of phscall allowed quantum clonng attacks. Ke-Wods: - Quantum Cptogaph Quantum Clonng Quantum Infomatonal Dstance Intoducton Quantum cptogaph s an emegng technolog that offes new foms of secut potecton howeve the quantum clonng ased attacks aganst the potocol wll pla a cucal ole n the futue [ ]. We dentf the quantum clonng ased attacks n the quantum channel and fnd potental and effcent solutons fo the detecton n secet quantum communcatons. The ncoheent and coheent attacks aganst quantum cptogaph ae ased on mpefect quantum clones. The tpe of quantum clone used depends on the quantum cptogaph potocol. Aganst the Fou-state (BB84) Eve the eavesdoppe uses the phasecovaant clone whle fo the Sx-state potocol the optmal esults can e acheved the unvesal quantum clone (UCM) [8 9 0 ]. We use an effcent computatonal geometc method to analze the quantum nfomaton theoetcal mpacts of phscall allowed attacks on the quantum channel. Ou goal s to measue the level of quantum clonng actvt on the quantum channel usng fast computatonal geometc methods. Ou pape s oganzed as follows. Fst we dscuss the asc facts aout computatonal geomet and quantum nfomaton theo. Then we explan the man elements of ou secut analss and we show the applcaton of ou theo fo the secut analss of eavesdoppe detecton on the quantum channel. Fnall we summaze the esults.. Clonng Attacks n Quantum Cptogaph The ncoheent quantum clonng ased attack s the most common eavesdoppng stateg [8 9] thus n ou geometcal ased secut analss we stud the ncoheent attack ased attacke model. The secut of QKD schemes eles on the noclonng theoem []. Conta to classcal nfomaton n a quantum communcaton sstem the quantum nfomaton cannot e coped pefectl. If Alce sends a nume of photons ISSN: Issue Volume 9 Mach 00

2 Laszlo Gongos Sando Ime N though the quantum channel an eavesdoppe s not nteested n copng an ata state onl the possle polazaton states of the attacked QKD scheme. To cop the sent quantum state an eavesdoppe has to use a quantum clone machne and a known lank state 0 onto whch the eavesdoppe would lke to cop Alce s quantum state. If Eve wants to cop the -th sent photon she has to appl a unta tansfomaton U whch gves the followng esult: U 0 () fo each polazaton states of qut. A photon chosen fom a gven set of polazaton states can onl e pefectl cloned f the polazaton angles n the set ae dstnct and ae all mutuall othogonal [ 7]. The unknown non-othogonal states cannot e cloned pefectl the clonng pocess of the quantum states s possle onl f the nfomaton eng cloned s classcal hence the quantum states ae all othogonal. The polazaton states n the QKD potocols ae not all othogonal states whch makes t mpossle an eavesdoppe to cop the sende s quantum states []. In the ncoheent-tpe attacks Eve mpefectl clones the sende s quantum state usng he quantum state poe she sends one cop to Bo and keeps the othe cop. We denote Eve s quantum state E and the unta opeaton whch desces the nteacton etween the sent qut and Eve s state s denoted U thus the whole tansfomaton can e desced as [6]: U E 0 E00 0 E0 () U E E 0 E 0 whee E j denotes Eve s cloned quantum state and E can e wtten as matx whose elements ae Eve s states E j. We measue the nfomaton theoetcal mpact of quantum clonng actvt n the quantum channel whee Alce s and Bo s sde can e modeled andom vaales X and Y. Ou geometcal secut analss s focused on the cloned mxed quantum state eceved Bo. Alce s pue state s denoted A Eve s clone modeled an affne map and Bo s mxed nput state s denoted A B. The geneal model fo the quantum clone ased attack fo quantum cptogaph s llustated n Fg.. H X H X Y HX H X Y Alce s pue qut A Random state Eve s quantum clone Quantum Clone Cloned state Bo s mxed nput state A Fg.. The analzed attacke model and the entopes. We measue n a geometcal epesentaton the nfomaton whch can e tansmtted n the pesence of an eavesdoppe on the quantum channel. We seek to maxmze H X and mnmze H X Y n ode to maxmze the adus of the smallest enclosng all of Bo whch desces the maxmal tansmttale nfomaton fom Alce to Bo n the attacked quantum channel: max H X H X Y. () all possle x To compute the adus of the smallest nfomatonal all of quantum states we use the von Neumann entop and quantum elatve entop. Geometcall the pesence of an eavesdoppe causes a detectale mappng to change fom a noseless one-to-one elatonshp to a stochastc map [6 7]. Phscall Allowed Tansfomatons The map of the quantum clone compesses the Bloch-all as an affne map. Ths affne map must e a complete postve tace pesevng map whch shnks the Bloch all along the x and z dectons. The quantum nfomaton theoetcal analss of the eavesdoppe s clonng machne ndcates how much the eavesdoppe clones the sent pue quantum states. In ou model due to eavesdoppe actvt the sent pue quantum states ecome mxed states. Eve s output s epesented a denst matx and he opeaton s a tace-pesevng completel postve (CP) map. We denote Eve s map whch s tace pesevng f T T fo all denst matces and postve f the egenvalues of ae nonnegatve wheneve the egenvalues of ae non-negatve. Eve s map has to e CP thus I s a postve map fo all n whee n I n s the dentt map on n n matces [7]. B ISSN: Issue Volume 9 Mach 00

