Quantumizing Classical Games:
An Introduction to Quantum Game Theory

Sowmitra Das
MSc, Applied Mathematics
Imperial College London

Email<EMAIL_ADDRESS>
As submission of Course-Project for :
MATH70007: Dynamics of Learning & Iterated Games
Autumn 2022

###### Abstract

We give a concise and self-contained introduction to the theory of Quantum
Games by reviewing the seminal works [1, 2, 3, 4] which initiated the study of
this field. By generalizing this body of work, we formulate a protocol to
_Quantumize_ any finite classical $n$-player game, and use a novel approach of
describing such a Quantum Game in terms of commuting Payoff Operators. We
describe what advantages can be gained by players by quantumizing such a game,
particularly, what additional Nash Equilibria the players can achieve and the
Pareto-Optimality of these additional equilibria.

###### Contents

  1. 1 Introduction
  2. 2 Review of Quantum Mechanics
  3. 3 Review of Game Theory
  4. 4 Quantum vs. Classical Players
  5. 5 Quantum vs. Quantum Players
  6. 6 The Density Matrix Formalism
  7. 7 Game Quantumization Protocol
  8. 8 A Mixed Strategy Quantum Game
  9. 9 Alternate Quantumization Protocol
  10. 10 Discussions and Conclusion

##  1 Introduction

Quantum Information Theory is the study of Information Processing on Quantum
Systems, utilizing the effects of Quantum Mechanics. This encompasses novel
domains such as Quantum Computation [5], Quantum Communication [6, 7], Quantum
Metrology [8], etc. Particularly, the principles of of Quantum Information
could be applied to any domain which uses computation, communication or
information-processing of any sort to perform a specific task. This brings us
to the study of Game Theory and its extension to Quantum Games.

Game Theory concerns itself with the strategic behaviour of rational agents
who are interacting with each other to achieve some payoff from a finite pool
of resources. The study of Quantum Games is a reasonable successor to this
theory, if the agents have to manipulate quantum systems or have access to
quantum effects while deciding on their strategies. such effects include
Superposition, Entanglement and/or the randomness of Quantum Measurements.
This analysis becomes particularly worthy of interest if these effects and
correlations result in strategies that cannot be emulated by classical means,
and if these _quantum_ strategies result in better payoff for the players.

We embark on our study of Quantum Games because over the past 2 decades,
researchers have shown just that. In 1999, in a seminal paper [1] Meyer showed
that, it is possible to construct a 2-player Quantum Game, where, if one
player is restricted to classical strategies and the other has access to
Quantum strategies, the Quantum player always wins. Then in 2000, Eisert,
Wilkens and Lewenstein [9, 2] quantumized another 2-player game, in which they
showed that if both players have access to a restricted set of Quantum
Strategies, they can achieve a Nash Equilibrium that is Pareto-Optimal to the
Classical Nash Equilibrium. Dahl [10], and later Landsburg [4, 11] generalized
this to arbitrary 2-player games, and show that the payoff resulting from both
players using Quantum Strategies is always at least as great as that obtained
from using Classical Strategies, and one can construct scenarios in which it
is greater.

In this expositional article, we review all of these results - by first laying
down the principles of Quantum Mechanics and Classical Game Theory (in a form
that is suitable for our purposes of generalization), and then provide a
framework to combine these two. In this journey, we quantumize the Penny-Flip,
Prisoner’s Dilemma and the Battle of Sexes games. At the end, we generalize
this body of work, and provide a protocol to quantumize any finite $n$-player
game. We also introduce a slightly novel device of _Payoff Operators_ to
describe the $n$-player Quantum Games. We hope this short introduction will be
helpful to anyone hoping to enter into research in this field.

##  2 Review of Quantum Mechanics

First, we review the basic principles of Quantum Mechanics and Quantum
Information Theory in a form that is suitable for the formulation of Quantum
Games. In what follows, we give a concise description of the postulates used
in the Quantum-Mechanical description of natural systems.

  1. 1.

