Abstract
The competitive multiarmed bandit (CMAB) problem is related to social issues such as maximizing total social benefits while preserving equality among individuals by overcoming conflicts between individual decisions, which could seriously decrease social benefits. The study described herein provides experimental evidence that entangled photons physically resolve the CMAB in the 2arms 2players case, maximizing the social rewards while ensuring equality. Moreover, we demonstrated that deception, or outperforming the other player by receiving a greater reward, cannot be accomplished in a polarizationentangledphotonbased system, while deception is achievable in systems based on classical polarizationcorrelated photons with fixed polarizations. Besides, random polarizationcorrelated photons have been studied numerically and shown to ensure equality between players and deception prevention as well, although the CMAB maximum performance is reduced as compared with entangled photon experiments. Autonomous alignment schemes for polarization bases were also experimentally demonstrated based only on decision conflict information observed by an individual without communications between players. This study paves a way for collective decision making in uncertain dynamically changing environments based on entangled quantum states, a crucial step toward utilizing quantum systems for intelligent functionalities.
Introduction
Unique physical attributes of photons have been intensively studied for information processing to solve computationally demanding problems such as timeseries prediction using photonic reservoir computing^{1}, combinatorial optimization based on coherent Ising machines^{2}, and deep learning employing nanophotonic circuits for cognition^{3}. Decision making is another important branch of research where the objective is to identify decisions that will maximize benefits in dynamically changing uncertain environments^{4,5}, with direct applications for reinforcement learning. In this context, the multiarmed bandit (MAB) problem is one of the important fundamental problems in decision making, where the objective is to maximize the rewards obtained from multiple slot machines, whose reward probabilities are unknown^{4}, in contrast with the prisoner problem^{6}. To solve the MAB problem, it is necessary to explore better slot machines. However, too much exploration may result in excessive loss, whereas too quick of a decision or insufficient exploration may lead to missing the best machine. We previously successfully solved the MAB problem, by employing excitation transfer via nearfield coupling^{7}, single photons^{8,9}, and chaotic lasers^{10,11}. This type of decisionmaking problem becomes even more difficult when the number of decision makers, i.e. the number of individuals who join the game or simultaneously play the slot machines, is multiple; then the problem is referred to as a competitive multiarmed bandit (CMAB) problem^{12,13}, which is the focus of the study described herein. In collective decision making, social values are highlighted, such as the maximization of the total social benefits, guarantee of equality among individuals, and so on^{12,13,14,15}. The CMAB problem is important in practical applications ranging from traffic control, where everyone choosing the same road may lead to a traffic jam^{16} to resource allocation in infrastructures, such as communications^{12,17} where everyone wanting to communicate at the same time leads to congestion for example. A fundamental question asked in the study described herein was whether quantum entanglement^{18,19} could bring improvement for reinforcement learning applications^{20,21} or resolve the difficulties of the CMAB problem. The usefulness of entangled photons is addressed in the quantum game literature^{22,23,24,25} regarding resolving Nash equilibrium in noncooperative games formulated by payoff matrices in game theory^{26}. The study described herein was focused on the CMAB problem, which differs from the nonzerosum game^{24} in the sense that the reward in the CMAB problem is not given deterministically, unlike in conventional game theory, but rather probabilistically; thus, one can lose even when the choice is correct, and vice versa. Hence, it is not possible to address the CMAB problem using the payoff matrix formulation alone.
This paper theoretically and experimentally demonstrates the usefulness and superiority of quantumentangled photons for collective decision making and physically solving the MAB problem on the social level, for example, maximizing the total benefits while preserving equality among individuals by overcoming conflicts between individual decisions. Moreover, we demonstrate that deception, or greedily outperforming the other player by trying to receive a greater reward than him, is impossible in a polarizationentangledphotonbased system, while such greedy action is achievable in systems based on classical polarizationcorrelated photons with fixed polarizations. Autonomous and dynamic alignment schemes for polarization bases, which are necessary for CMAB applications, are also experimentally demonstrated based only on decision conflict information observed by an individual without communications between players. To the best of our knowledge, such a detailed analysis of the polarization basis, which is also necessary for quantum games, has not been provided elsewhere. In the Discussion, we consider the case of randomly crosspolarized photon pairs and compare its performance with that of entangled photon pairs. The physical limit of photon sources for collective decision making is also discussed. Finally, we study the influence of using a statistical mixture of entanglementdegreemodulated photons and nonentangled photons on both social welfare and individual freedom for players. Although the following discussion is restricted to photonic entangled states, the transposition to any other type of entanglement system is straightforward, giving a broad generality to the present study.
Decision Making
System architecture
For the simplest case that preserves the essence of the CMAB problem, we consider two players (called Players 1 and 2 hereafter), each of whom selected one of two slot machines (Machines A and B hereafter), with the goal of maximizing the total social reward. The reward probabilities of Machines A and B are denoted as P_{A} and P_{B}, respectively. The amount of reward that could be dispensed by each slot machine per play is assumed to be unity even when multiple players choose that same machine. Possibly, the two players make the same decision at the same time, causing conflict between their decisions. In that case, the reward is divided into two halves, which are allocated to the two players. In terms of reward, this penalty is intended to favour collective, i.e. not conflictual, choices with respect to individual interests. From the viewpoint that the two individuals playing the casino act as a team, when a player chooses the best slot machine, the other one should select the other machine to maximize the sum of their rewards. This example manifests itself as players easily becoming locked in a local minimum due to conflict between their decisions, since everyone wants more rewards and tries to select the higherrewardprobability slot machine, whereas the total team rewards could be increased if they cooperated^{12}.
