User:Aannulis/Quantum refereed game

Bell state Quantum Refereed Game:

In a Bell State Quantum Refereed Game, there are three participants, Alice, Bob, and the Referee. The game consists of three doors. Behind each door is either an x or an o (spin up state or spin down state). The referee gives Alice and Bob three conditions about what is behind each of the doors. For example, the conditions could be: 1) Doors1 and 2 have the same. 2) Doors 2 and 3 have the same. 3) Door 1 and 3 are different.

The aim of this game is for Alice and Bob to find a matching pair behind the doors. In quantum terms, this means that Alice and Bob produce matching density states. During the game, Alice and Bob are not allowed to communicate, but they are allowed to strategize before the game begins. They do this by sharing an entangled pair of photons. Strategizing allows for Alice and Bob to maximize their changes of winning. Without strategizing, Alice and Bob have a 2/3 chance of winning. By strategizing, Alice and Bob's probability of producing matching quantum states increases from 2/3 to 3/4.

CHSH Refereed Game:

One example of a quantum refereed game is a CHSH quantum refereed game. A CHSH game uses binary form to represent outputs (i.e 0's and 1's). In this game Alice and Bob are playing together against a hypothetical house and are not allowed to communicate with one another. Alice and Bob are each given a random qubit from the referee. To win the game they need to provide outputs (a and b) that maximize a⊕b = xy.

Classical Analysis of a CHSH Refereed Game: The best strategy for Alice and Bob is to always return a state of 0 to the referee. If they do this as evidenced by the chart they win with a probability of 75%. Due to these probabilities the house decides that if Alice and Bob win they gain 1 point but if they lose they lose 3 points. This gives a probability payout of $$1*(3/4)-3*(1/4)=0 $$. Which ensures that everyone breaks even.

Quantum Analysis of a CHSH Quantum Refereed Game

Alice and Bob can use entangled states to rig the game to their favor. Alice and Bob both receive a photon in the entangled state |ψ⟩ = √2 [|0⟩|0⟩ + |1⟩|1⟩]. If Alice receives x=1 she can apply the Unitary Û(π/4) to her qubit and then measure it, if she receives x=0 all she needs to do is measure her qubit. If Bob gets y=0 he applies Û(π/8) and measures it and if he receives y=1 he applies Û(-π/8) and then measures it. Using this strategy allows for them to have a maximum winning probability of S= $$2\sqrt{2}$$, which allows for Alice and Bob to win every time.

The referee:

The role of the referee is to pass along qubits to players Alice and Bob. It is the referee's job to entangle the qubits, which is argued to be essential in quantum games. When Alice and Bob return the qubits to the referee, the referee then checks their final states.

Additions to definition of quantum refereed game:

A language L is considered to have a refereed game with error ε if it has a polynomial time verifier satisfying these conditions: for each string x∈L Alice (the yes prover) can convince the referee to accept x with probability of at least 1-ε regardless of Bob's strategy (the no prover) and for each string x$$\not\in$$ L Bob can convince the referee to reject x with a probability of at least 1-ε regardless of Alice's strategy.

One turn Quantum refereed games:

One turn quantum refereed games are a sub set of quantum refereed games (QRG) where there are two unbounded players (Alice and Bob) and a computationally bounded referee. They are called one turn games or QRG1 because there is only one turn per game. The game works by having each player send a density matrix to the referee who then plugs those states into his quantum circuit. The winner of the game is decided by the outcome of the circuit where, Alice wins the majority of times when a "yes" or |1> state is produced by the circuit and Bob wins the majority of the time when a "no" or |0> state is produced by the circuit. A turn consists of the referee sending a message to the prover (Alice or Bob) and then Alice or Bob sending a response back to the referee. The order of the game goes as follows: Alice sends the referee her density matrix, then the referee processes Alice's state and sends a state to Bob, Bob then measures the state and sends a classical result back to the referee, the referee then checks Bob's measurement and either produces a "yes" meaning Alice wins or produces a "no" and Bob wins.

https://arxiv.org/pdf/cs/0412102.pdf