Photonic implementation of the quantum Morra game

In this paper, we study a faithful translation of a two-player quantum Morra game, which builds on previous work by including the classical game as a special case. We propose a natural deformation of the game in the quantum regime in which Alice has a winning advantage, breaking the balance of the classical game. A Nash equilibrium can be found in some cases by employing a pure strategy, which is impossible in the classical game where a mixed strategy is always required. We prepared our states using photonic qubits on a linear optics setup, with an average deviation less than 2% with respect to the measured outcome probabilities. Finally, we discuss potential applications of the quantum Morra game to the study of quantum information and communication.


I. INTRODUCTION
Modern game theory, initially developed by J. Nash and J. Neumann [1,2] for analysing economic strategies among rational players in a mathematically consistent model, has found widespread applications in various fields, including biology [3], politics [4], choice-theory [5] and computer science [6].However, the emergence of quantum information in the 80's and 90's enabled researchers to study game theoretic models in the quantum regime and it has turned out that superposition and entanglement allow new winning strategies that previously did not exist [7,8].Quantum games (QG) have already been used to design quantum cryptographic protocols with enhanced security [9,10] and optimise networks for frequency allocation [11].Moreover, studying QG's has revealed new insights on non-cooperative games such as prisoners dilemma where a new equilibrium strategy was found and improved the payoff to all players [12,13].Quantum games can now be implemented on plethora of systems, such as small-scale quantum processors [14], ion traps [15] or nitrogen vacancies in diamonds [16] however the most common platform has been photonics [17][18][19][20].This has the benefit that the games can be played between remote players, and that they can then be used as primitives for higher level distributed quantum communication tasks [21].
Here we consider one of the oldest known games that is still studied today, the Morra game [22,23].Morra is a non-cooperative game in which players hide a maximum number of coins (or fingers), and each player attempts to guess the total number.Players are ordered a priori, and the rule is that a player cannot repeat the guess of the previous ones.This rule apparently gives the first player an advantage over the rest, but unexpectedly everyone has an equal chance of winning.The Morra is a closely related game to the Spanish Chinos game, which has been studied extensively [24,25].The Morra's game has applications in modelling financial markets, and information transmission [24].
Quantum versions of Morra's game have been proposed in the past [26,27], with the aim of studying the equilibrium strategies between the two players.The basic idea is to replace the act of drawing a coin by that of applying an operator on a quantum state shared by all players, say a boson, or a hard-core boson (fermions), a qubit, or two qubits.These operations create a state in which one measures an observable that is the classical analogue of the total number of coins.In some situations quantum effects may lead to the breaking of the classical balance of the players.
Here we show a novel implementation of the quantum Morra game (QMG) using qutrits that, unlike the previous proposals [26,27], can reproduce the classical game faithfully.This version of the game allows us to deform the underlying rules, which we shall define later, thereby generating new effects.This implementation makes use of a three level system as shared resource for players to act on, similar to the Aharonov three quantum boxes game [16].We implement this game employing entangled photonic qubits and obtained good agreement between the theory and the experimental results.

A. Classical Morra Game
We shall consider the simplest version of the Morra game that involves two players, Alice and Bob, who can draw each from 0 to 1 coins and guess the total number of coins with the restriction that Bob cannot repeat the result predicted by Alice.This game can be generalised to more players and coins [26].We distinguish between pure strategies, which involve players selecting a specific number of coins, and mixed strategies, where players randomize their coin choices based on probabilities assigned to each possible number of coins.A non-cooperative game which allows mixed strategies is guaranteed to have at least one Nash equilibrium [28], a situation where neither player can improve their payoff by unilaterally changing their strategy, as long as their opponent does not change their strategy.We refer to a strategy as optimal when it leads to a Nash equilibrium [2].
The optimal strategy for Alice is to choose at random c A = 0, 1 coins and to guess always g A = 1, so as not to reveal information to Bob [26].This is based on the fact that with four possible outcomes, the most probable value of the sum is 1.Bob's optimal strategy is to choose c B = 0, 1 coins at random and make his guess g B in a rational way [29].
A rational player seeks to maximise their expected payoff or benefit [30].For example, users competing for bandwidth in a radio network [31], or prisoners minimising their jail sentence [12].In the context of the Morra game, if Bob chooses c B = 0, then by reason he must exclude the option g B = 2, and if he chooses c B = 1, then he must exclude g B = 0. Playing the optimal strategy, each player wins on average half of the time, resulting in a symmetric game with equal winning probabilities.
This strategy is also Pareto optimal [32], since no player can improve their own payoff without reducing their opponents payoff.In game theory, this scenario is called a zero-sum game [1].This result is also valid for two players and a general number of coins [26].

