Strong isomorphism in Eisert-Wilkens-Lewenstein type quantum games

0.11 MB·

by  Piotr Frackiewicz

Strong isomorphism in Eisert-Wilkens-Lewenstein type quantum
games
PiotrFra¸ckiewicz
InstituteofMathematics
PomeranianUniversity,Poland
January23, 2017
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1 Abstract
0
2 Theaimofthispaperistobringtogetherthenotionsofquantumgameandgameisomorphism.Theworkisintended
as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a
n
quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two
a
J isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are
concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in
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strategicform.
1

]
T 1 Introduction
G
. Sixteenyearsofresearchonquantumgameshavegivenusmanyideasofhowquantumgamescouldbedescribed. For
s
c example, we have learned from [1] that players who are allowed to use some specific unitary operators may gain an
[ advantageovertheplayerswhouseonlyclassicalstrategies. Theschemesintroducedin[2]and[3]giveustwodifferent

ways of describing quantum2 2 games. The paper [4], in turn, providesus with a quantumscheme for the Cournot
1
×
v duopolygame.Whatconnectstheseprotocolsisthecapabilitytoobtaintheclassicalgame.Thisappearstobeagenerally
3 acceptednecessaryconditionimposedonaquantumscheme. Onecanalsofindotherandmoresubjectiveguidelinesfor
3 quantumgameschemes.Paper[5]showshowtogeneralizetheschemeintroducedin[3]byassumingthatthenewmodel
6
should output the classical game (up to the order of players’ strategies) if the initial state is one of the computational
5
basis states. In addition, the work of Bleiler [6] distinguishes between proper and more strict complete quantization.
0
. Roughlyspeaking,thefirstnotionconcernsquantumschemeswherethecounterpartsofclassicalpurestrategiescanbe
1
found in pure quantum strategies. The second one requires the quantumstrategy set to include the counterpartsof the
0
mixedclassicalstrategies. With thesenotionstheMarinatto-Weber(MW) [3] schemeturnsoutto benotevena proper
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1 quantization. TheEisert-Wilkens-Lewenstein(EWL)[2]scheme,inturn,isacompletequantization,andthisisthecase
: as long asthe players’ quantumstrategiesincludethe one-parameterunitaryoperatorsU(θ,0,0)(see formula(11)). In
v
i particular,wecanfindalotofpaperswheretheEWLschemewasstudiedwiththetwo-parameterunitarystrategies[7],
X
[8],[9],[10]. However,asitwasnotedin[11]theset U(θ,α,0) appearsnottoreflectanyreasonablephysicalconstraint
{ }
r asthissetisnotclosedundercomposition.Moreover,[12]showedthatdifferenttwo-parameterstrategyspacesintheEWL
a
schemeimplydifferentsetsofNashequilibria. Inthispaperweexplainwhythesetoftwo-parameterunitaryoperators
may not be reasonablefrom the game theory viewpoint. Our criterion is formulatedin terms of isomorphicgames. If
we assume thatboth classical gamesare the same with respectto gametheoreticaltools, we requirethe corresponding
quantumgamesto beequivalentin the sameway. Itis worthnotingthatquantumgameschemesintroducedin [2], [3]
and the refined MW scheme defined in [13] preserve the so-called strategic equivalence. We recall what this means.
The following definition can be found in [14]. See the Preliminary section for the definition of strategic form game
(N,(S ) ,(u) )anditscomponents.
i i N i i N
∈ ∈
Definition1 Two games in strategic form (N,(S ) ,(u) ) and (N,(S ) ,(v) ) with the same set of players and
i i N i i N i i N i i N
∈ ∈ ∈ ∈
the same sets of pure strategies are strategically equivalentif for each player i N the function v is a positive affine
i
transformationofthefunctionu. Inotherwords,thereexistα >0andβ Rsuc∈hthat
i i i
∈
v(s)=αu(s)+β, foreach s S . (1)
i i i i i
∈
Yi N
∈
ItisclearthatgiventwostrategicallyequivalentgamesΓandΓ,theplayeri’spayoffoperatorsM andM inthequantum
′ i i′
gamesareconnectedbyequation M = αM +β. Then,bylinearityoftrace,thequantumpayofffunctionssatisfy(1),
i′ i i i
1
i.e., tr(ρ M ) = αtr(ρ M)+β. Virtually,strategicallyequivalentgamesΓandΓ describethesamegame-theoretical
fin i′ i fin i i ′
problem.Inparticular,everyequilibrium(pureormixed)ofthegameΓisanequilibriumofthegameΓ.
′
The strategy equivalence can be extended to take into account different orders of players’ strategies. This type of
equivalenceisincludedinthedefinitionofstrongisomorphism. Clearly,ifforexample,twobimatrixgamesdifferonly
in the order of a player’s strategies we still have the games that describe the same problem from the game theoretical
viewpoint.Givenaquantumscheme,itappearsreasonabletoassumethattheresultingquantumgamewillnotdependon
thenumberingofplayers’strategiesintheclassicalgame. Asaresult,ifthereisastrongisomorphismbetweengames,
werequirethatthequantumcounterpartsofthesegamesarealsoisomorphic.
