Quantum Number Tricks

Transcription

1 J Softwar Enginring & Applications, 00, : doi:046/jsa0009 Publishd Onlin March 00 ( Quantum umbr Trics Taashi Mihara Dpartmnt of Information Scincs Arts, Toyo Univrsity, Kawago, Japan mihara@toyonttoyoacjp Rcivd ovmbr rd, 009; rvisd Dcmbr st, 009; accptd Dcmbr 9 th, 009 ABSTRACT Som rsults indicat that quantum information basd on quantum physics is mor powrful than classical on In this papr, w propos nw trics basd on quantum physics Our trics ar mthods inspird by th stratgis of quantum gam thory In ths trics, magicians hav th ability of quantum physics, but spctators hav only classical on W propos quantum trics such that, by manipulating quantum coins quantum cards, magicians guss spctators valus Kywords: Quantum Tric, Entangld Stat, Gam Thory Introduction Th studis on quantum information hav succdd in such as quantum computation, quantum cryptography, quantum communication complxity, so on For xampl, Shor s quantum factoring algorithm is on of rprsntativ rsults in ths filds [] In addition, quantum gam thory has bn also proposd it has bn shown that quantum gam thory is mor powrful than classical on In 998, for a coin flipping gam, Myr proposd a quantum stratgy for th first tim showd that th quantum stratgy has an advantag ovr classical ons [] Morovr, h also showd th importanc of a rlationship btwn quantum gam thory quantum algorithms Aftr that, othr typs of quantum stratgis hav bn also proposd For xampl, Eisrt t al proposd a quantum stratgy with ntangld stats for a famous two-playr gam calld th Prisonr s Dilmma [] (also s Du t al [4,5], Eisrt Wilns [6], Iqbal Toor [7]) For anothr famous two-playr gam calld th Battl of th Sxs, Marinatto t al also proposd a quantum stratgy with ntangld stats [8] For ths gams, thy showd quantum ash quilibriums diffrnt from classical ons In this papr, w propos quantum trics basd on mthods inspird by th stratgis of quantum gam thory Magicians hav th ability of quantum physics, but spctators hav only classical on By manipulating quantum coins quantum cards, magicians guss spctators valus For xampl, w propos trics such that by using ntangld stats, a magician transmits a spctator s valu to anothr magician without communicating btwn thm Th rmaindr of this papr has th following organization In Sction, w dfin notations basic oprations usd in this papr In Sction, w propos quantum coin trics In Sction 4, w propos quantum card trics Finally, in Sction 5, w provid som concluding rmars Prliminaris First, w dnot som basic notations Lt 0 Z 0 n Z n B, n, n for a positiv intgr n Lt a b b intgrs W say that a is congrunt to b to modulus n if n is a divisor of a b dnot by a b(mod n), w dnot an innr product modulo of a b by ab Finally, lt b an xclusiv-or oprator, g, ( 0 0) ( 0 0) (0 0) xt, w dfin som basic quantum notations As stats of qubit, lt 0 ( 0) T (0 ) T, whr is Dirac notation A T is th transposd matrix of matrix A Throughout this papr, w ta B q 0 as a computational basis a masurmnt basis Morovr, w dnot an n -qubit basis stat by b b b b bbnbbbn n, whr is a tnsor product b i B q ( i,,, n ) In addition, w dnot a basis in an -dimnsional systm,