II. QUANTUM MORRA GAME
In our version of the QMG, first we will associate the total number of coins, namely 0, 1, 2, to three orthogonal states | 0⟩, | 1⟩, | 2⟩ that form the basis of a qutrit.To produce these states, the players have two unitary operators, O 0 and O 1 , that correspond to the number of coins they have in their hands at the start of each roll.The joint state created by Alice and Bob is given by where |ϕ⟩ is an initial state and note |ψ 0,1 ⟩ and |ψ 1,0 ⟩ are identical states (a relative phase between them is unobservable).Using this equation the basis states are obtained as Choosing O 0 as the identity operator we find that |ϕ⟩ = | 0⟩ and The operator O 1 has to be unitary.A solution compatible with (2) is O 1 = X, where X is the Pauli matrix for qutrits, that satisfies X 3 = I.In the quantum game, the operator X 3 never arises because the maximum number of X operators is 2.However, we shall consider that having three coins in the box is the same as having none.
The outcome of a round of the game is determined by measuring the observable The state after such measurement will be | 0⟩, | 1⟩, or | 2⟩, thereby revealing the number of coins.The winner of the round is the player whose guess matches the number of coins.Our game can be generalised to to n-players, which we show in Appendix E. The scalability of potential moves improves within this encoding scheme compared to previous implementations [26,27] due to the binary choice of the coin operator.The original game employed four operators [26] and the single qubit game uses three operators [27].Consequently, our game scales as 2 N , where N is the number of players, compared to the previous 4 N and 3 N for the respective games.

A. Quantum Deformation
In this section we will precisely define what a deformation is in the context of our game.We are specifically referring to the players encoding operator (3) that controls the unitary evolution of the shared state, and the underline probability distribution of the game.To go beyond the classical setting we use a parameter θ to smoothly deform the operator away from (3).The physical meaning of this deformation is that the players are able to toss a certain superposition of the number of coins instead of just the classical options.
We have translated the classical Morra game into a quantum game by means of a one-to-one correspondence between classical and quantum objects.The quantum realm allows us to define the superposition of coin states, in analogy to the superposition of cat states.For this we use the quantum Fourier transform Measuring the number of coins in each of the states (5) yields the values 0, 1 or 2 with equal probability 1/3.Moreover, applying the operator X to (5), that is adding a quantum coin, just multiply these states by a global phase, X| ĵ⟩ = ω −j | ĵ⟩.
Along with the Pauli matrix X for qutrits (3), there is a matrix Z defined as Z = diag(1, ω, ω 2 ) that satisfies ZX = ωXZ and is related to X by the Fourier transform where . This connection will allow us to deform the quantum game from the phase space.
The two players game can be deformed by replacing Z in ( 6) by Z θ = diag 1, e iθ , e 2iθ yielding a modified X operator, where The states created by the action of X θ are where we used that X 2 θ = X 2θ .These states are normalised but are not orthogonal except for θ = 2π/3 and 4π/3.We have found a general two-qubit quantum circuit for the deformation unitary (7) using eight local unitaries and three CNOT gates [33,34], as outlined in the Appendix D. Regarding the physical interpretation of the deformation, (9) tells us that, now, there exists the possibility of one player tossing two coins with a certain probability although in the classical version they only have one coin in their hands.
We recall that the probabilities are given by, where |n⟩ are the states | 0⟩, | 1⟩, | 2⟩ and |ψ a,b ⟩ are the common states generated by Alice and Bob using the operators X θ and X 2 θ .In particular the average probabilities of Alice are given by P a