2 Preliminaries
In order to make our paper self-contained we give the important preliminaries from game theory and quantum game
theory.
2.1 Strong isomorphism
Firstwerecallthedefinitionofstrategicformgame[14].
Definition2 AgameinstrategicformisatripleΓ=(N,(S ) ,(u) )inwhich
i i N i i N
∈ ∈
N = 1,2,...,n isafinitesetofplayers.
• { }
S isthesetofstrategiesofplayeri,foreachplayeri N.
i
• ∈
u : S S S Risafunctionassociatingeachvectorofstrategies s = (s) withthepayoffu(s)to
i 1 2 n i i N i
• × ×···× → ∈
playeri,foreveryplayeri N.
∈
Thenotionofstrongisomorphismdefinesclassesofgamesthatarethesameuptonumberingoftheplayersandtheorder
ofplayers’strategies. Thefollowingdefinitionsaretakenfrom[15](seealso[16],[17]and[18]). Thefirstonedefinesa
mappingthatassociatesplayersandtheiractionsinonegamewithplayersandtheiractionsintheothergame.
Definition3 Given Γ = (N,(S ) ,(u) ) and Γ = (N,(S ) ,(u) ), a game mapping f from Γ to Γ is a tuple
i i N i i N ′ i′ i N ′i i N ′
f =(η,(ϕ) )whereηisabijecti∈onfrom∈N toN andforanyi ∈N,ϕ i∈sabijectionfromS toS .
i i N i i η(i)
∈ ∈
Example1 Letusconsidertwobimatrixgames
l r l r
′ ′
t (a ,b ) (a ,b ) t (a ,b ) (a ,b )
00 00 01 01 and ′ ′00 ′00 ′01 ′01 . (2)
b (a10,b10) (a11,b11)! b′ (a′10,b′10) (a′11,b′11)!
Then, N = 1,2 and S = t,b , S = l,r , S = t ,b , S = l,r . As an example of a game mapping let
{ } 1 { } 2 { } 1′ {′ ′} 2′ {′ ′}
f =(η,ϕ ,ϕ ),
1 2
η=(1 2,2 1), ϕ =(t l,b r ), ϕ =(l b,r t ). (3)
1 ′ ′ 2 ′ ′
→ → → → → →
Sinceϕ : S S andϕ : S S ,itfollowsthat f maps(s ,s ) S S to(ϕ (s ),ϕ (s )). From(3)weconclude
1 1 → 2′ 2 2 → 1′ 1 2 ∈ 1× 2 2 2 1 1
that
f =((t,l) (b,l),(t,r) (t,l),(b,l) (b,r ),(b,r) (t,r )). (4)
′ ′ ′ ′ ′ ′ ′ ′
→ → → →
In general case, mapping f from (N,(S ) ,(u) ) to (N,(S ) ,(u) ) identifies player i N with player η(i) and
i i∈N i i∈N i′ i∈N ′i i∈N ∈
maps S to S . This means that strategy profile (s ,...,s ) S S is mapped into profile (s ,...,s ) that
i η(i) 1 n ∈ 1 × ···× n ′1 ′n
satisfiesequations =ϕ(s)fori N.
′η(i) i i ∈
Thenotionofgamemappingisabasisfordefinitionofgameisomorphism. Dependingonhowrichstructureofthe
gameisto bepreservedwe candistinguishvarioustypesofgameisomorphism. Onethatpreservesthe players’payoff
functionsiscalledthestrongisomorphism.Theformaldefinitionisasfollows:
Definition4 Given two strategic games Γ = (N,(S ) ,(u) ) and Γ = (N,(S ) ,(u) ), a game mapping f =
i i N i i N ′ i′ i N ′i i N
(η,(ϕ) ) is called a strong isomorphism if relation u∈(s) = u∈ (f(s)) holds for eac∈h i N∈ and each strategy profile
i i∈N i ′η(i) ∈
s S S .
1 n
∈ ×···×
FromtheabovedefinitionitmaybeconcludedthatifthereisastrongisomorphismbetweengamesΓandΓ,theymay
′
differmerelybythenumberingofplayersandtheorderoftheirstrategies.
2
Example2 Let f beagamemappingdefinedinExample1. Bydefinition, f becomesthestrongisomorphismifcondition
u(s)=u (f(s))isimposedonthepayoffsin(2). Thisgives
i ′η(i)
a =b , a =b , a =b , a =b ,
00 ′10 01 ′00 10 ′11 11 ′01 (5)
b =a , b =a , b =a , b =a ,
00 ′10 01 ′00 10 ′11 11 ′01
where, for instance, a = b follows from equation u ((t,r)) = u ((t,l)). Substituting (5) into (2) we conclude that
01 ′00 1 ′2 ′ ′
games
l r l r
′ ′
t (a ,b ) (a ,b ) t (b ,a ) (b ,a )
00 00 01 01 and ′ 01 01 11 11 . (6)
b (a10,b10) (a11,b11)! b′ (b00,a00) (b10,a10)!
areisomorphic. Inthiscase,thegamesdifferbythenumberingofplayersandtheorderofstrategiesofplayer2. Indeed,
inthesecondgameof(6)player1and2choosenowbetweencolumnsandrows,respectively. Moreover,player1’sfirst
(second)strategystillguaranteesthepayoffa ora (a ora )whereasplayer2’sstrategiesareinterchanged:thefirst
00 01 10 11
oneimpliesnowthepayoffb orb .
