# Decoherence effects in the quantum qubit flip game using Markovian approximation

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## Abstract

We are considering a quantum version of the penny flip game, whose implementation is influenced by the environment that causes decoherence of the system. In order to model the decoherence, we assume Markovian approximation of open quantum system dynamics. We focus our attention on the phase damping, amplitude damping and amplitude raising channels. Our results show that the Pauli strategy is no longer a Nash equilibrium under decoherence. We attempt to optimize the players’ control pulses in the aforementioned setup to allow them to achieve higher probability of winning the game compared with the Pauli strategy.

## Keywords

Lindblad master equation Decoherence effects Quantum games Open quantum systems## 1 Introduction

Quantum information experiments can be described as a sequence of three operations: state preparation, evolution and measurement [1]. In most cases, one cannot assume that experiments are conducted perfectly; therefore, imperfections have to be taken into account while modeling them. In this work, we are interested in how the knowledge about imperfect evolution of a quantum system can be exploited by players engaged in a quantum game. We assume that one of the players possesses the knowledge about imperfections in the system, while the other is ignorant of their existence. We ask a question of how much the player’s knowledge about those imperfections can be exploited by him/her for their advantage.

We consider implementation of the quantum version of the penny flip game, which is influenced by the environment that causes decoherence of the system. In order to model the decoherence, we assume Markovian approximation of open quantum system dynamics. This assumption is valid, for example, in the case of two-level atom coupled to the vacuum, undergoing spontaneous emission (amplitude damping). The coherent part of the atom’s evolution is described by one-qubit Hamiltonian. Spontaneous emission causes an atom in the excited state to drop down into the ground state, emitting a photon in the process. Similarly, phase damping channel can be considered. This channel causes a continuous decay of coherence without energy dissipation in a quantum system [2].

The paper is organized as follows: in the two following subsections, we discuss related work and present our motivation to undertake this task. In Sect. 2, we recall the penny flip game and its quantum version; in Sect. 3, we present the noise model; in Sect. 4, we discuss the strategies applied in the presence of noise and finally in Sect. 5, we conclude the obtained results.

### 1.1 Related work

Imperfect realizations of quantum games have been discussed in literature since the beginning of the century. Johnson [3] discusses a three-player quantum game played with a corrupted source of entangled qubits. The author implicitly assumes that the initial state of the game had passed through a bit-flip noisy channel before the game began. The corruption of quantum states in schemes implementing quantum games was studied by various authors, *e.g.,* in [4], the authors study the general treatment of decoherence in two-player, two-strategy quantum games; in [5], the authors perform an analysis of the two-player prisoners’ dilemma game; in [6], the multiplayer quantum minority game with decoherence is studied; in [7, 8], the authors analyze the influence of the local noisy channels on quantum Magic Squares games, while the quantum Monty Hall problem under decoherence is studied first in [9] and subsequently in [10]. In [11], the authors study the influence of the interaction of qubits forming a spin chain on the qubit flip game. An analysis of trembling hand-perfect equilibria in quantum games was done in [12]. Prisoners’ dilemma in the presence of collective dephasing modeled by using the Markovian approximation of open quantum systems dynamics is studied in [13]. Unfortunately, the model applied in this work assumes that decoherence acts only after the initial state has been prepared and ceases to act before unitary strategies are applied. Another interesting approach to quantum games is the study of relativistic quantum games [14, 15]. This setup has also been studied in a noisy setup [16].

### 1.2 Motivation

- 1.
The entangled state is prepared,

- 2.
It is transferred through a noisy channel,

- 3.
Players’ strategies are applied,

- 4.
The resulting state is transferred once again through a noisy channel,

- 5.
The state is disentangled,

- 6.
Quantum local measurements are performed, and the outcomes of the games are calculated.

## 2 Game as a quantum experiment

In this work, our goal is to follow the work done in [11] and to discuss the quantum penny flip game as a physical experiment consisting in preparation, evolution and measurement of the system. For the purpose of this paper, we assume that preparation and measurement, contrary to noisy evolution of the system are perfect. We investigate the influence of the noise on the players’ odds and how the noisiness of the system can be exploited by them. The noise model we use is described by the Lindblad master equation, and the dynamics of the system is expressed in the language of quantum systems control.

