Properly Quantized History Dependent Parrondo Games, Markov Processes, and Multiplexing Circuits
PProperly Quantized History Dependent Parrondo Games,Markov Processes, and Multiplexing Circuits
Steven A. Bleiler ∗ , Faisal Shah Khan † November 4, 2018
Abstract
In the context of quantum information theory, “quantization” of various mathemat-ical and computational constructions is said to occur upon the replacement, at variouspoints in the construction, of the classical randomization notion of probability dis-tribution with higher order randomization notions from quantum mechanics such asquantum superposition with measurement. For this to be done “properly”, a faithfulcopy of the original construction is required to exist within the new “quantum” one,just as is required when a function is extended to a larger domain. Here proceduresfor extending history dependent Parrondo games, Markov processes and multiplexingcircuits to their ”quantum” versions are analyzed from a game theoretic viewpoint, andfrom this viewpoint, proper quantizations developed.
For the most part, mathematicians view games as functions, a point of view that allows theenlargement of the sets of possible strategies, outcomes and solutions in a game withoutnecessarily eliminating the players abilities to play the original game in the new context.One way this is achieved is by identifying a game with its payoff function and then extend-ing this function’s domain. Since an extended domain necessarily restricts to the originalone, the original game can be recovered from the new “extended” game when appropriaterestrictions are introduced. This allows meaningful comparison between the game theoreticproperties of the two versions of the game. The use of domain extension is ubiquitous in ∗ Portland State University, Portland, Oregon 97207-0751. † Portland State University, Portland, Oregon 97207-0751. Email: [email protected] a r X i v : . [ qu a n t - ph ] O c t ame theory and is most commonly recognized in the form of mixed strategies , that is,randomizations between the so-called pure strategic choices of a player.To elaborate, recall that a key goal in the study of multi-player, non-cooperative gamesis the identification of potential Nash equilibria. Informally, a Nash equilibrium occurswhen each player chooses to play a strategy that is a best reply to the choice of strategiesof all the other players. In other words, unilateral deviation from the choice of strategy at aNash equilibrium by any player cannot improve that player’s payoff in the game. However,Nash equilibria need not be optimal and in other cases they may not even exist. In suchsituations, games are frequently “enlarged” via the definition of an extended set of strategicchoices and an analysis of the extended game performed. As an example for finite games,passing to mixed strategies often gives rise to Nash equilibria in the “mixed game” that sim-ply do not exist in the original game. Formally, the mixed game results from an extensionof the domain of the payoff function to include randomization between the pure strategiesin the form of probability distributions over the pure strategies. That mixed strategies arisefrom domain extension is also clear from the fact that a faithful copy of every pure strategyset sits inside the corresponding mixed strategy set by considering a pure strategy as playedwith certainty. The so-called mixed strategy equilibria sit outside the collection of purestrategies within the mixed ones, and are thus considered as “new” equilibria of the originalgame.About a decade ago, Meyer [12] proposed the extension of the domain of a game’s pay-off function so as to include quantum mechanical operations. A concrete example of suchan extension was provided soon after by Eisert, Wilkens, and Lewenstein [5], and appliedto the game Prisoner’s Dilemma. The area of study arising from these ideas has come tobe known as quantum game theory. Typically, research in the subject looks for differentthan usual behavior of the payoff function of an n player game under quantization, thatis as mentioned above the replacement at various points in the payoff functions definitionof probability distribution by quantum superposition and measurement. This typically in-volves the replacement of strategic choices or of a family of outcomes by qudits, that isquantum systems having d ”pure” quantum states. Also typically, quantum operations oneach qudit are then considered as a set of quantum strategies for the players. The differentthan usual behavior is often the occurrence of Nash equilibria that were unavailable in theoriginal game. Following these heuristics produces a quantized game which is referred toas a quantization of the original game.Because of the lack of explicit reference to any mathematically formal approach ofdomain extension, these heuristics sometimes produce quantizations that are not true exten-2ions. In such cases, it is impossible to meaningfully compare any game-theoretic resultsthat these quantization generate, such as Nash equilibria, with the results from the originalgame. Indeed, such quantizations truly ”change the game”. On the other hand, proper quan-tizations are true extensions and necessarily restrict to the original game, making possiblemeaningful comparison between the results of the original game and the quantized one. Aformal approach to game quantization via generalizing mixtures developed by one of theauthors [2] is utilized herein to develop proper quantizations of history dependent Parrondogames.It should be noted that games are not the only informatic or computational constructionsand processes currently undergoing extension and analysis via quantization. Two prominentareas of study are Markov processes [10] and the so-called multiplexing circuits [9]. Con-cerns regarding the existence of faithfully embedded versions of the classical object withinthe quantized one have also arisen in these areas.The problems here are also more subtle, because in these areas it is stochasticity, asopposed to probability distribution, serving as the classical component of the constructionbeing replaced by quantum mechanical operations. In particular, the frequently consid-ered replacement quantum concept of completely positive operators and measurement forMarkov processes [10] is also more general than just (normalized) quantum superpositionand measurement, forming what in game theoretic language might be termed mixed quan-tum superposition, i.e. a non-trivial probability distribution over the collection of superpo-sitions. By jumping to this most general form of quantum probability, the existence issue offaithfully embedded copies of the classical process becomes clouded. As illustrated here,and discussed in more detail in a subsequent publication [3], clarity on this issue is gainedby initially restricting consideration to quantum Markov processes obtained through the re-placement of stochasticity by (normalized) quantum superposition and measurement, andsubsequently following to the more general situation.As for quantum multiplexing circuits, motivated by the similarity between the informa-tional behavior of classical