A Study of Quantum Strategies for Newcomb s Paradox

Transcription

1 Bune, 00, : 4-50 do:0.436/b Publhed Onlne March 00 ( A Study of Quantum Stratege for ewcomb Paradox Taah Mhara Department of Informaton Scence and Art, Toyo Unverty, Satama, Japan. Emal: mhara@toyonet.toyo.ac.jp Receved Augut 4 th, 009; reved October th, 009; accepted ovember 3 rd, 009. ABSTRACT ewcomb problem a game between two player, one of who ha an ablty to predct the future: let have an ablty to predct Alce wll. ow, prepare two boxe, Box and Box, and Alce can elect ether Box or both boxe. Box contan $. Box contan $,000 only f Alce elect only Box ; otherwe Box empty($0. Whch better for Alce? Snce Alce cannot decde whch one better n general, th problem called ewcomb paradox. In th paper, we propoe quantum tratege for th paradox by havng quantum ablty. Many other reult ncludng quantum tratege put empha on fndng out equlbrum pont. On the other hand, our reult put empha on whether a player can predct another player wll. Then, we how ome potve oluton for th problem. Keyword: Game Theory, ewcomb Paradox, Quantum Strategy, Meyer Strategy. Introducton Quantum mechanc ha been ncorporated nto many feld called nformaton. The mot famou reult are a quantum factorng algorthm by Shor [] and a quantum databae earch algorthm by Grover []. Moreover, the tude on quantum nformaton have ucceeded n uch a quantum computaton, quantum crcut, quantum cryptography, quantum communcaton complexty, and o on. Recently, game theory baed on quantum mechanc, quantum game theory, ha been alo propoed and t ha been hown that quantum game theory more powerful than clacal one. Game theory one of the mot famou decon mang method and ha been ued n many tuaton both theoretcally and practcally. There had exted the bac concept wth repect to thee game nce early tme but n corporaton wth Morgentern, von eumann [3] frtly contructed the theory ytematcally. owever, the man prncple of th theory wa baed on clacal phyc although he wa famlar wth quantum mechanc. In 998, for a con flppng game, Meyer [4] propoed a quantum trategy for the frt tme and howed that the quantum trategy ha an advantage over clacal one. Th game called PQ Penny Flp. PQ Penny Flp: The tarhp Enterpre facng ome mmanent and apparently necapable calamty when Q appear on the brdge and offer to help, provded Captan Pcard can beat hm at penny flppng: Pcard to place a penny head up n a box, whereupon they wll tae turn (Q, then Pcard, then Q flppng the penny (or not, wthout beng able to ee t. Q wn f the penny head up when they open the box. Th game a two-player zero-um game and the probablty that each player wn at mot / wth clacal tratege. Meyer howed a quantum trategy wth whch Q can alway wn by ung a uperpoton of quantum tate effectvely. In th game, Pcard contraned to play clacally. The quantum trategy then executed n the followng way. Let 0 and repreent the head and tal of the penny, repectvely. Frt, Pcard prepare 0 and Q apple a Walh-adamard operaton defned n the next ecton to the tate: 0 ( 0 ext, Pcard decde clacally whether he flp t or not. owever, the tate doe not change even f Pcard flpped t. Fnally, Q apple to the tate and alway obtan 0. Th mean that Q alway wn. Moreover, he alo howed the mportance of a relatonhp between quantum game theory and quantum algorthm. Later, other type of quantum tratege have been alo propoed. In ther tratege, all the player can ue quantum operaton. For example, Eert et al. [5] propoed a quantum trategy wth entanglement for a fa- Copyrght 009 ScRe B

