Noise Effects in Quantum Magic Squares Game PIOTR GAWRON JAROSL AW MISZCZAK Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, ul. Ba ltycka 5 8 0 Gliwice, 44-100, Poland 0 gawron,miszczak @iitis.gliwice.pl 2 { } n JAN SL ADKOWSKI a Institute of Physics, University of Silesia, ul. Bankowa 14 J 1 Katowice, 40-007, Poland 3 [email protected] ] h February 3, 2008 p - t n Abstract a u Inthearticleweanalysehownoisinessofquantumchannelscaninfluencethemagicsquaresquantum q pseudo-telepathygame. Weshowthattheprobabilityofsuccesscanbeusedtodeterminecharacteristics [ of quantumchannels. Therefore thegame deserves more careful studyaiming at its implementation. 1 v 1 Introduction 8 4 8 Quantumgametheory,thesubclassofgametheorythatinvolvesquantumphenomena[1,2],liesatthecross- 4 roadsof physics,quantum informationprocessing,computer andnaturalsciences. Thanks to entanglement, . 1 quantum players1 can sometimes accomplish tasks that are impossible for their classical counterparts. In 0 this paper we present the detailed analysis of one of the pseudo-telepathy games[3, 4]. These games provide 8 simple,yetnontrivial,examplesofquantumgamesthatcanbeusedtoshowtheeffectsofquantumnon-local 0 correlations. Roughlyspeaking,agamebelongstothepseudo-telepathyclassifitadmitsnowinningstrategy : v for classical players, but it admits a winning strategy provided the players share the sufficient amount of i X entanglement. Thisphenomenoniscalledpseudo-telepathy,becauseitwouldappearasmagicaltoaclassical player, yet it has quantum theoretical explanation. r a Our main goal is to study the connection between errors in quantum channels and the probability of winning in magic squares game. The magic square game is selected because of its unique features: it is easy to show that there is no classicalwinning strategy, simple enoughfor a laymanto follow its course and feasible. Thepaperisorganizedasfollows. Wewillbeginbypresentingthemagicsquarespseudo-telepathygame. The we will introduce tools we are using to analyse noise in quantum systems. Then we will attempt to answer the following question: What happens if a quantum game is played in non-perfect conditions because of the influence of quantum noise? The results showing the connection between the noise level and the probability of winning will be given in Section 4. Finally we will point out some issues that yet should be addressed. 1By a quantum player we understand a player that, at least in theory, can explore and make profits from the quantum phenomenainsituationsofconflict, rivalryetc. 1 2 Magic square game The magic square is a 3 3 matrix filled with numbers 0 or 1 so that the sum of entries in each row is even × and the sum of entries in each column is odd. Although such a matrix cannot exist2 one can consider the following game. There are two players: Alice and Bob. Alice is given the number of the row, Bob is given the number of the column. Alice has to give the entries for a row and Bob has to give entries for a column so that the parity conditions are met. In addition, the intersection of the row and the column must agree. Alice and Bob can prepare a strategy but they are not allowed to communicate during the game. There exists a (classical) strategy that leads to winning probability of 8. If parties are allowed to share 9 a quantum state they can achieve probability 1. In the quantum version of this game[5] Alice and Bob are allowed to share an entangled quantum state. The winning strategy is following. Alice and Bob share entangled state 1 Ψ = (0011 1100 0110 + 1001 ). (1) | i 2 | i−| i−| i | i Depending on the input (i.e. the specific row and column to be filled in) Alice and Bob apply unitary operators A I and I B , respectively, i j ⊗ ⊗ i 0 0 1 i 1 1 i 1 1 1 1 − − − 1 0 i 1 0 i 1 1 i 1 1 1 1 1 A1 = √2 0 −i 1 0  A2 = 12 −i 1 −1 i  A3 = 2 1 1 −1 1  (2) − − −  1 0 0 i   i 1 1 i   1 1 1 1     − −   − − −  i i 1 1   1 i 1 i   1 0 0 1  − − 1 i i 1 1 1 i 1 i 1 1 0 0 1 B1 = 2 −1 −1 i −i  B2 = 21 1 i 1 −i  B3 = √2 −0 1 1 0  (3) − −  i i 1 1   1 i 1 i   0 1 1 0   −   − − −   −        where i and j denote the corresponding inputs. Thefinalstateisusedtodeterminetwobitsofeachanswer. Theremainingbitscanbefoundbyapplying parity conditions. 