01 11
RelabelingplayersortheirstrategiesdoesnotaffectagamewithregardtoNashequilibria. If f isastrongisomorphism
between games Γ and Γ, one may expect that the Nash equilibria in Γ map to ones in Γ under f. We will prove the
′ ′
followinglemmaasitisneededthroughoutthepaper.
Lemma1 Let f beastrongisomorphismbetweengamesΓandΓ. Strategyprofiles =(s ,...,s ) S S isa
′ ∗ ∗1 ∗n ∈ 1×···× n
NashequilibriumingameΓifandonlyif f(s ) S S isaNashequilibriuminΓ.
∗ ∈ 1′ ×···× n′ ′
Proof Theproofisbasedonthefollowingobservation.Since f((s ,...,s ))=(s ,...,s )wheres =ϕ(s),itfollows
1 n ′1 ′n ′η(i) i i
that f((s,s ))maybewrittenas s ,s . As f isanisomorphism,wehaveu(s)=u (f(s))foreachstrategyprofile
i i ′η(i) ′η(i) i ′η(i)
s. Thus − (cid:16) − (cid:17)
ui(s∗)=u′η(i) s∗1′,...,s∗n′ (7)
(cid:16) (cid:17)
and
ui si,s∗i =u′η(i) f si,s∗i =u′η(i) s′η(i),s∗′η(i) . (8)
Thisallowsustoconcludethattheineq(cid:0)ualit−y(cid:1) (cid:0) (cid:0) −(cid:1)(cid:1) (cid:16) − (cid:17)
u(s ) u(s,s ) (9)
i ∗ ≥ i i ∗−i
holdsforeachi N andeachstrategys S ifandonlyif
i i
∈ ∈
u′η(i)(cid:16)s∗1′,...,s∗n′(cid:17)≥u′η(i)(cid:16)s′π(i),s∗−′η(i)(cid:17) (10)
foreachη(i) N andeachstrategys S . Thisfinishestheproof. (cid:4)
∈ ′i ∈ i′
2.2 Eisert-Wilkens-Lewenstein scheme
LetusconsiderastrategicgameΓ=(N,(S ) ,(u) )withS = si,si foreachi N. ThegeneralizedEisert-Wilkens-
LewensteinapproachtogameΓisdefinedibyi∈Ntriplei iΓ∈N =(Ni,(Dn)0 ,1(oM) ),wh∈ere
EWL i i N i i N
∈ ∈
D isasetofunitaryoperatorsfromSU(2).ThecommonlyusedparametrizationforU SU(2)isgivenby
i
• ∈
eiαcosθ ieiβsin θ
U(θ,α,β)= 2 2 , θ [0,π], α,β [0,2π). (11)
ie−iβsin2θ e−iαcos2θ ! ∈ ∈
Then D isassumedtoincludeset U(θ,0,0): θ [0,π] . ElementsU D playtheroleofplayeri’sstrategies.
i i i
{ ∈ } ∈
Theplayers,bychoosingU D,determinethefinalstate Ψ accordingtothefollowingformula:
i i
∈ | i
n
1
Ψ = J U(θ,α,β) J0 n where J = 1 n+iσ n (12)
(1istheidentitymatrixof|siize2a†ndOi=σ1 isithiePiauliima|triix⊗ X). √2(cid:16) ⊗ ⊗x (cid:17)
x
M isanobservabledefinedbytheformula
i
•
M = ai j ...j j ... j . (13)
i j1...jn| 1 nih 1 n|
j1,...X,jn∈{0,1}
Thenumbersai aretheplayeri’spayoffsinΓsuchthatai =u si ,...,si . Theplayeri’spayoffu inΓ
j1...jn j1...jn i j1 jn i EWL
isdefinedastheaveragevalueofmeasurementM,i.e., (cid:16) (cid:17)
i
n
u U(θ,α,β) ≔ ΨM Ψ . (14)
i i i i i  h | i| i
Oi=1 
3
3 Strong isomorphism in Eisert-Wilkens-Lewenstein quantum games
HavingspecifiedthenotionofstrongisomorphismandthegeneralizedEisert-Wilkens-Lewensteinschemewewillnow
checkiftheisomorphismbetweentheclassicallyplayedgamesmakesthecorrespondingquantumgamesisomorphic.We
firstexaminethecasewhentheplayers’unitarystrategiesdependontwoparameters.ThequantumgameΓ with
EWL
D = U(θ,α,0): θ [0,π],α [0,2π) (15)
i i i i i i
{ ∈ ∈ }
isparticularlyinterested. ThatsettingwasusedtointroducetheEWLscheme[2]andhasbeenwidelystudiedinrecent
years(see,forexample,[7],[8],[9],[10]). WebeginwithanexampleofisomorphicgamesthatdescribethePrisoner’s
Dilemma.