_Postulate 1: States_
The fundamental object of interest of a system in Quantum Mechanics is its
“state”. The first postulate asserts that, the quantum state of an _isolated_
system is a (generally infinite-dimensional) vector in a complex Hilbert space
$\mathcal{H}(\mathbb{C})$. However, for our purposes, we will be solely
focusing on finite dimensional systems. Of particular interest will be systems
of dimension 2, called qubits - aptly named because they are the quantum
analogue of their classical counterpart “bits”. So, in Dirac’s famous bra-ket
notation, the state of a qubit $\ket{\psi}$ is represented as -

$\ket{\psi}=\begin{bmatrix}\alpha\\\ \beta\end{bmatrix}$

where, $\alpha$ and $\beta$ are complex numbers. However, $\alpha$ and $\beta$
cannot be arbitrary, and have to obey certain constraints which will be
outlined in subsequent postulates.

Particular states of qubits are given special names. The canonical vectors
$\begin{bmatrix}1\\\ 0\end{bmatrix}$ and $\begin{bmatrix}0\\\ 1\end{bmatrix}$
are called the $\ket{0}$ and $\ket{1}$ states respectively, thus named because
they are generally used to reflect the “0” and “1” states of a classical bit.
However, as is evident, a general quantum state of a qubit can be a complex
linear combination of these two states, i.e, in general -

$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$

This is called a “ _superposition_ ” state, which is one of the defining
features of a quantum system, and which has no classical analogue.

  2. 2.

_Postulate 2: Measurement_
A measurement of an observable $O$ of a Quantum System,

$\mathcal{M}_{o}=\\{(o_{1},\Pi_{1}),(o_{2},\Pi_{2}),\ldots,(o_{n},\Pi_{n})\\}$

is described by a set of outcomes $\\{o_{1},o_{2},\ldots o_{n}\\}$ and a
corresponding complete set of orthonormal projection operators
$\\{\Pi_{1},\Pi_{2},\ldots,\Pi_{n}\\}$, such that,

$\Pi_{i}\Pi_{j}=\delta_{ij}\Pi_{i}\qquad\text{and}\qquad\sum_{i}\Pi_{i}=\mathbb{I}$

where, $\delta_{ij}$ is the Kronecker Delta and $\mathbb{I}$ is the Identity
Operator.

One of the most intriguing (and mysterious) features of the Quantum-Mechanical
description of nature is that the measurement postulate asserts, the outcome
of a Quantum Measurement is not deterministic, rather probalistic. If a
measurement of $O$ is performed on a quantum system in a state $\ket{\psi}$,
and the measurement outcome $o_{i}$ has a corresponding projection operator
$\Pi_{i}$, then the _probability_ that the outcome $o_{i}$ will be obtained
upon measurement is given by -

$p(o_{i})=\bra{\psi}\Pi_{i}\ket{\psi}$

Here, $\bra{\psi}$ is the Hermitian Conjugate of the vector $\ket{\psi}$, i.e,

$\ket{\psi}^{\dagger}=(\ket{\psi}^{\textsf{T}})^{*}=\bra{\psi}$

Due to this probabilistic description, a quantum system (in general) does not
have a specific _a priori_ value for an observable $O$, and any of the
outcomes $o_{i}$ can be obtained as a measurement result.

Another consequence of this postulate is that it provides a constraint on the
form of $\ket{\psi}$. Since the rules of probability dictate that, the
probabilities of all outcomes must sum to 1, we obtain -

$1=\sum_{i}p(o_{i})=\sum_{i}\bra{\psi}\Pi_{i}\ket{\psi}=\bra{\psi}\left(\sum_{i}\Pi_{i}\right)\ket{\psi}=\bra{\psi}\mathbb{I}\ket{\psi}=\bra{\psi}\ket{\psi}$

i.e, $\bra{\psi}\ket{\psi}=1$. This implies that, $\ket{\psi}$ is a normalised
vector. Particularly, if we have a qubit state
$\ket{\psi}=\begin{bmatrix}\alpha\\\ \beta\end{bmatrix}$, then,

