Constructing Entanglers in 2-Players--N-Strategies Quantum Game

In quantum games based on 2-player--$N$-strategies classical games, each player has a quNit (a normalized vector in an $N$-dimensional Hilbert space ${\cal H}_N$) upon which he applies his strategy (a matrix $U \in$ SU(N)). The players draw their payoffs from a state $|\Psi \ra=J^\dagger U_1 \otimes U_2 J|\Psi_0 \ra \in {\cal H}_N \otimes {\cal H}_N $. Here $|\Psi_0 \ra$ and $J$ (both determined by the game's referee) are respectively an {\it unentangled} 2-quNit (pure) state and a unitary operator such that $|\Psi_1 \ra \equiv J|\Psi_0 \ra \in {\cal H}_N \otimes {\cal H}_N$ is {\it partially entangled}. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of $|\Psi_1 \ra$. Hence, it is practical to design the entangler $J=J(\beta)$ to be dependent on a {\it single} real parameter $\beta$ that controls the degree of entanglement of $|\Psi_1 \ra$, such that its von-Neumann entropy $ {\cal S}_N(\beta)$ is continuous and obtains {\it any value} in $[0, \log N]$. Moreover, an efficient control of $ {\cal S}_N(\beta)$ is possible only if $|\Psi_1 \ra$ appears in a Schmidt decomposed form. Designing $J(\beta)$ for $N=2$ is quite standard. Extension to $N>2$ is not obvious, and here we suggest an algorithm to achieve it.

The theory of quantum games is an evolving discipline that, similar to quantum information, explores the implications of quantum mechanics to fields outside physics proper, such as economics, finance, auctions, gambling etc. [1][2][3][4][5][6][7]. One way of constructing a quantum game is to start from a standard (classical) game and to "quantize" it by formulating appropriate rules and letting the players employ quantum tools such as qubits and quantum strategies (gates). This procedure has been applied on classical strategic games that describe an interactive decision-making in which each player chooses his strategy only once, and all choices are taken simultaneously. A simple example is a quantum game based on 2-player-2-strategies classical game usually defined by a game table (for example, the prisoner dilemma). We refer to it briefly as a 2-2 game. There is an extensive work on the quantized version of classical strategic 2 − 2 games, most of them are based on the protocol specified in Ref. [3]. It requires application of an entanglement (unitary) operator J(β) (where β is a real parameter), that acts on a non-entangled 2-qubit (pure) state resulting in an entangled state whose degree of entanglement is measured by its von-Neumann entropy S 2 (β). A desired property of J(β) is that S 2 (β) is a continuous function of β that varies (preferably monotonically) between 0 (no entanglement) and log 2 (maximal entanglement). The reason for exploring partially entangled 2-qubit states is that the existence of pure strategy Nash equilibrium in the 2-2 quantum game crucially depends on the degree of entanglement (see below). Controlling the entropy by a single parameter such that all values between 0 and log 2 are obtained is referred to here as single parameter completeness. Another practical property required from J(β) is that it can easily be constructed from the classical strategies. In a 2-2 game, the classical strategy of a player is iσ y ∈ SU(2), and an appropriate construction is then J(β) = e −i β 2 σy⊗σy . Its action on an unentangled 2-qubit state (e.g |00 ) yields, where the left (right) factor in the Kronecker product refers to player 1 (2). In this way, |Ψ 1 = J(β)|00 appears in a Schmidt decomposed form, enabling an easy computation of the corresponding entanglement entropy of the 2-qubit state on the RHS as, Thus, S 2 (β) is a continuous function of β and gets all values in [0, log 2] namely, J(β) as defined in Eq. (1) satisfies single parameter completeness. Other properties (of less significance) are that S 2 (β) is periodic with period π, symmetric about the mid-point π 2 , with S 2 (0) = 0 and S 2 ( π 2 ) = log 2. The entangler defined in Eq. (1) has a property referred to as classical commensurability, [J(β), σ y ⊗ σ y ] = 0. Following the rules of the game specified in Ref. [3] it means that players in a 2-2 quantum game can, if they wish, use their classical strategies as a special case and if they do so, they collect the corresponding classical payoffs. In most cases, however, the classical strategies do not constitute a pure-strategy Nash Equilibrium (NE) (defined below). Generically, as we explain below, classical commensurability does not hold for N > 2 (see however Ref. [8]).
In the present work we examine the issue of constructing J(β) for a 2 − N quantum game based on a 2-players-N-strategies classical game. We suggest a natural extension of Eq. (1) for constructing an operator J(β) that turns a non-entangled 2-quNit states into an entangled one. For N = 3, 4 the corresponding von-Neumann entropy S N (β) varies continuously between 0 and log N so that single parameter completeness is satisfied. Unfortunately, this method does not work for N > 4 because in that case S N (β) < log N . To alleviate this deficiency we suggest another method (albeit less intuitive) to design J(β) that satisfies single parameter completeness for any N . In what follows we will first introduce the classical 2-3 game using quantum information language and formulate its quantum version (extension to N > 3 is straightforward). Most of this introductory exposition is well established and is included here merely for self consistence. Then, in the second step, we shall address the issue of constructing J(β).
