aa r X i v : . [ qu a n t - ph ] N ov Three-player quantum Kolkata restaurant problem under decoherence
M. Ramzan
Department of Physics Quaid-i-Azam UniversityIslamabad 45320, Pakistan (Dated: June 13, 2018)Effect of quantum decoherence in a three-player quantum Kolkata restaurant problemis investigated using tripartite entangled qutrit states. Amplitude damping, depolarizing,phase damping, trit-phase flip and phase flip channels are considered to analyze thebehaviour of players payoffs. It is seen that Alice’s payoff is heavily influenced by theamplitude damping channel as compared to the depolarizing and flipping channels. However,for higher level of decoherence, Alice’s payoff is strongly affected by depolarizing noise.Whereas the behaviour of phase damping channel is symmetrical around 50 % decoherence.It is also seen that for maximum decoherence ( p = 1) , the influence of amplitude dampingchannel dominates over depolarizing and flipping channels. Whereas, phase dampingchannel has no effect on the Alice’s payoff. Therefore, the problem becomes noiselessone at maximum decoherence in case of phase damping channel. Furthermore, the Nashequilibrium of the problem does not change under decoherence. Keywords: Decoherence; qutrit channels; Kolkata restaurant problem.
I. INTRODUCTION
Game theory a branch of applied mathematics, was initially developed for use in economicsby von Neumann and Morgenstern [1] and important contributions were given by John Nash [2].It attempts to capture mathematical behavior in strategic situations, in which an individual’ssuccess of making a choice depends on the choice of the other players. It is usually used to modelthe behavior of biological, economical and computer systems. During last few years, a number ofclassical games has been converted into the realm of quantum mechanics [3-18]. Recently, Miszczaket al. [19] has studied a qubit flip game on a Heisenberg spin chain. They have shown that beingwell aware of the dimensionality of the system, a player can achieve a mean payoff equal to almost1. More recently, Sharif et al. [20] has proposed the quantum solution to a three-player KolkataRestaurant problem. The Kolkata Paise Restaurant (KPR) [21] is a repeated game similar to theMinority games, played between a large number of agents having no interaction among themselves.Since quantum minority games [22-25] has attracted much attention in recent years. They have alsobeen analyzed under the influence of decoherence by Flitney and Hollenberg [26]. It is therefore,important to study the behaviour of quantum restaurant problem in the presence of environmentalinfluences.Since, it is not possible to completely isolate a quantum system from its environment. Therefore,it is important to study the system-environment dynamics in the presence of environmental effects.Quantum games may provide a feasible platform for implementing quantum information processingin physical systems [27] and can be used to probe the influence of decoherence in such systems[6, 28-31]. In this connection, quantum channels provide a natural theoretical framework for thestudy of decoherence in noisy quantum communication systems. Quantum error correction [32-33]and entanglement purifications [34] can be employed to avoid the problem of decoherence.In this paper, the effect of quantum decoherence in a three-player quantum Kolkata restaurantproblem is studied using entangled qutrit states. By considering different noisy qutrit channelsparameterized by decoherence parameter p such that p ∈ [0 , II. DECOHERENCE AND QUANTUM KOLKATA RESTAURANT PROBLEM
In the Kolkata Paise Restaurant (KPR) problem, N non-communicating agents have to choosebetween n choices. The agents receive a gain in their utility if their choice is not too crowded,i.e. the number of agents that made the same choice is under some threshold limit. The choicescan also have different values of utility associated with them, accounting for a preference profileover the set of choices. Therefore, in KPR, N prospective customers choose from N restaurantseach evening in a parallel decision mode. Each restaurant have identical price but different rank k (agreed by the all the N agents) and can serve only one customer. If more than one agents arriveat any restaurant on any evening, one of them is randomly chosen and is served and the rest donot get dinner that evening.For the sake of simplicity, let the three agents, Alice, Bob and Charlie have three possiblechoices: security 0, security 1 and security 2. They receive a payoff $ = 1 if their choice is unique,otherwise they receive $ = 0. Therefore, the game is a one shoot game, that is, it is a non-iterative,and the agents have no information from previous rounds. Since the agents cannot communicate,therefore, there is nothing left to do other than randomizing between the choices. Randomizationgives the agent i an expected payoff of E c ($) = 4 /