multiplexing circuits and certain quantum logic circuits, Shendeet al coined the term quantum multiplexer for the latter in [15]. To be precise, some of thebits in a multiplexing circuit are acted upon by appropriate logic gates under the controlof the logical values of some other bits in the circuit. Quantum circuits exhibiting a sim-ilar structure, such as the one for the controlled-NOT gate, are also considered under thisformulation as quantum multiplexers. However, this definition of a quantum multiplexeris far too informal, allowing for the possibility that a given classical multiplexer may beidentified with a whole class of distinct quantum multiplexers. Thus the relation between3 multiplexer and its ”quantizations” is not functional but relational, and the question ofpreservation of a faithful copy of the original multiplexer in the quantized one becomes illdefined. A functional relation between the original and quantum versions of a particularmultiplexer that arises in the context of history dependent Parrondo games is established insection 5, that at the same time establishes the notion of proper quantization for multiplex-ers. In particular within the quantum versions of the multiplexer lies faithfully embeddedcopies of the classical to which the quantum multiplexer could be restricted.For the Markov processes and multiplexing circuits considered here, the focus of atten-tion on (normalized) quantum superposition and measurement also allows the successfulresolution of the question of what constitutes an appropriate evaluative quantum analogueof the stable state of a Markov process or as expressed in the context of quantum multiplex-ers what constitutes the appropriative evaluative initial state. As mentioned above, furtherdiscussion of the issues expressed in the previous paragraphs and an answer to the initialstate question for the more general contexts appears in a subsequent publication [3]. Parrondo et. al first formulated such games in [14]. The subject of Parrondo games has seenmuch research activity since then. Parrondo games typically involve the flipping of biasedcoins and yield only expected payoffs. A Parrondo game whose expected payoff is positiveis said to be winning . If the expected payoff is negative, the game is said to be losing , andif the expected payoff is , the game is said to be fair .Parrondo games are of interest because sequences of such games occasionally exhibitthe Parrondo effect ; that is, when two or more losing games are appropriately sequenced,the resulting combined game is winning. Frequently, this sequence is randomized whichmeans that the game played at each stage of the sequence is chosen at random with respectto a particular probability distribution over the games being sequenced. A comprehensivesurvey of Parrondo games and the Parrondo effect by Harmer and Abbott can be found in[7]. A special type of Parrondo games is the history dependent Parrondo game, introducedin [14] by Parrondo et al. This game is again a biased coin flipping game, where nowthe choice of the biased coin depends on the history of the game thus far, as opposed tothe modular value of the capital. A history dependent Parrondo game B (cid:48) with a two stagehistory is reproduced in Table 1.As above, let X ( t ) be the capital available to the player at time t . At stage t , this capital4efore last Last Coin Prob. of gain Prob. of loss t − t − at t at t gain gain B (cid:48) p − p gain loss B (cid:48) p − p loss gain B (cid:48) p − p loss loss B (cid:48) p − p Table 1: History dependent game B (cid:48) .goes up or down by one unit, the probability of gain determined by the biased coin used atthat stage. Obtain a Markov process by setting Y ( t ) = (cid:32) X ( t ) − X ( t − X ( t − − X ( t − (cid:33) . (1)This allows one to analyze the long term behavior of the capital in game B (cid:48) via the station-ary state of the process Y ( t ) . The transition matrix for this process is X = p p − p − p p p − p − p (2)The stationary state can be computed from the following equations p π + p π = π (1 − p ) π + (1 − p ) π = π p π + p π = π (1 − p ) π + (1 − p ) π = π and is given by s = π π π π = 1 N p p p (1 − p ) p (1 − p )(1 − p )(1 − p ) (3)5fter setting the free variable v = (1 − p )(1 − p ) and normalization constant N = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) j =1 ( π j ) = (cid:113) ( p p ) + 2 [(1 − p ) p ] + [(1 − p )(1 − p )] which simplifies to N = (1 − p )(2 p + 1 − p ) + p p . Consequently, the probability of gain in a generic run of the game B (cid:48) is p B (cid:48) gain = 1 N (cid:88) j =1 π j p j = p ( p + 1 − p )(1 − p ) (2 p + 1 − p ) + p p (4)where π j is the probability that a certain history j , represented in binary format, will occur,while p j is the probability of gain upon the flip of the last coin corresponding to history j .The expression for p B (cid:48) gain simplifies to p B (cid:48) gain = 1 / (2 + x/y ) (5)with y = p ( p + 1 − p ) > (6)for any choice of the probabilities p , . . . p , and x = (1 − p )(1 − p ) − p p . (7)Therefore, game B (cid:48) obeys the following rule: if x < , B (cid:48) is winning, that is, has positiveexpected payoff; if x = 0 , B (cid:48) is fair; and if x > , B (cid:48) is losing, that is, has negativeexpected payoff. Consider now the two stage history dependent game obtained by randomly sequencing thegames B (cid:48) and B (cid:48)(cid:48) where each of B (cid:48) and B (cid:48)(cid:48) are history dependent Parrondo games with twostage histories. This can be formally considered as a real convex linear combination of thegames B (cid:48) and B (cid:48)(cid:48) , where the coefficients on B (cid:48) and B (cid:48)(cid:48) are given by r , the probability thatthe game B (cid:48) is played at a given stage, and (1 − r ) , the probability that the game B (cid:48)(cid:48) is playedat a given stage. This is because the transition matrix of the Markov process associated to6he randomized sequence is obtained from the transition matrices T (cid:48) and T (cid:48)(cid:48) for the games B (cid:48) and B (cid:48)(cid:48) , respectively, by taking the real convex combination rT (cid:48) + (1 − r ) T (cid:48)(cid:48) . Explicitly,let T (cid:48) = α α − α − α α α − α − α (8)and T (cid:48)(cid:48) = β β − β − β β β − β − β . (9)with α j , β j ∈ [0 , representing the probability of gain for the j coin in games B (cid:48) and B (cid:48)(cid:48) respectively. Then the transition matrix rT (cid:48) + (1 − r ) T (cid:48)(cid:48) of the Markov process forthe randomized sequence of B (cid:48) and B (cid:48)(cid:48) consists of entries t j = rα j + (1 − r )( β j ) and − t j = r (1 − α j ) + (1 − r )(1 − β j ) in the appropriate locations. Call this randomizedsequence of games B (cid:48) and B (cid:48)(cid:48) the history dependent game B (cid:48) B (cid:48)(cid:48) with probability of gain t j . The stable state, computed in exactly the same fashion as the stable state for the game B (cid:48) in section 2 above, has form τ = τ τ τ τ = 1 R t t t (1 − t ) t (1 − t )(1 − t )(1 − t ) (10)with R = (cid:80) j =1 τ j a normalization constant. Using the stable state, the probability of gainin the game B (cid:48) B (cid:48)(cid:48) is