2 Quantum umbr Trics 4 a basis of qudit stats, by, whr ( ) is an intgr W call x a quantum rgistr Finally, w dfin som unitary matrics usd for quantum trics in this papr Lt I b th idntity matrix This opration mans no opration A Walsh-Hadamard opration H is Z q H x xz ( H 0 ( )( 0 ) H ( )( 0 )) ot that H=H - This opration is usd whn w ma a suprposition of stats An opration usd whn a coin is flippd is X, 0 X 0 ( X 0 X 0) Morovr, w dfin an opration btwn two qubits A Controlld ot gat, COT, is COT ( COT ct ct c, whr th first bit c is th controlld bit th scond bit t is th targt bit) W dnot th opration by COT (ij) whn th i-th bit is th controlld bit th j -th bit is th targt bit Entangld stats can b mad by using H COT For xampl, H 00 ( 0 ) 0 COT (0 0 ) Finally, w dfin two matrics for -stat transition Lt x Z A quantum Fourir transform [], QFT, is QFT x y xy y0 xy QFT y x Quantum Coin Trics y0 In this sction, w show som quantum coin trics using quantum stats Throughout this papr, w us Alic Bob as nams of magicians, us Carol Davis as nams of spctators participating in a magic show Morovr, Alic Bob can cooprat but cannot communicat with ach othr during ach show First, w show a simpl coin tric using a two-qubit ntangld stat Coincidnc: First, Alic prpars on coin put it in a box Th box is a containr such that no on cannot s th stat of th coin but can oprat it xt, Carol flips th coin or not Thn, Bob gusss th stat of th coin, i, ithr had (H) or tail (T) Mthod of Coincidnc W dnot H T by 0, rspctivly ) Bforh, Alic Bob shar an ntangld stat (0 0 ) whr Alic has th first qubit Bob has th scond qubit Alic s qubit is in a box ) Carol flips Alic s coin or not This mans that Carol applis X to Alic s qubit if sh wants to flip th coin; othrwis sh applis I to it Thn, if sh flips it, th stat bcoms ( 0 0 ) ) Alic Bob apply H to th stat Thn it bcoms (0 0 ) if Carol flippd th coin; othrwis th stat dos not chang, i, (0 0 ) 4) Bob masurs his qubit announcs th valu(h or T) to Carol 5) Carol opns th box confirms that hr valu is sam as Bob s valu This tric can b asily xtndd to multipl coins by prparing th ntangld stats ( )( 00 ) corrsponding to th numbr of coins xt, w show a tric gussing th numbr of Carol flipping coins Flip-Flop: First, Alic prpars coins in all th coins bing had xt, Carol flips som coins such that th stat of coins is m B Alic flips som coins Carol flips som coins Alic flips som coins Thn, Carol finds that th stat of final coins is m Copyright 00 SciRs

3 4 Quantum umbr Trics Mthod of Flip-Flop ) Alic prpars a stat 0 (all th coins ar had), xhibits it to Carol, puts it in a box ) Carol flips som coins th stat bcoms m ) Alic applis H x0 to it th stat bcoms ( ) mx x 4) Carol flips som coins th stat bcoms whr r B 5) Alic applis x0 H y0 x0 mx ( ) x r to it th stat bcoms ( ) ( ) mx ( xr) y y ry ( my) x ( ) ( ) y y0 x0 rm ( ) m 6) Carol opns th box confirms m Finally, w show a tric modifying Flip-Flop Flip-Flop: First, Alic prpars coins in all th coins bing had xt, Carol flips som coins such that th stat of coins is m B Alic flips som coins Carol flips som coins such that th addd stat of coins is m B Alic flips som coins Carol flips som coins Alic flips som coins Thn, Alic gusss th valu of m if Carol announcs th valu of m ; othrwis, Alic gusss th valu of m if Carol announcs th valu of m Mthod of Flip-Flop ) Alic prpars a stat 0, xhibits it to Carol, puts it in a box ) Carol flips som coins th stat bcoms m ) Alic dos not flip thm in hr turn 4) Carol flips som coins th stat bcoms mm 5) Alic applis H to it th stat bcoms ( mm) x ( ) x x0 6) Carol flips som coins th stat bcoms ( mm) x ( ) x r x0 whr r B 7) Alic applis H to it th stat bcoms ( mm) x ( xr) y ( ) ( ) y y0 x0 ry ( mmy) x ( ) ( ) y0 x0 ry ( ) mm y Thn, Alic masurs it obtains m m 8) Carol announcs ithr m or m Thn, Alic gusss m if Carol