As the decision is based on simultaneous photon detection, the “twoplayer” configuration requires photon pairs to be generated. An overview of the dedicated experimental setup is shown in Fig. 1. It is based on a standard Sagnac loop architecture^{27} used to generate the photon states by spontaneous parametric down conversion (SPDC), and analyse them for selecting Machine A or Machine B. The details are described in the Methods section. In the branch corresponding to Player 1, each signal photon goes through a halfwave plate (HW_{1}) and is subjected to a polarizing beamsplitter (PBS_{1}). If the photon is detected by the avalanche photodiode corresponding to the horizontally polarized light (APD1), the decision of Player 1 is to choose Machine A, whereas if the photon is detected by the avalanche photodiode corresponding to the vertically polarized light (APD2), then the decision of Player 1 is to choose Machine B. The same hold for player B by exchanging 1 by 3 and 2 by 4. Note that the two slot machines are externally arranged: we emulate the slot machines in a computer using pseudorandom sequences (see Methods for details).
Decision making by a single player
We start with the singleplayer situation, in which either Player 1 or Player 2 attacks the casino. This is essentially equivalent to the experimental demonstration described in ref.^{8}. In this case, it is desirable for the player to choose the higherrewardprobability slot machine, since a larger reward is desired. In order to specify the player action, we now introduce several notations to describe the system. The input photon state for the decision making of Player i (\(i=1,2\)) is denoted as \({\theta }_{i}\rangle \), θ_{i} being the polarization angle; it is delivered by the photon source and is therefore not controllable by the players. The action of Player i is to rotate HW_{i} by an angle \({\theta }_{{{\rm{HW}}}_{i}}\) which modifies the photon state polarization. The roles of HW_{i} and PBS_{i} are given respectively by
and
where \({H}_{i}\rangle \) and \({V}_{i}\rangle \) indicate photon states with horizontal and vertical polarization propagating in orthogonal directions beyond PBS_{i}^{28}. Therefore, the probabilities of photon measurement by APD1 and APD2, for example, which determine whether Player 1 decides to select Machine A or Machine B, are given by \({\cos }^{2}(2{\theta }_{{{\rm{HW}}}_{1}}{\theta }_{1})\) and \({\sin }^{2}(2{\theta }_{{{\rm{HW}}}_{1}}{\theta }_{1})\), respectively. Using the tugofwar principle described in ref.^{8}, which is also summarized in the Sec. 1 of Supplementary Information, the wave plate angle is controlled toward the higherrewardprobability slot machine. The reward probabilities of Machines A and B are chosen as P_{A} = 0.2 and P_{B} = 0.8, respectively, for the first 50 plays. In the next 50 plays, the reward probabilities are swapped, i.e. P_{A} = 0.8 and P_{B} = 0.2, to emulate a variable environment (Fig. 2a). Therefore, from the standpoint of individual players, selecting Machine B is the correct decision in the first 50 plays since it is highly likely to provide a greater reward. Likewise, choosing Machine A is correct for the next 50 plays. The adaptive decision making is implemented by updating the wave plate orientation toward the higherrewardprobability slot machine by revising the polarization adjuster (PA) values^{8}. First, only Player 1 plays the casino. Specifically, Player 1 conducts 100 consecutive slot plays, and this set of plays is repeated 10 times. The red curve shown in Fig. 2b,i represents the correct decision ratio (CDR) defined as the ratio of the number of selections of the machine yielding a higher reward probability over the number of trials at cycle t. This ratio quickly approaches unity, meaning that Player 1 effectively chooses the higherrewardprobability machine, i.e. Machine B. At cycle 51, the CDR drops due to the flip of the reward probabilities. However, the CDR gradually returns to unity as time elapses, which clearly indicates that Player 1 detects the change in the environment and revises the decision to the higherrewardprobability machine, i.e. Machine A. The red curve in Fig. 2b,ii shows the evolution of the accumulated reward averaged over 10 repetitions, which almost linearly increases with time. Its growth is attenuated after cycle 50 due to the reward probability change. Note that the accumulated reward of Player 1 at cycle 100 is about 66.
The blue curves in Fig. 2c,i,ii show the CDR and accumulated reward when only Player 2 played the slot machines. The behaviour is similar to the case of Player 1. The accumulated reward at cycle 100 is 67, which is almost equivalent to that in the case of Player 1. This finding demonstrates the successful decision making by single players as well as the validity of the strategy adopted for solving the asymmetry between APD collection efficiencies (see Methods).
Decision making by two noncooperative players: evidence of interest conflict
Let us now consider the case of 2 Players simultaneously using the casino, using the configuration described in Sec. 2 of the Supplementary Information. Suppose now that both Players 1 and 2 independently play the slot machines; in other words, in a noncooperative manner. The red and blue curves in Fig. 2d,i are the CDRs of Players 1 and 2, respectively, both of which exhibiting traces similar to those in the singleplayer cases (Fig. 2b,c). Actually, both Players 1 and 2 succeed in finding the higherrewardprobability slot machine over time. However, this result points toward conflicts between their decisions; hence, the accumulated rewards of Players 1 and 2 shown by the red and blue curves, respectively, in Fig. 2d,ii are seriously decreased, i.e. nearly half of those in the singleplayer cases. The summation of the accumulated rewards of Players 1 and 2, referred to as the team reward, is depicted by the green curve in Fig. 2d,ii and is 70.9 at cycle 100, which is only slightly larger than in the singleplayer cases. Indeed, the conflict ratio, which is defined as the number of times that the decisions of Players 1 and 2 are identical over the 10 repetitions, exhibits high values close to unity, as shown by the red curve in Fig. 2d,iii. This result indicates that conflicts between decisions occur very frequently during the slot plays.