3 Laszlo Gongos Sando Ime We use a computatonal geometcal method to analze the clonng actvt on the quantum channel and we use the Bloch all epesentaton. The actvt of an eavesdoppe on a sngle-qut n the Bloch sphee epesentaton can e gven an affne map as E A (4) whee A s a eal matx s a theedmensonal vecto s the ntal Bloch vecto of the sent pue quantum state and E s the Bloch vecto of the cloned state. In dealstc UCM and phase-covaant ased attacks the eavesdoppe s actvt does not change the cente of the Bloch all [] thus 0 and A s dagonal matx wth entes x z whch chaactezes the tetahedon. The entes of matx A specf the tetahedon n the paamete space of x z whee f x z. (5) The tetahedon s the convex hull of the ponts epesentng I and the thee otatons thus eve tansfomaton coespondng to a pont n the tetahedon can e desced as a statstcal mxtue of the I x and z extemal tansfomatons [7 8]. Thus f all the ponts n can e desced tansfomatons I x and z. Fg.. llustates the tetahedon. Fg.. The tetahedon epesentaton of phscall allowed clonng tansfomatons. The vetces of tetahedon epesent the I x and z Paul-tansfomatons whee I s the dentt tansfomaton and x z ae otatons aout the x and z axes.. Geomet of Quantum Clonng Based Eavesdoppng The quantum clone map compesses the Bloch-all as an affne map. Ths affne map must e a complete postve tace pesevng map whch shnks the Bloch all along the x and z dectons []. The vetces of tetahedon coespond wth the fou extemal maps whch can e desced as A A (6) j0 j j j whee A 0 s the dentt matx I and j we have Aj j and 0. The geneal tansfomaton of Eve s quantum clone can e desced as a convex sum of these maps as xz (7) whee and ae non-negatve paametes x z ae the Paul-tansfomatons and I the dentt tansfomaton. The vetces of epesent a unta map fo whch onl one opeato s equed n the opeato sum epesentaton whle the edges of epesent the two opeato maps and the faces of epesents the maps wth thee opeatos. The ponts nsde eque all the fou opeatos [7 8].. Attacke Model fo BB84 and Sx State Potocol In quantum cptogaph the most effectve eavesdoppng attacks use quantum clonng machnes [7 8 9]. Howeve an eavesdoppe can not measue the state of a sngle quantum t snce the esult of that measuement s one of the sngle quantum sstem s egenstates. The measued egenstate gves onl ve poo nfomaton to the eavesdoppe aout the ognal state [ 7]. The pocess of clonng pue states can e genealzed as (8) Q a x ac whee s the state n the Hlet space to e coped s a efeence state and Q s the anclla state [7]. As Wootes and Zuek showed an unknown quantum state 0 cannot e cloned pefectl [7] howeve t has susequentl een shown that an unknown quantum state can e cloned appoxmatel [ 8 9]. A clonng machne s ISSN: Issue Volume 9 Mach 00

4 Laszlo Gongos Sando Ime called smmetc f at the output all the clones have the same fdelt and asmmetc f the clones have dffeent fdeltes [8 9]. The effect of the eavesdoppe s quantum clone smpl shnks the Bloch all wth gven poalt p. The geneal model of the eavesdoppe s clonng machne s shown n Fg.. x x 0 0 Refeence state nput Eve s quantum clone Eve s outputs output output anclla Fg.. The geneal model of Eve s quantum clone. a c ac The nput qut state s denoted x whch s ntall n an entangled state wth a efeence qut denoted Bell state. x Afte the clonng tansfomaton the oveall sstem conssts of the thee outputs and the efeence quantum state thus output state ac can e wtten as a supeposton of doule Bell states [ ]: a c v z (9) x whee x z and v ae complex ampltudes wth x z v. The qut pas a and c ae Bell mxtues wth x p x p z p z and v p. The equaton v x z desces a theedmensonal suface n the space whee each pont x z epesents paametes x p p and z p z. Ths suface s an olate ellpsod and we denote the coodnates [8] of the ellpsod x z. As we wll see n Secton 5 the ellpsod has pola adus x whle the equatoal adus s z [8 0]. The tpe of the quantum clone machne depends on the actual potocol. Fo BB84 Eve chooses the phase-covaant clone whle fo the Sx-state potocol she uses the unvesal quantum clone (UCM) machne [8 9]. x. Clonng Machne Based Attacks Eve has a quantum clone machne and she nteacts wth the quantum channel connectng two the legtmate uses Alce and Bo. The effect of the eavesdoppe s smmetc quantum clone smpl shnks the Bloch all wth gven poalt p. In the BB84 potocol [7] Eve uses phasecovaant clonng machne thus Eve clones onl equatoal states: 0 e. (0) In the Sx-state potocol Eve consdes unvesal clonng [7] and clones all the states: () Usng the ncoheent-tpe clonng ased attack Eve apples the same unta tansfomaton to each sent. Eve does not ntoduce coelaton among the quantum state copes and she measues he state afte she cloned t [8]. Alce Bo and Eve mmedatel measue the quantum states snce the pates have no alt to stoe the quts... Unvesal Clonng If Eve uses a unvesal quantum clone then the value of paamete F Eve wll e ndependent of nput quantum state. The quantum clonng tansfomaton optmal [8 9] f hence the maxmal fdelt of optmal unvesal clonng s 5 UCM FEve and the maxmal adus Eve s 6 UCM Eve. () The quantum nfomaton theoetcal adus can e defned as UCM UCM S () Eve whee S s the von Neumann entop of the coespondng quantum state wth a adus length UCM Eve. Unvesal clonng has dect applcatons to eavesdoppng stateges n Sx-state quantum cptogaph. The map of UCM clone ased smmetc ncoheent attack fo the Sx-state potocol on pue nput state can e gven the followng completel postve map: Eve ISSN: Issue Volume 9 Mach 00

5 Laszlo Gongos Sando Ime 4p p (4) whee s the dentt tansfomaton. In Fg. 4. we show the quantum clone ased attacke model fo the Sx-state potocol. Alce s pue qut A Sx state QKD Eve s ncoheent attack UCM Cloned state Bo s mxed nput state A Fg. 4. The UCM clone ased attacke model fo the Sx-state potocol. In the UCM ased attacke model Eve has a statendependent quantum clone whee the poaltes p ae px p pz thus px p pz... Phase-covaant Clonng The est-known example of a state-dependent quantum clonng machne s the phase-covaant clonng machne [8 ]. The phase-covaant clonng machnes have a emakale applcaton n quantum cptogaph snce the ae used n the optmal stateg fo eavesdoppng [8 9 0]. In the Fou-state (BB84) quantum cptogaph potocol the optmal eavesdoppng attack s done a phase-covaant clonng machne whch clones the x equato. The mpotance of equatoal quts les n the fact that Fou-state quantum cptogaph eques these states athe than the states that span the whole Bloch sphee [9]. In phase-covaant clonng the clonng tansfomatons wee estcted fo pue nput states of 0 e fom whee the paamete 0 epesents the angle etween the Bloch vecto and the x-axs. These quts ae called equatoal quts ecause the z-component of the Bloch vecto s zeo. The phase-covaant quantum clone [9] can clone ata equatoal quts and the clone mantans the qualt of the copes fo all equatoal quts [6 7]. The educed denst opeato of the copes at the output can e expessed as [9] B out 8 8 (5) whee s othogonal to state. Thee the optmal fdelt of to phase-covaant clonng tansfomaton s gven phasecov. F (6) If Eve has a phase-covaant quantum clone then phasecov. the maxmal value of he adus Eve s phasecov. Eve. 8 (7) The quantum nfomaton theoetcal adus phasecov. Eve of the phase-covaant clone can e defned as phasecov. phasecov. S (8) Eve whee S s the von Neumann entop of the coespondng quantum state wth a adus length of phasecov. Eve. The map of the phase-covaant clone ased attack fo BB84 potocol on nput state can e gven the followng completel postve map: p p p 4 (9) p 4 wth px pz p 0 whee p. 8 In Fg. 5. we llustated the quantum clone ased attacke model fo the BB84 potocol. Alce s pue qut A BB84 potocol Eve Eve s ncoheent attack Phase covaant Cloned state Bo s mxed nput state A Fg. 5. The phase-covaant clone ased attacke model fo the BB84 potocol. B ISSN: Issue Volume 9 Mach 00