Example3 ThegeneralizedPrisoner’sDilemmagameandoneofitsisomorphiccounterpartsmaybegivenbythefol-
lowingbimatrices:
l r l r
′ ′
t (R,R) (S,T) t (S,T) (R,R)
Γ: and Γ : ′ , (16)
b (T,S) (P,P)! ′ b′ (P,P) (T,S)!
whereT > R > P > S. Notethatthegamesarethesameuptotheorderofplayer2’strategies. Letusnowexaminethe
EWLapproachtoΓandΓ definedbytriples
′
Γ = 1,2 ,( U(θ,α,0 : ) ,(M) ,
EWL i i i i 1,2 i i 1,2
{ } { } ∈{ } ∈{ } (17)
Γ =(cid:0) 1,2 ,( U (θ,α,0 ) ,(M ) (cid:1),
′EWL { } { i′ i′ ′i } i∈{1,2} i′ i∈{1,2}
(cid:0) (cid:1)
where
(M ,M )=(R,R)00 00 +(S,T)01 01 +(T,S)10 10 +(P,P)11 11,
1 2
| ih | | ih | | ih | | ih | (18)
(M ,M )=(S,T)00 00 +(R,R)01 01 +(P,P)10 10 +(T,S)11 11.
1′ 2′ | ih | | ih | | ih | | ih |
WefirstcomparethesetsofNashequilibriainΓ andΓ tocheckifthegamesmaybeisomorphic. Werecallfrom
EWL ′EWL
[2]thatthereistheuniqueNashequilibriumU (0,π/2,0) U (0,π/2,0)inΓ thatdeterminesthepayoffprofile(R,R).
1 2 EWL
⊗
WhenitcomestoΓ ,wesetn=2in(12)andreplace(13)byM andM from(18). Thenwecanrewrite(14)as
′EWL 1′ 2′
(u ,u ) U (θ ,α ,0) U (θ ,α ,0)
′1 ′2 1 1′ ′1 ⊗ 2 2′ ′2
(cid:0) θ (cid:1)θ 2
=(S,T) cos(α +α )cos 1′ cos 2′
′1 ′2 2 2!
θ θ θ θ 2
+(R,R) cosα cos 1′ sin 2′ +sinα sin 1′ cos 2′
′1 2 2 ′2 2 2!
θ θ θ θ 2
+(P,P) sinα cos 1′ sin 2′ +cosα sin 1′ cos 2′
′1 2 2 ′2 2 2!
θ θ θ θ 2
+(T,S) sin(α +α )cos 1′ cos 2′ sin 1′ sin 2′ . (19)
′1 ′2 2 2 − 2 2!
LetU (θ ,α ,0)beanarbitrarybutfixedstrategyofplayer2.Thenitfollowsfrom(19)thatstrategyU (θ ,α ,0)specified
2′ 2′ ′2 1′ 1′ ′1
byequation
U (θ ,3π/2 α ,0) ifα [0,3π/2],
U1′(θ1′,α′1,0)=U1′(θ2′,7π/2−α′2,0) ifα′2 ∈(3π/2,2π), (20)
 1′ 2′ − ′2 ′2 ∈
is player 1’s best reply to U2′(θ2′,α′2,0) as it yields player 1 the payoff T. Hence, a possible Nash equilibrium would
generatethemaximalpayoffforplayer1. Ontheotherhand,givenafixedplayer1’sstrategyU (θ ,α ,0),player2can
1′ 1′ ′1
obtainapayoffthatisstrictlyhigherthanS bychoosing,forexample,U (θ ,α ,0)withθ =0,α =2π α . Thismeans
2 2′ ′2 2′ ′2 − ′1
thattheplayer1wouldobtainstrictlylessthanT. Hence,thereisnopureNashequilibriuminthegamedeterminedby
Γ . Asaresult,wecanconcludebyLemma1thatgames(17)arenotstronglyisomorphic.
′EWL
TheexamplegivenaboveshowsthattheEWL approachwith thetwo-parameterunitarystrategiesmayoutputdifferent
Nashequilibriadependingontheorderofplayers’strategiesintheclassicalgame. Thisappearstobea strangefeature
sincegames(16)representthesamedecisionproblemfromagame-theoreticalpointofview.