III. EXPERIMENTAL IMPLEMENTATION
We experimentally realise the QMG with a linear optics circuit, using polarisation encoded photons to prepare a two-qubit state.We can write the qutrit state |ã⟩ as a two-photon state |i A i B ⟩ where (i A , i B = H, V ) are the horizontal and vertical polarisation states of each photon and the indices indicate a different photon mode.The qutrit to qubit transformation can be found in the Appendix B, where we arrive at the same encoding states, Figure 1 shows our experiment outlined in three stages; Alice and Bob input their coin choice to the device, the device prepares the shared state, and finally measures the outcome.We prepare our states ( 9) and (10) using the parameterised half-wave plate configuration (α 1 , α 2 , α 3 ).We used solutions that match the desired outcome probability of the shared state and preserve the entanglement entropy.We reframe the problem as an optimization problem and determine the solution with the L-BFGS-B algorithm [37] implemented in SciPy [38], as outlined in the Appendix C.These states require high fidelity and purity to recreate the game faithfully, so we take advantage of a bright photon-pair source using an aperiodically-poled potassium titanyl phosphate (apKTP) crystal, which creates spectrally pure entangled photon pairs via spontaneous parametric downconversion (SPDC) [39,40].

A. Results
Approximately 150 000 rounds are played for 34 evenly spaced values of θ in the interval [0,2π].The data collection methods are discussed in Appendix A. The normalised probabilities for states |1 θ ⟩ and |2 θ ⟩ can be found in Appendix C. To analyse how these new outcome probabilities affect the players strategies we have averaged Bob's two plays over each of Alice's choices using (12).The results have been plotted in Figure 2a when Alice plays zero coins, and Figure 2b when Alice plays one coin.The experimental values closely align with the theoretical expectation, deviating by a maximum of 2% and confirms a successful implementation of the QMG.We have measured a fidelity, compared to the ideal states | 0⟩, | 1⟩ and | 2⟩, of 97%.The main causes for the uncertainties in our experimental data are due to factors such as imbalances on the PDC photon generation loop, losses and depolarisation on the single-mode fibres and imperfections in the retardance of the half waveplates.
In Table I we summarise the most notable points in the Figure 2, where P i (G) is Alice's probability of winning by guessing G when playing i coins.
At θ values close to 2π/3, the classical regime is recovered, matching the values predicted by the theory in Figure 2. Similarly, at 4π/3, the classical probabilities are reproduced, but the guesses for coins 1 and 2 are flipped.This creates a classically impossible situation since Alice plays no coins and can expect a 50% probability of winning by guessing 2 coins.
Notably, for θ equalling 0 and 2π, the game undergoes a complete deformation, resulting in all coin states overlapping into the 0 coin outcome, as we can see in Table I.In this case, Alice will always win if she guesses 0 coins, regardless of the number of coins in play.
When θ = π, the probable outcome for each coin is identical regardless of the coin Alice picks.Even so, Alice still has an advantage over Bob as guessing | 0⟩ yields a winning result higher than 0.5.

IV. STRATEGIES
The deformation θ opens up the possibility for new strategies in which Alice's winning probability is higher than 50%.We focus on analysing the best strategies for each player when they play randomly, as well as characterising the optimal strategy as a function of the deformation.Each player's strategy is defined by the probability of playing zero or one coin, as well as their guesses for each situation.Since Alice plays first, we consider strategies in which her guess is independent of the number of coins she plays, in order to not reveal information, similarly to the classical case.The winning probability of Alice given a strategy where she plays a coins with probability P A a and guesses n coins, while Bob plays b coins with probability P B b is where p a,b is given in (11).On the contrary, Bob can change his guess n b based on the number of coins he plays, but it must be different from Alice's guess.The winning probability of Bob is We say that a player's strategy is stable when his probability of winning is higher than that of the other players even if they change their strategies.
In the classical game, it is possible to achieve a Nash equilibrium with a mixed strategy where each player wins half the time by choosing 0 or 1 coins at random.However, the deformation θ allows Alice to have an advantage over Bob when both playing randomly because she can make guesses that increase her winning probability or the draw probability as shown in Figure 3a.In general, these strategies are not stable, and both Alice and Bob can improve them, provided they know each other's strategies.Although Alice's winning probability is higher than Bob's, she can further increase it, as shown in Figure 3b.Bob can try to improve his strategy as well, but in some cases, he is unable to turn the situation around, and Alice's winning probability remains higher as shown in Figure 3c.In these cases, Alice's strategy is stable, although not optimal.
Figure 3 allows us to verify if there are values of θ that allow an optimal strategy when playing randomly apart from the classical case.This is possible for θ = 4π/3 since it is equivalent to the classical case under the exchange of states |1 θ ⟩ and |2 θ ⟩.Remarkably, for θ = π, it is also possible to achieve equilibrium by playing randomly, although Alice's winning probability is higher than Bob's.In this case, Alice must guess zero coins and Bob one or two.For other values of theta, Alice and Bob's improved strategies are pure.For instance, in the optimal strategy for θ = π/3, Alice consistently plays 0 coins and guesses 0, while Bob plays 1 coin and guesses 1.In this scenario, Alice's winning probability is 0.46, Bob's winning probability is 0.44, and the probability of a draw is 0.1.This case highlights the counterintuitive nature of the deformed game, as in classical terms, this strategy would allow Bob to win every time.