9, where the superscript c represents the classicalstrategy.In this problem, let Alice, Bob and Charlie share a general tripartite entangled qutrit state ofthe form ρ in = f | Ψ in i h Ψ in | + (1 − f )27 I (1)where the parameter f controls the degree of entanglement and | Ψ in i = 1 √ | i + | i + | i ) (2)In order to analyze the effect of entanglement, another general initial state is also considered asgiven below | Ψ in i = sin θ cos φ | i + sin θ sin φ | i + cos θ | i (3)where 0 ≤ θ ≤ π and 0 ≤ θ ≤ π. If we set θ = π/ , π/ φ = ± cos − (1 / √
3) in the aboveequation, the three-qutrit state becomes the maximally entangled state. The strategies of theplayers can be defined by the unitary operator U acting on the initial qutrit state of the problemgiven as [35] U = z ¯ ω ¯ z ω − ¯ z ω z ¯ ω ¯ z ω − ¯ z ω z ¯ ω ¯ z ω − ¯ z ω (4)where −→ z = sin θ cos φe iα sin θ sin φe iα cos θe iα (5)and −→ ω = cos χ cos θ cos φe i ( β − α ) + sin χ sin φe i ( β − α ) cos χ cos θ sin φe i ( β − α ) − sin χ cos φe i ( β − α ) − cos χ sin θe i ( β − α ) (6)where 0 ≤ χ ≤ π/ ≤ β , β ≤ π. After the action of players unitary operators the state ofthe game transform to ρ ´ f = ( U † A ⊗ U † B ⊗ U † C )( | Ψ in i h Ψ in | )( U A ⊗ U B ⊗ U C ) (7)The evolution of the state of a quantum system in a noisy environment can be described by thesuper-operator Φ in the Kraus operator representation as [1]˜ ρ f = Φ ρ f = X k E k ρ f E † k (8)where the Kraus operators E i satisfy the following completeness relation X k E † k E k = I (9)We have constructed the Kraus operators for the game from the single qutrit Kraus operators (asgiven in equations (9-11) below) by taking their tensor product over all n combination of π ( i )indices E k = ⊗ π e π ( i ) (10)where n is the number of Kraus operators for a single qutrit channel. The single qutrit Krausoperators for the amplitude damping channel are given by [36] E = √ − p
00 0 √ − p , E = √ p
00 0 00 0 0 , E = √ p (11)Similarly, the single qutrit Kraus operators for the phase damping channel are given as [36] E = p − p , E = √ p ω
00 0 ω , (12)where ω = e πi . The single qutrit Kraus operators for the depolarizing channel are given by [37] E = p − pI, E = r p Y, E = r p Z, E = r p Y , E = r p Y Z (13) E = r p Y Z, E = r p Y Z , E = r p Y Z , E = r p Z (14)where Y = , Z = ω
00 0 ω (15)The single qutrit Kraus operators for the phase flip channel are given by E = √ − p
00 0 √ − p , E = √ p
00 0 00 0 0 , E = √ p (16)and the single qutrit Kraus operators for the trit-phase flip channel are given by E = r − p , E = r p e πi e − πi ,E = r p e − πi
00 0 e πi , E = r p e πi
00 0 e − πi (17)where p = 1 − e − Γ t represents the quantum noise parameter usually termed as decoherence param-eter. Here the bounds [0 ,
1] of p correspond to t = 0, ∞ respectively. The final state of the gameafter the action of the channel can be written as ρ f = Φ p ( ρ f ) (18)where Φ α is the super-operator realizing the quantum channel parametrized by the real number p (decoherence parameter). The payoff operator for i th player (say Alice) can be written as P A = X x , x , x =0 | x x x i h x x x | , x = x = x + X x , x , x =0 | x x x i h x x x | , x = x = x (19)The payoff of i th player can be calculated as E i ($) = Tr { P A ˜ ρ f } (20)where Tr represents the trace of the matrix. The optimal strategy for players is found to be U opt given by U opt ( θ, φ, χ, α , α , α , β , β ) = ( π , cos − (1 / √ , π , π , π , π , π , π p = 0.In order to interpret the effect of decoherence on the three-player quantum Kolkata restaurantproblem, different graphs has been plotted as a function of decoherence parameter. In figure (1),Alice’s payoff is plotted as a function of decoherence parameter p for (a) f = 0 . , (b) f = 0 . , (c) f = 1 and θ = π , φ = cos − (1 / √
3) for amplitude damping, depolarizing, phase damping,trit-phase flip and phase flip channels, where AD, Dep, PD, TPF and PF represent the amplitudedamping, depolarizing, phase damping, trit-phase flip and phase flip channels respectively. It isseen that Alice’s payoff is heavily influenced by the amplitude damping channel as compared tothe depolarizing and flipping channels. In figures (2 and 3), Alice’s payoff is plotted as a functionof θ and φ for p = 0 . p = 0 . θ and φ for p = 1 (a) amplitude damping, (b)phase damping, (c) depolarizing and (d) trit-phase flip channels, respectively. It is shown that formaximum decoherence i.e. p = 1 , amplitude damping channel dominates over the depolarizing andflipping channels having considerable reduction in the payoff. Whereas, phase damping channel hasno effect on the Alice’s payoff. In case of phase damping channel, the problem becomes noiselessat maximum decoherence. However, maximal entanglement gives the maximum payoff (6 / p = 0) and it reduces as one changes the degree of entanglement from its maxima or introducesthe value of decoherence parameter p >