computed to be p B (cid:48) B (cid:48)(cid:48) gain = 1 R (cid:88) j =1 τ j t j = t ( t + 1 − t )(1 − t ) (2 t + 1 − t ) + t t . (11)Just as in case of the game B (cid:48) , the expression for p B (cid:48) B (cid:48)(cid:48) gain reduces to p B (cid:48) B (cid:48)(cid:48) gain = 1 / (2 + x (cid:48) /y (cid:48) ) (12)7ith y (cid:48) = t ( t + 1 − t ) > (13)for any choice of the probabilities t , . . . t , and x (cid:48) = (1 − t )(1 − t ) − t t . (14)The game B (cid:48) B (cid:48)(cid:48) therefore behaves entirely like the game B (cid:48) , following the rule: if x (cid:48) < , B (cid:48) B (cid:48)(cid:48) is winning, that is, has positive expected payoff; if x (cid:48) = 0 , B (cid:48) B (cid:48)(cid:48) is fair; and x (cid:48) > , B (cid:48) B (cid:48)(cid:48) is losing, that is, has negative expected payoff.It is therefore possible to adjust the values of the α j and β j in games B (cid:48) and B (cid:48)(cid:48) sothat they are individually losing, but the combined game B (cid:48) B (cid:48)(cid:48) is now winning. This is theParrondo effect. In the present example, the Parrondo effect occurs when (1 − α )(1 − α ) > α α (15) (1 − β )(1 − β ) > β β (16)and (1 − t )(1 − t ) < t t . (17)The reader is referred to [8] for a detailed analysis of the values of the parameters whichlead to the Parrondo effect in such games.Restricting to the original work of Parrondo et al, a special case occurs when we con-sider one of the games in the randomized sequence to be of type A . That is, flipping asingle biased coin which on the surface appears to have no history dependence. However,note that such a game may be interpreted as a history dependent Parrondo game with a twostage history where the coin used in A is employed for every history. Call such a historydependent game A (cid:48) . The transition matrix for A (cid:48) takes the form ∆ = p p − p − p p p − p − p . (18)Now, forming randomized sequences of games A (cid:48) and B (cid:48) is seen to agree with the formingof convex linear combinations mentioned above. In particular, as analyzed in [14] if games A (cid:48) and B (cid:48) are now sequenced randomly with equal probability, the Markov process for the8andomized sequence is given with transition matrix containing the entries q j = ( α j + p ) and − q j = [(1 − α j ) + (1 − p )] in the appropriate locations (recall that the probabilityof win for game A is p ), and has stationary state ρ = ρ ρ ρ ρ = 1 M q q q (1 − q ) q (1 − q )(1 − q )(1 − q ) (19)Denote this randomized sequence of games A (cid:48) and B (cid:48) by A (cid:48) B (cid:48) . The probability of gain inthe game A (cid:48) B (cid:48) is p A (cid:48) B (cid:48) gain = 1 M (cid:88) j =1 ρ j q j = q ( q + 1 − q )(1 − q ) (2 q + 1 − q ) + q q (20)As in the more general case of the game B (cid:48) B (cid:48)(cid:48) , it is now possible to adjust the values ofthe parameters p and p j ’s in games A (cid:48) and B (cid:48) so that they are individually losing, but thecombined game A (cid:48) B (cid:48) is now winning. This happens when − p > p (21) (1 − α )(1 − α ) > α α (22)and (1 − q )(1 − q ) < q q . (23)Parrondo et al show in [14] that when p = − (cid:15) , α = − (cid:15) , α = α = − (cid:15) , α = − (cid:15) , and (cid:15) < , the inequalities (21)-(23) are satisfied. This is Parrondo et al’soriginal example of the Parrondo effect for history dependent Parrondo games.Next we review pertinent features of the formal approach to games developed by Bleilerthat puts game quantization in the context of domain extension. We start with a formal definition.
Definition 3.0.1.
Given a set { , , · · · , n } of players, for each player a set S i ( i = 1 , · · · , n ) of so-called pure strategies , and a set Ω i ( i = 1 , · · · , n ) of possible outcomes , a normal orm game G is a vector-valued function whose domain is the Cartesian product of the S i ’sand whose range is the Cartesian product of the Ω i ’s. In symbols G : n (cid:89) i =1 S i −→ n (cid:89) i =1 Ω i The function G is referred to as the payoff function .Here a play of the game is a choice by each player of a particular strategy s i the collec-tion of which forms a strategy profile ( s , · · · , s n ) whose corresponding outcome profile is G ( s , · · · , s n ) = ( ω , · · · , ω n ) , where the ω i ’s represent each player’s individual outcome.Note that by assigning a real valued utility to each player which quantifies that player’spreferences over the various outcomes, we can without loss of generality, assume that the Ω i ’s are all copies of R , the field of real numbers.In game theory, a rational players’ concern is the identification of a strategy that guar-antees a maximal utility. For a fixed ( n − -tuple of opponents’ strategies then, rationalplayers seek a best reply , that is a strategy s ∗ that delivers a utility at least as great, if notgreater, than any other strategy s . When every player can identify such a strategy, the result-ing strategy profile is called a Nash equilibrium or occasionally just an equilibrium of thenormal form game G . Other ways of expressing this concept include the observation that noplayer can increase his or her payoffs by unilaterally deviating from his or her equilibriumstrategy, or that at equilibrium all of a player’s opponents are indifferent to that player’sstrategic choice.However, normal form games need not have Nash equilibria amongst the pure strategyprofiles. As remarked above, game theoretic formalism now calls upon the theorist to extendthe normal form game G by enlarging the domain and extending the payoff function. Ofcourse, the question of if and how a given function extends is a time honored problem inmathematics and the careful application of the mathematics of extension is what will drivethe formalism for quantization. In the classical theory, the standard extension at this point isconstructed by allowing each player to randomize between his strategic choices, a processreferred to as mixing . A mixed strategy for player i is an element of the set of probability distributions over the setof pure strategies S i . Formally, for a given set X , denote the probability distributions over X by ∆( X ) and note that when X is finite, with k elements say, the set ∆( X ) is just the10igure 1: Extension of the game G to G mix . k − dimensional simplex ∆ ( k − over X , i.e., the set of real convex linear combinationsof elements of X . Of course, we can embed X into ∆( X ) by considering the element x asmapped to the probability distribution which assigns 1 to x and 0 to everything else. For agiven game G , denote this embedding of S i into ∆( S i ) by e i .Let p = ( p , . . . , p n ) be a mixed strategy profile. Then p induces the product distri-bution over the product (cid:81) S i . Taking the push out by G of the product distribution (i.e.,given a probability distribution over strategy profiles, replace the profiles with their imagesunder G ) then gives a probability distribution over the image of G , Im G , considered as amulti-set. Following this by the expectation operator E , we obtain the expected outcome ofthe profile p . Now our game G can be extended to a new, larger game G mix . Definition 3.1.1.