$\absolutevalue{\alpha}^{2}+\absolutevalue{\beta}^{2}=1$

The second part of the measurement postulate dictates how the state of the
system changes under measurement. Similar to the outcome of the measurement,
this is also random, and it depends on the outcome. If an outcome of $o_{i}$
is obtained from a measurement of $O$ on a quantum system in a state
$\ket{\psi}$, then the post-measurement state given $o_{i}$ is -

$\ket{\psi_{i}}=\frac{\Pi_{i}\ket{\psi}}{\sqrt{p(o_{i})}}$

i.e, upon measurement, the state $\ket{\psi}$ instantaneously (and randomly)
“collapses” to the state $\ket{\psi_{i}}$ with probability $p(o_{i})$. This is
famously termed as “the collapse of the wave-function”.

At the risk of repitition, we reiterate that, this measurement procedure is
“intrinsically” random and does not represent any ignorance or faulty
equipment on the side of the experimenter. In a Quantum-Mechanical description
of nature, there is no dynamical equation that can deterministically predict
the state to which a collapse will occur. We can only measure probabilities.
(Although, there is an evident caveat. Clearly, if the probability of a
certain outcome is _1_ , we _can_ predict with full confidence the state to
which the collapse will occur after the measurement.)

We demonstrate the measurement postulate with the following example. Suppose,
a qubit is in the state -

$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$

We perform a measurement on this state with outcomes and projection operators
-

$\mathcal{M}=\\{(0,\outerproduct{0}{0}),(1,\outerproduct{1}{1})\\}$

Then, the probabilities of getting the outcomes 0 and 1 are, respectively,

$p(0)=\bra{\psi}\big{(}\outerproduct{0}{0}\big{)}\ket{\psi}=\absolutevalue{\bra{0}\ket{\psi}}^{2}=\absolutevalue{\alpha}^{2}$
$p(1)=\bra{\psi}\big{(}\outerproduct{1}{1}\big{)}\ket{\psi}=\absolutevalue{\bra{1}\ket{\psi}}^{2}=\absolutevalue{\beta}^{2}$

If the measurement outcome is $0$, the post measurement state is -

$\ket{\psi_{0}}=\frac{(\outerproduct{0}{0})\ket{\psi}}{\sqrt{p(0)}}=\frac{\alpha\ket{0}}{\absolutevalue{\alpha}}\sim\ket{0}$

up to a global phase factor $e^{i\theta}$ which is not important in Quantum
Mechanics. Similarly, we can show that,

$\ket{\psi_{1}}=\ket{1}$

This type of measurement using the projectors $\outerproduct{0}{0}$ and
$\outerproduct{1}{1}$ is also called a “measurement in the
$\\{\ket{0},\ket{1}\\}$ basis”. In short, if a state
$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}$ is measured in the
$\\{\ket{0},\ket{1}\\}$ basis, then, $\ket{\psi}$ collapses to $\ket{0}$ with
probability $\absolutevalue{\alpha}^{2}$ and it collapses to $\ket{1}$ with
probability $\absolutevalue{\beta}^{2}$. Here, $\alpha$ and $\beta$ are called
the _probability amplitudes_ corresponding to $\ket{0}$ and $\ket{1}$
respectively.

  3. 3.

_Postulate 3: Evolution_
The third postulate describes how a state evolves with time. It dictates that,
the evolution of the state of an _isolated_ quantum system (i.e, one which is
not interacting with the outside world, or, one which is not being measured)
is described by a “linear” operator, given that normalization is preserved.
So, if a state $\ket{\psi}$ evolves into a state $\ket{\psi^{\prime}}$, then,

$\ket{\psi^{\prime}}=U\ket{\psi}$

for some linear operator $U$. In addition, if $\ket{\psi}$ is normalized, then
$\ket{\psi^{\prime}}$ also has to be normalized. So, we have,

$1=\bra{\psi^{\prime}}\ket{\psi^{\prime}}=\bra{\psi}U^{\dagger}U\ket{\psi}$

Since, this needs to be true for all normalized vectors $\ket{\psi}$, we have,

$U^{\dagger}U=\mathbb{I}$

Operators which satisfy this condition are called _Unitary_ Operators. So, in
short, time evolution of an isolated quantum system is described by a Unitary
Operator.

