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Extra info for Investigations in Quantum Games using EPR-Type Setups

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We suggest that the following two constraints should be imposed on this association. 54 (c1) The players choose their moves (or actions) from the same set in both the classical and the quantized game. (c2) The players agree on explicit expressions for their payoffs which must not be modified when switching between the classical and the quantized version of the game. Games with these properties are expected to be immune to the criticism raised above. In the new setting the only ‘parameter’ is the input state on which the players act; its nature will determine the classical or quantum character of the game.

He then makes an important observation that when the homogeneous space is identified with Hilbert space projectors this identification induces another metric on the space which is locally convex. Now EPR’s definition of element of physical reality [33] forces the square of the locally convex metric equal to the metric determined by the hidden variable measure. But, in fact, it is impossible because the square of a locally convex metric cannot be a metric. Recently, Fine and Malley [69, 70] have argued that violations of local realism can be found in all those situations where non-commutative observables are involved, without the necessity of sophisticated correlation experiments.

30) AB (B) j is i,j |cij |2 = 1. The state is called direct product (or separable) state if it is possible to factor it into two normalized states from the Hilbert spaces HA and HB . Assume that ψ (A) cj span the space of the composite system made of the two subsystems is a linear combination of the product basis states |i |ψ A |j B ) |j B A (A) = i ci |i A and ψ (B) are the two normalized states from HA and HB , respectively. 31) Now a state in HA ⊗ HB is called entangled if it is not a direct product state.

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