Assigning the expected outcome to each mixed strategy profile we obtainthe extended game G mix : (cid:89) ∆( S i ) → (cid:89) Ω i Note G mix is a true extension of G as G mix ◦ Π e i = G ; that is, the diagram in Figure1 is commutative.Having placed a game G and the corresponding game G mix in the domain extensioncontext, the next natural step is to place the notions of mediated communication and cor-related equilibrium [1, 13] in a similar context. However, since the latter have no directrelevance tot he topic of this article, we simply refer the reader to [2] for details. Classically, probability distributions over the outcomes of a game G (the image of G ) wereconstructed. Now the goal is to pass to a more general notion of randomization, that ofquantum superposition. Begin then with a Hilbert space H that is a complex vector space11quipped with an inner product. For the purpose here assume that H is finite dimensional,and that there exists a finite set X which is in one-to-one correspondence with an orthogonalbasis B of H . When the context is clear as to the basis to which the set X is identified,denote the set of quantum superpositions for X as Q S ( X ) . Of course, it is also possible todefine quantum superpositions for infinite sets, but for the purpose here, one need not be sogeneral. What follows can be easily generalized to the infinite case.As mentioned above, the underlying space of complex linear combinations is a Hilbertspace; therefore, we can assign a length to each quantum superposition and, up to phase,always represent a given quantum superposition by another that has length 1.For each quantum superposition of X we can obtain a probability distribution over X by assigning to each component the ratio of the square of the length of its coefficient tothe square of the length of the combination. This assignment is in fact functional, and isabusively referred to as measurement. Formally: Definition 3.2.1.
Quantum measurement with respect to X is the function q measX : Q S ( X ) −→ ∆( X ) given by αx + βy (cid:55)−→ (cid:32) | α | | α | + | β | , | β | | α | + | β | (cid:33) Note that geometrically, quantum measurement is defined by projecting a normalizedquantum superposition onto the various elements of the normalized basis B . Denote quan-tum measurement by q meas if the set X is clear from the context.Now given a finite n -player game G , suppose we have a collection Q , . . . , Q n of non-empty sets and a protocol , that is, a function Θ : (cid:81) Q i → Q S (ImG) . Quantum measure-ment q meas ImG then gives a probability distribution over
ImG . Just as in the mixed strategycase we can then form a new game G Θ by applying the expectation operator E . Definition 3.2.2.
Assigning the expected outcome to each probability distribution over Im G that results from quantum measurement, we obtain the quantized game G Θ : (cid:89) Q i → (cid:89) Ω i Call the game G Θ thus defined to be the quantization of G by the protocol Θ . Callthe Q i ’s sets of pure quantum strategies for G Θ . Moreover, if there exist embeddings e (cid:48) i : S i → Q i such that G Θ ◦ (cid:81) e (cid:48) i = G , call G Θ a proper quantization of G . If there12igure 2: Extension of the game G to G Θ . exist embeddings e (cid:48)(cid:48) i : ∆( S i ) → Q i such that G Θ ◦ (cid:81) e (cid:48)(cid:48) i = G mix , call G Θ a complete quantization of G .This formal approach to games, and in particular game quantization, is summed up inthe commutative diagram of Figure 2. Note that for proper quantizations, the original gameis obtained by restricting the quantization to the image of (cid:81) e (cid:48) i . For general extensions, theGame Theory literature refers to this as “recovering” the game G .Though the following plays no role here, it is worth noting that nothing prohibits usfrom having a quantized game G Θ play the role of G in the classical situation and by con-sidering the probability distributions over the Q i , create a yet larger game G m Θ , the mixedquantization of G with respect to the protocol Θ . For a proper quantization of G , G m Θ isan even larger extension of G . The game G m Θ is described in the commutative diagramof Figure 3. In abstract quantum mechanics, one can access this more general notion ofa mixed quantum operation directly via the consideration of completely positive operatorson a quantum system, and this approach can be used to create quantum games directly.However in this more direct construction the importance and true role of embeddings of theoriginal and mixed games is obscured, and the existence of subgames identical to the origi-nal and mixed games becomes problematic. This is exactly what happens in the context ofMarkov processes, see [10]. 13igure 3: Extension of the game G Θ to G m Θ . Figure 4:
Proper quantization of a one player game with strategy space S via the protocol Θ andquantum strategy space Q . In many cases, the Q i of the quantization protocols are expressed as quantum operations.These operations require a state to “operate” on. In this situation the definition of protocoladditionally requires the definition of an “initial state” together with the family of quantumoperations which act upon this state, along with a specific definition of how these quantumoperations are to act. As exemplified in the following sections, different choices for theinitial state can give rise to very different protocols sharing a common selection and actionof quantum operations. When a protocol Θ depends on a specific initial state I , the protocolis then denoted by Θ I .In subsequent sections, a version of the formalism adapted to one player games will beutilized to construct quantizations of history dependent Parrondo games that are in fact do-main extensions. The underlying quantization paradigm being the replacement of probabil-ity distributions by the more general notion of quantum superposition followed by measure-ment. The functional diagram for proper quantization that will be utilized is given in Figure4 where the commutativity of the diagram requires that E ◦ ( q Im Gmeas ) ◦ Θ ◦ e = G Θ ◦ e = G .14ncorporating the discussion above, when games G s and protocols Θ I depend on a giveninitial states s and I , respectively, the initial states s and I are regarded as part of the singleplayer’s strategic choice. In these cases, the embedding e of S into Q additionally requiresthe mapping of the initial state s of G s to the initial state I of the protocol Θ I . The resultingquantum game is denoted by G Θ I s . A major insight about quantized games that results from the formal domain extension ap-proach to quantum games in section 3 is that for the quantization of a game to be game-theoretically