6 Laszlo Gongos Sando Ime The optmal clonng tansfomaton fo the BB84 states can e wtten as follows [8 9]: U 0 0 X (0) U 0 X The phase-covaant quantum clonng tansfomaton poduces two copes of the equatoal qut wth optmal fdelt. Geometcal Descpton of Clonng Attacks Usng the adus vecto x z of the sent pue qut the adus vecto E x z of the cloned quantum state s gven as M C () j E j j jx z whee M j denotes the components of a matx M and C j ae the thee components of a constant eal column vecto C. Eve s clonng tansfomaton opeato on the sent pue quantum state A n the smmetc ncoheent attack wth BB84 and wth a Sx-state potocol can e desced as () A B. The effect of clonng tansfomaton A can e gven the affne map E M C j E j j jx z j s the j-th component of Alce s adus whee and M j ae the nne components of the eal matx M whle C j s the j-th element of column vecto C 0 whch s gven sn cos sn sn 0 C sn sn snsn 0. () cos snsn 0 Accodng to matx M Eve s quantum clone maps the cloned state onto a maxmall mxed state f an paamete n takes the value. vecto x z Thus the effect of Eve s clonng tansfomaton can e desced the affne map [7 8]: E j Mj. (4) jx z In a geometcal epesentaton Eve s quantum clonng tansfomaton maps the Bloch all onto a compessed Bloch all [7 8] wth adus E. The matx M contans a comnaton of otatons and contactons of the vectos on the Bloch sphee whle the vecto C coesponds to a shft n the ogn of the Bloch sphee. Eve s UCM-ased attack clonng tansfomaton n the Sx-state potocol can e desced the adus vecto whee matx E M M UCM can e expessed as: M UCM cos cos 0 0 (5) 0 coscos cos cos whee the paametes ae the fee paametes of the quantum clonng tansfomaton [8 0 ]. In the BB84 potocol the phase-covaant clonng-ased attack can e desced wth paametes 0 0 and matx M phasecov. as: M phasecov whee 0. The affne tansfomaton of ths map can e desced matx M whee epesents the tetahedon : M x z The dagonal entes of the matx take values such that the effect of Eve s quantum clonng ISSN: Issue Volume 9 Mach 00

7 Laszlo Gongos Sando Ime tansfomaton can e epesented geometcall a tetahedon as we have defned t. In ths geometcal epesentaton each pont of x z whch les nsde s an allowed set of dagonal elements fo the affne tansfomaton of the quantum clone defned matx M.. Geometcal Repesentaton of Incoheent Attacks n QKD Potocols Usng the tetahedon appoach Eve s clonng actvt can e desced x z. In the BB84 potocol wth a phase-covaant clone Eve has to mnmze wth x z. (6) In the Sx-state potocol Eve uses the UCM tansfomaton whch can e desced x z. (7) The poalt that Alce and Bo choose the same ass ut get a dffeent t s D whle the poalt that the get the same t n the same ass s F. The dstuance D can e gven fo an ass E E E E D. (8) The fdelt F of the clone s [0] E E E E F D. (9) In the BB84 potocol wth a phase-covaant clone the fdelt of the clonng tansfomaton s E00 E E E00 (0) whle n the Sx-state potocol wth UCM we have E E 00 E E 00. () If Alce and Bo shae the same ass and same t and Eve guesses coectl the value of the shaed t Eve has to guess whethe he poe s n state E00 E 00 o E E. If Eve uses an F F optmal measuement to guess ths t value he success poalt [] p c s p E c 00 E. () F Eve has to maxmze p c gven an allowed dstuance level wth mn and a maxmum mn allowed dstuance gven Dmax. To educe E00 E Eve has to mnmze F E00 E and n the phase-covaant cloneased attack ths mnmum can e eached fo mnmn mn. () In the Sx-state potocol ths mnmum can e eached fo mn mn mn. (4) In the next sectons we wll show the tetahedon ntepetaton of clonng-ased attacks and the phscall allowed effects n the BB84 and Sx-state potocols... Modelng Phscall Allowed Attacks - Sx- State Potocol The optmal unvesal quantum clonng machneased smmetc ncoheent attack n the Sx-state quantum cptogaph potocol can e epesented as a educed tetahedon fomed matx M UCM. The vetces of the ognal tetahedon coespond to a sngle opeato map the edges ae two opeato maps whle the fou faces epesent all thee opeato maps [8 9]. The ponts n the nteo of the tetahedon ae all fou opeato maps of A A. (5) j0 j j j In Fg. 6 we show the tetahedon epesentaton of the optmal unvesal quantum clonng machneased attack n the Sx-state quantum cptogaph potocol. Fg. 6. The optmal unvesal quantum clonng machne-ased attack n the Sx-state quantum cptogaph potocol. The clonng tansfomaton can e epesented n the educed tetahedon fomed M on UCM ISSN: Issue Volume 9 Mach 00

