On maximally entangled Eisert-Lewenstein-Wilkens quantum games
arXiv:1410.7706v1 [quant-ph] 28 Oct 2014
Katarzyna Bolonek-Laso´ n1 Faculty of Economics and Sociology, Department of Statistical Methods, University of Lodz, Poland.
Piotr Kosi´ nski2 Faculty of Physics and Applied Informatics, Department of Informatics, University of Lodz, Poland. Abstract Maximally entangled Eisert-Lewenstein-Wilkens games are analyzed. The general conditions are derived which allow to determine the form of gate operators leading to maximally entangled games. Some examples are presented.
I
Introduction
The seminal papers of Eisert, Lewenstein and Wilkens (ELW) [1], [2] opened new field of intensive research called the theory of quantum games [3]÷[47]. 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. 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 1 2
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1
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 operator which allows quantum correlations to influence the outcome of the game. In the original ELW proposal the gate operator depends on one arbitrary parameter. For N-strategies games one can construct the gate operators depending on N2 parameters.
The strength of quantum correlations is measured by the quantum entangle-
ment. It is not surprising that the so called maximally entangled games play a distinguished role. They do not admit nontrivial pure Nash equilibria [8], [36], [41] and exhibit additional structures [36], [41]. Whether the game is maximally entangled or not depends on the choice of the gate operator. In the present paper we derive the general conditions on the parameters entering the gate operator for the game to be maximally entangled. The paper is organized as follows. In sec. II we remind the definition of the general ELW game. Then, in Sec. III we derive the conditions which must be imposed on gate operator to yield maximally entangled game. Sec. IV is devoted to some examples. Finally, Sec. V contains some concluding remarks.
II
Quantum ELW game
Let us describe the general setting for the quantum ELW game. The starting point is a classical two-players symmetric game. Each player has N strategies at his/her disposal (the original Eisert et al. proposal corresponds to N = 2). The classical game is completely defined by N × N payoff matrices P A,B with matrix elements
A,B B ′ A Pσ,σ ′ , σ, σ = 1, ..., N; Pσ,σ ′ (Pσ,σ ′ ) denotes Alice (Bob) payoff in the case Alice and
Bob choose the strategies σ and σ ′ , respectively. In order to construct the quantum counterpart of the game we ascribe to any
2
player a N-dimensional Hilbert space H spanned by the vectors 0 1 0 0 |1i = .. , · · · , |Ni = .. . . . 1 0
(1)
The Hilbert space of the game is H ⊗ H. We start with the vector |1i⊗|1i. The key
element of the definition of quantum game is the choice of an unitary gate operator J which introduces quantum entanglement. The initial state of the game is defined as |Ψi i = J (|1i ⊗ |1i) .
(2)
Once the initial state is defined Alice and Bob perform their moves represented by unitary matrices UA,B ∈ SU(N). Then the final measurement is made which yields the final state |Ψf i = J + (UA ⊗ UB ) J (|1i ⊗ |1i) .
(3)
This allows us to compute the players expected payoffs $A,B =
N X
σ,σ′ =1
A,B ′ Pσ,σ ′ |hσ, σ |Ψf i|
2
(4)
where |σ, σ ′ i ≡ |σi ⊗ |σ ′ i. In order to construct the gate operator J we assume that all classical pure strategies are contained in the set of pure quantum ones. It has been shown in Ref. [48] that one can construct a multiparameter family of gate operators J obeying this condition. To this end we define first the unitary matrix V by [48] 1 Vα,β = √ ε(α−1)(β−1) , N where ε = exp
2iπ N
α, β = 1, ..., N
(5)
is the first primitive root from unity. Let Λi , i = 1, ..., N − 1,
be any basis in the Cartan subalgebra of SU(N) ( consisting of diagonal traceless hermitean matrices). Define N −1 X
N −1 i X e λk (Λk ⊗ Λk ) + µkl (Λk ⊗ Λl + Λl ⊗ Λk ) J = exp i 2 k6=l=1 k=1
3
!
(6)
with λk , µkl real and µkl = µlk . Then the relevant gate operator reads J = (V ⊗ V ) Je V + ⊗ V + .
We see that J depends on N − 1 +
III
N −1 2
=
N 2
(7)
free parameters.