Decision making by cooperative players: polarizationcorrelated versus polarizationentangled photons
Toward realizing collective decision making, two conditions must be fulfilled. The first method of avoiding conflicts between decisions is to introduce correlations between the two photons, thereby statistically linking the decisions of Players 1 and 2. To this end, we deliver polarizationorthogonal photon pairs denoted by \({\theta }_{1},{\theta }_{2}\rangle \), where
to the two players as input photon states. In practice, θ_{1} = 0 and θ_{2} = π/2, corresponding to a horizontal polarization for Player 1 and a vertical polarization for Player 2 in the PBS polarization basis. The underlying idea of using orthogonal polarizations is to promote the players to select distinct machines. The actions of the two players are again to rotate the waveplates and to analyse the photon states through the polarization beam splitter, represented by Eqs (1) and (2). The probability amplitudes of observing photons at one of APD1 and APD2, and at one of APD3 and APD4 are as follows:
where M denotes the operator describing the action of wave plates and polarization beam splitters. The coincidence of observing photons at APD1 and APD3, and at APD2 and APD4 according to Eqs (4) and (7), respectively, indicates conflict between the decisions made by Players 1 and 2.
Correlated photon polarization is not a sufficient condition to prevent conflict: in view of both players acting as a team, they must also perform coherent choices. This insufficiency led to the second requirement, namely, the use of correlated wave plate angles. Here, we represent this condition by the rotation of both wave plates by the same amount; that is:
By subjecting Eqs (3) and (8) to Eqs (4–7), the probability of conflict between decisions is
which can be obtained by summing the squared moduli of Eqs (4) and (7), while the probability of no conflict between decisions is:
From Eq. (9), \({\theta }_{{{\rm{HW}}}_{1}}\) should be configured as \(\frac{{\theta }_{1}}{2}+\frac{\pi }{4}\times N\), with N being a natural integer to avoid conflict between decisions. Note that this is technically impossible if the photon state θ_{i} randomly varies from photon to photon, as it would be the case with a nonpolarized classical source of light. Furthermore, even when conflict between decisions is successfully avoided, i.e. by using fixed θ_{i} as is the case for signal and idler photons produced by SPDC, the resulting decision is biased toward a specific machine, leading to a reward distribution that favours a specific player. It means that the equality between the players decreases.
To avoid conflict between decisions, the probability amplitudes in Eqs (4) and (7) must always vanish, requiring a second contribution exactly cancelling the oscillating sine and cosine terms. To this end, we utilize a coherent superposition of states corresponding to entangled states. Due to the symmetry of the equations, a natural choice is to exchange the roles of θ_{1} and θ_{2} and to introduce a π phase shift, or, in other words, to use a specific entangled state known as the maximally entangled singlet photon state and given by
where θ_{1} and θ_{2} are orthogonal to each other as specified in Eq. (3). Usually, maximally entangled photons are represented by the forms such as \(\frac{1}{\sqrt{2}}(HV\rangle VH\rangle )\) or \(\frac{1}{\sqrt{2}}(0,1\rangle 1,0\rangle )\) unlike Eq. (11). The reason behind the utilization of Eq. (11) is that we keep coherent notations among single photons (\({\theta }_{1}\rangle ,{\theta }_{2}\rangle \)) and polarizationcorrelated photons (\({\theta }_{1},{\theta }_{2}\rangle \)) introduced above based on the polarization angles of θ_{1} and θ_{2}. Also, by representing polarization angles of input photons, the role of halfwave plates is clearly grasped as discussed below. For such reasons, we introduce the form of entangled photons by Eq. (11).
The probability amplitude originating from the second term of Eq. (11) can be derived according to the following equations:
The probability of photodetection at both APD1 and APD3, meaning that both Players 1 and 2 select Machine A, is then given by the squared modulus of the coherent sum of Eqs (4) and (11), both multiplied by \(1/\sqrt{2}\), which leads to:
This probability always yields zero regardless of the values of θ_{i} and \({\theta }_{{{\rm{HW}}}_{i}}\) as long as the conditions of Eqs (3) and (8) apply. Likewise, the probability that both Players 1 and 2 select Machine B is given by Eq. (16) and therefore is also always zero. Thus, conflicts between decisions never occur, leading to the maximum overall social reward. Conversely, the probability of observing photons at APD1 and APD4 or at APD2 and APD3 can be expressed as
which is always 0.5 when Eqs (3) and (8) are satisfied. Thus, both players have equal opportunities to select each slot machine, which is the foundation of the equality provided by polarizationentangled photons.
Figure 3 characterizes the details of the collective decisionmaking performance with respect to the polarization basis. Figures 3a,i,b,i depict the accumulated reward at cycle 100 as a function of the common orientation of the halfwave plates, which corresponds to the common polarization basis, regarding the decision making based on polarizationcorrelated and polarizationentangled photon pairs, respectively. The red squares, blue diamonds, and green circles correspond to the rewards received by Player 1, Player 2, and the team, respectively.