Onewaytomakegames(17)isomorphicistoreplaceplayeri’sstrategyset(15)withthealternativetwo-parameter
strategyspace
F = U(θ,0,β): θ [0,π],β [0,2π) (21)
i i i i i i
{ ∈ ∈ }
every time player i’s strategies are switched in the classical game. In the case of games (17) this means that quantum
games
Γ = 1,2 ,(D ,D ),(M) , Γ = 1,2 ,(D ,F ),(M ) (22)
EWL { } 1 2 i i∈{1,2} ′EWL { } ′1 2′ i′ i∈{1,2}
(cid:0) (cid:1) (cid:0) (cid:1)
4
are isomorphic. Indeed, define a gamemap f˜ = (η,ϕ˜ ,ϕ˜ ) with η(i) = i for i = 1,2 and bijectionsϕ˜ : D D and
1 2 1 1 → ′1
ϕ˜ : D F satisfying
2 2 → 2′
ϕ˜ (U (θ ,α ,0))=U (θ ,α ,0), ϕ˜ (U (θ ,α ,0))=U (π θ ,0,π α ). (23)
1 1 1 1 1′ 1 1 2 2 1 1 2′ − 2 − 2
Themapϕ˜ shouldactuallydistinguishcasesα [0,π)andα [π,2π)tobeawell-definedbijectionasitwasdonein
2 2 2
∈ ∈
equation(20).Tosimplifytheproofwesticktotheform(23)throughoutthepaperbearinginmindthatforπ α <[0,2π)
2
−
wecanalwaysfindtheequivalentangle3π α [0,2π).Wehavetoshowforgames(22)that
2
− ∈
u(U U )= ΨM Ψ = Ψ M Ψ =u(f˜(U U )) (24)
i 1⊗ 2 h | i| i h ′| i′| ′i ′i 1⊗ 2
for i = 1,2, where Ψ = J (U U )J00 and Ψ = J (f˜(U U ))J00 . First, note that U (π θ ,0,π α ) =
| i † 1 ⊗ 2 | i | ′i † 1 ⊗ 2 | i 2′ − 2 − 2
iσ U (θ ,α ,0). Hence,weobtain
x 2′ 2 2
Ψ = J f˜(U (θ ,α ,0) U (θ ,α ,0))J00
′ † 1 1 1 2 2 2
| i ⊗ | i
= J (U (θ ,α ,0) U (π θ ,0,π α ))J00
† 1′ 1 1 ⊗ 2 − 2 − 2 | i
=(1 ( iσ ))J (U (θ ,α ,0) U (θ ,α ,0))J00
⊗ − x † 1′ 1 1 ⊗ 2′ 2 2 | i
=(1 ( iσ ))Ψ . (25)
x
⊗ − | i
Applicationof(25)finallyyields
Ψ M Ψ = Ψ(1 iσ )M (1 ( iσ ))Ψ = ΨM Ψ . (26)
h ′| i′| ′i h | ⊗ x i′ ⊗ − x | i h | 1| i
In similar way we can provea more generalfact. Namely, if F is player2’s strategyset in one of games(17) and D
2 2
isinthetheotheronethengames(17)becomestronglyisomorphic. ThisobservationsuggeststhattheEWLschemeis
robustwithrespecttochangingtheorderofplayers’strategiesintheclassicalgameiftheplayerscanusestrategiesfrom
D F,orequivalentlyfromthesetSU(2).Beforestatingthegeneralresultwestudyaspecificexample.
i i
∪
Example4 Letusconsiderthefollowingthree-persongames:
l r l r
t (a ,b ,c ) (a ,b ,c ) t (a ,b ,c ) (a ,b ,c )
v 000 000 000 010 010 010 w 001 001 001 011 011 011
b (a100,b100,c100) (a110,b110,c110)! b (a101,b101,c101) (a111,b111,c111)!
and
l r l r
′ ′ ′ ′
t (a ,b ,c ) (a ,b ,c ) t (a ,b ,c ) (a ,b ,c )
v ′ 000 000 000 010 010 010 w ′ 001 001 001 011 011 011 .
′ b′ (a100,b100,c100) (a110,b110,c110)! ′ b′ (a101,b101,c101) (a111,b111,c111)!
Thegamesare(strongly)isomorphicviagamemapping f =(η,ϕ ,ϕ ,ϕ )suchthat
1 2 3
η=(1 2,2 3,3 1),
→ → → (27)
ϕ =(t l,b r ), ϕ =(l w,r v), ϕ =(v b,w t ).
1 ′ ′ 2 ′ ′ 3 ′ ′
→ → → → → →
Weseefrom(27)thattheisomorphismmapsstrategyprofilesasfollows:
f(t,l,v)=(b,l,w), f(t,r,v)=(b,l,v), f(b,l,v)=(b,r ,w),
′ ′ ′ ′ ′ ′ ′ ′ ′
f(b,r,v)=(b,r ,v), f(t,l,w)=(t,l,w), f(t,r,w)=(t ,l,v) (28)
′ ′ ′ ′ ′ ′ ′ ′ ′
f(b,l,w)=(t,r ,w), f(b,r,w)=(t,r ,v).