Two-Players-Three-Strategies Classical Games: Trits Consider a two-players classical game with three strategies for each player. For example, two prisoners may have three options, marked as three values of a trit C=|1 ,S=|2 and D=|3 for Confess, Stay quiet or Don't confess. The two prisoner "system" can be found in nine two-trit states |ij , i = 1, 2, 3 j = 1, 2, 3, corresponding to the nine entries of the game table. The protocol of the classical game with 2-players and 3-strategies is as follows: The referee (judge) calls the players (prisoners) and tells them he assumes that they are in an initial two-trit state |11 meaning (C,C) namely both confess. He then asks them to decide whether to leave their respective trit state as it is on |1 or to change it either to |2 (meaning S) or to |3 (meaning D). These replacement operations (specified explicitly below) are the players classical strategies. If the initial state suggested by the referee is |11 the strategies of the two players include 1 3 ( leaving the trit at |1 as it is), S 12 (swapping of |1 and |2 namely, replacing C by S) and S 13 (swapping |1 and |3 namely replacing C by D). These three operations generate the group S 3 of permutations of three elements 123 ijk . Explicitly, 1 3 =(123), S 12 =(213), S 13 =(321), S 13 S 12 S 13 =S 23 =(132) S 13 S 12 =S 312 =(312), S 12 S 13 =S 231 =(231). We shall indicate below that a quantum strategy is a gate represented by an SU(3) matrix. A reasonable requirement for the procedure of "quantizing" a classical game is that the classical strategies obtain as special case of the quantum ones. For that purpose we need to construct a representation of the permutation group S 3 in terms of unitary matrices with unit determinant. This can be achieved by choosing, For example, suppose player 1 and 2 choose respective strategies s 1 = S 12 and s 2 = S 13 . This brings the system into a state s 1 ⊗ s 2 |11 = |23 = |SD . Then the respective payoff of player k, k = 1, 2 will be u k (s 1 , s 2 ) = u k (S,D) = u k (2, 3) where u k (i, j) is the payoff of player k at entry (i, j) of the game table. The analogous Quanum Game: We now briefly explain the structure of the corresponding quantum game. Its main ingredients are qutrits, quantum strategies, and entanglement operations. Both versions use the same game table but the payoff rules are somewhat different. 1 and 2 qutrit states Consider the three dimensional Hilbert space H 3 with orthonormal basis vectors |1 , |2 and |3 . A qutrit is a vector A maximally entangled two-qutrit state is written as, given in a Schmidt decomposed form. Its entanglement degree is measured by the von Neumann entropy The referee suggests an initial non-entangled two-qutrit state |Ψ 0 (e.g the analog of the classical two trit state |Ψ 0 =|11 ). Before letting the players apply their quantum strategies, the referee operates on |Ψ 0 with a unitary operator J(β) such that |Ψ 1 = J(β)|Ψ 0 is entangled (otherwise the game remains classical). Construction of the operator J(β) (our central goal) is detailed below. At this stage of the game, the players apply their respective strategies U 1 ⊗ U 2 . Finally, the referee applies the operator J † (β) leading to the final state where γ k is the octet of 8 Euler angles defining the SU(3) matrix U (γ k ) (that is the strategy of player k = 1, 2). The payoff P k of player k = 1, 2 is given by, where u k (i, j) are the payoffs at entry (i, j) of the classical game table. Like in the classical game, each player choses a strategy with the goal of maximizing his payoff.
Pure Strategy Nash Equilibrium (NE) Because the set of 8 Euler angles γ uniquely determines the player's strategy U (γ) ∈ SU(3), a pure strategy NE in the 2-3 quantum game is a pair of strategies (γ * 1 , γ * 2 ) (each represents 8 angles), such that The question of whether pure strategy NE exists in 2-2 quantum game, and its relation to the degree of entanglement (controlled by β) has been discussed in numerous works [9][10][11][12][13][14][15]. In brief, if there is NE in the classical game that is not Pareto efficient [16], then there is a critical value β c above which there is no pure strategy NE in the quantum game. As β approaches β c from below, the respective payoffs in the quantum game at NE approach the Pareto point of cooperation [11,15]. This is the main reason why, right from the onset, we stressed the relevance of partially entangled 2-quNit states where S N (β) < log N . Absence of classical commensurability We now explain why, in a 2-3 quantum game there is no classical commensurability [8]. Recall that classical commensurability means that if the players use classical strategies they respectively get their classical payoffs. For a classical strategyγ we have U (γ) ∈ S 3 . From Eq. (6) This is possible only if J is a function of A ⊗ A where A is a 3 × 3 matrix satisfying [A, S 12 ] = [A, S 13 ] = 0, and A is not just a multiple of 1. But this is impossible because S 12 and S 13 generate an irreducible representation of the permutation group S 3 and hence, according to Schure's lemma A is just a multiple of 1 3 . These arguments naturally hold for any N > 2.