  4. 4.

_Postulate 4: Composite Systems_
So far, we have considered the states of systems which are isolated. This
postulate provides a means to combine systems, or from another perspective,
how to describe systems which are interacting with each other (but are
combinedly isolated from the rest of the universe).

Suppose, a system $A$ is interacting with a system $B$ (the combined system
$AB$ being isolated). If, the state of the isolated system $A$ is described by
the Hilbert Space $\mathcal{H}_{A}$ and the state of the isolated system $B$
is described by the Hilbert Space $\mathcal{H}_{B}$, then the state of the
combined interacting system $AB$ is described by the Tensor-Product Hilbert
Space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where,

$\mathcal{H}_{A}\otimes\mathcal{H}_{B}=\text{span}\left\\{\ket{\psi}_{A}\otimes\ket{\phi}_{B}\Big{|}\ket{\psi}_{A}\in\mathcal{H}_{A},\ket{\psi}_{B}\in\mathcal{H}_{B}\right\\}$

i.e, $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ consists of linear combinations
of normalized tensor-product states of the form
$\ket{\psi}_{A}\otimes\ket{\phi}_{B}$ over the field of complex numbers
$\mathbb{C}$.

There are 2 classes of states in the space
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. States of the form
$\ket{\psi}_{A}\otimes\ket{\phi}_{B}$ are called “Product States”, and they
describe a situation in which the system $A$ is in the state $\ket{\psi}_{A}$
and the system $B$ is in the state $\ket{\phi}_{B}$. However, infinitely more
interesting, is a second class of states which cannot be factorized into
seperate states for the individual systems. A simple example is the following
2-qubit state -

$\ket{\Phi}_{AB}=\frac{1}{\sqrt{2}}\left(\ket{0}_{A}\otimes\ket{0}_{B}+\ket{1}_{A}\otimes\ket{1}_{B}\right)$

As can be checked easily, the state $\ket{\Phi}_{AB}$ cannot be decomposed
into any state of the form $\ket{\psi}_{A}\otimes\ket{\phi}_{B}$. These type
of states are called “ _Entangled_ ” states and they represent a scenario in
which the combined system $AB$ has a state, but the individual systems do not
have a state of their own. Entangled states form the basis of the essential
“quantumness” in Quantum Information Theory, and all of the recent
breakthroughs in Quantum Computing. As we will soon see, they are also at the
heart of the advantages of Quantum Strategies over Classical Strategies in
game theoretic settings.

##  3 Review of Game Theory

Before moving on to analyzing the advantages of Quantum Strategies, we review
the basic notions of Game Theory and solution concepts in a form that is
suitable for generalization to the Quantum Domain.

A Pure-Strategy $n$-player “Game” G$(\mathcal{S},\pi)$ is described by a tuple
of “strategy” spaces -

$\mathcal{S}=(\mathcal{S}_{1},\mathcal{S}_{2},\ldots,\mathcal{S}_{n})$ (1)

and a tuple of payoff functions -

$\pi=(\pi_{1},\pi_{2},\ldots,\pi_{n})$ (2)

Each of the sets $\mathcal{S}_{i}$ represents the collection of ‘strategies’
player $i$ has access to in the game. This can be the set of moves that player
$i$ can play, or the decisions that he/she might take to influence the outcome
of the game. The set $\mathcal{S}_{i}$ can be written as follows -

$\mathcal{S}_{i}=\\{s_{i}^{1},s_{i}^{2},\ldots,s_{i}^{\alpha},\ldots\\}$ (3)

where, the $s_{i}^{\alpha}$ are the pure strategies that player $i$ has access
to. Henceforth, we consider “finite” games, i.e, games in which each of the
sets $\mathcal{S}_{i}$ are finite. We define a “ _play_ ” $\mathcal{P}$ of the
Game G as an $n$-tuple of strategies