announcd m ; othrwis sh gusss m Lt b th numbr of coins Thn, th complxity of ths mthods mntiond in this sction is in O ( ) tim bcaus ach opration of X, H, COT can b xcutd in O() tim 4 Quantum Card Trics In this sction, w show som quantum card trics using quantum stats Magicians Alic Bob gusss th numbrs slctd by spctators Carol Davis Throughout this sction, lt arithmtic oprations b xcutd to modulus a prim intgr First, lt (Alic, Carol) (Bob, Davis) b two pairs Thn, w show trics such that Alic gusss Davis s numbr Bob gusss Carol s numbr Tlpathy: First, Alic prpars a card writtn a numbr, puts in a box Th numbr of this card can b rwrittn xt, Carol multiplis it by m adds a rom r to it, whr mr Z Finally, Bob prpars th numbrd cards Carol opns th box obtains a numbr By turning ovr Bob s card corrsponding th numbr, Carol confirms that th rvrs sid of th card is m Mthod of Tlpathy ) Bforh, Alic Bob shar th following ntangld stat x0 x x whr Alic has th first rgistr Bob has th scond rgistr Alic s rgistr is put in a box ) Carol multiplis Alic s rgistr by m, adds r to it Thn, th stat bcoms mx r x x0 ) Alic Bob apply QFT to it th stat bcoms Copyright 00 SciRs

4 Quantum umbr Trics 4 ( mxr) y xy y y y0 y0 x0 ry ( myy) x y y y0 y0 x0 ry y y myy0(mod ) Thn, Bob masurs it obtains my y 0(mod ) 4) Bob prpars a st of pairs (my ) my y 0(mod ) That is, h writs y to th surfac of a card writs m to th rvrs sid H mas cards corrsponding to all th possibl pairs of ( my ) Thn, h xhibits th st of th cards to Carol 5) Carol opns th box nows Thn, sh turns ovr Bob s card writtn y confirms that th valu of th rvrs sid is m Mutual Tlpathy: Lt Alic Carol b on pair, Bob Davis b anothr pair First, Alic prpars a card writtn a numbr, puts in a box Bob also prpars a card writtn a numbr, puts in anothr box xt, Carol multiplis it by m, adds a rom r to it Davis multiplis it by m, adds a rom r to it Hr, mmrr Z Finally, Bob prpars th numbrd cards Carol opns th box obtains a numbr By turning ovr Bob s card corrsponding th numbr, Carol confirms that th rvrs sid of th card is m In addition, Alic prpars th numbrd cards Davis opns th box obtains a numbr By turning ovr Alic s card corrsponding th numbr, Davis confirms that th rvrs sid of th card is Mthod of Mutual Tlpathy ) Bforh, Alic Bob shar th following ntangld stat x0 x x whr Alic has th first rgistr Bob has th scond rgistr Alic s rgistr is put in a box, Bob s rgistr is put anothr box ) Carol multiplis Alic s rgistr by m adds r to it Davis multiplis Bob s rgistr by m adds r to it Thn, th stat bcoms y mx rmx r x0 y m In addition, Davis announcs m to Bob ) Alic Bob apply QFT to it th stat bcoms ( ry ry) ( my my) x y0 y0 x0 y y ( ry ry ) y y my my0(mod ) Thn, Alic Bob masur it obtain y y, rspctivly, my my 0(mod ) 4) Bob prpars a st of pairs (m y) my my 0(mod ) That is, h writs y to th surfac of a card writs m to th rvrs sid H mas cards corrsponding to all th possibl pairs of ( m y) Thn, h xhibits th st of th cards to Carol 5) Carol opns th box nows y Thn, sh turns ovr Bob s card writtn y confirms that th valu of th rvrs sid is m ot that Alic can also now m hr 6) Alic also prpars a st of pairs (m,y ) my my 0(mod ), Davis can find th corrct pair (m,y ) xt, w show a card tric similar to Flip-Flop Prdiction: First, Alic prpars a card writtn 0, puts it in a box xt, Carol adds m Z to it Alic xcuts som opration Carol multiplis it by m adds a rom r to it, whr m r Z Alic xcuts som opration, opns th box, obtains a numbr Finally, Alic prpars th numbrd cards By turning ovr Alic s card corrsponding m, Carol confirms that th rvrs sid of th card is m Mthod of Prdiction ) Alic prpars a stat 0, xhibits it to Carol, puts it in a box ) Carol adds m to it th stat bcoms m ) Alic applis QFT to it th stat bcoms mx x0 x 4) Carol multiplis it by m adds r to it Thn, th stat bcoms mx mxr x0 5) Alic applis QFT to it th stat bcoms Copyright 00 SciRs