1. Furthermore, the Nash equilibrium of the problem doesnot change under decoherence.
III. CONCLUSIONS
Quantum three-player Kolkata restaurant problem is investigated in the presence of decoherenceusing tripartite entangled qutrit states using amplitude damping, depolarizing, phase damping, trit-phase flip and phase flip channels. It is seen that for lower level of decoherence, amplitude dampingchannel heavily influences the payoffs as compared to the depolarizing and flipping channels. How-ever, for higher level of decoherence, the payoff is strongly affected by depolarizing noise. It isalso seen that for maximum level of decoherence, amplitude damping channel dominates over thedepolarizing and flipping channels. Whereas, phase damping channel has no effect on the Alice’spayoff at p = 1. Therefore, the problem becomes noiseless at maximum decoherence for phasedamping channel only. Furthermore, the Nash equilibrium of the problem does not change underdecoherence. [1] von Neumann J and Morgenstein O, The theory of games and economic behaviour, Princeton UniversityPress, Princeton, NJ (1944).[2] Nash J, Equilibrium points in n-person games, Proc. National Academy of Science 36 p. 48 (1950).[3] Meyer D A 1999 Phys. Rev. Lett. [11] Flitney A P and Abbott D 2002 Phys. Rev. A , 030301(R).[23] Chen Q, Wang Y, 2004 Physics Letters A, A , 98,102[24] A. P. Flitney, A. Greentree, (Elsiever Science, feb 2008).[25] C. Schmid, A.P. Flitney, (arXiv:0901.0063v1 [quant-ph], 2008).[26] Flitney A P and Hollenberg L C L 2007 Quantum Inf. Comput. (7), 2513[28] Ramzan M, Nawaz A, Toor A H and Khan M K 2008 J. Phys. A: Math. Theor. , 667[32] Steane A 1996 Phys. Rev. Lett. Figures captions
Figure 1 . (Color online). Alice’s payoff is plotted as a function of decoherence parameter p for (a) f = 0 . , (b) f = 0 . , (c) f = 1 and θ = π , φ = cos − (1 / √
3) for amplitude damping, depolarizing,phase damping, trit-phase flip and phase flip channels.
Figure 2 . (Color online). Alice’s payoff is plotted as a function of θ and φ for p = 0 . Figure 3 . (Color online). Alice’s payoff is plotted as a function of θ and φ for p = 0 . Figure 4 . (Color online). Alice’s payoff is plotted as a function of θ and φ for p = 1 (a) amplitudedamping, (b) phase damping, (c) depolarizing and (d) trit-phase flip channels for the state ofequation.0 p P ay o ff (a) f=0.2 ADDepPDTPFPF 0 0.2 0.4 0.6 0.8 10.450.50.550.6 p P ay o ff (b) f=0.5 ADDepPDTPFPF0 0.2 0.4 0.6 0.8 10.450.50.550.60.650.7 p P ay o ff (c) f=1 ADDepPDTPFPF 0 0.2 0.4 0.6 0.8 10.450.50.550.60.650.7 p P ay o ff (d) θ = π /4 & φ =cos(1/sqrt(3)) − ADDepPDTPFPF
FIG. 1: (Color online). Alice’s payoff is plotted as a function of decoherence parameter p for (a) f = 0 . , (b) f = 0 . , (c) f = 1 and θ = π , φ = cos − (1 / √
3) for amplitude damping, depolarizing, phase damping,trit-phase flip and phase flip channels. θ (a) Amplitude Damping φ P ay o ff θ (b) Phase damping φ P ay o ff θ (c) Depolarizing φ P ay o ff θ (d) Trit−phase flip φ P ay o ff FIG. 2: (Color online). Alice’s payoff is plotted as a function of θ and φ for p = 0 . θ (a) Amplitude Damping φ P ay o ff θ (b) Phase damping φ P ay o ff θ (c) Depolarizing φ P ay o ff θ (d) Trit−phase flip φ P ay o ff FIG. 3: (Color online). Alice’s payoff is plotted as a function of θ and φ for p = 0 . θ (a) Amplitude Damping φ P ay o ff θ (b) Phase damping φ P ay o ff θ (c) Depolarizing φ P ay o ff θ (d) Trit−phase flip φ P ay o ff FIG. 4: (Color online). Alice’s payoff is plotted as a function of θ and φ for pp