In 2011, Lloyd, Maccone et al derived dynamical equations that a chronology-respecting system interacting with a closed time-like curve “CTC” would experience. They discuss the possibility of time travel in the absence of general-relativistic closed-timelike curves, and investigate the implications of P-CTCs for enhancing the power of computation.[1]
Now Maccone, Lloyd, and myself are considering game playing using this mathematical “time-machine” model in order to remove bad Nash equilibria in so-called Prisoner’s dilemma-type problems. The use of projective closed time-like curves (aka “P-CTCs”) in game theoretic settings may seem overpowered, but, as quantum physicists, it is a fun exercise to consider this question, one that involves cooperation with yourself, lending your most central decisions over to a quantum experiment. The age old philosophical question of free-will is one of central relevance, should it be possible to implement these closed time like curves. Experiments using photons in a lab in Turin, Italy have already been done. [2]
A scenario is the following, which I call post-selected games (for the moment). Alice and Bob play a game in which the rewards/penalties are only distributed if the result of a quantum measurement made at the end of the game is positive. To participate in the game, they ante up (i.e., pay a fee in order to play), so that they have an incentive for the post-selecting measurement turning out positive. The post-selection allows effective P-CTCs: that is, conditioned on the measurement at the end being positive, A knows what B will do when she picks her own strategy, and vice versa. In the case of prisoner’s dilemma, this removes the bad Nash equilibrium, since if A defects, she knows that B will defect as well, while if A cooperates she knows that B will also cooperate.
To this end, I have been looking at the quantum private query (QPQ) protocol primitive, since these quantum private queries are indeed natural maps for implementing anonymous quantum queries, whereby the agent being queried does not see or store the question by definition of the construction of the quantum search.[3] I believe that using more general n-GHZ states (“quantum treasure maps”) we could implement a contract that executes only if (and only when) all the involved parties decide to
cooperate. This protocol might be able to find a use-case application in the smart contracts of Ethereum. We could perhaps propose a quantum smart contract algorithm as a result of the joint measurements of local quantum memories i.e. quantum random access memories “q-RAMs”.
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