′ ′ ′ ′ ′ ′
Let us now define the EWL quantum extensions Γ and Γ for the three-player game where we identify the
EWL ′EWL
players’firstandsecondstrategieswithvalues0and1,respectively.Thatis,
Γ =(N,(D) ,(M) ) and Γ =(N,(D) ,(M ) ) (29)
EWL i i N i i N ′EWL ′i i N i′ i N
∈ ∈ ∈ ∈
whereN = 1,2,3 ,D = D =SU(2)foreachi N,
{ } i ′i ∈
(M ,M ,M )= (a ,b ,c )P , (30)
1 2 3 j1j2j3 j1j2j3 j1j2j3 j1j2j3
j1,jX2,j3=0,1
(M ,M ,M )= (c ,a ,b )P , (31)
1′ 2′ 3′ j1j2j3 j1j2j3 j1j2j3 f(j1j2j3)
j1,jX2,j3=0,1
5
where P = j j j j j j . Given f = (η,(ϕ) )letusdefineamapping f˜= (η,(ϕ˜) )suchthatϕ˜ : D D
j1j2j3 | 1 2 3ih 1 2 3| i i∈N i i∈N i i → ′η(i)
fori N and
∈
ϕ˜ (U (θ ,α ,β ))=U (θ ,α ,β ),
1 1 1 1 1 2′ 1 1 1
ϕ˜ (U (θ ,α ,β ))=U (π θ ,2π β ,π α ), (32)
2 2 2 2 2 3′ − 2 − 2 − 2
ϕ˜ (U (θ ,α ,β ))=U (π θ ,2π β ,π α ).
3 3 3 3 3 1′ − 3 − 3 − 3
Then, f˜inducesabijectionfromD D D toD D D suchthat
1⊗ 2⊗ 3 ′1⊗ ′2⊗ ′3
f˜(U U U )=(ϕ˜ (U (θ ,α ,β )),ϕ˜ (U (θ ,α ,β )) ϕ˜ (U (θ ,α ,β )))
1 2 3 3 3 3 3 3 1 1 1 1 1 2 2 2 2 2
⊗ ⊗ ⊗
=U (π θ ,2π β ,π α ) U (θ ,α ,β ) U (π θ ,2π β ,π α ).
1′ − 3 − 3 − 3 ⊗ 2′ 1 1 1 ⊗ 3′ − 2 − 2 − 2
AccordingtotheEWLscheme,thepayofffunctionsforΓ andΓ areasfollows:
EWL ′EWL
u(U U U )= ΨM Ψ , where Ψ = J (U U U )J000
i 1 2 3 i † 1 2 3
⊗ ⊗ h | | i | i ⊗ ⊗ | i (33)
u(U U U )= Ψ M Ψ , where Ψ = J (U U U )J000
′i 1′ ⊗ 2′ ⊗ 3′ h ′| i′| ′i | ′i † 1′ ⊗ 2′ ⊗ 3′ | i
fori N. InordertoprovethatΓ andΓ areisomorphicwehavetocheckif
∈ EWL ′EWL
u(U U U )=u (f˜(U U U ))fori N. (34)
i 1⊗ 2⊗ 3 ′η(i) 1⊗ 2⊗ 3 ∈
Withoutlossofgeneralitywecanassumethati=1. Letusfirstevaluatestate Ψ ,
′
| i
Ψ = J U (π θ ,2π β ,π α ) U (θ ,α ,β ) U (π θ ,2π β ,π α ) J000 . (35)
| ′i † 1′ − 3 − 3 − 3 ⊗ 2′ 1 1 1 ⊗ 3′ − 2 − 2 − 2 | i
(cid:16) (cid:17)
Notethat
U (π θ ,2π β ,π α ) U (θ ,α ,β ) U (π θ ,2π β ,π α )
1′ − 3 − 3 − 3 ⊗ 2′ 1 1 1 ⊗ 3′ − 2 − 2 − 2
=( σ 1 σ )(U (θ ,α ,β ) U (θ ,α ,β ) U (θ ,β ,α )) (36)
− x⊗ ⊗ x 1′ 3 3 3 ⊗ 2′ 1 1 1 ⊗ 3′ 2 2 2
and
U (θ ,α ,β ) U (θ ,α ,β ) U (θ ,α ,β )
1′ 3 3 3 ⊗ 2′ 1 1 1 ⊗ 3′ 2 2 2
=S U (θ ,α ,β ) U (θ ,α ,β ) U (θ ,α ,β ) S , (37)
η 2′ 1 1 1 ⊗ 3′ 2 2 2 ⊗ 1′ 3 3 3 η†
(cid:16) (cid:17)
whereS isapermutationmatrixthatchangestheorderofqubitsaccordingtoη,
η
S = 000 000 + 001 010 + 010 100 + 011 110
η
| ih | | ih | | ih | | ih |
+ 100 001 + 101 011 + 110 101 + 111 111. (38)
| ih | | ih | | ih | | ih |
Using (36), (37), the fact that [J†, σx 1 σx] = [J†,Sη] = [J,Sη] = 0 and Sη†000 = 000 we may write Ψ′ as
− ⊗ ⊗ | i | i | i
follows:
Ψ = (σ 1 σ )S J U (θ ,α ,β ) U (θ ,α ,β ) U (θ ,α ,β ) J000
| ′i − x⊗ ⊗ x η † 2′ 1 1 1 ⊗ 3′ 2 2 2 ⊗ 1′ 3 3 3 | i
= (σ 1 σ )S Ψ(cid:16) (cid:17) (39)
x x η
− ⊗ ⊗ | i
Note that j1j2j3Sη = Sη† j1j2j3 †. This means that Sη is the inverse operation when acting on dual vectors. This
h | | i
observationtogetherwith(cid:16) thefactth(cid:17)at f changesthestrategyorderforplayer1and3leadustoconclusionthatoperator
(σ 1 σ )S canbeviewedas f 1 inthesenseofthefollowingequality:
x x η −
± ⊗ ⊗
j j j ( σ 1 σ )S = f 1(j j j ). (40)
1 2 3 x x η − 1 2 3
|h | − ⊗ ⊗ | |h |
Letusnowconsiderterm Ψ P Ψ for Ψ givenby(35). From(39)and(40)itfollowsthat
h ′| f(j1j2j3)| ′i | ′i
Ψ P Ψ = f(j j j )Ψ 2 = f(j j j )(σ 1 σ )S Ψ 2
h ′| f(j1j2j3)| ′i |h 1 2 3 | ′i| |h 1 2 3 | x⊗ ⊗ x η| i|
= j j j Ψ 2 = ΨP Ψ . (41)
|h 1 2 3| i| h | j1j2j3| i
Hence,
u (f˜(U U U ))= Ψ M Ψ = ΨM Ψ =u (U U U ). (42)
η(1) 1⊗ 2⊗ 3 h ′| η′(1)| ′i h | 1| i 1 1⊗ 2⊗ 3
Similarreasoningappliestothecasei=2,3. Wehavethusprovedthatgamesgivenby(29)areisomorphic.