2 A Study of Quantum Stratege for ewcomb Paradox 43 mou two-player game called the Proner Dlemma (alo ee Du et al. [6,7], Eert and Wlen [8], and Iqbal and Toor [9]. In ther trategy, entanglement play an mportant role. For another famou two-player game called the Battle of the Sexe, Marnatto et al. [0] alo propoed a quantum trategy wth entanglement. For thee game, they howed quantum ah equlbrum dfferent from clacal one. Furthermore, there are many reult beng related to game uch a the Monty all problem by D Arano et al. [], Fltney and Abbott [], and L et al. [3], Parrondo game by Fltney et al. [4], game n economc by Potrow and Sładow [5 7], ewcomb paradox by Potrow and Sładow [8], and o on. In th paper, we tudy ewcomb problem. ewcomb problem a thought experment between two player, Alce and. Alce a common human beng. On the other hand, may be a wzard havng an ablty to predct the future, or not. can predct Alce wll f he a wzard. Then, the problem a follow: ewcomb problem: prepare two boxe, Box and Box, and Alce can elect ether Box or both boxe. Box contan $. Box contan $,000 only f Alce elect only Box ; otherwe Box empty($0. Whch better for Alce? o one now the anwer except. amely, there ext no bet clacal trategy. Therefore, th problem called ewcomb paradox. We how ome quantum tratege for ewcomb paradox by ung entanglement. It thought that entanglement eental a the man power of quantum nformaton and many reult mentoned above alo have ued entanglement effectvely. Frt, we how ome bac quantum tratege wth entanglement. In the other related tude mentoned above, each player operate only each agned qubt although the tate are entangled. On the other hand, our propoed tratege operate not only one qubt but alo tate between two qubt. Conequently, we how that our quantum tratege wth entanglement are more powerful than clacal one. Fnally, we how ome quantum tratege for ewcomb paradox. Potrow and Sładow howed a quantum oluton for ewcomb paradox by ung Meyer trategy [8]. ewcomb paradox whether a player can predct another player wll. We alo tudy th problem by applyng our tratege. Then, we obtan potve reult. That, n ome cae, a player can predct another player wll. The remander of th paper ha the followng organzaton. In Secton, frt, we defne notaton and bac operaton ued n th paper. Moreover, a the tool of our quantum tratege, we how two fundamental lemma wth relaton to entanglement. In Secton 3, we denote two type of two-player zero-um game. We then how that n thee game, each player cannot wn wth certanty wth clacal tratege but one de player can wn wth certanty wth quantum one. In Secton 4, we tudy ewcomb paradox. We modfy th problem and how ome quantum tratege to t by ung the reult n Secton 3. Fnally, n Secton 5, we provde ome concludng remar.. Prelmnare In th ecton, frt, we defne ome notaton ued n th paper. Let be a btwe excluve-or operator,.e., Let a be the negaton of a bt for a 0,.e., a a. Moreover, let bc a n bc be the nner product of b and c, where b ( bb... bn and c ( cc... cn for b c 0 (... n. ext, we defne ome bac operaton. Let 0 ( 0 T and (0 T, where Drac T notaton and A the tranpoed matrx of a matrx A. Let I be the dentty matrx. Th operaton mean no operaton. A Walh-adamard operaton ( 0 ( ( 0 and ( ( 0. ote that. Th operaton ued when we mae a uperpoton of tate. A an operaton ued when they flp a con clacally, player ue an operaton X, 0 X 0 ( X 0 and X 0 phae-hft operaton ( S( 00 S( by S( 0 0 e. We alo defne a and S( e, where. Moreover, we defne an operaton between two qubt. ere, we denote an n -qubt tate by b b bn bb bn, where a tenor product. Then, let COT be a Controlled ot gate, COT Copyrght 009 ScRe B

3 44 A Study of Quantum Stratege for ewcomb Paradox ( COT ct ct c, where the frt bt c the controlled bt and the econd bt t the target bt. We denote the operaton by COT when the -th bt ( j the controlled bt and the j -th bt the target bt. Fnally, we how two bac reult operatng entangled tate. Thee reult can be ued a tool of mang a pecfc tate n order that one de player alway wn game by ung our quantum tratege mentoned n the followng ecton. Lemma. Let ( (... ( bb b bb... b be a -qubt entangled tate, where b 0 0 (... and. Then, ( bc ( cc... c c when we apply the Walh-adamard operaton to ( all the qubt of. ( Proof. When we apply to all the qubt of, (( c... c ( ( c... c ( bc ( b c c 0 c 0 bc c ( ( ( ( c... c c 0 c 0 bc ( cc... c c where (.... Then, the tatement of th lemma then atfed. Th lemma mean that a player can obtan a tate c c atfyng c 0 f he mae the tate (0 and that he can obtan a tate c c at- ( fyng c f he mae the tate. ext, we how a reult ued a a player quantum trategy when another player flp con clacally. ( Lemma. Let be the -qubt entangled tate of Lemma., and let ( ( bc ( cc... c c ow, let the operaton X be appled to ome qubt ( of,.e., Alce b b a a b a b a a b a b Fgure. Two player payoff matrx ( c x c x... c x ( ( bc x c where x 0 (.... ote that x f X ha been executed. Then, ( ( xb (... ( ( x x bb b b b... b Proof. When we apply to all the qubt of (, x ( (... d ( bc ( cx d x d d d 0 d 0 c xd ( bd c ( ( d d... d d 0 d 0 c ( xb (... ( ( x... bb b b b b where the lat expreon obtaned by notng that except for the tate correpondng to bd 0 and b d, the tate vanh. The tatement of th lemma then atfed. 