8 Laszlo Gongos Sando Ime the lne etween the dentt tansfomaton and the maxmall mxed state map [5 6]. Usng ths esult the affne tansfomaton of Eve s quantum clonng tansfomaton s gven M wth condtons x z z x and z x whee x cos cos cos cos and z coscos. The thee dagonal entes of M have to ensue the complete postvt of the map thus the allowed egon n the space of x and z foms a tetahedon wth vetces at and. Usng tetahedon to epesent matx M onl the vetces and edges of the tetahedon ae touched and each pont x z whch les nsde s an allowed set whle no pont on the face o an face s contaned. The educed tetahedon can e vsualzed as the tetahedon wth each face of the tetahedon scooped out and of depth extendng all the wa to the centod. In a gaphcal epesentaton of all the vetces and edges of ae contaned whle no othe pont on an face s contaned... BB84 Potocol The phase-covaant quantum clonng-ased attack aganst BB84 cannot e modeled the educed tetahedon fomed matx M UCM. In the tetahedon epesentaton the phase-covaant clone-ased smmetc ncoheent attack on BB84 can e epesented on the face of the tetahedon fomed M. phasecov. The phase-covaant ased attack on BB84 can e modeled geometcall on the lne etween the dentt tansfomaton and the md-pont of the edge lng etween ponts fom the Z and X- tansfomatons. We cannot use tetahedon to desce the eavesdoppe clonng tansfomaton M phasecov n BB84 ecause the quantum opeatos emoved fom le on the faces of. The phase-covaant quantum clonng-ased attack aganst BB84 n the tetahedon epesentaton s llustated n Fg. 7. Fg. 7. The phase-covaant quantum clonng-ased attack aganst BB84 n the tetahedon epesentaton. The ponts emoved fom ae equed to desce the clonng tansfomaton M thus phasecov. the educed tetahedon fomed M UCM cannot desce all possle outcomes. 4 Poposed Infomaton Theoetcal Model fo Quantum Clonng Detecton In ou secut analss the dstances etween quantum states ae defned the quantum elatve entop of quantum states. The elatve entop of quantum states measues the nfomatonal dstance etween quantum states []. The Shannon entop H p of quantum states s gven the von Neumann entop S whch s a genealzaton of classcal entop to quantum states [ ]. The entop of quantum states can e gven n the followng wa: S T log. (6) The elatve entop of quantum states measues the nfomatonal dstance etween quantum states usng the negatve entop of quantum states [ 5] as the geneato functon F : F S T log. (7) The elatve quantum entop etween denst matces and can e desced the stctl convex and dffeentale functon F as: D F F F (8) whee T s the nne poduct of quantum states and F s the gadent. In Fg. 8 we vsualze the quantum nfomatonal dstance D as the vetcal dstance etween ISSN: Issue Volume 9 Mach 00

9 Laszlo Gongos Sando Ime the geneato functon F and H the hpeplane tangent to F at. The ntesecton H s denoted pont at quantum state on H. The quantum elatve entop etween two mxed quantum states depends on the lengths of the Bloch vectos and the angle etween them as llustated n Fg Fg. 9. The quantum elatve entop depends on the lengths of the vectos and the angle etween them. Fg. 8. Vsualzng the geneato functon as negatve von Neumann entop. The quantum nfomatonal dstance s not smmetc no does t satsf the tangula nequalt of metcs. The sphecal Delauna tangulaton etween pue states and etween pue and mxed states can e smpl otaned as the D Eucldean Delauna tessellaton estcted to the Bloch sphee [ ]. 4. Quantum Relatve Entop Between Mxed Quantum States The quantum elatve entop of a geneal x z and mxed state quantum state x z wth ad x z s gven x z and D log log 4 (9) log log 4 whee xx zz. Fo a maxmall mxed state xz 000 and 0 the quantum elatve entop can e expessed as D log 4 (40) log log. 4 In the poposed Delauna tangulaton method we appl quantum elatve entop as a dstance measue onl fo mxed states snce the Delauna tangulaton of pue states s dentcal to the conventonal sphecal Delauna dagam []. 4. Geometcal Backgound If the set of quantum states s denoted n the Voono cell vo fo quantum state s gven j vo x d x d x (4) d s the dstance functon. The whee ccumccle of the gven quantum states s the ccle that passes though the quantum states and of the edge and endponts and of the tangle. The tangle t s sad to e Delauna when ts ccumccle s empt. c Delauna tesselaton Quantum states on the Bloch all Fg. 0. The tangle of quantum states coesponds to the vetex c (a) and Delauna tessellaton on the Bloch sphee (). Fo an empt ccumccle the ccle passng though the quantum states of a tangle t T encloses no othe vetex of the set. In ou secut analss we use the fact that the Voono dagam vo of a set of quantum states ISSN: Issue Volume 9 Mach 00

10 Laszlo Gongos Sando Ime and the Delauna tangulaton Del ae dual to each othe [4]. The quantum Delauna tangulaton of a set of Del s the quantum states denoted geometc dual of quantum Voono dagams vo. The quantum Voono dagams can e fst-tpe o ght sded dagams. Smlal we can deve two tangulatons fom quantum Voono dagams. The fst-tpe quantum nfomatonal all ccumscng an smplex of quantum Delauna tangulaton Del s empt. Fg.. The empt all popet fo quantum Delauna tangulaton. If we choose a suset of at most d states n n then the convex hull of the assocated quantum states s a smplex of the quantum tangulaton of f thee exsts an empt quantum nfomatonal all passng though the. The fst-tpe and second-tpe quantum dagams fo quantum states whch have non-equal ad dffe. The quantum dagams etween these states ae not equal to Eucldean dagams. In Fg. (a) we llustate the dual-delauna dagam fo pue states wth. 4 The quantum dagam fo pue states s equvalent to the odna Eucldean dagam. In Fg. () we llustate the fst-tpe and second-tpe dagams fo mxed states wth ad n Bloch all epesentaton. The fsttpe quantum dagam s llustated old lnes 4 the dashed lnes show the dual cuved second-tpe dagam. Fg.. Dual-Delauna dagam fo pue states (a) and fo mxed states (). The odna quantum Voono dagam gves the egons that ae neaest to a set of gven states. The futhest Voono dagams ae the opposte of odna Voono dagams. The futhest quantum Voono dagams dentf the egons whch have the geatest dstance fom gven ponts. If we have a classcal Voono dagam of a set of quantum states n then the cells detemne the egons that contan the closest ponts to the stes. We can defne a smla stuctue fo futhest ponts and such a dagam s called the futhest-pont Voono dagam [8]. In Fg. we llustate the dffeence etween classcal quantum Voono dagams and futhest quantum Voono dagams fo a set of quantum states n the Bloch all epesentaton. We can conclude that the futhest quantum dual-delauna dagam dffes fom the odna Voono dagam and has an empt cell. Fg.. Compason of classcal quantum Voono dagams and futhest quantum Voono dagams fo a set of quantum states n the Bloch all epesentaton. In Fg. 4 we compae the odna Delauna tangulaton and the futhest Delauna tangulaton. The futhest pont Delauna edges do not ntesect and the futhest Delauna tangulaton of detemnes the convex hull and the cente of the smallest enclosng all. ISSN: Issue Volume 9 Mach 00