$\mathcal{P}(\mathbf{\alpha}):=(s_{1}^{\alpha_{1}},s_{2}^{\alpha_{2}},\ldots,s_{n}^{\alpha_{n}})$

where, player $i$ chooses to play the strategy $s_{i}^{\alpha_{i}}$
($\alpha_{i}\in\\{1,2,\ldots,\absolutevalue{\mathcal{S}_{i}}\\}$). Then,
depending on this play, each player $j$ receives a reward that is computed by
the pay-off function $\pi_{j}$ -

$\pi_{j}(\mathcal{P})=\pi_{j}(s_{1}^{\alpha_{1}},s_{2}^{\alpha_{2}},\ldots,s_{n}^{\alpha_{n}})$

In general, the pay-off that player $j$ receives, depends not only on his/her
own strategy $s_{j}^{\alpha_{j}}$ but also on the strategies that the other
players have played. The analysis of games played by rational players is
carried out under the assumption that, each player tries to maximize his/her
own payoff.

In games of “ _Complete Information_ ”, each player knows the strategy spaces
$\mathcal{S}_{i}$ and payoff functions $\pi_{i}$ of all the other players.
However, a player $i$ doesn’t know what strategy player $j$ will use until the
game is played and all the players have revealed their strategies. Every
player only knows the set of possible strategies that each player can draw
from.

Within this framework, it is interesting to analyze what strategies the
players might use. These play of strategies that rational players might use in
a game are called ‘solutions’ of the game. One such solution concept is that
of a _Dominant Strategy_. Player $i$ has a dominant strategy $s_{i}^{D}$, if

$\pi_{i}(s_{1}^{\alpha_{1}},\ldots,s_{i-1}^{\alpha_{i-1}},s_{i}^{D},s_{i+1}^{\alpha_{i+1}},\ldots
s_{n}^{\alpha_{n}})\geq\pi_{i}(s_{1}^{\alpha_{1}},\ldots,s_{i-1}^{\alpha_{i-1}},s_{i}^{\alpha_{i}},s_{i+1}^{\alpha_{i+1}},\ldots
s_{n}^{\alpha_{n}})$ $\forall
s_{j}^{\alpha_{j}}\in\mathcal{S}_{j},\;j=1,2,\ldots,i,\ldots,n\qquad\text{for
a fixed }i$

Simply put, a dominant strategy $s_{i}^{D}$ maximizes the payoff of player $i$
for _any_ play of strategies from the other players. The strategy $s_{i}^{D}$
is called _Strictly Dominant_ if the above inequality is strict.

In general, there exist games where some or none of the players have a
dominant strategy. However, if any player does have such a strategy,
rationality dictates that the player will play the dominant strategy since it
maximizes his payoff no matter what the other players do. If _each_ player $j$
has a dominant strategy $s_{j}^{D_{j}}$, then the play of dominant strategies
$\mathcal{P}^{D}$ -

$\mathcal{P}^{D}=(s_{1}^{D_{1}},s_{2}^{D_{2}},\ldots,s_{n}^{D_{n}})$

can describe a “solution” to the game, as in rational players will tend to
play this strategy.

However, as stated earlier, in a general scenario some or none of the players
may have a dominant strategy. What can be the solution to the game in that
scenario ? We need a more general solution concept to describe these settings.
One such solution concept is the famous _Nash Equilibrium_ [12].