Designing J(β) The main result of the present study concerns the analysis and construction of an entanglement operator J(β) for N > 2. We carry it out for N = 3 and then extend it straightforwardly to any N . For N = 3 we require that J(β)|11 yields an entangled 2-qutrit state with β specifying the degree of entanglement that achieves any value between 0 and log 3. Following the 2-2 game framework specified in Eq. (1), we try to construct J by exponentiating a combination of classical strategies. In order to get the "diagonal" 2-qutrit states |22 and |33 from the qutrit state |11 we have to operate on |11 with Z ≡ S 12 ⊗ S 12 + S 13 ⊗ S 13 . Therefore, we define J(β) = e i β 2 Z = e i β 2 (S12⊗S12+S13⊗S13) .
Calculation of the exponent yields (11) Maximal entanglement obtains when the absolute values of all three coefficients are equal, namely, Here S 3 (β) raises monotonically from 0 to its first maximum log 3, hence we have found the desired entanglement operator that satisfies single parameter completeness. Fig. 1(a) displays the von Neumann entropy S 3 (β) of the entangled 2qutrit state (11) as function of β. Here again it possesses other properties, namely, S 3 (β) is a periodic function of β with period 4π 3 and it is symmetric about the mid-point 2π 3 where it has a local minimum. It has two maxima for β = 4π 9 , 8π 9 where it equals log 3. Inspecting Eq. (10), we see that, in quantum games, the entanglement is not obtained in terms of spin rotations but, rather, in terms of permutation exponentials that are SU(3) matrices (for N = 2 these are the same). Extension to arbitrary N: Let S ij ∈ S N denote the N ×N matrix representing the permutation i ↔ j and let |ij , i, j = 1, 2, . . . , N be an unentangled 2-quNit state. To get an entangled state from |11 we define and assert that To proceed, let us define the absolute value squared of the coefficients, It is easy to verify that 1) 0 N and symmetric about the mid-point 2π N . The entanglement entropy is Maximal entanglement S N (β) = log N obtains for β 0 that is the solution of the equality For N = 3 the two solutions are specified in Eq. (12). For N = 4, there is a single solution at β 0 = π 2 , as shown in   16), and maximal entanglement is not achieved. It might be argued that 2-N quantum games with N > 4 are much rarer than those with smaller N but we believe that the construction of J(β) that satisfies single parameter completeness also for N > 4 is useful in other areas, (outside the ballpark of quantum game theory), so we carry it out for the sake of completeness. The method suggested here is not based on permutation exponentials as in Eq. (10). It consists of the following steps.
that is a maximally entangled state.
2. Diagonalize R as R = V ΛV −1 where V is the matrix of eigenvectors of R, and Λ=diag{e iη1 , e iη2 , . . . , e iηN }, is the diagonal matrix of (unimodular) eigenvalues of R with eigenphases {η i }.  Generically, the eigenphases are not rational multiples of π so that S N (β) is not periodic, but this lack of periodicity is of no special significance.

Now consider the matrix
Illustration for N = 5 : A convenient way to build an appropriate unitary N × N matrix R is to start from a simple non-singular matrix A and then orthogonalize it within the Grahm-Schmidt procedure. For example, Proceeding with the list of steps prescribed above we can easily construct J(β) and compute the von Neumann entropy of the state |Ψ 1 = J(β)|11 upon which the players apply their strategies according to the game protocol specified in Eq. (6). The result is given in Fig. 2. As explained in the figure's caption, the degree of entanglement is controlled by a single parameter and S N (β) is a continuous function of β reaching any value in the interval [0, log N ]. Thus we have achieved our goal of constructing an entangler J(β) that turns a non-entangled 2-quNit state into an entangled one given in a Schmidt decomposed form with single parameter completeness satisfied.
Summary: In conclusion, we suggest two methods to design an entanglement operator J(β) that turns a non-entangled 2-quNit state to a partially entangled state whose von Neumann entropy is fully controlled by a single real parameter. The first method is intuitively clear and simple, based on exponential of classical strategies, Eq. (10), and results in a the von Neumann entropy, as displayed in Figure 1. This method does not work for N > 4 because the resulting entropy does not reach the maximally entangled value log N . For that reason we suggest another method that is somewhat less transparent but works for any N . The resulting entropy as function of β, is displayed in Figure 2.