11 Laszlo Gongos Sando Ime Fg. 4. Compason of odna Delauna tangulaton and futhest Delauna tangulaton. The quantum dagams of pue quantum states and of mxed quantum states wth equal ad ae equvalent to odna Eucldean dagams. The quantum dagams of mxed states wth dffeent ad ae equvalent to quantum nfomatonal dagams. 4. Computatonal Geomet n Clonng Detecton We would lke to compute the adus of the smallest enclosng all of the cloned mxed quantum states thus fst we have to seek the cente c of the set of quantum states. The set of quantum n states s denoted. The dstance d etween an two quantum states of s measued the quantum elatve entop thus the mnmax mathematcal optmzaton s appled to quantum elatve entop-ased dstances to fnd the cente c of the set. We denote the quantum elatve entop fom c to the futhest pont of dc max dc. (4) Usng mnmax optmzaton we can mnmze the maxmal quantum elatve entop fom c to the futhest pont of c ag mn c d c. (4) In Fg. 5 we llustate the ccumcente c of fo the Eucldean dstance and fo quantum elatve entop []. Fg. 5. Ccumcente fo Eucldean dstance and quantum elatve entop alls. The nfomatonal theoetcal effect of the eavesdoppe s clonng machne s desced the adus of the smallest enclosng quantum nfomatonal all. The quantum nfomatonal theoetcal adus s equal to the maxmum quantum nfomatonal dstance fom the cente and can e expessed as: mn max D. (44) In ou geometcal appoach we compute the smallest enclosng nfomaton all Delauna tessellaton whch s the fastest known tool to seek the cente of a smallest enclosng all of ponts [4 5]. Fo UCM and phase-covaant clonng the connecton etween nfomaton theoetcal adus and Bloch vecto Bloch can e defned as: S Bloch (45) whee S s the von Neumann entop of the coespondng quantum state wth maxmum length vecto Bloch. The nfomatonal theoetcal adus of UCM and phase-covaant clones ae denoted and. UCM phasecov. 4.. Laguee Dagam fo Quantum States We use the Laguee Delauna dagam [4 4 5] to compute the adus of the smallest enclosng all. In geneal the Laguee dstance fo geneatng ponts x wth weght n a Eucldean space s defned dl x x. (46) The Delauna dagam fo the Laguee dstance s called the Laguee-Delauna dagam. Fo the Laguee secto of two thee-dmensonal Eucldean alls B P and B Q centeed at quantum states and we can wte the equaton x Q P 0. (47) In a Eucldean space the Laguee dstance dl x wth weght can e ntepeted as the squae of the length of the lne segment statng at and tangent to the ccle centeed at x wth adus. Thus the ccle centeed at x wth adus s the ccle assocated wth x [4]. We show a new method fo devng the quantum elatve entop-ased Delauna tessellaton on the Bloch all to detect eavesdoppng actvt on the quantum channel. In ou algothm we pesent an effectve soluton to seek the cente c of the set ISSN: Issue Volume 9 Mach 00

12 Laszlo Gongos Sando Ime of smallest enclosng quantum nfomaton all usng Laguee dagams [5]. Ou geometcal-ased secut analss has two man steps:. We constuct Delauna tangulaton fom Laguee dagams on the Bloch all.. We seek the cente of the smallest enclosng all. 4.. Quantum Delauna Tangulaton fom Laguee Dagams As we have seen n a Eucldean space the Laguee dstance of a pont x to a Eucldean all s defned as dl x x and fo n alls whee n the Laguee dagam [4] of s defned as the mnmzaton dagam of the coespondng n dstance functons d L x x. (48) In Fg. 6 we show the odna tangulaton of quantum elatve entop-ased Voono dagam. Quantum states on the Bloch all detemne the smallest enclosng all of alls usng coe-sets. The coe-sets have an mpotant ole n ou calculaton and appoxmate method. We appl the appoxmaton algothm pesented Badou and Clakson howeve n ou algothm the dstances etween quantum states ae measued quantum elatve entop [5 9]. The -coe set s a suset of the set such that fo the ccumcente c of the mnmax all [5] dc (50) whee s the adus of the smallest enclosng quantum nfomaton all of the set of quantum states [5 9]. The appoxmatng algothm fo a set of quantum states S s sn and ccumcente c fst fnds the fathest pont s m of all set B and moves c towads s m n dn tme n eve teaton step. The algothm seeks the fathest pont n the all set B Ball c n Ball cn n maxmzng the quantum nfomatonal dstance fo a cuent ccumcente poston c as max D F c n. Usng equaton max x D F c x DF c S we get max D F c n (5) max D c S. n F In Fg. 7 we llustate the smallest enclosng all of alls n the quantum space. Ball c Fg. 6. Regula tangulaton on the Bloch all. We use the esult of Auenhamme to constuct the quantum elatve entop-ased dual dagam of the Delauna tessellaton usng the Laguee dagam of the n Eucldean sphees of equatons [5] x x F. (49) The most mpotant esult of ths equvalence s that we can effcentl constuct a quantum elatve entop-ased Delauna tangulaton on the Bloch sphee usng fast methods fo constuctng classcal Eucldean Laguee dagams [5 6]. 4.4 Cente of the Quantum Infomatonal Ball In ou secut analss we use an appoxmaton algothm fom classcal computatonal geomet to c S Quantum states Fg. 7. The smallest enclosng all of a set of alls n the quantum space. We denote the set of n d-dmensonal alls B n whee BallS S s the cente of all and s the adus of the -th all. The smallest enclosng all of set B n s Ball c wth mnmum the unque all adus and cente c [6]. ISSN: Issue Volume 9 Mach 00