A play of strategies
$\mathcal{P}^{NE}=(s_{1}^{e_{1}},s_{2}^{e_{2}},\ldots,s_{n}^{e_{n}})$ is
called a Nash Equilibrium if and only if -

$\pi_{i}(s_{1}^{e_{1}},\ldots,s_{i-1}^{e_{i-1}},s_{i}^{e_{i}},s_{i+1}^{e_{i+1}},\ldots
s_{n}^{e_{n}})\geq\pi_{i}(s_{1}^{e_{1}},\ldots,s_{i-1}^{e_{i-1}},s_{i}^{\alpha_{i}},s_{i+1}^{e_{i+1}},\ldots
s_{n}^{e_{n}})$ $\forall s_{i}^{\alpha_{i}}\in\mathcal{S}_{i}\;\text{for a
fixed }i$

And this holds for all players $i=1,2,\ldots,n$. This means that, _unilateral_
deviance of any _single_ player from $\mathcal{P}^{NE}$ cannot increase his
payoff. So, none of the players have an incentive to deviate from the
equilibrium solution. If the above inequality is strict, then the solution is
called a _Strict Nash Equilibrium_. A game might have multiple Nash
Equilibria.

As can be seen clearly, if $\mathcal{P}^{D}$ is a dominant solution to a game
G, then it is also a Nash Equilibrium. However, Pure-Strategic games do not
always admit a Nash Equilibrium solution. In order to formulate a solution
concept that always exists, we have to extend our notion of pure strategies to
“mixed” strategies. A _Mixed Strategy_ game
$\textbf{G}_{M}(\mathbf{P}_{\mathcal{S}},\bar{\pi})$ corresponding to a pure-
strategy game G$(\mathcal{S},\pi)$ is described by a tuple of probability
distribution spaces -

$\mathbf{P}_{\mathcal{S}}=(\mathbf{P}_{\mathcal{S}_{1}},\mathbf{P}_{\mathcal{S}_{2}},\ldots,\mathbf{P}_{\mathcal{S}_{n}})$

where,

$\mathbf{P}_{\mathcal{S}_{i}}=\left\\{\mathbf{p}_{i}=[p_{i}^{\alpha_{1}}\;p_{i}^{\alpha_{2}}\;\ldots\;p_{i}^{\alpha_{\absolutevalue{\mathcal{S}_{i}}}}]\,\Big{|}\,p_{i}^{\alpha_{i}}\geq
0,\sum_{\alpha}p_{i}^{\alpha_{i}}=1\right\\}$

is the set of all probability distributions over the strategy space
$\mathcal{S}_{i}$. A play of $\mathbf{G}_{M}$ now consists of a tuple of
probability vectors $\mathbf{p}_{i}$ -

$\mathcal{P}_{\mathbf{G}_{M}}:=(\mathbf{p}_{1},\mathbf{p}_{2},\ldots,\mathbf{p}_{n})$

where each $\mathbf{p}_{i}$ is a probability distribution over the strategies
$s_{i}^{\alpha_{i}}$.
The payoff functions $\pi_{i}$ are now replaced by _expected_ payoff functions
$\bar{\pi}_{i}$, where,

$\bar{\pi}_{i}=\sum_{\alpha_{1},\alpha_{2},\ldots,\alpha_{n}}(p_{1}^{\alpha_{1}}\cdot
p_{2}^{\alpha_{2}}\cdot\ldots\cdot
p_{n}^{\alpha_{n}})\cdot\pi_{i}(s_{1}^{\alpha_{1}},s_{2}^{\alpha_{2}},\ldots,s_{n}^{\alpha_{n}})$

which can be succinctly written as -

$\bar{\pi}_{i}=\sum_{\mathcal{P}_{\mathbf{G}}}p(\mathcal{P_{\mathbf{G}}})\cdot\pi_{i}(\mathcal{P_{\mathbf{G}}})$

where the sum is over all plays $\mathcal{P}_{\mathbf{G}}(\mathbf{\alpha})$ of
the _pure-strategy_ game $\mathbf{G}$, and