3 Quantum noise A interesting question arises: what happens if a quantum game is played in non-perfect (real-world) condi- tions because of the presence of quantum noise.[7, 8] In the most general case quantum evolution is described by superoperator Φ, which can be expressed using Kraus representation[6]: Φ(ρ)= E ρE †, (4) k k k X where E †E =I. k k k In following we will consider typical quantum channels, namely P depolarizing channel: 1 3αI, ασ , ασ , ασ , • − 4 4 x 4 y 4 z nq o p p p 1 0 0 √α amplitude damping: , , • 0 √1 α 0 0 (cid:26)(cid:20) − (cid:21) (cid:20) (cid:21)(cid:27) phase flip, bit flip and bit-phase flip with Kraus operators • √1 αI,√ασ , √1 αI,√ασ and √1 αI,√ασ respectively. z x y − − − 2Th(cid:8)ereforetheadjectiv(cid:9)em(cid:8)agicisused. (cid:9) (cid:8) (cid:9) 2 Realparameterα [0,1]representsheretheamountofnoiseinthechannelandσ ,σ ,σ arePaulimatrices. x y z ∈ In our scheme, the Kraus operators are of the dimension 24. They are constructed from one-qubit operators e by taking their tensor product over all n4 combinations of π(i) indices k E = e , (5) k π(i) π O where n is the number of Kraus operator for a single qubit channel. 3.1 The comparison of channels Although one can assign physical meaning to the parameter α, this meaning can be different for different channels. Therefore we are using channel fidelity[9] to compare quantum channels. We define channel fidelity as: ∆(Φ)=F(J(Φ),J(I)), (6) 2 where J is Jamio lkowskiisomorphism and F is the fidelity defined as F(ρ1,ρ2)=tr( √ρ1ρ2√ρ1) . p 4 Results In this section we are analysing the influence of the noise on success probability and fidelity of non-perfect (mixed) quantum states in the case when the noise operator is applied before the game gates. 4.1 Calculations Thefinalstateofthisschemeisρ =(A B )Φ (Ψ Ψ)(A† B†),whereΦ isthesuperoperatorrealizing f i⊗ j α | ih | i⊗ j α quantum channel parametrized by real number α. Probability P (α) is computed as the probability of i,j measuring ρ in the state indicating success f P (α)=tr ρ ξ ξ , (7) i,j f i i | ih |! i X where ξ are the states that imply success. i | i 4.2 Success probability We compute success probabilityP (α) for different inputs (i,j 1,2,3 )anddifferent quantum channels. i,j ∈{ } Our calculations show that mean probability of success, P(α)= P (α), heavily depends on the i,j∈{1,2,3} i,j noise level α. The game results for each combination of gates A ,B for depolarizing, amplitude damping, i j P phase damping, phase, bit and bit-phase flip channels are listed in Fig. 1. Fig. 2 presents mean success probability P(α) as the function of error rate. In the case of depolarizing channel, the success probability as the function of noise amount is the same for all the possible inputs. In the case of amplitude and phase damping channels, wan can observe three different types of behaviour. These functions are non-increasing for those channels. The bit, phase and bit-phase flip functions reach their minima for α = 1/2 and are symmetrical. This means that high error rates influence the game weakly. One can easily see that in case of input (1,3) the phase-flip channel does not influence the probability of success. The same is true for input (2,3) and bit flip channel and also for input (3,3) and bit-phase flip channel. Therefore it is possible to distinguish those channels by looking at success probability of magic-squares game. The graphical representation of dependency between mean success probability and channel fidelity is presented in the form of parametric plot in Fig. 3. 