13 Laszlo Gongos Sando Ime The algothm does teatons to ensue an appoxmaton thus the oveall cost of the dn algothm s [5]. The smallest enclosng all of all set B can e wtten as mn F c c B (5) whee FB X d X B max d X B n and the dstance functon d measues the elatve entop etween quantum states [9]. The mnmum all of the set of alls s unque thus the ccumcente c of the set of quantum states s c ag mn F c. c B The man steps of ou algothm can e summazed as: Algothm.. Select a andom cente c fom the set of quantum states c S fo do. Fnd the fathest pont s of wt. quantum elatve entop S ag max s DF s c. Update the ccumccle: c F Fc FS. 4. Retun c At the end of ou algothm the adus of the smallest enclosng all wth espect to the quantum nfomatonal dstance s equal to the nfomatonal theoetcal fdelt of the clonng tansfomaton. Usng the nfomaton theoetcal adus mn max D the adus of the est cloned state can e expessed as: S Bloch (5) whee S s the von Neumann entop of quantum state wth maxmum length vecto Bloch. 5 Fttng the Smallest Quantum Ball Geometcall the smallest quantum nfomatonal all can e computed fom the ntesecton of contous of the quantum elatve entop all wth the ellpsod of the secet channel whch ellpsod s geneated the eavesdoppe s clone machne. The maxmum length adus can e detemned an teatve algothm usng the quantum elatve entop as a dstance measue. In Fg. 8(a) the smallest quantum nfomatonal all wth adus Dmax ntesects the channel ellpsod at magntude m of the Bloch vecto. The Eucldean dstance etween the ogn and cente c s denoted m. Smlal the Eucldean dstance etween the ogn and state s denoted m. In ou geometcal teaton algothm we would lke to detemne the locaton of vecto nsde the channel ellpsod such that the lagest possle contou value Dmax touches the channel ellpsod suface and the emande of the D max contou suface les entel outsde the channel ellpsod. The pont on the channel ellpsod suface s defned as the set of channel output. The vecto s defned n the nteo of the ellpsod as the convex hull of the channel ellpsod. To detemne the optmal length of the adus the algothm moves pont. fom the optmum As we move vecto poston a lage contou coespondng to the lage value of the quantum elatve entop D wll ntesect the channel ellpsod suface thee max D wll ncease. We can conclude that vecto should e adjusted to mnmze max D as llustated n Fg. 8(). Fg. 8. Intesecton of adus of smallest enclosng quantum nfomatonal all and channel ellpsod (a). The optmal all s shown n lght-ge (). The computed adus s equal to the adus of the smallest quantum nfomatonal all hence the quantum nfomatonal adus can e used to deve the fdelt of the eavesdoppe s quantum clone machne. The vecto should e adjusted to ISSN: Issue Volume 9 Mach 00

14 Laszlo Gongos Sando Ime mnmze the value of max D. To fnd the optmal value of vecto n ou geometcal analss we choose a stat pont fo vecto n the nteo of the ellpsod. In Fgue 9(a) we show the ntal stat pont nsde the channel ellpsod chosen the algothm. The vecto of state s denoted. In the next step the algothm detemnes the set of ponts to the vecto on the ellpsod suface most dstant fom usng the quantum elatve entop as dstance measue. In Fgue 9() the new state s notated. that the teaton conveges to the optmal max D. ecause the algothm mnmzes Fg. 0. The algothm makes a step towads the found suface pont vecto and updates the vecto. At the end of the teaton pocess the adus of the smallest quantum nfomatonal all can e expessed as mn max D. (55) In Fg. we compae the smallest quantum nfomatonal all and the odna Eucldean all. Fg. 9. The algothm detemnes the ponts on the ellpsod suface most dstant fom the pont usng the quantum elatve entop as dstance measue. The maxmum dstance etween the states can e expessed as max D. We choose a andom Bloch sphee vecto fom the maxmal set of ponts accodng to vecto. The selected pont s denoted. The algothm makes a step fom towads the suface pont vecto n the Bloch sphee space. In ths step the algothm updates vecto to (54) whee denotes the sze of the step. In Fg. 0(a) the updated state and the vecto of the state ae denoted and. The cente of the quantum nfomatonal all s denoted. In Fg. 0() we llustate the quantum nfomatonal dstance etween the fnal cente pont and the maxmal dstance state usng the notaton max D. Usng the updated vecto the algothm contnues to loop untl max D no longe changes. We conclude Fg.. The maxmum dstance states of the smallest alls dffe fo the quantum nfomatonal dstance and Eucldean dstance. We can conclude that the quantum states and whch detemne the Eucldean smallest enclosng all ae dffeent fom the states of the quantum nfomatonal all. 5. Smallest Quantum Infomatonal Ball fo UCM-Based Clonng The UCM clone-ased ncoheent attack n the Sxstate potocol can e detected f the adus of mpefect UCM clonng s equal o geate than the adus UCM of the dealstc UCM all n ellpsod epesentaton thus UCM x z. (56) ISSN: Issue Volume 9 Mach 00

15 Laszlo Gongos Sando Ime Ths suface s an olate ellpsod and can e expessed x z xxz z. The ellpsod x z has pola adus x whle the equatoal adus s z. The dstance to the ogn s x z px p pz thus the closest pont to the ogn s at the pole of the ellpsod and can e expessed as. (57) Usng the ellpsod epesentaton we can model the effects of Eve s quantum clone. The clonng tansfomaton wll e detected Bo f the pont x z epesentng the qualt of the clonng tansfomaton les on o outsde the optmal UCM all epesented ellpsodal adus UCM. In Fg. we llustate the adus UCM of the UCM all and the adus of the coespondng mpefect UCM clonng tansfomaton n the Sxstate potocol. The ogn of epesents zeo clonng actvt n the channel. 0 thus n ths case Bo has a quantum nfomatonal all wth adus. In Fg. we show the nfomaton theoetcal ad and UCM. The smallest enclosng quantum all of the mpefect UCM clone has adus 4 S (59) whle the adus of the dealstc UCM-ased clonng attack n the Sx-state potocol can e expessed as 4 UCM UCM S. (60) The smallest quantum nfomatonal all wth adus s shown n ge the all of the dealstc UCM clone wth adus UCM s shown n lght ge. Fg.. The smallest enclosng quantum nfomatonal all of optmal and mpefect unvesal clone. Fg.. Compason of optmal UCM and mpefect unvesal clonng n Sx-state potocol. In ou quantum nfomatonal dstance-ased geometcal secut analss Bo wll detect the quantum clone f UCM ecause n ths case we can gve the followng condton fo the adus of hs smallest enclosng quantum nfomatonal all: 4 4 UCM S S (58) whee S s the von Neumann entop. In ths geometcal epesentaton - f thee s no quantum clone on the quantum channel - then We can conclude that f UCM then UCM hence the nfomatonal theoetcal adus wll e smalle. 5. Smallest Quantum Infomatonal Ball fo Phase-covaant Based Attack In the phase-covaant ased smmetc ncoheent attack n the BB84 quantum cptogaph potocol the clonng actvt can e detected Bo f the adus of the mpefect phase-covaant clone s equal o geate than the adus phasecov of the phase-covaant all n the ellpsod epesentaton phasecov. Usng the ellpsod epesentaton we can model the effects of Eve s phase-covaant quantum clone-ased attack. The mpefect clonng tansfomaton s denoted pont x0 z whch les on o outsde the optmal phase-covaant all. ISSN: Issue Volume 9 Mach 00

