Note on maximally entangled Eisert-Lewenstein-Wilkens quantum games

Maximally entangled Eisert-Lewenstein-Wilkens games are analyzed. For a general class of gate operators defined in the previous papers of the first author the general conditions are derived which allow to determine the form of gate operators leading to maximally entangled games. The construction becomes particularly simple provided one does distinguish between games differing by relabelling of strategies. Some examples are presented.


I Introduction
The seminal papers of Eisert, Lewenstein and Wilkens (ELW) [2], [3] opened new field of intensive research called the theory of quantum games [4]÷ [48]. They proposed a method of constructing a quantum counterpart of a given non-cooperative classical game.
Eisert, Lewenstein and Wilkens pointed out that there is an intimate connection between the theory of quantum games and quantum communication. They speculated also that games of survival are being played on molecular level ruled by the laws of quantum mechanics. 1 kbolonek@uni.lodz.pl 2 pkosinsk@uni.lodz.pl Their approach constitutes one of the paradigms of the theory of quantum games. This is because it provides simple yet subtle scheme which allows to study the influence of non-classical correlations which break Bell-like inequalities on the properties (in particular the efficiency in the presence of restricted resources) of classical games. In fact, the ELW game may be viewed as a strightforward generalization of classical symmetric noncooperative game where the players take advantage from the fact that quantum probability distributions violate, in general, such inequalities.
The original proposal concerned the quantization of symmetric 2-players game with two strategies at each player's disposal. It can be generalized to the case of arbitrary number N of admissible strategies. For general N the structure of the game becomes much richer. The key role in construction of quantum game is played by the gate which allows quantum correlations to influence the outcome of the game. In the original ELW proposal the gate depends on one arbitrary parameter.
For N-strategies games one can construct the gate depending on N 2 parameters [49]÷ [52].
The strength of quantum correlations is measured by the quantum entanglement. It is not surprising that the so called maximally entangled games play a distinguished role. They do not admit nontrivial pure Nash equilibria [9], [37], [42] and exhibit additional structures [37], [42]. They are particularly worth of being studied in more detail.
Whether the game is maximally entangled or not depends on the choice of the gate. In the present paper we derive the general conditions on the parameters entering the gates introduced in Refs. [49]÷ [52] for the game to be maximally entangled. They appear to be quite straightforward and manageable. What is also important they are applicable to the family of gates which seem to exhaust all interesting cases and which provide a natural generalization of the gate introduced by Eisert et al. In order to construct a quantum counterpart of the game we ascribe to any player a N-dimensional Hilbert space H spanned by the vectors The Hilbert space of the game is H ⊗ H. We start with the vector |1 ⊗ |1 . The key element of the definition of quantum game is the choice of an unitary operator (the gate) J which introduces quantum entanglement. The initial state of the game is defined as Before proceeding further let us comment briefly on the choice of initial state. One of the main points in construction of ELW game is that any entanglement is introduced and controlled by the gate J. Therefore, we start with an unentangled vector (i.e. with Schmidt number 1) which can be written as tensor product of Alice and Bob states; so, in general, |Ψ ∈ H being an arbitrary normalizable state. Now, there exists an unitary operator U such that Note that the phase of |Ψ is irrelevant (cf. eq. (6) below) so it can be adjusted such that U ∈ SU (N). Moreover, U is not unique; in fact, it can be multiplied from the right by any element of S (U(1) × U(N − 1)). By virtue of eqs. (3) and (4) |Ψ may be always replaced by |1 provided simultaneously the replacements This allows us to compute the players expected payoffs where |σ, σ ′ ≡ |σ ⊗ |σ ′ .
As it was mentioned above it is the form of the gate J which determines the properties of the game. If J is trivial (i.e. a product of two local unitary matrices) the game reduces to the classical one. Any pure quantum strategy corresponds to some, in general mixed, classical one (actually, there is some overcounting, i.e. a number of pure quantum strategies correspond to the same classical strategy). The situation changes if J ceases to be the product of local unitary operators. Then the initial state of the game becomes entangled. The degree of entanglement plays a crucial role in the study of game properties. Roughly speaking the more entangled the state is the more the outcome probabilities differ from the values allowed by inequalities of Bell type resulting in properties not shared by the classical game one starts with. In view of this it is natural to ask about the maximally entangled case.
As we have already mentioned in the Introduction maximally entangled games are distinguished by their properties. One of us has shown [52] that the key element here is that the stability subgroup of initial state, i.e. the subgroup of the group SU(N) × SU(N) of all strategies which consists of elements which leave invariant the initial state is isomorphic to the diagonal subgroup of SU(N) × SU(N). This has, for example, deep impact on the structure of Nash equilibria. It is known [1], [23] that the validity of Nash theorem extends to the quantum domain; as in the classical games the Nash equilibria correspond, in general, to mixed strategies.
However, for maximally entangled case one finds an additional property: even if the classical payoff matrix admits pure Nash equilibria, they cease to exist in the quantized version (except some trivial cases). Moreover, the study of mixed strategies corresponding to Nash equilibria simplifies considerably. This is due to the fact that, again as a consequence of the form of stability subgroup, the final outcome depends essentially only on the product of the matrices representing players moves [52]. This property has been successfully used in the case N = 2 to provide the classification of mixed Nash equilibria [37], [38].
In order to construct the gate operator J we assume that the quantum game is still symetric and all classical pure strategies are contained in the set of pure quantum ones. It has been shown in Refs. [49], [52] that one can construct a multiparameter family of gates J obeying this condition. To this end we define first the unitary matrix V by [49] where ε = exp 2iπ N is the first primitive root from unity.
be any basis in the Cartan subalgebra of SU(N) ( consisting of diagonal traceless hermitean matrices). Define with λ k , µ kl real and µ kl = µ lk . Then the relevant gate reads We see that J depends on N − 1 + N −1 2 = N 2 free parameters. The above definition is quite general. In fact, it seems that this is the only freedom left if one assumes that all classical strategies are properly included is related to the choice of phases for the basic vectors (1).