3. Quantum Stratege Ung Entangled State 3. Strategc Game We denote a trategc game a ( S u, where the et of player, S the et of tratege of player, and u the payoff functon of player,.e., u SSS R (the et of real number. The game can be gven alo by a matrx hown n Fgure. For more detal, we hall refer the reader to the boo by, e.g., reference [,3]. For two player, ={Alce, }, the et of Alce tratege S, and the et of tratege a a a S b b b. Then, each value of Alce payoff functon ua( a j b aj, and each value of payoff functon ub( aj b bj, where j {}. If bj aj for any j { }, b j can be omtted. In certan crcumtance, we repreent a value of payoff functon not a a real number but a ome player wn or lo, and o on. In addton, quantum tratege are permtted any untary matrce,.e., we can mae a uper- Pcard Q Unchanged Changed I Q wn Q loe X Q wn Q loe Fgure. PQ penny Flp Copyrght 009 ScRe B

4 A Study of Quantum Stratege for ewcomb Paradox 45 Alce Soccer Move Soccer 3,5 -,- Move -,- 5,3 Fgure 3. Battle of the exe poton of ome fundamental tratege by quantum tratege. ow, let u formalze PQ Penny Flp. The et of player PcardQ, the et of Pard tratege Sp IX, the et of Q tratege Sq = any untary matrce. Operaton I mean no con flp, and operaton X mean a con flp. The payoff matrx hown n Fgure. Unchanged / Changed mean that the tate of the con fnally unchanged/changed. Meyer howed a quantum trategy that Q alway wn [4]. ext, we how a mple oluton for the Battle of the Sexe ung an entangled tate. Th dea lead to the reult of the followng ubecton. The payoff matrx hown n Fgure 3. Alce prefer move to occer, and prefer occer to move. owever, both prefer havng a date. The trategy a follow. Alce and hare the followng entangled tate: (00 where Alce ha a frt qubt and ha a econd qubt. In decdng ether occer or move, both meaure the tate. Alce/ elect occer when the outcome of the bt 0, otherwe he/he elect move. ote that f Alce outcome 0, outcome alo 0, and vce vera. amely, the probablty of electng (Soccer, Soccer/(Move,Move /, and the probablty of electng (Soccer,Move/(Move,Soccer 0. Therefore, they can have a date wth certanty. Moreover, for example, f the payoff of (Soccer,Soccer (4,6 ntead of (3,5, the probablty of electng (Soccer,Soccer become greater than that of (Move,Move by preparng c 00 c a the entangled tate, where c c complex number atfyng c c and c c. 3. -Con Even-Odd Game Frt, we denote a zero-um game ung con between two player, ={Alce, }. Throughout th paper, uppoe that Alce contraned to play clacally,.e., Sa I X. Then, we how that nether Alce nor can wn the game wth certanty. Wth only clacal tratege but can wn t wth certanty wth our quantum trategy. ere, let and T Predct Predct even odd Alce even wn loe odd loe wn Fgure 4. -Con even-odd chec repreent the head and tal of a con, repectvely. -Con Even-Odd Chec: Frt, Alce prepare con and put them nto a box n the tate of all the con beng head,.e., (,,,. Suppoe that any player cannot ee the nde of the box. ext, flp ome con(or not, Alce flp ome con(or not, and flp ome con(or not. Fnally, they open the box. Alce wn f the number of odd; otherwe wn f the number of even. Becaue we can regard th problem a whether can predct Alce trategy, the payoff matrx can be alo hown a Fgure 4. It obvou that f they ue only clacal tratege, nether Alce nor can wn the game wth certanty. owever, f ue a quantum trategy, he can wn the game wth certanty. ow, we how a quantum trategy for th game wth whch wn wth certanty. Theorem 3. For -Con Even-Odd Chec, there ext a quantum trategy wth whch wn wth certanty. Proof. We denote and T by 0 and, repectvely. Th mean that Alce prepare Frt, execute the followng operaton. I ( COT( ( ext, Alce flp the con ung the operaton X and I, becaue he can only execute clacal tratege. Then, the tate become ( bb... b bb... b where b 0 (.... Fnally, by ung Lemma., obtan bc ( c c... c c 0 Thu, can obtan the bt c 0 c c c atfyng. Th mean that the number of even and he can wn the game wth certanty. We can prove the ame theorem by ung the Meyer quantum trategy [4] for con: Copyrght 009 ScRe B