### 2.1 Penny flip game

In order to provide classical background for our problem, let us consider a classical two-player game, consisting in flipping over a coin by the players in three consecutive rounds. As usual, the players are called Alice and Bob. In each round, Alice and Bob performs one of two operations on the coin: flips it over or retains it unchanged.

At the beginning of the game, the coin is turned heads up. During the course of the game the coin is hidden and the players do not know the opponents actions. If after the last round, the coin tails up, then Alice wins, otherwise the winner is Bob.

*non-flipping strategy*and \(F\) to the

*flipping strategy*. Bob’s pay-off table for this game is presented in Table 1. Looking at the pay-off tables, it can be seen that utility function of players in the game is balanced; thus, the penny flip game is a zero-sum game.

Bob’s pay-off table for the penny flip game

\(NN\) | \(FN\) | \(NF\) | \(FF\) | |
---|---|---|---|---|

\(N\) | \(1\) | \(-1\) | \(-1\) | \(1\) |

\(F\) | \(-1\) | \(1\) | \(1\) | \(-1\) |

A detailed analysis of this game and its asymmetrical quantization can be found in [17]. In this work it was shown that there is no winning strategy for any player in the penny flip game. It was also shown, that if Alice was allowed to extend her set of strategies to quantum strategies she could always win. In Miszczak et al. [11] it was shown that when both players have access to quantum strategies the game becomes fair and it has the Nash equilibrium.

### 2.2 Qubit flip game

*Lindblad*form as

*Lindblad*operators, representing the environment influence on the system [2] and \(\rho \) is the state of the system.

For the purpose of this paper we chose three classes of decoherence: *amplitude damping*, *amplitude raising* and *phase damping* which correspond to noisy operators \(\sigma _{-}=| 0 \rangle \langle 1 |\), \(\sigma _{+}=| 1 \rangle \langle 0 |\) and \(\sigma _z\), respectively.

Let us suppose that initially the quantum coin is in the state \(| 0 \rangle \langle 0 |\). Next, in each round, Alice and Bob perform their sequences of controls on the qubit, where each control pulse is applied according to Eq. (3). After applying all of the nine pulses, we measure the expected value of the \(\sigma _z\) operator. If \(\mathrm{tr}(\sigma _z\rho (T))=-1\) Alice wins, if \(\mathrm{tr}(\sigma _z\rho (T))=1\) Bob wins. Here, \(\rho (T)\) denotes the state of the system at time \(T=9\Delta t\).

Alternatively we can say that the final step of the procedures consists in performing orthogonal measurement \(\{O_\mathrm{tails}\rightarrow | 1 \rangle \langle 1 |,O_\mathrm{heads}\rightarrow | 0 \rangle \langle 0 |\}\) on state \(\rho (T)\). The probability of measuring \(O_\mathrm{tails}\) and \(O_\mathrm{heads}\) determines pay-off functions for Alice and Bob, respectively. These probabilities can be obtained from relations \(p(\mathrm{tails})=\langle 1 |\rho (T)| 1 \rangle \) and \(p(\mathrm{heads})=\langle 0 |\rho (T)| 0 \rangle \).

### 2.3 Nash equilibrium

*Pauli strategy*, which is mixed and gives Nash equilibrium [11]; therefore, this strategy is a reasonable choice for the players. According to the Pauli strategy, each player chooses one of the four unitary operations \(\{{1\!\!1}, \mathrm{i}\sigma _{x}, \mathrm{i}\sigma _{y}, \mathrm{i}\sigma _{z}\}\) with equal probability. Thus, to obtain the Pauli strategy, each player chooses a sequence of control parameters \((\alpha _1^\square , \alpha _2^\square , \alpha _3^\square )\) listed in Table 2. The symbol \(\square \) can be substituted by \(A_1,B,A_2\). It means that in each round, one player performs a unitary operation chosen randomly with a uniform probability distribution from the set \(\{ {1\!\!1}, \mathrm{i}\sigma _x, \mathrm{i}\sigma _y, \mathrm{i}\sigma _z \}\).