$p(\mathcal{P}_{\mathbf{G}}(\mathbf{\alpha}))=p_{1}^{\alpha_{1}}\cdot
p_{2}^{\alpha_{2}}\cdot\ldots\cdot p_{n}^{\alpha_{n}}$

is the probability that the play $\mathcal{P}_{\mathbf{G}}(\mathbf{\alpha})$
will occur in the Mixed-Strategy game $\mathbf{G}_{M}$.
As described above, the probability distribution $p(\mathcal{P}_{\mathbf{G}})$
is separable, and each player chooses his/her strategy (viz. probabilities)
independently of the other players. We can easily envision probability
distributions for $p(\mathcal{P}_{\mathbf{G}})$ which are not separable, and
contemplate whether this results in additional properties of the game or
additional equilibria. Such games where $p(\mathcal{P})$ is not separable are
called (classically) “Correlated Games”. The emphasis on the _classical_
correlation is because, in the formulation of Quantum Games, we will see that
Quantum Mechanics, particularly Quantum Entanglement, can allow for much more
exotic correlations which cannot be mimicked by classical
means111Interestingly, this topic was the subject of the 2022 Nobel Prize in
Physics..

A remarkable result in the field of Game Theory, proved by Nash [12], is that
in any finite mixed-strategy game, there exists at least one Nash Equilibrium.
Thus, we have finally arrived at our desired solution concept.

Before moving on to the formulation of Quantum Games, we review another game-
theoretic concept called _Pareto-Dominance_ that is relevant in describing the
advantage of Quantum Strategies. A play $\mathcal{P}$ of a game $\mathbf{G}$
(mixed or pure) is _Pareto-Comparable_ to another play $\mathcal{P}^{\prime}$
if -

$\pi_{i}(\mathcal{P})\geq\pi_{i}(\mathcal{P}^{\prime})\;\forall
i\qquad\text{or}\qquad\pi_{i}(\mathcal{P})\leq\pi_{i}(\mathcal{P}^{\prime})\;\forall
i$

In general, two plays $\mathcal{P}$ and $\mathcal{P^{\prime}}$ are _not_
Pareto-Comparable. However, if they _are_ Pareto-Comparable, then, the play
$\mathcal{P}$ is said to _Pareto-Dominate_ the play $\mathcal{P}^{\prime}$ if

$\pi_{i}(\mathcal{P})\geq\pi_{i}(\mathcal{P}^{\prime})\;\forall i$

We write this succinctly as -

$\mathbf{\pi}(\mathcal{P})\succeq\pi(\mathcal{P}^{\prime})$

If the inequality is strict, then $\mathcal{P}$ _strictly_ Pareto-dominates
$\mathcal{P}^{\prime}$, and we write it as

$\pi(\mathcal{P})\succ\pi(\mathcal{P}^{\prime})$

A play $\mathcal{P}_{o}$ is called _Pareto-Optimal_ if, for any other play
$\mathcal{P}$ comparable to $\mathcal{P}_{o}$,

$\pi(\mathcal{P}_{o})\succeq\pi(\mathcal{P})$

This implies that, if the play $\mathcal{P}_{o}$ is Pareto-Optimal, no other
play of strategies can increase the payoff of one player without decreasing
the payoff of another.
Pareto-optimality captures the concept of “social welfare”, in a way that, it
ranks plays in an order which maximizes the overall payoff of all the players.
As a neutral judge of a game, we would like to have a solution of a game which
is a Nash Equilibrium and which is also Pareto-Optimal. However, a certain
solution might not always possess these two criteria at the same time.

##  4 Quantum vs. Classical Players

We begin our discussion of Quantum Game Theory by reviewing one of the seminal
results of this field, which initiated the study of Quantum Games. In 1999,
David Meyer first proposed that one can construct 2-player games with quantum
systems in which, if one player is restricted to Classical Strategies, while
the other player has access to Quantum Strategies, the Quantum player will
always win [1]. What is meant by these Classical and Quantum strategies, and
the game that Meyer used to demonstrate this result is described below.

#### The (Classical) Penny-Flip Game

This is a 2-player game consisting of players $Q$ and $C$, and a coin. The
coin starts out in the state Heads. First, Player Q is asked to select between
one of 2 moves or strategies - Flip (F) or No-Flip (N), i.e, Q can choose to
either Flip the coin or leave it as it is. Next, Player C is asked to do the
same, i.e, choose between F or N, without revealing the coin to him/her.
Finally, Player Q is again asked to choose between one of the 2 moves. The
coin is revealed to both players at the end of these 3 moves. (The coin
remains hidden until the 3 moves are over. Each player can only perform
operations on the coin, but cannot know its state during the game.) If at the
end of the 3 moves, the state of the coin is Heads, Q wins. If it is Tails, C
wins.