5 46 A Study of Quantum Stratege for ewcomb Paradox ( ( 0 X or I(Alce (0 ( Therefore, we next denote a generalzed even-odd game uch that cannot wn wth certanty by ung only Meyer trategy but can wn wth certanty wth our trategy. -Con Even-Odd Chec(G: Frt, Alce prepare con and put them nto a box n the tate of (,D,, D, where D { T} ( Suppoe that any player cannot ee the nde of the box and that doe not now D. ext, flp ome con(or not, Alce flp ome con(or not, and flp ome con(or not. Fnally, they open the box. Alce wn f the number of odd; otherwe wn f the number of even. The payoff matrx of th problem can be hown a ame a Fgure 4. Alo n th cae, t obvou that f they ue clacal tratege, nether Alce nor can wn the game wth certanty. Moreover, becaue Meyer trategy ue the property of: X (0 (0, mut now the ntal tate of all the con n order to wn the game. owever, f he ue our quantum trategy, can wn the game wth certanty. ow, we how a quantum trategy for th game wth whch wn wth certanty. Theorem 3. For -Con Even-Odd Chec(G, there ext a quantum trategy wth whch wn wth certanty. Proof. Alo n th cae, we denote and T by 0 and, repectvely. Therefore, Alce prepare 0 d... d, where d 0 ( Frt, execute the followng operaton: I 0 d... d ( 0 d... d COT( ( 0 d... d d... d ext, Alce flp the con ung the operaton X and I. Then, the tate become ( bb... b bb... b Fnally, by ung Lemma., obtan c 0 bc ( c c... c Thu, can obtan the bt cc c atfyng c 0. Th mean that the number of even and he can wn the game wth certanty Con Flppng Game ext, we denote zero-um game between two player modfyng the game n the prevou ubecton and how that nether Alce nor can wn the game wth certanty wth clacal tratege but can wn them wth certanty wth our quantum trategy. -Con Flp: Frt, Alce prepare con and put them nto a box n the tate of all the con beng head,.e., (,,,. Suppoe that any player cannot ee the nde of the box. ext, flp ome con(or not, Alce flp m con( m 0... under eepng the value of m ecret from, and flp ome con(or not. Fnally, they open the box. Then, wn f all the con are n the followng tate: the tate of the con (,,, f m even, or the tate of the con (T,,, f m odd. Otherwe Alce wn. We how the payoff matrx n Fgure 5. We can regard alo th problem a whether can predct Alce trategy, It obvou that f they ue clacal tratege, nether Alce nor can wn the game wth certanty. Moreover, by the ame reaon n the prevou ubecton, cannot alo wn the game wth certanty even f Meyer trategy ued. owever, f ue our quantum trategy, he can wn the game wth certanty. ow, we how a quantum trategy for th game wth whch wn wth certanty. Theorem 3.3 For -Con Flp, there ext a quantum trategy wth whch wn wth certanty. Proof. Alce prepare Frt, execute the followng operaton. I ( COT( ( cc... c Alce c0 Predct even Predct odd even (,,, other tate odd other tate (T,,, Fgure 5. -Con Flp Copyrght 009 ScRe B

6 A Study of Quantum Stratege for ewcomb Paradox 47 Alce Predct even even (,D,, D Predct odd other tate odd other tate (T,D,, D Fgure 6. -Con Flp(G ext, Alce flp m con ung the operaton X. Then, the tate become c x c x c x, c 0 where x 0 (... and By ung Lemma., obtan x m. ( ( x... Moreover, he execute the followng operaton. ( ( x... COT( I ( 0 ( x x Then, x 0 f even; otherwe m x f m odd. Therefore, can wn the game wth certanty. ext, we denote a generalzed -con flppng game. -Con Flp(G: Frt, Alce prepare con and put them nto a box n the tate of (,D,, D, where D T ( Suppoe that any player cannot ee the nde of the box and that doe not now D. ext, flp ome con(or not, Alce flp m con( m0... under eepng the value of m ecret