The team reward is very high, about 100, when the polarization is 0° and 90°, even in the case of correlated photon pairs. For example, a correlated photon pair given by \({H}_{1},\,{V}_{2}\rangle \) is a photon pair corresponding to the polarization angle θ_{1} = 0°. In this case, Player 1 always detects a \({H}_{1}\rangle \) photon, leading to the decision to choose Machine A, whereas Player 2 always detects a \({V}_{2}\rangle \) photon. It indicates that the decision is to select Machine B. Therefore, from the viewpoint of correct decision making, Player 2 achieves a higher CDR in the first 50 cycles and a lower CDR in the second 50 cycles due to the reward probability flipping, as clearly demonstrated in Fig. 3a,ii [0]. A similar tendency is observable with the input photon pair described by \({V}_{1},{H}_{2}\rangle \) that corresponds to the polarization angle θ_{1} = 90°, as shown in Fig. 3a,ii [90]. The conflict ratio remains lower in these specific polarization cases than others, as shown by the red squares in Fig. 3a,iii.
However, in terms of equality (or fairness) this scenario is highly inefficient since a player can select either machine. Indeed, with 0° polarization, only Player 2 earns a greater reward in the first 50 plays, and the imbalance between Players 1 and 2 is significant. More specifically, the equality depicted by the diamonds in Fig. 3a,iii is significantly decreased; it is defined as the average ratio between the numbers of times that the higherrewardprobability machine was selected by Players 1 and 2. The exact definition of equality is provided in Sec. 3 of Supplementary Information.
On the contrary, with the use of entangled photons, the team reward always reaches the theoretical maximum (100) regardless of the common polarization basis, as shown in Fig. 3b,i. It is due to the maximally entangled state that is invariant upon rotation of the basis, provided that the bases are the same for both players. The CDRs of Players 1 and 2 always randomly fluctuate around 0.5, as shown in Fig. 3b,ii. This fluctuation agrees with the fact that nearly identical rewards were received by Players 1 and 2, as can be seen in Fig. 3b,i. The conflict rate, shown by the red squares in Fig. 3b,iii, is always small regardless of the polarization basis, though nonzero due to experimental imperfections. Finally, the equality remains always high for all of the common polarization bases, clearly showing that the entangled states yielded results superior to those achievable using the correlated states in terms of all of the investigated social properties.
To summarize the figures of merit of all of the decisionmaking strategies, the total rewards resulting from using singlephoton decision making for single players and noncooperative and collective decision making are compared in Fig. 4. The orange and green bars depict the experimental and simulation results, respectively, which agree well throughout the experiments. For the twoplayer games, the experimentally obtained rewards of the individual players are also shown: red for Player 1 and purple for Player 2. The diagonal and vertical stripes areas indicate the rewards accumulated during the first 50 and second 50 plays, respectively, emphasizing the effective equality or inequality between the two players. The error bars show the maximum and minimum observed values. Clearly, the maximum team reward is achieved by using entangled photons. Furthermore, the individual rewards in the entangled photon case are higher than those in the case of two noncooperative players, indicating that nonconflict and equal opportunities not only lead to the social maximum, but also benefit the individual players.
In order to check the sensitivity of the previous results on the reward probabilities P_{A} and P_{B}, the latter are changed to P_{A} = 0.4 and P_{B} = 0.6. The data is shown in Sec. 4 and Fig. S2 in the Supplementary Information. It appears that finding the higherrewardprobability machine is more difficult in that case due to the smaller difference between the reward probabilities than in the former cases. As a consequence, the total reward is substantially lower. These differences are due to the longer time needed to reach stable selection of the higherrewardprobability machine. On the contrary, with correlated and entangled photons, the team reward does not change, and the entangled photons again provide the maximum total reward. This finding clearly demonstrates that collective decision making based on entangled photons ensures that the social maximum reward will be achieved regardless of the difficulty of the given problem. This has strong implications in terms of allocation resources as for example in network communications as the maximized efficiency is ensured whatever the actual qualities of the two channels which may fluctuate in time.
Deception or greedy action
An important condition for establishing the social maximum in the CMAB solution by using polarizationentangled photons is sharing of the polarization basis among the players. However, one of the players could have his/her basis misaligned, for instance upon trying to increase his/her own reward on the detriment of the other player’s – an action called deception, or greedy action. In the following, we theoretically investigate how polarizationcorrelated and polarizationentangled photons allow or inhibit “deception”, characterized by \({\theta }_{{\rm{HW}}2}\ne {\theta }_{{\rm{HW}}1}\) (i.e. Eq. (8) no longer holds).
Greedy action in correlatedphotonbased systems
Using polarizationcorrelated photon pairs characterized by Eqs (4–7), the expected reward received by Player 1 in a single play is
The multiplication factor 0.5 in the second and third terms on the righthand side indicates that the reward is halved due to the conflict between decisions. Eq. (18) can be reduced to
Likewise, the expected reward received by Player 2 is
From Eqs (19) and (20), the expected amount by which the reward of Player 2 exceeds that of Player 1 is given by:
It means that the expected reward can be biased toward a particular player depending on the difference between the polarization bases and the incoming photon polarization. In addition, this characteristic implies that no matter what θ_{1} and \({\theta }_{{{\rm{HW}}}_{1}}\) are, it is possible for Player 2 to receive a reward greater than (or at least equal to) that received by Player 1 by configuring
where N is a natural integer. Thus, deception, or greedy action by a player in the system to gain a greater reward than the other, is generally achievable when the system is governed by correlated photons.