16 Laszlo Gongos Sando Ime In Fg. 5 we compae an dealstc phasecovaant clone quantum all and an mpefect phase-covaant clone quantum all. Fg. 4. The ellpsodal ad fo optmal phase-covaant clonng and mpefect clonng actvt. The local coodnate sstem x 0 z epesents the qualt of the clonng tansfomaton and the eavesdoppng actvt wll e detected Bo f 4 x 0 z 8. (6) In the quantum clone-ased attack n BB84 Bo wll detect the quantum clone f phasecov whee s the adus epesentng the mpefect phase-covaant clonng attack. In ths case we can gve the followng condton fo the nfomaton theoetcal adus of hs smallest quantum nfomatonal all S (6) phasecov S S 8 whee S s the von Neumann entop. The smallest quantum nfomatonal all wth adus s shown n ge the maxmal all of the phase-covaant clone s shown n lght ge. In the fgues the nfomaton theoetcal adus s denoted. The quantum all of the mpefect phasecovaant clone s llustated wth adus S (6) the dealstc phase-covaant clone s denoted adus phasecov phasecov S. (64) Fg. 5. The smallest enclosng quantum nfomatonal all of optmal and mpefect phase-covaant clone. It can e concluded that the nfomatonal theoetcal ad fo dealstc and mpefect phasecovaant clonng ae dffeent. 5. Compason of UCM and Phasecovaant Based Attacks In the thee-dmensonal ellpsod epesentaton the adus phasecov of the phase-covaant cloneased attack s smalle than adus UCM. Fo the adus of the UCM and phase-covaant all n the ellpsodal epesentaton phasecov UCM (65) 4 whee phasecov x z. 8 In Fg. 6 we llustate phasecov and UCM n the thee-dmensonal ellpsodal epesentaton. Fg. 6. Compason of UCM and phase-covaant ased attack n ellpsodal epesentaton. ISSN: Issue Volume 9 Mach 00

17 Laszlo Gongos Sando Ime Usng the esults deved n Secton. the followng connecton holds etween ad UCM and phasecov of the smallest enclosng quantum nfomatonal alls of UCM and phase-covaant clonng-ased attack: 4 UCM UCM S (66) phasecov phasecov S. In Fg. 7 we llustate the ad UCM and phasecov of the smallest enclosng quantum nfomatonal all fo a UCM-ased attack and fo BB84 n the Bloch sphee epesentaton. dagam fo cloned equatoal states n the BB84 potocol. The sent pue quantum states cloned Eve s phase-covaant quantum clone ae denoted and 4. Fg. 8. Dual Delauna dagam of cloned equatoal states n the BB84 potocol. Usng Delauna tessellaton we compute the convex-hull of the cloned equatoal states and 4. In Fg. 9 we llustate the convex-hull of cloned states n two- and thee-dmensonal Bloch all epesentatons. Fg. 7. Compason of smallest enclosng quantum nfomatonal all of UCM and phase-covaant clones. It can e concluded that the est qualt of the two outputs smultaneousl can e ealzed wth a UCM. If an eavesdoppe uses a phase-covaant clone one of the two outputs should have ette fdelt whle the fdelt of the second output wll e lowe. 6 Applng Ou Method to Quantum Cptogaph Usng the esults deved n Sectons 5. and 5. the quantum channel n the BB84 and Sx-state potocols s secue f phasecov. and UCM. In ou geometcal method we compute the adus of the smallest enclosng quantum nfomatonal all to detemne the secut of the quantum communcaton. Fg. 9. The convex hull of cloned mxed states. The convex hull computed Delauna tangulaton. Fom the convex set we can compute the smallest enclosng quantum nfomatonal all and ts adus. In Fg. 0 we have llustated the Eucldean smallest enclosng all the dashed ccle and the quantum elatve entop all. 6. BB84 and Phase-covaant Clonng In ths secton we llustate the quantum nfomatonal alls fo the analzed quantum clones. In Fg. 8 we llustate the dual Delauna Fg. 0. The smallest enclosng quantum nfomatonal alls. ISSN: Issue Volume 9 Mach 00

18 Laszlo Gongos Sando Ime Fom the smallest enclosng quantum nfomatonal all we can detemne the adus whch desces the nfomatonal theoetcal mpact of the eavesdoppe clonng machne. The cente of the smallest enclosng quantum nfomatonal all s denoted c. 6. Sx State Potocol and Unvesal Clonng In Fg. (a) we have llustated the Voonocells fo the cloned states and the thee-dmensonal convex hull (lght-ge) of cloned states 4 5 and 6. The cloned states geneated Eve s unvesal quantum clone machne usng the Sx-state quantum cptogaph potocol. Fom the convex hull we compute the smallest enclosng quantum nfomatonal all. In Fg. () we have llustated the smallest quantum nfomatonal all and ts adus. Fg.. The convex hull (a) and the smallest quantum all () of cloned mxed states n Sx-state potocol. 7 Optmzaton The quantum elatve entop-ased algothm at the -th teaton gves an -appoxmaton of the eal ccumcente thus to get an appoxmaton ou algothm eques a tme dn d d (67) fst samplng n ponts. Based on the computatonal complext of the smallest enclosng all the appoxmaton of the fdelt of the eavesdoppe clonng machne can e computed n a tme d. (68) In ths secton we mpove ou method to get a d (69) tme -appoxmaton algothm n quantum space. In Fg. we llustate the mpoved algothm on a set of quantum states. The appoxmate all has adus the enclosng all has adus. The appoxmate cente c s denoted n lack the coeset ae denoted ge ccles. The optmal adus etween the cente c and the fathest quantum state s denoted [9]. Coe-set In Fg. we show an example of a twodmensonal smallest enclosng quantum nfomatonal all and ts nfomatonal theoetcal adus. Fathest quantum state c Fg.. The appoxmate (lght) and enclosng quantum all (dake). Fg.. The smallest enclosng quantum nfomatonal all. The cente pont s c and the adus of the smallest enclosng quantum nfomatonal all s In the poposed algothm the optmal adus s etween and the pocess temnates as n at most teatons. The man steps of the mpoved appoxmaton algothm ae [9]: ISSN: Issue Volume 9 Mach 00

