Bisynchronous Games and Factorizable Maps

Synchronous games with $*$-isomorphic game algebras

“…Every synchronous game with n inputs and k outputs is * -equivalent to a bisynchronous game with n inputs and nk outputs. The main idea of this construction is that players respond to questions with allowable answers and the question that they received [21]. However, the case of bisynchronous games with identical question and answer sets is much more interesting: any winning hereditary strategy corresponds to a quantum permutation over a (hereditary) unital * -algebra A, subject to orthogonality conditions imposed by the rule function of the game.…”

“…We recall that a quantum permutation over A is a collection of elements p ij ∈ A, where 1 ≤ i, j ≤ m, such that each p ij is self-adjoint; j p ij = i p ij = 1 for all i, j; and p ij p ik = δ jk p ij and p ij p ℓj = δ iℓ p ij , where δ jk denotes the usual Dirac delta. In this way, bisynchronous games on m inputs and m outputs are closely related to the quantum permutation group S + m [21]. In the setting of a bisynchronous game with m questions and m answers and a winning hereditary strategy given by projections {E a,x } m a,x=1 , the quantum permutation obtained is the matrix U = (E a,x ) m a,x=1 .…”

“…In the setting of a bisynchronous game with m questions and m answers and a winning hereditary strategy given by projections {E a,x } m a,x=1 , the quantum permutation obtained is the matrix U = (E a,x ) m a,x=1 . Indeed, each entry is a self-adjoint idempotent, and row and column sums are PVMs in the hereditary case [21]. Some caution is required: in the game algebra A(G) with generators e a,x , while m a=1 e a,x = 1 and e a,x e b,x = 0 for a = b and e a,x e a,y = 0 for x = y, there is no guarantee that m x=1 e a,x = 1; that is, the column sums of the matrix (e a,x ) m a,x=1 are 1, but the row sums might not be 1 in general.…”

“…Regarding these games, two problems were posed by V.I. Paulsen and M. Rahaman in [21]: In contrast, if one allows for |I| = |O|, then the answers to both (1) and ( 2) are known. Indeed, this is since every synchronous game G = (I, O, λ) with |I| = n and |O| = k is equivalent, in a sense, to a bisynchronous game with n inputs and nk outputs [21], and there are examples of a synchronous game violating (1) (see [11]) and a synchronous game satisfying (2) (see [12]), both without the requirement that |I| = |O|.…”

“…Paulsen and M. Rahaman in [21]: In contrast, if one allows for |I| = |O|, then the answers to both (1) and ( 2) are known. Indeed, this is since every synchronous game G = (I, O, λ) with |I| = n and |O| = k is equivalent, in a sense, to a bisynchronous game with n inputs and nk outputs [21], and there are examples of a synchronous game violating (1) (see [11]) and a synchronous game satisfying (2) (see [12]), both without the requirement that |I| = |O|. On the other hand, a key example of a bisynchronous game with |I| = |O| is the graph isomorphism game Iso(G, H) between two finite, simple, undirected graphs G and H with the same number of vertices.…”