3 Depolarizing channel: (i,j)i,j =1,2,3 P (α) = 1α4 2α3+3α2 2α+1. { | } i,j 2 − − Amplitude damping channel: (i,j) (1,1),(1,2),(2,3),(3,3) P (α) = 1α2 α+1 ∈{ } i,j 2 − (i,j) (1,3) P (α) = 2α2 2α+1 i,j ∈{ } − (i,j) (2,1),(2,2),(3,1),(3,2) P (α) = α2 3α+1 ∈{ } i,j − 2 Phase damping channel: (i,j) (1,1),(1,2),(2,3),(3,3) P (α) = 1α2 α+1 ∈{ } i,j 2 − (i,j) (1,3) P (α) = 1 i,j ∈{ } (i,j) (2,1),(2,2),(3,1),(3,2) P (α) = 1α+1 ∈{ } i,j −2 Phase flip: (i,j) (1,1),(1,2),(2,3),(3,3) P (α) = 8α4 16α3+12α2 4α+1 i,j ∈{ } − − (i,j) (1,3) P (α) = 1 i,j ∈{ } (i,j) (2,1),(2,2),(3,1),(3,2) P (α) = 2α2 2α+1 i,j ∈{ } − Bit flip: (i,j) (1,1),(1,2),(3,1),(3,2) P (α) = 2α2 2α+1 i,j ∈{ } − (i,j) (2,3) P (α) = 1 i,j ∈{ } (i,j) (1,3),(2,1),(2,2),(3,3) P (α) = 8α4 16α3+12α2 4α+1 i,j ∈{ } − − Bit-phase flip: (i,j) (1,1),(1,2),(2,1),(2,2) P (α) = 2α2 2α+1 i,j ∈{ } − (i,j) (3,3) P (α) = 1 i,j ∈{ } (i,j) (1,3),(2,3),(3,1),(3,2) P (α) = 8α4 16α3+12α2 4α+1 i,j ∈{ } − − Figure 1: Success probability for all combinations of magicsquares game inputs for depolarizing, amplitude damping, phase damping channels, phase, bit and bit-phase flip channels 4 1 1 amplitude damping channel dephasing channel )) )) αα classical treshold αα classical treshold (( 0.9 (( 0.9 PP PP yy yy tt tt bilibili 0.8 bilibili 0.8 aa aa bb bb oo oo rr 0.7 rr 0.7 pp pp ss ss ss ss ee ee cccc 0.6 cccc 0.6 uu uu SS SS 0.5 0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 EErrrroorr rraattee αα EErrrroorr rraattee αα 1 1 flipping channels phase damping channel )) )) αα classical treshold αα classical treshold (( 0.9 (( 0.9 PP PP yy yy tt tt bilibili 0.8 bilibili 0.8 aa aa bb bb oo oo rr 0.7 rr 0.7 pp pp ss ss ss ss ee ee cccc 0.6 cccc 0.6 uu uu SS SS 0.5 0.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 EErrrroorr rraattee αα EErrrroorr rraattee αα Figure 2: Dephasing and damping channels cause monotonic decrease of mean success probability in the function of noiseamount. Amplitude damping channelcausesthe success function to attain minimal proba- bility of success for error rate 3. Flipping channels give symmetrical functions with minimum for error rate 4 1. 2 5 1 channel ∆(Φ ( )) P(α) α • AD √1 α+1 8/256 (8α2 12α+9)/9 − − F (α 1)4 (32α4 64α3+56α2 24α+9)/9 0.9 (cid:0) − (cid:1) − − D (3α 4)4/256 (α4 4α3+6α2 4α+2)/2 α)α) PD ((√1 −α+1)4)/16 −(2α2 6α+−9)/9 (( − − PP yy tt 0.8 ii ll ii bb aa bb oo rr pp 0.7 ss ss ee cc cc uu amplitude damping (AD) SS 0.6 flipping channels (F) dephasing (D) phase damping (PD) 0.5 0 0.2 0.4 0.6 0.8 1 QQuuaannttuumm cchhaannnneell fifiddeelliittyy ∆∆((ΦΦ (( )))) αα •• Figure3: Theinfluence ofnoiseonsuccessprobability,incaseofdifferentquantumchannelsiscomparedby usingtheparametricplotofsuccessprobabilityP (α)versusquantumchannelfidelity∆(Φ(α))forα [0,1]. i,j ∈ Note that plots for flipping channels and phase damping channel overalpin range ∆(Φ(α)) 1 ,1 . ∈ 16 (cid:2) (cid:3) 6 5 Conclusions We have shown how the probability success in magic squares pseudo-telepathy game is influenced by dif- ferent quantum noisy channels. The calculations show that, by controlling noise parameter and observing probabilitiesofsuccess,itispossibletodistinguishsomechannels. Thuswehaveshownthatimplementation of magic square game can provide the example of channel distinguishing procedure. In case of all channels success probability drops, with the increase of noise, below classical limit of 8. 9 Therefore the physical implementation of quantum magic squares game requires high precision and can be a very difficult task. Wehavealsoshownthatifchannelfidelity ishigherthan1/10the probabilityofsuccessisalmostlinear. Therefore channel fidelity is good approximationof success probability for not very noisy channels. Acknowledgements This research was supported in part by the Polish Ministry of Science and Higher Education project No N519 012 31/1957. References [1] J. Eisert et al., Phys. Rev. Lett. 83 (1999) 3077. [2] E. W. Piotrowski and J. S ladkowski, An invitation to quantum game theory, Int. J. Theor. 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