19 Laszlo Gongos Sando Ime Algothm.. Select a andom cente c fom the set of quantum states. max DFc ;. max DFc ; 4. fo 5. do 6. S ag max D c ; c 7. Move Ball c on the geodesc untl t touches the fathest pont S; 8. smax DFc S ; 9. f s then else. ; 4. ; 4 4. untl. F 7. Rate of Convegence In ou expemental smulaton we have compaed the coe-set algothm and ou mpoved coe-set algothm to fnd the smallest enclosng quantum nfomaton all. We have analzed the appoxmaton algothms fo 0 cente updates and we have measued the qualt of the appoxmaton wth espect to quantum elatve entop. The esults of ou smulaton ae shown n Fg. 4. The x-axs epesents the nume of cente updates to fnd the cente of the smallest enclosng quantum nfomatonal all the -axs epesents the quantum nfomatonal dstance etween the found cente c and the optmal cente c. Quantum nfomatonal dstance Coe-set algothm Impoved algothm Nume of cente updates Fg. 4. The ate of convege of appoxmaton algothms. Fom the esults we can conclude that each algothm fnds the appoxmate cente c to the optmal cente c ve fast. The quantum elatve entop-ased appoxmaton algothms have a ve accuate convegence of c towads c howeve the mpoved coe-set algothm conveges faste wth a smalle nume of cente updates. 8 Conclusons In quantum cptogaph an eavesdoppe cannot clone the sent quts pefectl howeve the est eavesdoppng attacks ae ased on mpefect quantum clones. We have poposed a fundamentall new appoach to computng the nfomatonal theoetcal mpacts of an eavesdoppe s clonng machne n the quantum channel. The analzed ncoheent attacks ae the eavesdoppe s most geneal stateg howeve ou method can e extended fo dffeent tpes of attacks. The eavesdoppe s clonng actvt and the mpacts of he clonng tansfomaton can e measued geometcall. Ou method s ased on Laguee dagams wth quantum elatve entop used as dstance measue We showed that the geometc space can e dvded and can e computed ve effcentl usng Delauna tessellaton on the Bloch sphee n a easonale computatonal tme. As futue wok we would lke to extend ou method to othe potocols and to collectve and coheent attacks. We would lke to constuct a moe effectve algothm to compute the nfomatonal theoetcal mpacts of the eavesdoppe s clonng machne on a pvate quantum channel. Refeences: [] L. Gongos S. Ime: Computatonal Geometc Analss of Phscall Allowed Quantum Clonng Tansfomatons fo Quantum Cptogaph In Poceedngs of the 4th WSEAS Intenatonal Confeence on Compute Engneeng and Applcatons (CEA '0). Havad Unvest Camdge USA. 00. pp. -6. Pape 8. [] S. Ime F. Balazs: Quantum Computng and Communcatons An Engneeng Appoach Pulshed John Wle and Sons Ltd The Atum Southen Gate Chcheste West Sussex PO9 8SQ England 005 ISBN X 8 pages [] P. W. Lamet A. P. Majte A. Boas M. Casas and A. Plastno. Metc chaacte of the quantum Jensen-Shannon dvegence. Phscal ISSN: Issue Volume 9 Mach 00

20 Laszlo Gongos Sando Ime Revew A (Atomc Molecula and Optcal Phscs) (5):05. [4] F. Auenhamme and R. Klen. Voono Dagams. In J. Sack and G. Uuta (Eds) Handook of Computatonal Geomet Chapte V pp Elseve Scence Pulshng [5] J.-D. Bossonnat C. Womse and M. Yvnec. Cuved Voono dagams. In J.-D.Bossonnat and M. Tellaud (Eds) Effectve Computatonal Geomet fo Cuves and Sufaces 007 pp Spnge-Velag Mathematcs and Vsualzaton. [6] Cef N.J. M. Bouennane A. Kalsson and N. Gsn 00 Phs. Rev. Lett [7] D Aano G.M. and C. Macchavello 00 Phs. Rev. A [8] Acín A. N. Gsn L. Masanes and V. Scaan 004 Int. J. Quant. Inf.. [9] R. Pangah. Mnmum enclosng poltope n hgh dmensons. CoRR cs.cg/ [0] N. J. Cef Phs. Rev. Lett (000). [] Zhang W.-H. Yu L.-B. Ye L. Optmal asmmetc phase-covaant quantum clonng Phscs Lettes Secton A: Geneal Atomc and Sold State Phscs 56 () 006 pp [] Satosh Iama Masano Oha Mathematcal Chaactezaton of Quantum Algothm Poceedngs of the 4th WSEAS Intenatonal Confeence on Appled Mathematcs (MATH '09) 009. pp [] Masana Asano Masano Oha Quantum Telepotaton wth Non-Maxmal Entangled State Poceedngs of the 4th WSEAS Intenatonal Confeence on Appled Mathematcs (MATH '09) 009. pp [4] Yuj Hota A Categocal Appoach to Quantum Algothm Poceedngs of the 4th WSEAS Intenatonal Confeence on Appled Mathematcs (MATH '09) 009. pp [5] Ron Goldman Fou Open Mathematcal Polems Related to Compute Gaphcs and Geometc Modelng Poceedngs of the Intenatonal Confeence on Computatonal and Infomaton Scence 009 pp [6] Stanslaw P. Kaspeczuk Quantum Defomatons of Algeas Assocated wth Integale Hamltonan Sstems Poceedngs of the 5th Amecan Confeence on Appled Mathematcs 009 pp [7] Moslav Svtek Phscs of Infomaton Repesentaton Tansmsson and Pocessng Recent Advances On Data Netwoks Communcatons Computes Poceedngs of the 8th WSEAS Intenatonal Confeence on Data Netwoks Communcatons Computes (DNCOCO '09) 009. pp. 05- [8] Mak Bugn Mathematcal Theo of Infomaton Technolog Poceedngs of the 8th WSEAS Intenatonal Confeence on Data Netwoks Communcatons Computes (DNCOCO '09) 009. pp ISSN: Issue Volume 9 Mach 00