ThesameconclusioncanbedrawnforgameswitharbitrarybutfinitenumberN ofplayers.
6
Proposition1 LetΓ=(N,(S ) ,(u) )andΓ =(N,(S ) ,(u) )bestronglyisomorphicstrategicformgameswith
i i N i i N ′ i′ i N ′i i N
S = S = 2 and let Γ =∈ (N,(D∈) ,(M) ) and Γ∈ =∈(N,(D ) ,(M ) ) with D = D = SU(2) be the
c|oir|resp|oni′|dingquantumgaEmWLes.ThenΓ i i∈NandΓi i∈N arestr′oEWngLlyisomorp′ihii∈cN. i′ i∈N i ′i
EWL ′EWL
Proof The prooffollowsby the same methodas in Example4. Let f = (η,(ϕ) ) be a strongisomorphismbetween
i i N
Γ and Γ. Depending on ϕ : S S such that ϕ(si) = sη(i) for A = si,si∈and A = sη(i),sη(i) we construct
′ i i → η′(i) i k l i { 0 1} η(i) { 0 1 }
f˜=(η,(ϕ˜) )where
i i N
∈ U (θ,α,β) ifϕ si = sη(i)
ϕ˜(U(θ,α,β))= η′(i) i i i i k k (43)
i i i i i U (π θ,2π β,π α) ifϕ (cid:16)si(cid:17)= sη(i) .
Then f˜ N U = N U ,whereU =ϕ˜(Uη′(i))for−i=i 1,.−..Ni. S−inceiηisapie(cid:16)rmk(cid:17)utatiko⊕n21andU(π θ,2π β,π α)=
i=1 i i=1 i′ η′(i) i i − − −
iσxU((cid:16)θN,α,β),w(cid:17)ecaNnwriterelation(43)as
−
U (θ ,α ,β ) ifϕ si = sη(i)
ϕ˜ (U (θ ,α ,β ))= i′ η−1(i) η−1(i) η−1(i) i k k (44)
η−1(i) η−1(i) η−1(i) η−1(i) η−1(i)  iσ U (θ ,α ,β ) ifϕ (cid:16)si(cid:17)= sη(i) .
Asaresult, f˜maps N U onto N U asfollow−s: x i′ η−1(i) η−1(i) η−1(i) i(cid:16) k(cid:17) k⊕21
i=1 η−1(i) i=1 i′
N N
N N N 1 ifϕ si = sη(i)
f˜ U = V U (θ ,α ,β ), V = i k k (45)
 i i i′ η−1(i) η−1(i) η−1(i) i  iσ ifϕ (cid:16)si(cid:17)= sη(i) .
LetusnowconsideraOpi=e1rmutatioOni=1matriOxi=1S M thatrearrangestheorder−ofbxasisstait(cid:16)esk(cid:17)j k⊕210 , 1 inthetensor
η 2N i
product j j ... j . SinceS permutesthe∈elementsinasimilarwayas f˜,itisnotdifficu{l|ttio}s∈ee{|thia|t i}
1 2 N η
| i| i | i
N N
S U(θ,α,β)ST = U(θ ,α ,β ) (46)
η i i i i η i η−1(i) η−1(i) η−1(i)
Oi=1 Oi=1
Itisalsoclearthatσ N commuteswith N V andS andsodoes J = (1 N +iσ N)/√2. Thusthefinalstate Ψ =
⊗x i=1 i η ⊗ ⊗x | ′i
J f˜ N U(θ,α,β) J0 N maybewNrittenas
† i=1 i i i i | i⊗
(cid:16)N (cid:17)
N N N
Ψ = VS J U(θ,α,β) J0 N = VS Ψ . (47)
| ′i i η † i i i i  | i⊗ i η| i
Oi=1 Oi=1  Oi=1
Analysissimilartothatinequations(40)-(42)showsthat
N N
u f˜ U =u U , (48)
η(i)  i i i
 Oi=1  Oi=1 
whichisthedesiredconclusion. (cid:4)
Asthefollowingexampleshows,theconverseisnottrueingeneral.
