User:Jbradmitchell/Quantum game theory

History

In 1969, John Clauser, Michael Horne, Abner Shimony, and Richard Holt (often referred to collectively as "CHSH") wrote an often-cited paper describing experiments which could be used to prove Bell's theorem. In one part of this paper, they describe a game where a player could have a better chance of winning by using quantum strategies than would be possible classically. While game theory was not explicitly mentioned in this paper, it is an early outline of how quantum entanglement could be used to alter a game.

*paragraph already in history*

Since Meyer's paper, many papers have been published exploring quantum games and the way that quantum strategies could be used in games that have been commonly studied in classical game theory.

Multiplayer games

PQ Penny Flip Game

The PQ penny flip game involves two players: Captain Picard and Q. Q places a penny in a box, then they take turns (Q, then Picard, then Q) either flipping or not flipping the penny without revealing its state to either player. After these three moves have been made, Q wins if the penny is heads up, and Picard if the penny is face down.

The classical Nash Equilibrium has both players taking a mixed strategy with each move having a 50% chance of either flipping or not flipping the penny, and Picard and Q will each win the game 50% of the time using classical strategies.

Allowing for Q to use quantum strategies, namely applying a Hadamard gate to the state of the penny places it into a superposition of face up and down, represented by the quantum state

$|\psi _{1}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )$

In this state, if Picard does not flip the gate, then the state remains unchanged, and flipping the penny puts it into the state

$|\psi _{2}\rangle ={\frac {1}{\sqrt {2}}}(|1\rangle +|0\rangle )=|\psi _{1}\rangle$

Then, no matter Picard's move, Q can once again apply a Hadamard gate to the superposition which results in the penny being face up. In this way the quantization of Q's strategy guarantees a win against a player constrained by classical strategies.

This game is exemplary of how applying quantum strategies to classical games can shift an otherwise fair game in favor of the player using quantum strategies.

Quantum Card Game

A classically unfair card game can be played as follows: There are two players, Alice and Bob. Alice has three cards: one has a star on both sides, one has a diamond on both sides, and one has a star on one side and a diamond on the other side. Alice places the three cards in a box and shakes it up, then Bob draws a card so that both players can only see one side of the card. If the card has the same markings on both sides, Alice wins. But if the card has different markings on each side, Bob wins. Clearly, this is an unfair game, where Alice has a probability of winning of 2/3 and Bob has a probability of winning of 1/3. Alice gives Bob one chance to "operate" on the box and then allows him to withdraw from the game if he would like, but he can only classically obtain information on one card from this operation, so the game is still unfair.

However, Alice and Bob can play a version of this game adjusted to allow for quantum strategies. If we describe the state of a card with a diamond facing up as $|0\rangle$ and the state where the star is facing up as $|1\rangle$, after shaking the box up, we can describe the state of the face-up part of the cards as:

$|r\rangle =|r_{0}\ r_{1}\ r_{2}\rangle$

where each $r_{k}$ is either 0 or 1.

Now, Bob can take advantage of his ability to operate on the box by constructing a machine as follows: First, he has a unitary matrix defined as $U_{k}={\begin{bmatrix}1&0\\0&e^{i\pi r_{k}}\end{bmatrix}}$. This matrix is equal to $I$ if $r_{k}$ is 0 and $Z$ if $r_{k}$ is 1. He then creates his machine by putting this matrix between two Hadamard gates, so his machine now looks as follows:

$HU_{k}H={\frac {1}{2}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}1&0\\0&e^{i\pi r_{k}}\end{bmatrix}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1+e^{i\pi r_{k}}&1-e^{i\pi r_{k}}\\1-e^{i\pi r_{k}}&1+e^{i\pi r_{k}}\end{bmatrix}}$

This machine operating on the state $|0\rangle$ gives

$HU_{k}H|0\rangle ={\frac {1+e^{i\pi r_{k}}}{2}}|0\rangle +{\frac {1-e^{i\pi r_{k}}}{2}}|1\rangle =|r_{k}\rangle$

So if Bob inputs $|000\rangle$ to his machine, he obtains

$(HU_{k}H\otimes HU_{k}H\otimes HU_{k}H)|000\rangle =|r_{0}\ r_{1}\ r_{2}\rangle$

and he knows the state (i.e. the mark facing up) of all three of the cards. From here, Bob can draw one card, and then choose to either withdraw, or keep playing the game. Based on the first card that he draws, he can know from his knowledge of the face-up values of the cards whether or not he has drawn a card that will give him even chances of winning going forward (in which case he can continue to play a fair game) or if he has drawn the card that will guarantee that he loses the game. In this way, he can make the game fair for himself.

This is an example of a game where a quantum strategy can make a game fair for one player when it would be unfair for them with classical strategies.

Paradoxes

Quantum game theory also offers a solution to Newcomb's Paradox.

Take the two boxes offered in Newcomb's game to be coupled, as the contents of box 2 depend on if the ignorant player takes box 1. Quantum game theory enables a situation such that foreknowledge by otherwise omniscient player isn't required in order to achieve the situation. If the otherwise omniscient player operates on the state of the two boxes using a Hadamard gate, then sets up a device that operates on the state defined by the two boxes to operate again using a Hadamard gate after the ignorant player's choice. Then, no matter the pure or mixed strategy that the ignorant player uses, the ignorant player's choice will lead to it's corresponding outcome as defined by the premise of the game. Because choosing a strategy for the game, then changing it to fool to otherwise omniscient player (corresponding to operating on the game state using a NOT gate) cannot give the ignorant player an additional advantage, as the two Hadamard operations ensure that the only two outcomes are those defined by the chosen strategy. In this way, the expected situation is achieved no matter the ignorant player's strategy without requiring a system knowledgeable about that player's future.