5 44 Quantum umbr Trics mx ( mxr) y y y0 x0 ry ( mmy) x y y0 x0 y ry whr m my 0(mod ) Thn, Alic masurs it obtains y 6) Alic prpars a st of pairs (m m) m my 0(mod ) That is, sh writs m to th surfac of a card writs m to th rvrs sid sh mas cards corrsponding to all th possibl pairs of ( m m) Thn, sh xhibits th st of th cards to Carol 7) Carol turns ovr Alic s card writtn m confirms that th valu of th rvrs sid is m Finally, w show a tric such that Alic gusss th numbr slctd by Carol in a situation that Alic prpars a st of cards bforh Mindrading: Bforh, Alic prpars a st of cards Sh writs ach y Z to ach card writs ( y) to th rvrs sid, whr ( y) is a rom prmutation of y First, Carol slcts m Z announcs it to Alic Alic prpars a card, puts in a box xt, Carol adds a rom r Z to it Alic xcuts som opration Finally, Carol opns th box, obtains a numbr By turning ovr Alic s card corrsponding to th numbr, Carol confirms that th rvrs sid of th card is m Mthod of Mindrading 5) Carol opns th box, obtains w, confirms that th valu of th rvrs sid is m Lt cn ( ) b th tim complxity of arithmtic oprations, whr n is th siz of th input In addition, lt qn ( ) b th tim complxity of QF T It is now that both cn ( ) qn ( ) ar within polynomial of n Thn, th complxity of thir mthods mntiond in this sction is in Oc ((log) q(log )) tim 5 Conclusions In this papr, w proposd nw coin trics card trics basd on quantum physics In ths trics, magicians had th ability of quantum physics, but spctators had only classical on Thrfor, magicians could manipulat coins cards as quantum stats Morovr, by sharing ntangld stats, thy could transmit spctators valus without communicating btwn thm Sinc our trics ar simpl straightforward ons using quantum stats, thy ar somwhat clumsy Thrfor, it is a futur wor to construct polishd trics Morovr, in our trics, spctators had only classical powr Thrfor, it is an intrsting problm that w construct quantum trics whn spctators also hav quantum powr REFERECES [] P W Shor, Polynomial-tim algorithms for prim factorization discrt logarithms on a quantum computr, SIAM Journal on Computing, Vol 6, pp , 997 [] D A Myr, Quantum stratgis, Physical Rviw Lttrs, Vol 8, pp , 999 ) Carol slcts m announcs it to Alic [] J Eisrt, M Wilns, M Lwnstin, Quantum ) Alic prpars a stat gams quantum stratgis, Physical Rviw Lttrs, Vol 8, pp , 999 wx x [4] J Du, H Li, X Xu, M Shi, J Wu, X Zhou, R Han, x0 Exprimntal ralization of quantum gams on a quantum whr lt ( w) m This is put in a box computr, Physical Rviw Lttrs, Vol 88, 00 ) Carol adds a rom r to it, th stat bcoms [5] J Du, H Li, X Xu, X Zhou, R Han, Entanglmnt nhancd multiplayr quantum gams, Physical Rviw wx xr Lttrs, Vol 0, pp 9, 00 x0 [6] J Eisrt M Wilns, Quantum gams, Journal of QFT Modrn Optics, Vol 47, pp , 000 [7] A Iqbal A H Toor, Evolutionarily stabl stratgis wx ( xr ) y y in quantum gams, Physics Lttrs A, Vol 80, pp 49 56, 00 4) Alic applis to it th stat bcoms y0 x0 ry ( wy) x y0 x0 rw w y [8] L Marinatto T Wbr, A quantum approach to static gams of complt information, Physics Lttrs A, Vol 7, pp 9 0, 000 Copyright 00 SciRs