Maximally entangled games
We call the ELW game maximally entangled if the initial state (2) is maximally entangled. Let ρi = |Ψi i hΨi |
(8)
be the density matrix corresponding to the initial state. The state described by ρi is maximally entangled if the reduced density matrices are proportional to the unit matrix TrA ρi =
1 I, N
TrB ρi =
1 I. N
(9)
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 [36] and real Hilbert space [41]. 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 [8], [36]. As a result no nontrivial pure Nash equilibrium exists while the form of the mixed one is strongly restricted [24]. The existence of counterstrategies in the case of maximally entangled game can be established for any N [49]. It results in a simple way from the following property of such a game: the stability group of the initial state |Ψi i is (up to an automorphism) the diagonal subgroup of SU(N) × SU(N) [49]. 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 [50]. In the present section we derive the general conditions on the parameters λk and µkl which yield the gate operator for maximally entangled game. To this end let us write out explicitly the initial density matrix ρi . With the help of equation
4
(7) we find ρi = (V ⊗ V ) Je V + |1i ⊗ V + |1i (h1| V ⊗ h1| V ) Je+ V + ⊗ V + .
(10)
Let us note that Je is diagonal and can be written as Jeαβ,γδ = Jeαβ δαγ δβδ
(11)
e Using eqs. (9), (10) and (11) one easily where Jeαβ are the diagonal elements of J.
finds that the condition TrB ρi =
1 I N
can be written as
1 JJ+ = I Nee
(12)
where J is the N × N matrix defined by e
J αβ = Jeαβ .
(13)
e
The indices α and β on the left hand side number the matrix elements of J while e on the right hand side - the diagonal elements of J.
e
From eq. (12) we can derive a set of conditions on the parameters λk and µkl defining the gate operator. To this end we choose the basis in the Cartan subalgebra of SU(N). A convenient choice is (Λk )αβ = δkα δαβ − δk+1α δαβ
(14)
k = 1, ..., N − 1, α, β = 1, ..., N. Inserting this into eq. (6) one finds the explicit form of the matrix J in terms of the parameters λk and µkl : e
J αβ = exp (i (λα + λα−1 ) δαβ − iλα δα+1β − iλα−1 δαβ+1 + e
(15)
+i (µαβ + µα−1β−1 − µα−1β − µαβ−1 )) where, by definition, µ0α = µα0 = µN α = µαN = 0. Eqs. (12) and (15) provide the set of equations determining the parameters λk and µkl . In short, the necessary and sufficient condition for the gate operator J to define a maximally entangled game is that the matrix
√1 J , N e
with J defined by eq. (15), is unitary. e
5
IV
Some examples
The case N = 2 corresponds to SU(2) group. The Cartan subalgebra is onedimensional and is spanned by Λ = σ3 .
(16)
There is one parameter λ and eq. (12) yields eiλ e−iλ . J = e −iλ iλ e e
(17)
Eq. (12) leads to the following condition
e4iλ + 1 = 0
(18)
i.e. λ = π4 . Consider now the case N = 3. There are two basic elements of Cartan subalgebra of SU(3):
1 0 0 Λ1 = 0 −1 0 0 0 0
0 0 0 Λ2 = 0 1 0 0 0 −1
,
(19)
and three parameters λ1 , λ2 and µ12 = µ21 . Eqs. (12) and (15) yield a set of conditions which are equivalent to those derived in Ref. [51] (one has only to take into account the different choice of the Cartan subalgebra basis in [51]). It is shown in [51] that the resulting equations admit a discrete set of solutions. We omit the details here. Let us pass to the case N = 4. The gate operator is now parametrized by six quantities λ1 , λ2 , λ3 , µ12 = µ21 , µ13 = µ31 and µ23 = µ32 . According to the eqs. (12) and (15) demanding the maximal entanglement is equivalent to the unitarity of the matrix:
iλ1
i(−λ1 +µ12 )
i(µ13 −µ12 )
−iµ13
e e e e i(−λ1 +µ12 ) e ei(λ1 +λ2 −2µ12 ) ei(−λ2 +µ23 +µ12 −µ13 ) ei(µ13 −µ23 ) 1 1 √ J= 2 ei(µ13 −µ12 ) ei(−λ2 +µ12 +µ23 −µ13 ) Ne ei(λ2 +λ3 −2µ23 ) ei(−λ3 +µ23 ) e−iµ13 ei(µ13 −µ23 ) ei(−λ3 +µ23 ) eiλ3 6
.
