Shengyu Zhang The Chinese University of Hong Kong

Content Classical game theory. Quantum games: models. Quantum strategic games: relation to classical equilibria. Quantum advantage against classical players Quantum advantage with quantum players.

Quantum information processing Understanding the power of quantum  Computation: quantum algorithms/complexity  Communication: quantum info. theory  Cryptography: quantum cryptography, quantum algorithm breaking classical cryptosystems. Another field: game theory

Game: Two basic forms strategic (normal) form extensive form

Game: Two basic forms strategic (normal) form

Equilibrium Equilibrium: each player has adopted an optimal strategy, provided that others keep their strategies unchanged

Classical strategic games outpututility potential action

“quantum games” Non-local games EWL-quantization of strategic games* 1 Others  Meyer’s Penny Matching*2  Gutoski-Watrous framework for refereed game*3 *1. Eisert, Wilkens, Lewenstein, Phys. Rev. Lett., *2. Meyer, Phys. Rev. Lett., *3. Gutoski and Watrous, STOC, 2007.

EWL model ⋮ ⋮⋮⋮ |0|0 |0|0 What’s this classically? ⋮ States of concern outpututility potential action

EWL model Classically we don’t undo the sampling (or do any re- sampling) after players’ actions.  If a joint operation is allowed, then a simple swapping solves Prisoner’s Dilemma classically! ⋮ ⋮⋮ |0|0 |0|0 ⋮ ⋮ outpututility potential action

EWL model ⋮ ⋮⋮ |0|0 |0|0 ⋮ ⋮ outpututility potential action

Quantum strategic games ⋮ ⋮ ρ A simpler model, corresponding to classical games more precisely. CPTP ⋮ ⋮ outpututility potential action *1. Zhang, ITCS, 2012.

Other than the model Main differences than previous work in quantum strategic games: We consider general games of growing sizes.  Previous: specific games, usually 2*2 or 3*3 We study quantitative questions.  Previous work: advantages exist?  Ours: How much can it be?

Central question: How much “advantage” can playing quantum provide? Measure 1: Increase of payoff Measure 2: Hardness of generation

First measure: increase of payoff We will define natural correspondences between classical distributions and quantum states. And examine how well the equilibrium property is preserved.

Quantum equilibrium classical quantum Φ1Φ1 ΦnΦn ⋮ ⋮ s1s1 snsn ⋮ ρ u 1 (s) u n (s) ⋮ C1C1 CnCn ⋮ s1’s1’ sn’sn’ ⋮ p → s u 1 (s’) u n (s’) ⋮ classical equilibrium: No player wants to do anything to the assigned strategy s i, if others do nothing on their parts - p = p 1  …  p n : Nash equilibrium - general p: correlated equilibrium quantum equilibrium: No player wants to do anything to the assigned strategy ρ| Hi, if others do nothing on their parts - ρ = ρ 1  …  ρ n : quantum Nash equilibrium - general ρ: quantum correlated equilibrium

Correspondence of classical and quantum states classical quantum p: p(s) = ρ ss (measure in comp. basis) ρ p: distri. on S ρ p = ∑ s p(s) |s  s| (classical mixture) |ψ p  = ∑ s √p(s) |s  (quantum superposition)  ρ s.t. p(s) = ρ ss (general class) Φ1Φ1 ΦnΦn ⋮ ⋮ s1s1 snsn ⋮ ρ C1C1 CnCn ⋮ s1’s1’ sn’sn’ ⋮ p→sp→s

Preservation of equilibrium? classical quantum p: p(s) = ρ ss ρ p ρ p = ∑ s p(s) |s  s| |ψ p  = ∑ s √p(s) |s   ρ s.t. p(s) = ρ ss Obs: ρ is a quantum Nash/correlated equilibrium  p is a (classical) Nash/correlated equilibrium pNECE ρpρp |ψp|ψp gen. ρ Question: Maximum additive and multiplicative increase of payoff (in a [0,1]-normalized game, |S i | = n)?

Maximum additive increase classical quantum p: p(s) = ρ ss ρ p ρ p = ∑ s p(s) |s  s| |ψ p  = ∑ s √p(s) |s   ρ s.t. p(s) = ρ ss pNECE ρpρp 00 |ψp|ψp 0 1- Õ (1/log n) gen. ρ1-1/n max payoff increase, additive Question: Maximum additive and multiplicative increase of payoff (in a [0,1]-normalized game, |S i | = n)?

Maximum multiplicative increase classical quantum p: p(s) = ρ ss ρ p ρ p = ∑ s p(s) |s  s| |ψ p  = ∑ s √p(s) |s   ρ s.t. p(s) = ρ ss pNECE ρpρp 11 |ψp|ψp 1 Ω (n 0.585… ) gen. ρnn Question: Maximum additive and multiplicative increase of payoff (in a [0,1]-normalized game, |S i | = n)? max payoff increase, multipliative

Next Classical game theory. Quantum games: models. Quantum strategic games: relation to classical equilibria. Quantum advantage against classical players Quantum advantage with quantum players.

Meyer’s game: classical 0 0/1?

Meyer’s game: quantum

Meyer’s game: fairness issue

Quantization* 1 of strategic game: Penny Matching *1. Zu, Wang, Chang, Wei, Zhang, Duan, NJP, potential action classical outcome utility

Quantization of strategic game: Penny Matching potential action classical outcome utility

Quantum advantage in strategic games Entangled. Necessary? No entanglement. But has discord. 2. Wei, Zhang, manuscript, *2. Wei, Zhang, manuscript, potential action classical outcome utility

Games between quantum players After these examples, Bob realizes that he should use quantum computers as well. Question: Any advantage when both players are quantum? Previous correspondence results imply a negative answer for complete information games. But quantum advantage exists for Bayesian games!

Quantum Bayesian games potential actionclassical outcome utility

Quantum Bayesian games potential actionclassical outcome utility

Quantum Bayesian games potential actionclassical outcome utility

Quantum Bayesian games *1. Pappa, Kumar, Lawson, Santha, Zhang, Diamanti, Kerenidis, PRL, potential actionclassical outcome utility

Quantum Bayesian games *1. Pappa, Kumar, Lawson, Santha, Zhang, Diamanti, Kerenidis, PRL, potential actionclassical outcome utility

Viewed as non-locality Traditional quantum non-local games exhibit quantum advantages when the two players have the common goal.  CHSH, GHZ, Magic Square Game, Hidden Matching Game, Brunner-Linden game. Now the two players have conflicting interests. Quantum advantages still exist. Message: If both players play quantum strategies in an equilibrium, they can also have advantage over both being classical.

Summary Quantum strategic games: be careful with the model. Quantum vs. classical: quantum has advantage. Quantum vs. quantum: both have advantages compared to both being classical. (Experiments for the above two results.) The field of quantum game theory: call for more systematic studies in proper models.