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10 Sep 2019 Quantum Games: a Survey for Mathematicians Quantum games: a survey for mathematicians Vassili N. Kolokoltsov Department of Statistics, University of Warwick, email: [email protected] This is a draft of the final chapter 13 of the Second Edition of the book V. Kolokoltsov and O. Malafeyev ”Understanding game theory”, World Scientific, 2019 Abstract Main papers on quantum games are written by physicists for physicists, and the inevitable exploitation of physics jargon may create difficulties for mathematicians or economists. Our goal here is to make clear the physical content and to stress the new features of the games that may be revealed in their quantum versions. Some basic knowledge of quantum mechanics is a necessary prerequisite for studying quan- tum games. The most fundamental facts are collected in Section 1 describing closed finite-dimensional systems and complemented by the rules of quantum measure- ments in Section 2. These facts are sufficient for the main trend of our introductory exposition. However, for further developments of quantum games one needs some notions of open quantum systems. They are presented in Section 9, to which we refer occasionally, and which supplies the background that is necessary for reading modern research papers. The main sections 3 -7 build the foundations of quantum games via the basic examples. We omit sometimes the lengthy calculations (refer- ring to the original papers) once the physical part is sorted out and the problem is reformulated as pure game-theoretic problem of calculating the Nash or dominated equilibria. Section 8 is devoted to quantum games arising from the classical games arXiv:1909.04466v1 [math.OC] 10 Sep 2019 with infinite state space (like the classical Cournot duopoly). Section 10 touches upon general theory of finite quantum static games. Further links and references are provided in Section 11. 1 On finite-dimensional quantum mechanics Finite-dimensional quantum systems are described by finite-dimensional complex Eu- clidean or Hilbert spaces H = Cn, equipped with the standard scalar product (., .), which is usually considered to be anti-linear or conjugate linear with respect to the first argument, (au, v)=¯a(u, v) for a C, and linear with respect to the second one: ∈ (v,au) = a(v,u). For any orthonormal basis e , , e in H, the scalar product writes 1 ··· n down in coordinates as (u, v) = j u¯jvj, where u = j ujej, v = j vjej. The usual Euclidean norm is x = (x, x). These systems are referred to as qubits, qutrits in case k k P P P of n =2 or 3, and qunits for general n (or rather qudits with general d). p Pure states of a quantum system described by such space H are unit vectors in H. 1 Remark 1. More precisely, two vectors that differ by a multiplier are considered to de- scribe the same state, so that the state space is the projective space CP k−1 of the equiva- lence classes of vectors with equivalence defined as proportionality. The space of all n n matrices A =(Aij) can be considered as the space of all linear operators (Cn) in C×n. This correspondence is described equivalently either via the L action on the basis vectors as Aej = k Akjek, or in coordinates as (Au)j = k Ajkuk. (Cn) is a space of dimension n2 with the usual operator norm defined as L P P A = sup Ax . (1) k k kxk=1 k k m Similarly the space (H1,H2), H2 = C , of linear operators H1 H2 can be identified with the space of m Ln matrices. It becomes a norm space under→ the standard operator × norm (1), where x is the norm in H1 and Ax is the norm in H2. The subspacek ofk (Cn) consisting of self-adjointk k or Hermitian matrices, defined by the equation A∗ = A, whereL A∗ = A¯T (T for transpose and the bar for complex conjugation) will be denoted (Cn). The trace of A (Cn) is defined as tr A = A . It implies Ls ∈L j jj that tr(AB) = tr(BA), which in turn implies that tr(C−1AC)=trA for any invertible C. P Consequently one obtains two other equivalent expressions for the trace of A (Cn): ∈ Ls tr A = λ , where λ is the collection of all eigenvalues of A and j j { j} P tr A = (ej, Aej), j X where e is an arbitrary