A. Optimal strategies
We have calculated the winning probabilities for each player using the experimental data for the overlaps, demonstrating a strong agreement with the theory (see Figs. 4 and 3).One may wonder how the optimal strategy of each player changes depending on the θ deforma-tion.To find these strategies, we utilize an exhaustive search approach.Initially, we define Alice's strategy and identify the strategy for Bob to maximize his winning probability.Then, we check whether the best strategy for Alice, considering Bob's strategy, aligns with our initial choice.If this is the case, these strategies establish a Nash equilibrium; otherwise, we iterate the process by changing Alice's strategy.It is important to note that these strategies rely on information from both players.Therefore, it would not be possible for either of them to discover these strategies beforehand due to the lack of complete information.The winning probabilities for each player when following the optimal and stable strategy as a function of θ are shown in Figure 4.These strategies are mixed for θ ∈ [4π/9, 14π/9], while for the remaining values, they are pure.The transition between mixed and pure strategies occurs at non-trivial θ values, these are determined by the construction of the operator X(θ).Multiple applications of X(θ) will change the phase relation between the states, which can be seen in the exponent of 8, therefore this transition point is expected to be different with more players.

V. DISCUSSION AND CONCLUSION
In contrast to the classical game, the deformation θ gives Alice an advantage because she wins more often than Bob, and Bob cannot do anything to change it.Another significant aspect that sets it apart from the classical case is the possibility of both players losing.When playing optimally, the classical game has been shown to be a zero-sum game.This is not true for the quantum game and is more likely to occur at θ = π where there is a 20% chance that nobody wins.Moreover, the Nash equilibrium outside the classical case is no longer Paretooptimal except at θ = 0, 4π/3 and 2π.
The search for an optimum strategy using shared resources is closely aligned with other competitive games, such the spectrum scarcity problem [11].Our analysis has shown that sometimes pure or mixed strategies are available, but may come with different resource overheads.However, we did not consider these resource costs in this paper and leave it as an open problem for future work.
A natural extension of the current implementation is to a three player game, since the four possible outcomes can be encoded on just two photons.In fact, the number of qubits scales as ⌈log 2 (M + 1)⌉ (see Appendix E) where M represents the total number of coins, allowing for the efficient construction of a network of players.Moreover, the selection of basis states in this implementation remains arbitrary; we could have opted for three distinct Bell pairs to encode | 0⟩, | 1⟩ and | 2⟩.Using entangled states with minor modifications to the game, could open the possibility of non-local strategies enabled by quantum steering [41,42].This raises the question whether quantum games can be exploited within the network con-  text [21], that gain an advantage from multi-partite resources [36,43].
In summary, we have outlined a new construction of the two-player quantum Morra game, providing Alice with a winning advantage over Bob that surpasses the classical game.The deformation operator and theoretical states have been realized within a linear optics setup, achieving high fidelity and low standard deviation with respect to the measurement outcomes, which is less than 2%.
This formulation of the quantum Morra game has bridged the classical and quantum domains by incorporating the classical game as a particular instance within the extended quantum game.In analysing the strategies, we have identified a new Nash equilibrium that diverges from the classical game.This has enhanced our understanding of strategies in quantum game theory and provided insight into the realization of new quantum games.Surprisingly, it remains unknown whether the equivalent of a Kakutani fixed-point theorem exists in a complex space, such a discovery would guarantee a Nash equilibrium for all quantum games [44].Therefore, further theoretical development in quantum game theory is needed to support the implementation of more complex quantum games.
Finally, by extending the principles and techniques applied in this work to other quantum games, we can potentially evaluate the robustness and efficiency of quantum communication networks and deepen our understanding of quantum game theory.