significant, it must be proper. Previous work on the quantization of the historydependent Parrondo game by Flitney, Ng, and Abbott (FNA) [6] produced quantizationsthat are not proper. In this chapter, after recalling the basic facts regarding Parrondo gamesand the FNA quantization protocols, proper quantizations for the history dependent Par-rondo game and their randomized sequences are constructed.In [6], Flitney, Ng, and Abbott quantize the type A (cid:48) Parrondo game by considering theaction of an element of SU (2) on a qubit and interpret this as “flipping” a biased quantumcoin. They consider history dependent games with ( n − stage histories, and in the lan-guage of the Bleiler formalism, quantize these games via a family of protocols. In everyprotocol, n qubits are required and the unitary operator representing the entire game is a n × n block diagonal matrix with the × blocks composed of arbitrary elements of SU (2) . In the language of quantum logic circuits, this is a quantum multiplexer [9]. Thefirst ( n − qubits represent the history of the game via controls, as illustrated in Figure5 for a two stage history dependent game similar to the game B (cid:48) given in Table 1. Eachprotocol is defined as the action of the quantum multiplexer on the n qubits.The quantum multiplexer illustrated in Figure 5, where the elements Q . . . Q are ele-ments of SU (2) , operates as follows. When the basis of the state space ( C P ) ⊗ of threequbits is the computational basis B = {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} . Part of the quantization protocol for the history dependent Parrondo game. The first twowires represent the history qubits. the quantum multiplexer takes on the form of an × block diagonal matrix of the form Q = Q Q Q
00 0 0 Q , (24)where each Q j ∈ SU (2) . That is Q j = (cid:32) a j − b j b j a j (cid:33) (25)with a j , b j ∈ C satisfying | a j | + | b j | = 1 .For further description of the workings of the quantum multiplexer, the following con-vention, found in D. Meyer’s original work [12], will be used. Let a “win” or “gain” for aplayer be represented by the action “No Flip” which is the identity element of SU (2) . Forexample, in Meyer’s quantum penny flip game, the “quantum coin” is in the initial stateof “Head” represented by | (cid:105) and a gain for the player using the quantum strategies occurswhen the final orientation state of the coin is observed to be | (cid:105) . This is contrast to theconvention in FNA [6] where | (cid:105) represents a gain.Now the first two qubits of an element of B represent a history of the classical game,with | (cid:105) representing gain ( G ) and the | (cid:105) representing loss ( L ). The blocks Q j act on thethird qubit in the circuit under the control of the history represented by the binary configu-ration of the first two qubits. For example, if the first two qubits are in the joint state | (cid:105) ,the SU (2) action Q is applied to the third qubit. Similarly, for the other three basic initialjoint states of the first two qubits. This models the historical dependence of the game byhaving the history ( G, G ) correspond to the initial joint state | (cid:105) of the first two qubits, the16istory ( G, L ) correspond to the initial joint state | (cid:105) , the history ( L, G ) correspond to theinitial joint state | (cid:105) , and the history ( L, L ) correspond to the initial joint state | (cid:105) . Thus,an appropriate action is taken for each history.Recall from section 2 that the evaluation of the behavior of the classical history de-pendent Parrondo game requires more than just the Markov process. The evaluation alsorequires the stable state and a payoff rule. Note that the results of applying the quantummultiplexer depends entirely on the initial state on which it acts. That is, different initialstates result in differing final states. The payoff rule used by Abbott, Flitney, and Ng re-sembles that for the classical game in that the quantized versions are winning when theexpectation greater than (gain capital), fair if the expectation is equal to (break even),and losing if the expectation is less than (lose capital). Further, as in the classical gamethis question is decided by examining the probability of gain versus the probability of loss.In particular, if the probability of gain is greater than , the quantum game is winning. The FNA quantization protocols for the history dependent game attempt to replace theclassical biases of the coins in the game with arbitrary elements of SU (2) and the stablestate of Markov process describing the dynamics of the game with certain initial states ofthe qubits on which a quantum multiplexer, composed of the arbitrary elements of SU (2) ,acts. The problems with the FNA quantization protocols are two-fold. First, the attemptedembedding of the classical history dependent game into the quantized game by replacingthe biases of the classical coins with SU (2) elements, turns out to be relational ratherthan functional. That is, Equations (24) and (25) together give a large family of quantummultiplexers that the classical game maps could be mapped into, but no restrictions on thevarious choices for replacement of the biased coins that could give rise to an embedding.This relational mapping makes it impossible to recover the classical game by restricting thequantized game to the image of an embedded copy of the original. Therefore, the FNAquantization of the history dependent Parrondo game is not proper.A second problem arises from the choice of initial state. No attempt is made in FNAto produce an analog of the stable state of the corresponding Markov process. Instead,the authors merely note that different initial states can produce different results, and inparticular focus attention on two arbitrary initial states, one the maximally entangled state √ ( | (cid:105) + | (cid:105) ) , the other the basic state | (cid:105) . In the latter, the authors assert that thequantum game behaves like a classical game with fixed initial history ( L, L ) , according totheir convention in which | (cid:105) represents loss. Note that this is not a proper quantization of17ny classical history dependent game as it fails to incorporate the other histories representedin the stable state. For | (cid:105) = and when acted upon by the quantum multiplexer in Equation (24) produces the output a b which makes the failure of the protocol to incorporate the other histories apparent.A similar situation occurs where only the histories | (cid:105) and | (cid:105) are incorporated.This protocol is also not proper as only the histories ( L, L ) and ( G, G ) are non-triviallyrepresented in the initial state. For √ | (cid:105) + | (cid:105) ) = 1 √ √ a b − b a from which, again, the failure of the protocol to incorporate the other histories is apparent.Thus, both of the FNA quantization protocols fail to reproduce the Markovian dynamicsof the original history dependent Parrondo game and cannot be restricted to the payofffunction of the original game.Flitney et al also consider various “sequences” of the quantum games A (cid:48) and B (cid:48) , where B (cid:48) is played with three qubits and quantized using the maximally entangled initial state.These sequences are defined by compositions of the unitary operators defining the games.Indeed, these sequences now produce the results presented in [6]. These results are certainlynovel and perhaps carry scientific significance; however, they fail to carry game-theoreticsignificance as, with respect to the classical Parrondo games, each arises from a quantizationthat is not proper. In light of the Bleiler formalism discussed in section 3, constructing proper quantizationsof games is a fundamental problem for quantum theory of games. In this section, a properquantization paradigm is developed for both history dependent Parrondo games and ran-domized sequences of such.It is crucial at this stage to view the history dependent Parrondo game discussed in sec-tion 2 in the more formal game-theoretic context of domain extension discussed in section3. For this, consider the Parrondo games as one player games as a function, where theone player’s strategic choices in part correspond to the biases of the coins. For a historydependent Parrondo game with two historical stages, Parrondo et al refer to these choicesas a “choice of rules.” However, the mere choice of biases for the coins is not enough todetermine a unique normal form for these history dependent Parrondo games. In particular,19n initial probability distribution over the allowable histories is also required. Although anyspecific distribution suffices to uniquely determine such a normal form, as the structure ofthe game is given by a Markov process, there is a natural choice for this initial distribu-tion. Though this issue is not discussed by Parrondo et al, these authors immediately focuson this natural choice, namely, the distribution corresponding to the stationary state of theMarkov process representing the game.As functions, these history dependent Parrondo games now map the tuple ( P, s ) intothe element ( π p , π (1 − p ) , π p , π (1 − p ) , π p , π (1 − p ) , π p , π (1 − p )) of the probability payoff space [0 , × , where s = ( π , π , π , π ) ∈ ∆(hist G ) is thestationary state of the Markov process with transition matrix defined by P = ( p , p , p , p ) ∈ [0 , × , as in Equation (2). Formally, G s : [0 , × × ∆(hist G ) → [0 , × (26) G s : ( P, s ) (cid:55)→ ( π p , π (1 − p ) , π p , π (1 − p ) , π p , π (1 − p ) , π p , π (1 − p )) (27)The outcomes winning , breaking even , or losing to the player occur when p B (cid:48) gain > , p B (cid:48) gain = , and p B (cid:48) gain < , respectively.Note that in this more formal game-theoretic context for history dependent Parrondogames, the dependence of these games on the initial probability distribution s is made clear.This initial probability distribution plays the role of the initial state s for the classical game G s appearing in the proper quantization discussion in section 3.3.Consider the history dependent game B (cid:48) with only 2 histories. As in the FNA protocol,the quantization protocol for this game uses a three qubit quantum multiplexer with matrixrepresentation Q = Q Q Q
00 0 0 Q with each Q j ∈ SU (2) , together with an initial state.To reproduce the classical game, first embed the four classical coins that define the game B (cid:48) into blocks of the matrix Q corresponding to the appropriate history. The embedding is20ia superpositions of the embeddings of the classical actions of “No Flip” and “Flip” on thecoins into SU (2) given either by N = (cid:32) (cid:33) , F = (cid:32) − ηη (cid:33) (28)or by N ∗ = (cid:32) i i (cid:33) , F ∗ = (cid:32) − iηiη (cid:33) (29)with η = 1 . Call the embeddings in equations (28) basic embeddings of type 1 and theembedding in equations (29) basis embeddings of type 2 . Choosing the basic embedding oftype 1 embeds the j th coin into SU (2) as Q j = √ p j N + (cid:113) (1 − p j ) F = (cid:32) √ p j − (cid:112) − p j η (cid:112) − p j η √ p j (cid:33) (30)where p j is the probability of gain when the j th coin is played in the classical game B (cid:48) given in Table 1. Note that the probabilities p j of gaining are associated with the classicalaction N in line with Meyer’s original convention from [12] where | (cid:105) represents a gain.Hence, the elements of the subset W = ( | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) ) of B all represent possible gaining outcomes in the game. The probability of gain in thequantized game is therefore the sum of the coefficients of the elements of W that resultfrom measurement.Next, set the initial state I equal to (cid:113)(cid:80) nj =1 π j √ π √ π √ π √ π , (31)21here the π j are the probabilities with which the histories occur in the classical game, ascomputed from the stationary state of the Markovian process of section 2. The quantummultiplexer Q acts on I to produce the final state F I = 1 (cid:113)(cid:80) j =1 π j √ p π η (cid:112) (1 − p ) π √ p π η (cid:112) (1 − p ) π √ p π η (cid:112) (1 − p ) π √ p π η (cid:112) (1 − p ) π . (32)Measuring the state F I in the observational basis and adding together the resulting coeffi-cients of the elements of the set W (cid:48) gives the probability of gain in the quantized game tobe p QB (cid:48) gain = 1 (cid:80) j =1 π j (cid:88) j =1 p j π j = 1 N (cid:88) j =1 p j π j (33)which is equal to the probability of gain in the classical game.This proper quantization paradigm is based on the philosophy discussed in section3.3. That is, a proper quantization of a classical game G s that depends on an initialstate s requires that s be embedded into an initial state I on which the quantum multi-plexer acts. Here, the initial state s = ( π , π , π , π ) ∈ [0 , × embeds as the initialstate I ∈ ( C P ) ⊗ given in expression (31). The resulting game G Θ I s is the quanti-zation of the classical game G s by the protocol Θ I which maps the tuple ( Q, I ) , with Q = ( Q , Q , Q , Q ) ∈ [ SU (2)] × to F I ∈ ( C P ) ⊗ given in Equation (32). Formally, Θ I : [ SU (2)] × × ( C P ) ⊗ → ( C P ) ⊗ (34) Θ I : ( Q, I ) (cid:55)→ F I (35)By projecting on to the gaining basis W , one now gets a quantum superposition over theimage Im G of the game G . Finally, quantum measurement produces Im G . Call P roj thefunction that projects F I on to W , and denote quantum