(20)
As a result we arrive at the following equations ei(2λ1 −µ12 ) + ei(−2λ1 −λ2 +3µ12 ) + ei(λ2 +2µ13 −2µ12 −µ23 ) + ei(−2µ13 +µ23 ) = 0
(21)
ei(λ1 −µ13 +µ12 ) + ei(−λ1 +λ2 +µ13 −µ23 ) + ei(−λ2 −λ3 +2µ23 −µ12 +µ13 ) + ei(λ3 −µ23 −µ13 ) = 0 (22) ei(λ1 +µ13 ) + ei(−λ1 +µ12 −µ13 +µ23 ) + ei(λ3 −µ23 +µ13 −µ12 ) + ei(−µ13 −λ3 ) = 0
(23)
ei(−λ1 +2µ12 −µ13 ) + ei(λ1 +2λ2 −3µ12 −µ23 +µ13 ) + ei(−2λ2 −λ3 +3µ23 +µ12 −µ13 ) + ei(λ3 −2µ23 +µ13 ) = 0 (24) ei(−λ1 +µ12 +µ13 ) + ei(λ1 +λ2 −2µ12 −µ13 +µ23 ) + ei(−λ2 +λ3 +µ12 −µ13 ) + ei(µ13 −µ23 −λ3 ) = 0 (25) ei(2µ13 −µ12 ) + ei(−λ2 +2µ23 +µ12 −2µ13 ) + ei(λ2 +2λ3 −3µ23 ) + ei(−2λ3 +µ23 ) = 0
(26)
All the above equations have the same structure. Namely, denoting by ϕ1 , ϕ2 , ϕ3 and ϕ4 the arguments in the exponents one can write them as eiϕ1 + eiϕ2 + eiϕ3 + eiϕ4 = 0
(27)
ϕ1 + ϕ2 + ϕ3 + ϕ4 = 0;
(28)
obviously, the arguments ϕi are different in each case (21)÷(26). Up to renumbering, the general solution to eqs. (27), (28) reads ϕ1 = ϕ4 + (2m + 1) Π
(29)
ϕ2 = −ϕ4 − (2m + 1 + n) Π
(30)
ϕ3 = −ϕ4 + nΠ
(31)
where m and n are arbitrary integers. Let us note that we obtain, for fixed m and n, the one parameter family of solutions. For any of eqs. (21)÷(26) the variables ϕi are different functions of the initial parameters λk and µkl . By combining different solutions (29)÷(31) one obtains numerous solutions to the original equations (21)÷(26). We shall quote below few of them. An important point is that, contrary to the N = 2 and N = 3 cases, we are dealing here with the one parameter families of solutions. It is clear from the
7
derivation given above that one can expect the existence of such solutions for all N ≥ 4. Some solutions of equations (21)÷(26) are listed below:
λ1 = −β − π + πm λ2 = 2µ23 − π λ3 = −β + πm
λ1 = −β +
π 4
+ πm
λ2 = 2µ23 − π λ3 = −β +
π 4
+ πm
µ12 = µ23 − π µ12 = µ23 − π2 µ13 = β + µ23 µ13 = β + µ23 µ = 3π , 7π , −2β, π − 2β µ = π , 3π , π − 2β, 3π − 2β 23 23 2 2 2 2 4 4 λ1 = −β − π2 + πm λ1 = −β + 3π + πm 4 λ2 = 2µ23 − π λ2 = 2µ23 − π λ = −β + 3π + πm λ3 = −β + π2 + πm 3 4 µ12 = µ23 µ12 = µ23 + π2 µ13 = β + µ23 µ = β + µ 13 23 µ = 0, π, π − 2β, 3π − 2β µ = π , 5π , −2β, π − 2β 23 23 2 2 4 4 m − π2 n λ1 = −β + 5π λ1 = β + πm + πn 4 λ2 = π2 n λ2 = −4β + πm λ3 = −β + π4 m − π2 n λ3 = β + πm µ12 = 3π µ12 = −2β + π + πm + πn n+π 4 µ13 = −β + π2 m µ13 = β + π4 n µ23 = π4 n µ23 = −2β + π2 m
(32)
where β is a real parameter; m and n are integers.
V
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 operator (7) to be maximally entangled are given by eqs. (12) and (15). 8
In the N = 3 and N = 4 cases all equations resulting from our conditions have the same simple structure (cf. eqs. (27) and (28)). We expect this is also the case for general N. Namely, there will be N2 equations of the form N X
eiϕi = 0
(33)
i=1
N X
ϕi = 0
(34)
i=1
where in each set of the above equations ϕ’s are different linear combinations of the parameters entering the gate operator (6). For N = 2 and N = 3 we obtain a discrete set of solutions. For N = 4 the generic solutions depend on arbitrary parameter. We expect this situation will persist for N > 4.
Acknowledgment Research of Katarzyna Bolonek-Laso´ n was supported by NCN Grant no. DEC-2012/05/D/ST2/00754.
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