in terms of the basis |i
where The rows are labelled by 00, 01, 10, 11 and the columns by 0, 1, 2, 3.The matrix expressing the action of X 4 in the two qubit basis is given by Similarly, to find the action of the deformed unitary X θ on two qubits we first let it act trivially on the state | 3⟩, as in (B6), and perform the change of basis (B8) obtaining X 2×2 → X θ , x 0 (θ) x 1 (θ) The generalized quantum Morra game can be implemented with ⌈log 2 (r + 1)⌉ qubits.
Appendix F: A Quantum Coin Deposit It is well known that qubits belonging to a Bell pair are indistinguishable, this is a hallmark of entanglement.An intriguing consequence of our choice of encoding with a Bell pair results in the indistinguishably of the coins played by either player.This may have application in anonymous, secure Quantum Deposit system, wherein the choices of the participants remain concealed.This system operates independently of classical communication channels, offering a sharp departure from traditional games like Morra, which rely on some level of communication.The Quantum Deposit model could lead to new security protocols that require untraceable transactions or enhanced privacy.While the exploration of these applications is beyond the scope of this paper, future researchers could delve into the development of quantum cryptographic methods that capitalize on this work.

FIG. 1 .
FIG. 1.The QMG experimental implementation.(a) Alice and Bob's strategies are input into a device which prepares the joint state |ψ a,b ⟩ and subsequently measures it.(b) Alice and Bob play the QMG by secretly depositing coins cA and cB into the device which has some pre-determined parameter θ.Alice makes the first guess gA, followed by Bob gB.The device deforms the classical coins into quantum coins using the Unitary X(θ), performs a measurement and outputs the sum of the coins.(c) The linear optics circuit for state preparation begins with a photon pair source, constructed with a ap-KTP crystal housed in a Sagnac configuration[35,36] pumped with a ti:sapphire laser centered at 775 nm and prepared in H with a linear polariser (POL), producing degenerate photon pairs at 1550 nm, which are seperated from the pump with a dichroic mirror (DM).The states |ψ a,b ⟩ are encoded using the three half-wave plates (HWP) α1, α2, α3, outlined in the Appendix C. The measurement station consists of two polarising beam splitters (PBS) that project the two photons into the Z-basis, then coupled to superconducting nanowire single photon detectors (SNSPD), and 2-fold coincidences are identified using a time-tagging logic box.

FIG. 2 .
FIG.2.Alice's probability of winning versus θ, when playing 0 and 1 coin.The probabilistic outcome of obtaining (0,1,2) coins when Alice plays no coins (I) (a) and when she plays one coin (b) have been plotted on top of the theoretical curves.Errors bars have calculated assuming Poisson statistics, but found to be O(10 −4 ), therefore neglected.

FIG. 3 .
FIG. 3. (a)The winning probabilities of Alice (red) and Bob (green), as well as the probability of a draw (gray), when both players choose 0 or 1 coin at random but make their best guess.(b) Alice plays her best strategy knowing that Bob plays randomly.(c) Bob plays his best strategy knowing that Alice plays randomly.The experimental results have been plotted on top of the theoretical values.

FIG. 4 .
FIG.4.Optimal strategies for Alice and Bob.The winning probabilities for Alice (red) and Bob (green) are shown when each of them plays with their best strategies.The probability of a draw, where neither of them wins, is indicated in gray.The shaded area shows the values of θ for which the strategies are pure.The experimental results have been plotted on top of the theoretical values.

5 FIG. 6 .
FIG. 6.The coincidence counts on each of the 4 pairs of detectors coupled to the Transmitted (T) and Reflected (R) ports of the PBS to reproduce the overlaps for the states |1 θ ⟩ (a) and |2 θ ⟩ (b).Error bars have been calculated with Poissonian statistics, but omitted due to size.