orthonormal basis in Cn. { j} The space (Cn) is a Hilbert space with respect to the scalar product L ∗ (A, B)HS = tr(A B)= A¯ijBij, (2) i,j X with the corresponding norm called the Hilbert-Schmidt norm A = [tr(A∗A)]1/2. k kHS For Hermitian operators it simplifies to tr(A∗B) = tr(AB). n Any A s(C ) is diagonizable, meaning that there exists a unitary U such that ∗ ∈ L A = U DU, where D = D(λ1, ,λn) is diagonal with the eigenvalues λj of A on the diagonal. Then A is defined··· as the positive operator A = U ∗ D U{ with} D = | | | | | | | | D( λ1 , , λn ), the diagonal operator with the numbers λj on the diagonal. The functional| | ··· A| | tr( A ) defines yet another norm on (Cn),{| the|}trace norm: 7→ | | Ls A = tr( A )= λ . k ktr | | | j| j X The key point is that this norm is dual to the usual operator norm with respect to the duality provided by the trace: for A (Cn), ∈Ls A tr = sup tr(AB) , A = sup tr(AB) . (3) k k kBk=1 | | k k kBktr =1 | | 2 To show these equations it is handy to work in the basis where A is diagonal: A = D(λ , ,λ ). Then tr(AB)= λ B . Choosing B to be diagonal with B equal the 1 ··· n j j jj ij sign of λj it follows that P sup tr(AB) λ = A . | | ≥ | j| k ktr kBk=1 j X On the other hand, sup tr(AB) = sup λjBjj λj max Bjj λj . | | | | ≤ | | j | | ≤ | | kBk=1 kBk=1 j j j X X X Therefore the first equation in (3) is proved. The second equation is proved similarly. For two spaces H = Cn and H = Cm with orthonormal bases e , , e and f , f 1 2 1 ··· n 1 ··· m the tensor product space H1 H2 can be defined as the nm-dimensional space generated by vectors denoted e f , so⊗ that any ψ H H can be represented as j ⊗ k ∈ 1 ⊗ 2 n m ψ = a e f . (4) jk j ⊗ k j=1 X Xk=1 The tensor product of any two-vectors u = j ujej, v = k vkfk is defined as the vector P P u v = u v e f . ⊗ j k j ⊗ k Xj,k Similarly, the tensor product is defined for several Hilbert spaces. Namely, the product H = H1 H2 HK of spaces of dimensions n1 , nK can be described as the ⊗ ⊗···⊗ ··· 1 K n = nj-dimensional space generated by vectors denoted ej1 ejK and called the 1 K m m ⊗···⊗ tensor products of ej1 , , ejK , where e1 , , enm is an orthonormal basis in Hm. AnyQ ψ H H can··· be represented{ by··· the so-called} Schmidt decomposition ∈ 1 ⊗ 2 min(n,m) ψ = λ ξ η , (5) j j ⊗ j j=1 X where λ 0, and ξ and η are some orthonormal bases in H and H . In fact, one j ≥ { j} { j} 1 2 just has to write the singular decomposition of the matrix A =(ajk) from (4), namely to T represent it as A = UDV , where D = D(λ1,λ2, ) is diagonal of dimension m n and 2 ··· × ∗ U, V are unitary matrices. Here λj are the (common) eigenvalues of the matrices A A and AA∗ with λ 0. Then the vectors ξ = U e and η = V f form orthonormal bases j ≥ j lj l k lj l in H1 and H2 and ψ = U λ V e f jl l kl j ⊗ k Xj,k,l equals (5). Pure states of the tensor product ψ H H are called entangled (the term introduced ∈ 1⊗ 2 by E. Schr¨odinger in 1935) if they cannot be written in the product form ψ = u v with some u, v. It is seen that ψ is not entangled if and only if its Schmidt decomposition⊗ has only one nonzero term. On the other hand, a pure state ψ H H is called maximally ∈ 1 ⊗ 2 3 entangled, if its Schmidt decomposition has the maximal number of nonvanishing λj and they all are equal, and thus they equal 1/ min(n, m). In physics one usually works in Dirac’s notations. In these notations usual vectors p u = u e H are referred to as ket-vectors, are denoted u and are considered to be j j j ∈ | i column vectors with coordinates uj. The corresponding bra-vectors are denoted u and are P h | considered to be row vectors with coordinatesu ¯j. These notations are convenient, because they allow to represent both scalar and tensor products as usual matrix multiplications: for two ket-vectors u and v we have | i | i u v = u . v =(u, v)= u¯ v , h | i h | | i j j j X v u = v u¯ = v u¯ e e . | ih | ⊗ j k j ⊗ k Xj,k Therefore the latter product is often identified with the n n-matrix ρ with the entries × ρ = v u¯ . As matrices, they act on vectors w = w in the natural way: jk j k | i ρw = v u w , (ρw) = ρ w = v u¯ w = v u w .
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