Greedy action in polarizationentangledphotonbased systems
Following the same procedure for polarization entangled photons, the expected reward received by Player 1 for a single play is:
or equivalently:
The expected amount of reward received by Player 2 also results in Eq. (24). That is, no matter how Player 2 configures \({\theta }_{{{\rm{HW}}}_{2}}\), the rewards allocated to Players 1 and 2 are the same. Thus, even if Player 2 knows the higherrewardprobability machine and can rotate the wave plate with the intention of receiving a greater reward, such deception is impossible if the system is governed by polarizationentangled photons. Moreover, the expected total reward received by Players 1 and 2, given by two times Eq. (24), i.e.:
is less than its maximum value, given by P_{A} + P_{B}, if the halfwave plate alignment is disrupted \(({\theta }_{{{\rm{HW}}}_{1}}\ne {\theta }_{{{\rm{HW}}}_{2}})\) unless \({\theta }_{{{\rm{HW}}}_{2}}={\theta }_{{{\rm{HW}}}_{1}}+N\times \pi /2\) with N being a natural integer. That is, in addition to the inhibition of deception, the total social benefits are decreased if a selfish action is performed by one of the players.
Experimental investigation of Greedy action
We now investigate a scenario in which one of the players (here Player 2) is greedy, and tries to deceive the other player to obtain a greater reward by rotating his/her halfwave plate in the direction of the higherrewardprobability machine. The orientation of the halfwave plate is controlled toward the higherrewardprobability slot machine by revising the polarization control (PC) value^{8}. The PC value is limited to a maximum and minimum of 10 and −10. Essentially, larger (positively large) and smaller (negatively large) PC values indicate that the halfwave plate is rotated so that the polarization of the photon is toward the horizontal and vertical directions, respectively. (The details of the PC values are described in Sec. 1 of Supplementary Information.)
Figure 5 summarizes the total reward obtained with polarizationentangled photons at cycle 100 for Player 1, Player 2, and the team. The error bars indicate the maximum and minimum rewards. Clearly, the preservation of equality between players and decrease of the team reward obtained for polarizationentangled photons (Fig. 5a) agree with the theoretical analysis. In contrast, for polarizationcorrelated photons (Fig. 5b), Player 2 achieves deception through this greedy action, thereby destroying the equality, with almost no effect on the total team reward. This is the worst configuration, as selfish action only benefits to its author and not at least indirectly to the team.
Autonomous polarizationbasis alignment
As discussed above, polarizationbasis alignment between the players is crucial to realize the maximal social benefits. However, the optical system may suffer from certain environmental disturbances during the decisionmaking operations that degrade its performance. Therefore, online calibration that does not interrupt the decisionmaking operation is important and should be performed by the players and not by the photon provider. Here we discuss the resolution of these issues by two different methods considering that the goal is to configure \({\theta }_{{{\rm{HW}}}_{2}}\) with respect to the unknown \({\theta }_{{{\rm{HW}}}_{1}}\) using an adaptation algorithm.
Assumption I: no prior information about the polarization basis
We first investigate the possibility of aligning the polarization bases without any prior information, exploiting the fact that a halfreward event indicates conflict between decisions. Simultaneously, when the polarization bases are aligned, the probability of conflict between decisions, i.e. \({\sin }^{2}[2({\theta }_{{{\rm{HW}}}_{1}}{\theta }_{{{\rm{HW}}}_{2}})]\), is zero. Therefore, an alignment strategy is as follows.
[K0] If the receipt of a halfreward is observed, update \({\theta }_{{{\rm{HW}}}_{2}}\) by \({\theta }_{{{\rm{HW}}}_{2}}+{{\rm{\Delta }}}_{a}\).
Here, Δ_{a} is a constant employed to change \({\theta }_{{{\rm{HW}}}_{2}}\) gradually. If Δ_{a} is sufficiently small, by repeating [K0], the difference between the halfwave plate angles \({\theta }_{{{\rm{HW}}}_{1}}{\theta }_{{{\rm{HW}}}_{2}}\) should eventually become small; hence, the probability of conflict between decisions should decrease.
In the experimental demonstration, \({\theta }_{{{\rm{HW}}}_{2}}\) is initially −22.5° and Δ_{a} is 12.5°. \({\theta }_{{{\rm{HW}}}_{2}}\) should be made equal to \({\theta }_{{{\rm{HW}}}_{1}}\), which is 0°. The evolution of \({\theta }_{{{\rm{HW}}}_{2}}\) in each sequence is shown by the blue curves in Fig. 6a, while the target angle \({\theta }_{{{\rm{HW}}}_{1}}=0^\circ \) is depicted by the red lines. After applying [K0] twice, \({\theta }_{{{\rm{HW}}}_{2}}\) increases by +25°; hence, \({\theta }_{{{\rm{HW}}}_{2}}\) becomes 2.5°, which is sufficiently close to 0°. Even though \({\theta }_{{{\rm{HW}}}_{2}}\) cannot be exactly zero, it is evident from the evolution of \({\theta }_{{{\rm{HW}}}_{2}}\) shown in Fig. 6a that \({\theta }_{{{\rm{HW}}}_{2}}\) passes through the target angle and continues increasing. That is, even when \({\theta }_{{{\rm{HW}}}_{2}}{\theta }_{{{\rm{HW}}}_{1}}\) is very small, conflict between decisions cannot be perfectly avoided due to the imperfections of the experimental system (see Fig. S1 in Supplementary Information).
Assumption II: no prior information about the polarization basis but memorization of conflict allowed
To prevent such escape from the recalibrated angle due to error signals, one idea is to take the history into account. The revised calibration strategy is as follows.