from, and flp ome con(or not. Fnally, they open the box. Then, wn f all the con are n the followng tate: the tate of the con (,D,, D f m even, or the tate of the con (T,D,, D f m odd. Otherwe Alce wn. We how the payoff matrx n Fgure 6. Alo n th cae, t obvou that f they ue clacal tratege, nether Alce nor can wn the game wth certanty and that cannot wn the game wth certanty even f Meyer trategy ued. owever, f can ue our quantum trategy, he wn the game wth certanty. If whe to now only whether m even or odd, we can ealy contruct the followng protocol. COT( ( 0 d... d d d... d X or I(Alce where d x 0 COT( ( d x d x d x x d x d x (.... ow, we how a quantum trategy for th game wth whch wn wth certanty. Theorem 3.4 For -Con Flp(G, there ext a quantum trategy wth whch wn wth certanty. Proof. Alce prepare 0 d... d, where d 0 ( Frt, execute the followng operaton. I 0 d... d ( 0 d... d COT( (0 d... d d... d dc ( cc... c c 0 where d (0 d... d and c ( cc... c. ext, Alce flp m con ung the operaton X. Then, the tate become dc ( c x c x... c x c 0 where x 0 (... and x m. By ung Lemma., obtan ( xd ( 0... ( d d x d... d where x ( x x... x followng operaton. ( xd ( 0... ( x d d d... d COT( ( xd ( 0 ( x d... d I. Moreover, he execute the xd ( x d... d Then, x 0 f m even; otherwe x f m odd. Therefore, can wn the game wth certanty. Fnally, we denote a game combnng the Even-Odd game and the Flppng game. Copyrght 009 ScRe B

7 48 A Study of Quantum Stratege for ewcomb Paradox Alce Predct even even 0 odd Fgure 7. Extended Predct odd 0 -Con Flp Extended -Con Flp: Frt, Alce prepare con and put them nto a box n the tate of all the con beng head,.e., (,,,. Suppoe that any player cannot ee the nde of the box. ext, flp ome con(or not, Alce flp m con ( m0 under eepng the value of m ecret from, and flp ome con(or not, where for ome potve even number C, 0 m0 m and mcn for any nteger n and m0 m a nd mc(n for any nteger n Fnally, they open the box. Then, wn f all the con are n the followng tate: the number of even f m an element n 0, or the number of odd f m an element n. Otherwe Alce wn. We how the payoff matrx n Fgure 7, and contruct a quantum trategy uch that wn wth certanty. Theorem 3.5 For Extended -Con Flp, there ext a quantum trategy wth whch wn wth certanty. Proof. Alce prepare Frt, execute the followng operaton: I ( COT( ( S( C ( C e... ext, Alce flp m con ung the operaton X. Then, the tate become ( π/c... bb b e bb... b where b 0 ( and.... b m apple S( C to the tate. ( C bbb e bb b ( S( C ( mc mc e bbb e bb b mc mc e ( bbb e bb b Moreover, by ung Lemma., obtan f mc e e ( mc bc c 0 ; otherwe he obtan e e ( mc bc c c c c c c c f mc. Thu, m an element n and 0 c 0 f mc e ; otherwe m an element n and c f e πm/c. Therefore, can wn the game wth certanty. When C(n odd, can alway wn wthout operatng to con. On the other hand, our trategy ucceed even f C(n even. 4. Applcaton to ewcomb Paradox In th ecton, we tudy ewcomb paradox (Free Wll problem and how ome quantum tratege for th problem. A quantum oluton of th problem hown by ung Meyer quantum trategy by Potrow and Sładow [8]. We tudy th problem by ung our reult n prevou ecton. A problem a follow: ewcomb problem: Let have the ablty to predct Alce wll. ow, prepare two boxe, Box and Box, and Alce can elect ether Box or both boxe. Box contan $. Box contan $,000 only f Alce elect only Box ; otherwe Box empty($0. Whch better for Alce? The payoff matrx hown n Fgure 8. The focu of th problem whether can really predct Alce wll, or whether can control Alce wll. Obvouly, Alce trategy electng both boxe f cannot predct Alce wll. ow, we modfy th problem a mplfed problem. In addton, we oberve only tratege on Box. ewcomb: Frt, Alce decde whether he elect ether Box or not, but cannot now her electon. Box contan $,000 only f Alce elect Box ; otherwe Box empty($0. Can let Alce elect her frt wll even f Alce change her wll after the frt electon? Theorem 4. For ewcomb, there ext a quantum trategy that can be potvely olved. Proof. Alce elect a tate 0 f he elect Box ; otherwe he elect a tate. ote that doe not now the tate. To th tate, apple, Alce apple X f he change her wll, and apple. F- Alce Predct Both Predct Box Both $ $,00 Box $0 $,000 Fgure 8. ewcomb problem Copyrght 009 ScRe B