Control parameters for realizing the Pauli strategy

\(\alpha _1^\square \) | \(\alpha _2^\square \) | \(\alpha _3^\square \) | |
---|---|---|---|

\({1\!\!1}\) | \(0\) | \(0\) | \(0\) |

\(\mathrm{i}\sigma _x\) | \(\frac{\pi }{4}\) | \(-\frac{\pi }{2}\) | \(-\frac{\pi }{4}\) |

\(\mathrm{i}\sigma _y\) | \(0\) | \(-\frac{\pi }{2}\) | \(0\) |

\(\mathrm{i}\sigma _z\) | \(-\frac{\pi }{4}\) | \(0\) | \(-\frac{\pi }{4}\) |

## 3 Influence of decoherence on the game

### 3.1 Amplitude damping and amplitude raising

### 3.2 Phase damping

*i.e.,*the state evolves almost directly toward the maximally mixed state.

## 4 Optimal strategy for the players

Due to the noisy evolution of the underlying qubit, the strategy given by Table 2 is no longer a Nash equilibrium. We study the possibility of optimizing one player’s strategy, while the other one uses the Pauli strategy. It turns out that this optimization is not always possible. If the rate of decoherence is high enough, then the players’ strategies have little impact on the game outcome. In the low noise scenario, it is possible to optimize the strategy of both players.

In each round, one player performs a series of unitary operations, which are chosen randomly from a uniform distribution. Therefore, the strategy of a player can be seen as a random unitary channel. In this section \(\Phi _{A_1},\Phi _{A_2}\) denote mixed unitary channels used by Alice who implements the Pauli strategy. Similarly, \(\Phi _B\) denotes channels used by Bob.

### 4.1 Optimization method

*optimal*. In our case we assume that

### 4.2 Optimization setup

Our goal is to find control strategies for players, which maximize their respective chances of winning the game. We study three noise channels: the amplitude damping, the phase damping and the amplitude raising channel. They are given by the Lindblad operators \(\sigma _-\), \(\sigma _z\) and \(\sigma _+ = \sigma _-^\dagger \), respectively. In all cases, we assume that one of the players uses the Pauli strategy, while for the other player we try to optimize a control strategy that maximizes that player’s probability of winning. However, in our setup it is convenient to use the value of the observable \(\sigma _z\) rather than probabilities. Value 0 means that each player has a probability of \(\frac{1}{2}\) of winning the game. Values closer to 1 mean higher probability of winning for Bob, while values closer to -1 mean higher probability of winning for Alice.

### 4.3 Optimization results

#### 4.3.1 Phase damping

#### 4.3.2 Amplitude damping

#### 4.3.3 Amplitude raising

## 5 Conclusions

We studied the quantum version of the coin flip game under decoherence. To model the interaction with external environment, we used the Markovian approximation in the form of the Lindblad equation. Because of the fact that Pauli strategy is a known Nash equilibrium of the game, therefore, it was natural to investigate this strategy in the presence noise. Our results show that in the presence of noise, the Pauli strategy is no longer a Nash equilibrium. One of the players, Bob in our case, is always favoured by amplitude and phase damping noise. If we had considered a game with another initial state *i.e.,*, \(\rho _0=| 1 \rangle \langle 1 |\), Alice would have been favoured in this case. Our next step was to check if the players were able to do better than the Pauli strategy. For this, we used the BFGS gradient method to optimize the players’ strategies. Our results show that Alice, as well as Bob, are able to increase their respective winning probabilities. Alice can achieve this for all three studied cases, while Bob can only do this for the phase damping and amplitude damping channels.

## Notes

### Acknowledgments

The work was supported by the Polish Ministry of Science and Higher Education Grants: P. Gawron under the project number IP2011 014071. D. Kurzyk under the project number N N514 513340. Ł. Pawela under the project number N N516 481840.