[KM] If conflicts between decisions have not been detected in the past M plays, the detection of conflict between decisions in the current play is discarded. If there were K occurrences of conflicts between decisions in the past M plays, and the current play yields a conflict between decisions, then update \({\theta }_{{{\rm{HW}}}_{2}}\) by \({\theta }_{{{\rm{HW}}}_{2}}+{{\rm{\Delta }}}_{a}\) and register the occurrence of a conflict between decisions in the memory of the M most recent plays.
In the experimental implementation, M and K are 5 and 1, respectively, which we call [15], while Δ_{a} = 11.25°. As shown by the blue curves in Fig. 6b, the halfwave plate angle successfully approaches the target angle. With Δ_{a} = 11.25°, two position updates via [KM] perfectly resolve the initially imposed misalignment (−22.5°), so that the effect of memorizing past events is clear.
Indeed, although it is rare, the case of Fig. 6b shows the halfwave plate angle still passing through the target angle. By more severely restricting the condition of rotating \({\theta }_{{{\rm{HW}}}_{2}}\) in increasing M, the robustness against errors increases. As shown by the blue curves in Fig. 6c where M and K are 10 and 1, respectively, which is referred to as [110], the event of passing through the target angle is avoided; however, the adaptation is very slow. Specifically, too large of a memory (M = 10, [110]) provides robustness against errors but results in very slow responses, whereas no memory (M = 0, [10])) yields a fast response but reactions that are too sensitive to error signals. A moderate parameter choice ([15]) resolves both the error tolerance and alignment speed issues. The green, red, and blue curves in Fig. 6d summarize the evolution of the accumulated team rewards based on the [10], [15], and [110] calibration rules, respectively, where [15] is optimal for maximizing the total team rewards.
Discussion
As demonstrated herein, entangled photons enable the achievement of maximum social rewards, equality among individuals, and prevention of selfish actions in communities when solving the CMAB problem. Clear differences between polarizationcorrelated and polarizationentangled photons are also observed.
The correlated photon pairs mentioned so far do not share the same behaviour as the polarizationentangled photon pairs. In particular, photons from polarizationcorrelated pairs always have the same input polarization with this description, while entangled photons with state defined as \(\frac{1}{\sqrt{2}}(HV\rangle VH\rangle )\) do not have a fixed one. In that sense, a closer equivalence between correlated and entangled photon pairs is a series of photon pairs crosspolarized along random direction, each with a state of the form \({\theta },{\theta }+{\pi }/2\), with \(\theta \in [0,\,\mathrm{2pi}]\) taking random value for each pair. In this way, any given player still has equal probability on average to select one of the two machines whatever its waveplate angle is, though the relative angle values between players’ polarization bases will tune the conflict rate and thus influence the total reward.
Section 5 of Supplementary Information is dedicated to this case of study, from which three main observations can be made. The first is that individual and total rewards only depend on the relative angle between polarization bases, as is the case with entangled photons. Secondly, maximum total reward obtained with identical measurement bases is 12% lower than for entangled photon pairs. Finally, no deception strategy is able to make a player earn more reward for himself/herself only, as for the entangled photons case.
To summarize, such a system based on randomly crosspolarized photon pairs show the same dynamics at play with entangled photon pairs: individual and total rewards depend only on relative angles between players’ polarization bases, such that the action of a single player is sufficient to improve or reduce both players’ outcome. However, randomly crosspolarized photon pairs present weaker variations in comparison with entangled photon pairs, including lower maximum reward (88 compared with almost 100) and higher minimum reward when bases are at 90 degrees from each other (62 against 50). This kind of resource may then be of interest for applications where lower sensitivity to perturbations is needed, whereas entangled photon pairs are more interesting for maximum performance in lowperturbation conditions.
Now we consider the physical limit of photon sources in collective decision making. As far as the action of the players corresponds to the rotation of the waveplates, the players are fully independent from the photon source. There is no prior relation between the waveplate angles. However, the final decisions of the players depend on the specific photon states from the photon source, being either fixed polarizationcorrelated or polarizationentangled states. From this viewpoint, the photon source does influence players’ potential decisions. As an example, let us remind the situation where the same waveplate configuration is used for both players (with no rotation for simplicity) in the section of Decision making discussed above. For polarizationcorrelated photons, Player 1 always selects Machine A while Player 2 selects Machine B all the time, introducing a bias if the two machines do not have the same reward probability. For polarizationentangled photons, Player 1 and Player 2 randomly select Machine A and Machine B, ensuring equality. Namely, individual decisions cannot be specified by two polarizationentangled photons.
Another interesting feature of quantum entanglement is the robustness against thirdparty attack or source alteration. Indeed, as has been studied and shown for quantum key distribution with polarization entangled photon pair transmission through optical fibers^{29,30}, protocols can be elaborated to be able to detect any eavesdropping attack or alteration of the polarization entanglement of the photon pair source. These rely on tests where both players randomly rotate their waveplate at every measurement, then communicate through public channel which rotation angle they used at a given try and verify whether the conflict rate was below a certain limit for a given relative angle between their measurement bases: if the source is indeed sending entangled photon pairs and no one is eavesdropping, conflict rate should fall to almost zero, whereas a compromised source and/or communication channel would necessarily increase the conflict ratio measured by the players. This aspect may be of interest for applications such as sensitive resource allocation.