III Maximally entangled games
We call the ELW game maximally entangled if the initial state (2) is maximally entangled. Let be the density matrix corresponding to the initial state. The state described by ρ i is maximally entangled if the reduced density matrices are proportional to the unit matrix [53] Tr The maximally entangled game is distinguished by its properties. Consider first the N = 2 case. The game can be described in terms of quaternion algebra [37] and real Hilbert space [42]. What is more important, to any strategy of one player there exists an appropriate counterstrategy of the second player which leads to any outcome he/she desires [9], [37]. As a result no nontrivial pure Nash equilibrium exists while the form of the mixed one is strongly restricted [25].
The existence of counterstrategies in the case of maximally entangled game can be established for any N [50]; it results in a simple way from the following property of such a game: the stability group of the initial state |Ψ i is (up to an automorphism) the diagonal subgroup of SU(N) × SU(N) [50]. Therefore, for any N the maximally entangled games exhibit no nontrivial pure Nash equilibria. The structure of the mixed ones will be analyzed in a separate paper [54].
In the present section we derive the general conditions on the parameters λ k and µ kl which yield the gate for maximally entangled game. To this end let us write out explicitly the initial density matrix ρ i . With the help of equation (9) we find Let us note that J is diagonal and can be written as where J αβ are the diagonal elements of J. Using eqs. (11), (12) and (13) one easily finds that the condition Tr B ρ i = 1 N I can be written as where J is the N × N matrix defined by The indices α and β on the left hand side number the matrix elements of J while on the right hand side -the diagonal elements of J.
Eqs. (18) and (19) provide the general solution to our problem. The convenient strategy to solve them is to find first the solutions to the eqs. (18) and (20)
Let us pass to the case N = 4. The gate operator is now parametrized by six quantities λ 1 , λ 2 , λ 3 , µ 12 = µ 21 , µ 13 = µ 31 and µ 23 = µ 32 . According to the eqs. (14) and (17) demanding the maximal entanglement is equivalent to the unitarity of the matrix: Eqs. (18) and (19) take now the following form Generically, we have: By multiplying both sides of eq. (34) by exp − i 2 (ϕ 1 + ϕ 2 ) we get Thus, up to a renumbering Eq. (35) yields and we arrive at the one-parameter family of solutions which can be conveniently parametrized as where m and n are arbitrary integers. Let us note that we obtain, for fixed m and n, the one parameter family of solutions.