measurement by q meas . Then G Θ I s = q meas ◦ P roj ◦ Θ I : ( Q, I ) (cid:55)→ Im G (36)22igure 6: Proper Quantization, using the embedding e , of the History Dependent Game via thequantization protocol Θ I . is a proper quantization of the payoff function of the normal form of classical history de-pendent game G s given in Equations (26) and (27). Equation (36) can be expressed by thecommutative diagram of Figure 6, which the reader is urged to compare with Figure 4 insection 3.3.Note that by embedding s into I , the notion of randomization via probability distri-butions is generalized in the quantum game to the higher order notion of randomizationvia quantum superpositions plus measurement. In particular, the probability distribution P = ( p , p , p , p ) ∈ [0 , × that defines the Markov process associated with the gameis replaced with the quantum multiplexer Q = ( Q , Q , Q , Q ) ∈ [ SU (2)] × associatedwith the quantized game, and the stable state s of the Markov process is replaced with aninitial evaluative state I of the quantum multiplexer. Recall from section 2.1 that randomized sequences of games B (cid:48) and B (cid:48)(cid:48) are analyzed via aMarkov process with transition matrix equal to a real convex combination of the transitionmatrices of each game in which B (cid:48) is played with probability r and B (cid:48)(cid:48) with probability (1 − r ) . Moreover, such a sequence is considered to by an instance of a history dependentgame denoted as B (cid:48) B (cid:48)(cid:48) .Motivated by the discussion on proper quantization of the game Parrondo games B (cid:48) and B (cid:48)(cid:48) above, let us now consider a higher order randomization in the form of a quantumsuperposition of the quantum multiplexers used in the proper quantization of the the games B (cid:48) and B (cid:48)(cid:48) with the goal of producing a proper quantization of the game B (cid:48) B (cid:48)(cid:48) .As in section 5, associate the quantum multiplexer Q (cid:48) = ( Q (cid:48) , Q (cid:48) , Q (cid:48) , Q (cid:48) ) with the23ame B (cid:48) , where Q (cid:48) j = √ α j N + (cid:113) (1 − α j ) F = (cid:32) √ α j − (cid:112) − α j η (cid:112) − α j η √ α j (cid:33) , Next, associate the quantum multiplexer Q (cid:48)(cid:48) = ( Q (cid:48)(cid:48) , Q (cid:48)(cid:48) , Q (cid:48)(cid:48) , Q (cid:48)(cid:48) ) with the game B (cid:48)(cid:48) , where Q (cid:48)(cid:48) j = (cid:112) β j N ∗ + (cid:113) (1 − β j ) F ∗ = (cid:32) √ β j i − (cid:112) − β j ( iη ) (cid:112) − β j iη √ β j i (cid:33) . Now consider the quantum superposition
Σ = γ (cid:48) Q (cid:48) + γ (cid:48)(cid:48) Q (cid:48)(cid:48) (37) = γ (cid:48) Q (cid:48) + γ (cid:48)(cid:48) Q (cid:48)(cid:48) γ (cid:48) Q (cid:48) + γ (cid:48)(cid:48) Q (cid:48)(cid:48) γ (cid:48) Q (cid:48) + γ (cid:48)(cid:48) Q (cid:48)(cid:48)
00 0 0 γ (cid:48) Q (cid:48) + γ (cid:48)(cid:48) Q (cid:48)(cid:48) (38)of the quantum multiplexers Q (cid:48) and Q (cid:48)(cid:48) with ( γ (cid:48) ) + ( γ (cid:48)(cid:48) ) = 1 , (cid:12)(cid:12) γ (cid:48) (cid:12)(cid:12) = r, (cid:12)(cid:12) γ (cid:48)(cid:48) (cid:12)(cid:12) = (1 − r ) , γ (cid:48) γ (cid:48)(cid:48) − γ (cid:48)(cid:48) γ (cid:48) = 0 (39)and γ (cid:48) Q (cid:48) j + γ (cid:48)(cid:48) Q (cid:48)(cid:48) j = (cid:32) γ (cid:48) √ α j + γ (cid:48)(cid:48) (cid:112) β j i − (cid:0) γ (cid:48) (cid:112) − α j − γ (cid:48)(cid:48) (cid:112) − β j i (cid:1) η (cid:0) γ (cid:48) (cid:112) − α j + γ (cid:48)(cid:48) (cid:112) − β j i (cid:1) η γ (cid:48) √ α j − γ (cid:48)(cid:48) (cid:112) β j i (cid:33) (40)Set the evaluative initial state in this case equal to I = 1 (cid:113)(cid:80) nj =1 τ j √ τ √ τ √ τ √ τ (41)24here the τ j are the probabilities that form the stationary state of the classical game B (cid:48) B (cid:48)(cid:48) given in Equation (10). The claim is that the quantum multiplexer Σ in Equation (37)together with the evaluative initial state I in Equation (45) define a proper quantization ofthe classical game B (cid:48) B (cid:48)(cid:48) in which B (cid:48) is played with probability r and and B (cid:48)(cid:48) is played withprobability (1 − r ) .To check the validity of this claim, compute the output of Σ for the evaluative initialstate I in Equation (45): (cid:113)(cid:80) nj =1 τ j √ τ ( γ (cid:48) √ α + γ (cid:48)(cid:48) √ β i ) √ τ (cid:0) γ (cid:48) √ − α + γ (cid:48)(cid:48) √ − β i (cid:1) η √ τ ( γ (cid:48) √ α + γ (cid:48)(cid:48) √ β i ) √ τ (cid:0) γ (cid:48) √ − α + γ (cid:48)(cid:48) √ − β i (cid:1) η √ τ ( γ (cid:48) √ α + γ (cid:48)(cid:48) √ β i ) √ τ (cid:0) γ (cid:48) √ − α + γ (cid:48)(cid:48) √ − β i (cid:1) η √ τ ( γ (cid:48) √ α + γ (cid:48)(cid:48) √ β i ) √ τ (cid:0) γ (cid:48) √ − α + γ (cid:48)(cid:48) √ − β i (cid:1) η . The probability of gain produced upon measurement of this output is p QB (cid:48) B (cid:48)(cid:48) gain = 1 (cid:80) nj =1 τ j (cid:88) j =1 (cid:12)(cid:12)(cid:12) √ τ j ( γ (cid:48) √ α j + γ (cid:48)(cid:48) (cid:112) β j i ) (cid:12)(cid:12)(cid:12) (42)which simplifies to R (cid:88) j =1 τ j (cid:104)(cid:12)(cid:12) γ (cid:48) (cid:12)(cid:12) α j + (cid:12)(cid:12) γ (cid:48)(cid:48) (cid:12)(cid:12) β j + √ α j β j i (cid:0) γ (cid:48) γ (cid:48)(cid:48) − γ (cid:48)(cid:48) γ (cid:48) (cid:1)(cid:105) . (43)Using the conditions set up in Equation (39), the previous expression further simplifies togive p QB (cid:48) B (cid:48)(cid:48) gain = 1 R (cid:88) j =1 τ j [ rα j + (1 − r ) β j ] = 1 R (cid:88) j =1 τ j t j . which is exactly that given in Equation (11) in section 5.1 for the classical game B (cid:48) B (cid:48)(cid:48) .Again, note that this proper quantization paradigm requires mapping of the initial stateof the classical game B (cid:48) B (cid:48)(cid:48) , which is a probability distribution, into an initial state whichthe quantization protocol acts on, which is a higher order randomization in the form ofa quantum superposition which measures appropriately with respect to the observational25asis. The image of the normal form of the quantum game in [0 , agrees precisely with p QB (cid:48) B (cid:48)(cid:48) gain . Note that in this proper quantization of B (cid:48) B (cid:48)(cid:48) , not only is the initial state of theclassical game replaced by a quantum superposition, but also the probabilistic combinationof the transition matrices of the classical games is replaced with a quantum superpositionof the quantum multiplexers associated with each classical game. Recall from section 2.1 the classical analysis of the special case of the randomized sequenceof history dependent Parrondo games, with r = (1 − r ) = , in which one of the games is A (cid:48) . The game A (cid:48) has the property that regardless of history, game