Example5 Letusconsidertwo2 2bimatrixgamesthatdifferonlyintheorderofpayoffprofilesintheanti-diagonal,
×
i.e.,
l r l r
′ ′
t (a ,b ) (a ,b ) t (a ,b ) (a ,b )
Γ: 00 00 01 01 and Γ : ′ 00 00 10 10 . (49)
b (a10,b10) (a11,b11)! ′ b′ (a01,b01) (a11,b11)!
TheEWLquantumcounterpartsΓ andΓ forthesegamesarespecifiedbytriples(29),whereinthiscaseN = 1,2 ,
EWL ′EWL { }
D = D =SU(2)andthemeasurementoperatorstaketheform
i ′i
(M ,M )= (a ,b )P , (M ,M )= (a ,b )P , (50)
1 2 j1j2 j1j2 j1j2 1′ 2′ j1j2 j1j2 j2j1
j1,Xj2=0,1 j1,Xj2=0,1
where P = j j j j . Let us set a mapping f˜ = (η,(ϕ˜ ,ϕ˜ )) with η(i) = i and ϕ˜ (U(θ,α,β) = U (π θ,π/4
j1j2 | 1 2ih 1 2| 1 2 i i i i i i′ − i −
β,π/4 α))fori=1,2.Aneasycomputationshowsthat
i i
−
Ψ = J f˜(U U )J00
′ † 1 2
| i ⊗ | i
π π π π
= J U π θ , β , α U π θ , β , α J00
† 1 1 1 1 2 2 2 2
(cid:18) (cid:18) − 4 − 4 − (cid:19)⊗ (cid:18) − 4 − 4 − (cid:19)(cid:19) | i
=SFJ (U (θ ,α ,β ) U (θ ,α ,β ))J00 =SFΨ , (51)
† 1 1 1 1 2 2 2 2
⊗ | i | i
7
whereS hastheouterproductrepresentationS = 00 00 + 01 10 + 10 01 + 11 11 andF = 00 00 + 01 10 +
| ih | | ih | | ih | | ih | | ih | | ih |
10 01 11 11.Applicationofequation(51)gives
| ih |−| ih |
u f˜(U U ) = Ψ M Ψ = ΨFSM SFΨ = ΨM Ψ =u(U U ). (52)
′i 1⊗ 2 h ′| i′| ′i h | i′ | i h | i| i i 1⊗ 2
(cid:16) (cid:17)
Asaresult,gamesproducedbyΓ andΓ arestronglyisomorphic.Thisfact,however,isnotsufficienttoguarantee
EWL ′EWL
theisomorphismbetweenΓandΓ. Indeed,onecancheckthatthereisno f = (η,(ϕ ,ϕ ))tosatisfyu(s) = u (f(s))
′ 1 2 i ′η(i)
for each s t,b l,r and i = 1,2. Alternatively, given specific payoff profiles (a ,b ) = (4,4),(a ,b ) =
00 00 01 01
∈ { } × { }
(1,3),(a ,b ) = (3,1),(a ,b ) = (2,2), we can find three Nash equilibria in the game Γ and just one in the game
10 10 11 11
Γ. Hence,byLemma1games(49)arenotisomorphic.
′
4 Conclusions
Thetheoryofquantumgameshasnorigorousmathematicalstructure.Therearenoformalaxioms,definitionsthatwould
givecleardirectionsofhowaquantumgameoughttolooklike. Infact,onlyoneconditionistakenintoconsideration.It
saysthataquantumgameoughttoincludetheclassicalwayofplayingthegame. Asaresult,thisallowsustodefinea
quantumgameschemeinmanydifferentways. Theschemewehavestudiedinthepaperisdefinitelyingenious. Ithas
madeasignificantcontributiontoquantumgametheory.However,itleavesthefreedomofchoiceoftheplayers’strategy
sets. Our criterion for quantum strategic game schemes requires the quantum model to preserve strong isomorphism.
This specifies the strategy sets to be SU(2). We have shown that a proper subset of SU(2) in the EWL scheme may
imply different quantumcounterpartsof the same game-theoreticalproblem. In that case, the resulting quantum game
(inparticular,itsNashequilibria)dependsontheorderofplayers’strategiesintheinputbimatrixgame. Hence,givena
classicalgame,forexamplethePrisoner’sDilemma,wecannotsayanythingaboutthepropertiesoftheEWLapproach
withthetwo-parameterunitarystrategiesuntilwe specifyanexplicitbimatrixforthatgame. Thisisnotthecaseinthe
EWLschemewithSU(2)where,givenaclassicalbimatrixgameoritsisomorphiccounterpart,wealwaysobtainthesame
fromthegame-theoreticalpointofviewquantumgame.
Acknowledges
WewouldliketothanktheReviewersfordiscussionwhichundoubtedlyimprovedthequalityofourwork.
This work was supported by the Ministry of Science and Higher Education in Poland under the project Iuventus Plus
IP2014010973intheyears2015–2017.
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