It is worth stressing that our work exploits quantum entanglement as a fundamental resource for problems involving multiarmed bandits in competitive scenarios. While our proofofprinciple experiments are based on twophoton entanglement, it should be emphasized that this is not a fundamental limitation: regarding photonic states, one could alternatively exploit singlephoton entanglement with vacuum in a state like \({0\rangle }_{{\rm{A}}}{1\rangle }_{{\rm{B}}}{1\rangle }_{{\rm{A}}}{0\rangle }_{{\rm{B}}}\) where 0 and 1 denote the number of photons in modes A and B (as used with beam splitters for example), respectively. Therefore, a single photon could in principle do the same job as our two photons. However, working with vacuum states is not that easy: this would ultimately require certain homodyne measurements involving local oscillators interfering with single photons in order to develop unambiguous tests^{31} for examples.
If entangled photons turn out to be competitive in terms of social efficiency and equality, the freedom of simultaneous decision of Players 1 and 2 is indeed completely ruled out by the strong authority imposed here by the probability properties of entangled states: whatever the angle of a player’s waveplate, the selection probability for a given machine remains ½. In contrast, with two noncooperative players using single photons, although the total team reward is very poor because of the conflicts between decisions, the freedom of choosing machines is fully guaranteed, since a given machine selection probability follows a Malus law with respect to the waveplate angle. A mixture of (i) social decision making by using an entangledphotonbased decision maker for efficiency and equality within a team, and (ii) individualistic decision making by using a conventional singlephotonbased decision maker^{8,9} for freedom, is an interesting and important topic for future study, especially in dynamically changing uncertain environments. Simultaneously, the conflictavoidance nature of entangled photons may accelerate the exploration phase in finding higherrewardprobability selections among many alternatives, which is another topic requiring future research.
Social decision making and individualistic decisions may be weighted through the modulation of the degree of entanglement^{32} in the following form:
The parameters a, b, and ϕ are real numbers, so that an intended social metric is realized, rather than just maximally entangling the photons as done in this study according to Eq. (7). This can be achieved by adjusting HW_{E} and QW_{E}. A general mathematical formalism, including category theoretic approach^{33}, would facilitate the understanding of complex interdependencies of the entities.
Finally, the scalability of entangledphotonbased decision makers is another fundamental topic in view of many practical applications. It is indeed technologically challenging to realize entanglement among many photons^{34}. The issue of scalability could be addressed by employing for example novel material systems^{35,36,37} or integrated photonic circuits^{38}. It could also be addressed by considering entangled photons combined with a certain coding strategy in order to process many bits of information in a timemultiplexed manner^{11}. Hence, our pioneering results are anticipated to stimulate concrete implementation of entangledphoton (or more generally entangledexcitation)based quantum decision makers.
Conclusion
We have theoretically and experimentally demonstrated that entangled photons efficiently resolve the CMAB problem so that the total social reward is maximized, and social equality is accomplished, while also preventing deceptive or greedy actions. In solving competitive twoarmed bandit problems, two independent players using polarizedsinglephotonbased decision making find the higherrewardprobability machine, but the total reward is seriously decreased due to the conflicts of interest. Fixed polarizationcorrelated photon pairs are useful, to some extent, for deriving nonconflicting decisions, providing freedom of choice for players, and obtaining a greater total reward, but they cannot eliminate conflicts between decisions perfectly. Moreover, this method has difficulty to provide equality. In contrast, entangled photons both enable conflicts between decisions to be avoided and the theoretical maximum total reward to be obtained, while guaranteeing equality regardless of the players’ polarization bases. By highlighting the polarizationbasis requirement for maximum performance with entangled photons, we have investigated the issue of polarization and value alignment in decision making based on polarizationentangled photons. If polarizationentangledphotonbased decision making is employed, we find that deception, or preventing the other player receiving a greater reward by performing greedy actions in the twoarmed bandit problem, is impossible thanks to the physical properties of the polarization dependencies derived by quantum superposition of states. In other words, the reward is always equally shared on average among the players. Furthermore, the total common and individual rewards are decreased by greedy action in such a system, such that autonomous alignment schemes based only on interestconflict information were demonstrated, which can also be used to verify the integrity of the photon pair source and the communication channels. On the contrary, deception is achievable when the decisions are based on fixed polarizationcorrelated photons. Additionally, we have shown that deception prevention and guaranteeing equality between players is also achievable by using randomly crosspolarized photon pairs, at the cost of a lower maximum achievable reward and a lower sensitivity to misalignment between polarization bases. Entangledphotonbased systems are then more interesting for applications where maximum common performance is required or conflicts must be avoided, whereas randomly crosspolarizedphotonbased systems can be of interest if stability and lower sensitivity to perturbations are to be privileged.
The present work hence demonstrated that quantum entanglement, as verified with polarization entangled photon pairs, can be a powerful resource for achieving social maximum benefits as well as addressing key features such as preventing greedy actions when solving the CMAB problem. These features are the foundations of important applications, such as secured allocation of precious resources like energy or frequency bands in communication in the age of artificial intelligence.
Methods
We describe here the experimental setup used to generate photon pairs, as well as the emulation system for the slot machines. The output of an excitation laser passed through a polarizer, a halfwave plate (denoted HW_{E}), a quarterwave plate (QW_{E}), and a dichroic mirror (D), and was incident upon a polarization beam splitter (PBS_{L}) shown in Fig. 1. The horizontally and vertically polarized components of the incoming light travelled clockwise and anticlockwise, respectively, through a Sagnac loop containing a halfwave plate (HW_{L}) and type II quasiphasematched periodically poled KTiOPO_{4} (PPKTP) nonlinear crystal (Cr), where spontaneous parametric down conversion (SPDC) was induced^{39}. The entanglement of orthogonally polarized photons was generated in the PBS_{L}, where the two paths were recombined. The signal and idler photons corresponded to the outgoing components from the PBS_{L}; the signal photons were directed into the branch for the decision making of Player 1, whereas the idler photons travelled to the branch for the decision making of Player 2. Note that the signal and idler photons had distinct wavelengths and were spectrally selected to avoid contamination, which would have affected the final choices of the players.