8 A Study of Quantum Stratege for ewcomb Paradox 49 Alce Swerve Drve traght Swerve,, 3 Drve traght 3, 0, 0 Fgure 9. Chcen nally, the tate 0 f her frt electon Box ; otherwe t. Then, can let Alce elect her frt wll. Th trategy ue Meyer trategy. Moreover, we can alo how other proof by ung Theorem 3.3 or Theorem 3.4. For ewcomb, doe not permt Alce change. ext, by ung -Con Flp(G, we modfy th problem to problem uch that permt Alce change. ewcomb: Alce prepare con and put them nto a box n the tate of (,D,, D, where D T (,,,. ext, flp ome con(or not, Alce flp m con( m0... under eepng the value of m ecret from, and flp ome con(or not, where let Alce do not change her wll f m even; otherwe let he change her wll. Can now whether m even or odd? Theorem 4. For ewcomb, there ext a quantum trategy that can be potvely olved. Proof. Th reult mmedately obtaned by Theorem 3.4. We can regard th problem a ewcomb problem when the value of m her wll,.e., he elect Box when m even; otherwe he elect both boxe. Thu, can predct Alce wll. We can alo modfy ewcomb problem by ung -Con Flp and Extended -Con Flp. ext, we denote that we can alo modfy Alce wll n ewcomb to -player wll. -ewcomb: Frt, player, A A A, decde whether they elect Box or not. They prepare con and put them nto a box n the tate of (,D,, D, where D T ( 3..., and let player A (... deal wth the -th con. ow, flp ome con(or not. ext, each A (... flp h/her con(or not, where let the number of -player flp be m ( m0..., and let they do not change ther wll f m even; otherwe let they change ther wll. Fnally, flp ome con(or not. Can now whether m even or odd? The quantum trategy for ame a ewcomb. Fnally, we tudy the Chcen game. The payoff matrx hown n Fgure 9. In the Chcen game, the ah equlbrum pont are (Swerve, Drve traght and (Drve traght, Swerve. By clacal tratege, the probablty that each player elect ether Swerve or Drve traght /. Th mean that ether (Swerve, Swerve or (Drve traght, Drve traght may be elected. On the other hand, by our quantum tratege, ether (Swerve, Drve traght and (Drve traght, Swerve can be elected wth certanty becaue can predct Alce wll. 5. Concluon In th paper, we propoed quantum tratege wth entanglement for -con flppng game. Thee game are mult-qubt varaton of the quantum trategy by Meyer [4]. One player contraned to play clacally but the other player can ue a quantum trategy. We then howed that by ung a technque n quantum communcaton complexty theory, our quantum tratege have an advantage over clacal one. For a player ung the quantum trategy, the entanglement ued n order to obtan the nformaton of the enemy and the player can alway wn the game. Moreover, by rewrtng ewcomb paradox ncludng mult-player wll, we alo howed that we can ue our reult a t quantum tratege and that a player can have an ablty to predct another player wll. Can alway wn our game even f both player ue quantum tratege? Th anwer o. Meyer alo howed that Q cannot alway wn f Pcard can alo ue a quantum trategy [4]. In the ame reaon, cannot alway wn our game f Alce can alo ue a quantum trategy. Therefore, t an nteretng queton whether can alway wn the game n the cae when both player are contraned to only execute ome retrcted quantum operaton. REFERECES [] P. W. Shor, Polynomal-tme algorthm for prme factorzaton and dcrete logarthm on a quantum computer, SIAM Journal of Computng, Vol. 6, pp , 997. [] L. K. Grover, A fat quantum mechancal algorthm for databae earch, Proceedng of the 8th ACM Sympoum on Theory of Computng, pp. 9, 996. [3] J. von eumann and O. Morgentern, Theory of game and economc behavor, thrd edton, Prnceton Unverty Pre, Prnceton, 953. [4] D. A. Meyer, Quantum tratege, Phycal Revew Letter, Vol. 8, pp , 999. [5] J. Eert, M. Wlen, and M. Lewenten, Quantum game and quantum tratege, Phycal Revew Letter, Vol. 83, pp , 999. [6] J. Du,. L, X. Xu, M. Sh, J. Wu, X. Zhou, and R. an, Expermental realzaton of quantum game on a quan- Copyrght 009 ScRe B

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