V Solution for general N
It is quite easy to find the general form of the gate leading to maximally entangled game without solving eqs. (18) and (19). The price we have to pay is that some games are then equivalent in the sense described below.
According to the eqs. (13) and (14) the general form of the gate under consideration reads where W is some unitary matrix, W W + = I. However, W cannot be completely arbitrary. In order to preserve the symmetry of the game one must impose so W may be arbitrary unitary symmetric matrix. Deleting the trivial overall factor we assume that W is a symmetric element of SU(N). Such elements are generated by N − 1 traceless diagonal real matrices and N 2 off-diagonal real symmetric ones. On the level of the gate matrix J this implies that we should supply N 2 generators entering the exponent on the right hand side of eq. (8) by N − 1 generators of the form I ⊗ Λ k + Λ k ⊗ I, k = 1, ..., N − 1. However, denoting by J ′ a new gate one easily finds with the help of BCH formula Therefore, by relabelling the strategies, we arrive at the game differing only by classification of strategies.
Neglecting the above subtlety we conclude that the general gate operator leading to maximally entangled game is given by eqs. (9), (43) and (44).
Mathematically, the games differing only by relabelling of strategies equivalent in the sense that any question concerning one game (for example, the localization and classification of Nash equilibria, the existence of Pareto optimally strategies etc.) can be easily translated into equivalent question concerning the second game.
However, when one considers the physical realization of the game (like, for example in Ref. [14] where the game is implemented using two qubit nuclear magnetic resonance quantum computer) the games are no longer equivalent in the sense that the gate takes a particular form determined by the underlying physical mechanism (the same concerns the implementation of strategies). Referring to the above example described in [14] the single parameter characterizing the gate is determined, among others, by the spin-spin couplings between the nuclei.

VI Concluding remarks
We have considered the maximally entangled 2-players N-strategies games. The necessary and sufficient conditions for the game defined by the gate (9) to be maximally entangled are given by eqs. (14) and (17).
In order to find their explicit form we have to solve equations (18) and (19).
The most convenient choice is to find first the general solution to the equations (18) and (20) and then to solve eqs. (19) in terms of λ's and µ's. In this way we obtain a rich variety of solutions. The important point is that only for N=2 and 3 these solutions form the discrete sets. Starting from N=4 we obtain the continuous families of solutions with growing number of free parameters.
The admissible gate J may be multiplied by the product of arbitrary unitary matrices J A,B leading to new gate operator This amounts only to relabelling of Alice and Bob strategies If we, however, does distinguish between such games the construction of relevant gate becomes particularly simple (cf. eqs. (9), (43) and (44)).
All games considered above are symmetric with respect to the exchange of players. This is a natural situation from the point of view of game theory. However, we can admit a more general situation that, on the classical level, the number  8) and (9)). The condition for maximal entanglement reads now [53] Tr with One can now proceed as in the symmetric case arriving at the equations generalizing eqs. (18)÷(20).
As we have pointed out above the symmetric case seems to more natural from game-theoretic point of view. However, the asymmetric case is still very interesting.
Using the method presented in [52] and applying Schur's lemma one can find the stability subgroup of the initial state corresponding to the maximal entanglement.
It has quite nontrivial (although simple) structure which (as we discuss briefly in Sec. II) strongly influences the properties of the game. Therefore, the asymmetric case is worth of being studied in more detail.
ful discussion with P. Maślanka, K. Andrzejewski and J. and C. Gonera. We thank anonymous referee for interesting and useful remarks.
Appendix A Cartan subalgebras [55] We do not need the general definition of Cartan subalgebra for arbitrary Lie algebra. In the case of semisimple Lie algebras over the complex numbers the Cartan subalgebras are uniquely defined by the following properties: (i) they are maximal abelian subalgebras (ii) if an element X belongs to the Cartan subalgebra then adX is diagonalisable.
All Cartan subalgebras are related by inner automorphisms. The simultaneous diagonalization of all endomorphisms adX with X running over the basis of some Cartan subalgebra allows to give the full characterization of the structure of Lie algebra under consideration. In particular, the eigenspace corresponding to zero eigenvalue coincides with the Cartan subalgebra itself while the remaining eigenspaces are onedimensional.
The elements of Lie algebra of SU(N) group are the traceless hermitean N × N matrices. By virtue of (i) they commute so they can be diagonalised simultaneously.
Therefore, the maximal set of hermitean traceless diagonal matrices forms Cartan subalgebra of SU(N) algebra. The basis of this subalgebra can be chosen as in eq. (16).