A is always played. Sucha sequence was considered to by an instance of a history dependent game denoted by A (cid:48) B (cid:48) .In this section, a proper quantization of the randomized sequence is shown to follow as aspecial case of the proper quantization of the classical game B (cid:48) B (cid:48)(cid:48) developed in section 5.1above.As before, associate the quantum multiplexer Q (cid:48) = ( Q (cid:48) , Q (cid:48) , Q (cid:48) , Q (cid:48) ) , where Q (cid:48) j = √ p j N + (cid:113) (1 − p j ) F = (cid:32) √ p j − (cid:112) − p j η (cid:112) − p j η √ p j (cid:33) , with the game B (cid:48) . Now, first embed the game A into SU (2) using basic embeddings oftype 2. That is, A = √ pN ∗ + (cid:112) (1 − p ) F ∗ = (cid:32) √ pi −√ − p ( iη ) √ − piη √ pi (cid:33) . The transition matrix for the game A (cid:48) was given in Equation (18) and is reproduced here: ∆ = p p − p − p p p − p − p . The form of ∆ suggests that the quantum multiplexer Q (cid:48)(cid:48) = ( A, A, A, A ) should be associ-26ted with the game A (cid:48) . Now let γ (cid:48) = γ (cid:48)(cid:48) = √ in Equation (37) so that Σ = 1 √ (cid:48) + Q (cid:48) ) = 1 √ A + Q (cid:48) A + Q (cid:48) A + Q (cid:48)
00 0 0 A + Q (cid:48) (44)with A + Q (cid:48) j = (cid:32) √ pi + √ p j − (cid:0) √ − p ( iη ) + (cid:112) − p j η (cid:1) √ − piη + (cid:112) − p j η √ pi + √ p j (cid:33) = (cid:32) √ p j + √ pi − (cid:0)(cid:112) − p j − √ − pi (cid:1) η (cid:0)(cid:112) − p j + √ − pi (cid:1) η √ p j − √ pi (cid:33) . With the evaluative initial state I = 1 (cid:113)(cid:80) nj =1 ρ j √ ρ √ ρ √ ρ √ ρ (45)where the ρ j are the probabilities that form the stationary state of the classical game A (cid:48) B (cid:48) given in Equation (19), the quantum multiplexer Σ in Equation (37) defines a proper quan-tization of the classical game AB (cid:48) when both A and B (cid:48) are played with equal probability.27o see this, compute the output of Σ for the evaluative initial state I in Equation (45): (cid:113) (cid:80) nj =1 ρ j √ ρ ( √ pi + √ p ) √ ρ (cid:0) √ − p + √ − pi (cid:1) η √ ρ ( √ pi + √ p ) √ ρ (cid:0) √ − p + √ − pi (cid:1) η √ ρ ( √ pi + √ p ) √ ρ (cid:0) √ − p + √ − pi (cid:1) η √ ρ ( √ pi + √ p ) √ ρ (cid:0) √ − p + √ − pi (cid:1) η . The probability of gain produced upon measurement is p Q gain = 12 (cid:80) nj =1 ρ j (cid:88) j =1 (cid:12)(cid:12)(cid:12) √ ρ j ( √ pi + √ p j ) (cid:12)(cid:12)(cid:12) = 1 M (cid:88) j =1 ρ j (cid:18) p + p j (cid:19) = 1 M (cid:88) j =1 ρ j q j (46)which is exactly that given in equation (20) in section 2.1 for the classical game A (cid:48) B (cid:48) . A second proper quantization of the sequence B (cid:48) B (cid:48)(cid:48) can be constructed in a manner similarto that used to construct the proper quantization for B (cid:48) in section 5. Instead of forming aquantum superposition of the quantum multiplexers associated with each game, first embedthe classical coins used in the game B (cid:48) B (cid:48)(cid:48) into SU (2) as Y j = (cid:112) t j N + (cid:112) − t j F = (cid:32) √ t j − (cid:112) − t j η (cid:112) − t j η √ t j (cid:33) with t j = rα j + (1 − r ) β j and 1 − t j = r (1 − α j ) + (1 − r )(1 − β j ) Proper quantization of history dependent Parrondo games and their randomized se-quences. and associate the quantum multiplexer Y = ( Y , Y , Y , Y ) with the classical game B (cid:48) B (cid:48)(cid:48) .Set the initial state, as in section 5.1, equal to I = 1 (cid:113)(cid:80) nj =1 τ j √ τ √ τ √ τ √ τ where the τ j are the probabilities that form the stationary state of the classical game29 (cid:48) B (cid:48)(cid:48) given in Equation (10). The output state of this protocol is F I = 1 (cid:113)(cid:80) nj =1 τ j √ τ t (cid:112) τ (1 − t ) η √ τ t (cid:112) τ (1 − t ) η √ τ t (cid:112) τ (1 − t ) η √ τ t (cid:112) τ (1 − t ) η (47)which, upon measurement produces the probability of gain p QB (cid:48) B (cid:48)(cid:48) gain = 1 (cid:80) nj =1 τ j (cid:88) j =1 τ j t j which is exactly the probability of gain computed in Equation (20) of section 2.1 for theclassical game AB (cid:48) . Two approaches are used to properly quantize random sequences of Parrondo games A and B (cid:48) in which each game occurs with equal probability. One approach, discussed in section5, generalizes the notion of randomization between the two games via probability distri-butions to randomization between games via quantum superpositions. The other approach,discussed in section 5.3, embeds a probabilistic combination of the games into a quantummultiplexer directly rather than via quantum superpositions of the protocols for each game.In the former approach, note that it was crucial that game A was embedded into SU (2) using basic embedding of type 2 as this allowed for the use of the broader arithmeticalproperties, namely factorization, of complex numbers to reproduce the classical result. Inthe latter on the other hand, basic embedding of type 1 sufficed. The ideas developed in this article bring together formal game theory, Markov processes,and quantum information theory. Due to this multifaceted nature, the study of proper quan-30ization of games can potentially influence research in all three areas mentioned above.For instance, the proper quantization protocols developed for history dependent Parrondogames using a particular type of quantum multiplexer lend a game theoretic perspective tothe study of quantum logic circuits via quantum multiplexers. Indeed, the notion of theParrondo effect is now attached to quantum circuits and it is now natural to investigate thecharacterization of the “quantum Parrondo effect” in quantum circuits via a game theoreticperspective.Results in quantum logic synthesis show that an n qudit logic gate can be synthesizedvia a circuit consisting entirely of variations of the quantum multiplexer [9, 4]. Given theinterplay of game theory and quantum circuits in the quantization of history dependentParrondo games, it is also natural to ask how might an arbitrary quantum logic gate besynthesized via a quantum multiplexer circuit in a game theoretically meaningful way. Forexample, after assigning a fixed number of qubits in the circuit to each ”player”, for an arbi-trary quantum logic gate U , how might U be decomposed into a quantum multiplexer circuitand an initial state chosen such that a given game theoretic outcome might be realized?In an even broader context, to date there is no agreement in the literature on exactly whata quantum Markov process is. One difficulty lies with the formulation of an appropriatedefinition of the “quantum” analogue for the stable state of a classical process, an objecthere called the evaluative state. 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