For entangled photon generation, it was necessary for HW_{E} and QW_{E} to be installed properly to satisfy the condition of generating SPDC equally through both optical paths of the Sagnac loop^{27}. Thus, classical, which means not entangled, polarizationcorrelated photon pairs could also be generated easily by orienting the wave plates so that only the horizontally or vertically polarized component of the excitation laser was incident and travelled through the Sagnac loop either clockwise or anticlockwise. In addition to the benefits of the superior stability of generating SPDC by the Sagnac loop system^{27}, the difference between polarizationcorrelated and polarizationentangled photon pairs could easily be investigated using the same experimental architecture.
A schematic diagram of the experimental setup is shown in Fig. 1. A fibrepigtailed, diodepumped, solidstate laser (Obis, 405 FP) operated at a wavelength of 404 nm with an output power of 100 mW supplied excitation light through a quarterwave plate (QW_{E}) (Thorlabs, WPQ05M405) and halfwave plate (HW_{E}) (Thorlabs, WPH05M405) into a PPKTP crystal (Raicol, typeII colinear SPDC cut) in a polarization Sagnac loop built by a polarization beam splitter (PBS_{L}) (OptoSigma, PBSW12.73/7) and halfwave plate (HW_{L}) (Thorlabs, AHWP05M600)^{S }^{1}. The PPKTP crystal was mounted on a Peltier cooler (Raicol, Peltier controller) to hold the temperature at 313 K. The generated signal light was directed into the branch of Player 1 via a dichroic mirror (Thorlabs, BS011), while the idler light was sent to the branch of Player 2. Due to the limitations of the optical bench, 5mlong optical fibers (Thorlabs, P1780AFC5) were inserted for both branches, followed by halfwave plates (HW_{1} and HW_{2}) (Thorlabs, WPH05M808). In the singleplayer and twononcooperativeplayer cases, polarizers (P*) (Thorlabs, LPNIR050MP2) were used. The signal and idler light were then separately subjected to a grating installed in a spectrometer (Roper Scientific, SP2155 Monochromator) to obtain 805 nm and 812 nm light for the signal and idler, respectively. The signal light was incident upon PBS_{1} (Thorlabs, PBS251) and detected by either APD1 or APD2 (Excelitas, SPCMAQRH16). The idler light went to PBS_{2} (Thorlabs, PBS252) and was detected by either APD3 or APD4. The photon arrival time were evaluated using a 100psbinsize multipleevent time digitizer (timetodigital converter) (FAST ComTec, MCS6A), which was connected to a host computer (HP, Z400) with an Intel Xeon CPU (2.67 GHz), OS Windows 7 professional 64 bit. Three halfwave plates (HW_{E}, HW_{1}, and HW_{2}) and a quarterwave plate (QW_{E}) were mounted on motorized rotary positioners (Thorlabs, PRM1Z8) driven via DC servomotors and controlled by the host computer. LabVIEW (version 2012) was used to control the experimental system, including the slot machine emulation.
The slot machines were emulated in the host computer using pseudorandom numbers ranging from 0 to 1. If the random number was smaller than the reward probability of Machine A (P_{A}), a reward was dispensed. The same mechanism applied for Machine B.
The details of the following materials are shown in the Supplementary Information.

1.
Singleplayer and twononcooperativeplayer decisionmaking strategies

2.
Implementation of collective decision making

3.
Definition of equality

4.
Dependence of total rewards on casino setting

5.
Randomly crosspolarized photon pairs
Data Availability
Data used in this study is available upon reasonable request to the corresponding author.
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Acknowledgements
This work was supported in part by the CREST project (JPMJCR17N2) funded by the Japan Science and Technology Agency, the CoretoCore Program A. Advanced Research Networks and GrantsinAid for Scientific Research (A) (JP17H01277) funded by the Japan Society for the Promotion of Science and Agence Nationale de la Recherche, France, through the TWIN project (Grant No. ANR14CE26000101TWIN) and Placore project (Grant No. ANR13BS100007PlaCoRe), the Université Grenoble Alpes, France, through the Chaire IUA award granted to G.B. and the onemonth invited Professorship of M.N. We also acknowledge the Ph.D. grant to N.C. from the Laboratoire d’excellence LANEF in Grenoble (ANR10LABX5101). The authors thank the valuable contribution of Yannick Sonnefraud who had initiated the entangled photon experiment in the Grenoble group.
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M.N., H.H., A.D., S.H. and G.B. directed the project. M.N. and G.B. designed the system architecture. N.C., D.J., B.B. and G.B. designed and implemented the entangled photon system. N.C., D.J., M.N. and G.B. conducted the optical experiments. B.B., H.S., K.O., H.H., A.D., S.H., and G.B. investigated theoretical fundamentals. M.N., N.C., and G.B. analysed the data. M.N., N.C., B.B., S.H. and G.B. wrote the paper.
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Correspondence to Nicolas Chauvet.
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Chauvet, N., Jegouso, D., Boulanger, B. et al. Entangledphoton decision maker. Sci Rep 9, 12229 (2019) doi:10.1038/s41598019486477
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