## References

- 1.Heinosaari, T., Ziman, M.: The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
- 2.Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
- 3.Johnson, N.F.: Playing a quantum game with a corrupted source. Phys. Rev. A
**63**(2), 20302 (2001)CrossRefADSMathSciNetGoogle Scholar - 4.Flitney, A.P., Derek, A.: Quantum games with decoherence. J. Phys. A Math. Gen.
**38**(2), 449 (2005)CrossRefADSMathSciNetzbMATHGoogle Scholar - 5.Chen, J.-L., Kwek, L., Oh, C.: Noisy quantum game. Phys. Rev. A
**65**(5), 052320 (2002)MathSciNetCrossRefADSGoogle Scholar - 6.Flitney, A.P., Hollenberg, L.C.L.: Multiplayer quantum minority game with decoherence. Quantum Inf. Comput.
**7**(1), 111–126 (2007)MathSciNetzbMATHGoogle Scholar - 7.Gawron, P., Miszczak, J.A., Sładkowski, J.: Noise effects in quantum magic squares game. Int. J. Quantum Inf.
**6**(1), 667–673 (2008)CrossRefzbMATHGoogle Scholar - 8.Pawela, Ł., Gawron, P., Puchała, Z., Sładkowski, J.: Enhancing pseudo-telepathy in the magic square game. PLoS ONE
**8**(6), e64694 (2013)CrossRefADSGoogle Scholar - 9.Gawron, P.: Noisy quantum monty hall game. Fluct. Noise Lett.
**9**(1), 9–18 (2010)MathSciNetCrossRefGoogle Scholar - 10.Khan, S., Ramzan, M., Khan, M.K.: Quantum monty hall problem under decoherence. Commun. Theor. Phys.
**54**(1), 47 (2010)MathSciNetCrossRefADSzbMATHGoogle Scholar - 11.Miszczak, J.A., Gawron, P., Puchała, Z.: Qubit flip game on a Heisenberg spin chain. Quantum Inf. Process.
**11**(6), 1571–1583 (2012)MathSciNetCrossRefADSzbMATHGoogle Scholar - 12.Pakuła, I.: Analysis of trembling hand perfect equilibria in quantum games. Fluct. Noise Lett.
**8**(01), 23–30 (2008)CrossRefGoogle Scholar - 13.Nawaz, A.: Prisoners’ dilemma in the presence of collective dephasing. J. Phys. A Math. Theor.
**45**(19), 195304 (2012)MathSciNetCrossRefADSzbMATHGoogle Scholar - 14.Salman, K., Khan, M.K.: Relativistic quantum games in noninertial frames. J. Phys. A Math. Theor.
**44**(35), 355302 (2011)CrossRefADSzbMATHGoogle Scholar - 15.Goudarzi, H., Beyrami, S.: Effect of uniform acceleration on multiplayer quantum game. J. Phys. A Math. Theor.
**45**(22), 225301 (2012)CrossRefADSzbMATHGoogle Scholar - 16.Salman, K., Khan, M.K.: Noisy relativistic quantum games in noninertial frames. Quantum Inf. Process.
**12**(2), 1351–1363 (2013)CrossRefADSzbMATHGoogle Scholar - 17.Piotrowski, E.W., Sładkowski, J.: An invitation to quantum game theory. Int. J. Theor. Phys.
**42**(5), 1089–1099 (2003)CrossRefMathSciNetzbMATHGoogle Scholar - 18.Meyer, D.A.: Quantum strategies. Phys. Rev. Lett.
**82**(5), 1052 (1999)MathSciNetCrossRefADSzbMATHGoogle Scholar - 19.d’Alessandro, D.: Introduction to Quantum Control and Dynamics. CRC press, Boca Raton (2007)CrossRefzbMATHGoogle Scholar
- 20.Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.: The Mathematical Theory of Optimal Processes. Interscience Publishers, New York (1962)Google Scholar
- 21.Jirari, H., Pötz, W.: Optimal coherent control of dissipative \(N\)-level systems. Phys. Rev. A
**72**(1), 013409 (2005)CrossRefADSGoogle Scholar - 22.Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in FORTRAN 77, volume 1 of Fortran Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (1992)Google Scholar

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