The payoff function $\pi_{C}$ of player $C$ in the Penny-Flip game can be
presented in normal form as in Table 1.

|  | Q
---|---|---
|  | NN | NF | FN | FF
C | N | -1 | 1 | 1 | -1
F | 1 | -1 | -1 | 1
Table 1: Payoffs of Player C for different plays of the Penny-Flip Game.

For any play $\mathcal{P}$, the pay-off of player Q is
$\pi_{Q}(\mathcal{P})=-\pi_{C}(\mathcal{P})$. So, as it is currently
formulated, this is a _Zero-Sum_ game.

As can be easily checked for this game, none of the players have a dominant
strategy and there are no Nash-Equilibria in Pure Strategies. However, there
is a Nash Equilibrium in mixed strategies
$\mathcal{P}^{NE}(\mathbf{p}_{C},\mathbf{p}_{Q})$, where,

$\mathbf{p}_{C}=\left[\frac{1}{2}\;\frac{1}{2}\right]\qquad\mathbf{p}_{Q}=\left[\frac{1}{4}\;\frac{1}{4}\;\frac{1}{4}\;\frac{1}{4}\right]$

i.e, each player plays each of his/her pure strategies with equal probability
with an expected payoff of $\bar{\pi}_{P}=\bar{\pi}_{Q}=0$. None of the
players can do any better by unilaterally diverting from this strategy.

#### The Quantum Penny-Flip Game

Now we move on to “ _Quantumize_ ” this game. To do this, we will consider the
scenario where the penny in the _Quantum_ Penny-Flip game is not a classical
penny, but a quantum penny. That is, where a classical penny has two states H
(Heads) and T (Tails), a quantum penny is a two-dimensional quantum system
with orthonormal basis states $\ket{\textbf{H}}=\ket{0}$ and
$\ket{\textbf{T}}=\ket{1}$. The Quantum Penny allows for a much larger state
space, since its state $\ket{\psi}$ can be any superposition of the two basis
states -

$\ket{\psi}=\alpha\ket{\textbf{H}}+\beta\ket{\textbf{T}}$

The basis states $\ket{\mathbf{H}}$ and $\ket{\mathbf{T}}$ correspond to the
classical states $\mathbf{H}$ and $\mathbf{T}$. The classical strategies of
Flip and No-Flip now correspond to Quantum Operators F and N, such that,

$\textsf{F}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\qquad\textsf{N}=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ (4)

One can easily check that, F and N are Unitary and satisfy,

$\displaystyle\textsf{F}\ket{\mathbf{H}}=\ket{\mathbf{T}}$
$\displaystyle\textsf{N}\ket{\mathbf{H}}=\ket{\mathbf{H}}$
$\displaystyle\textsf{F}\ket{\mathbf{T}}=\ket{\mathbf{H}}$
$\displaystyle\textsf{N}\ket{\mathbf{T}}=\ket{\mathbf{T}}$

So, F and N indeed correspond to the classical strategies of Flip and No-Flip.
Our quantumization of the Penny-Flip game is complete.

Now, to demonstrate the advantage of Quantum Strategies, we will allow the
player Q (henceforth called the Quantum Player) to have access to _any_
unitary operation $U$ to apply to the quantum penny (in addition to F and N)
during his move, while the Player C (henceforth called the Classical Player)
is only restricted to the “classical” moves F and N. Thus, Q has access to the
entire suite of Quantum Operations, a.k.a strategies, on the quantum penny,
whereas, C has access to only classical strategies. In this framework, we will
now show that, the Quantum Player always wins, i.e, Q can select operators
such that, at the end of the 3 moves, the state of the penny is always
$\ket{\mathbf{H}}$ no matter what move C chooses to play.

The strategy that Q uses is as follows. The quantum penny starts out in the
state $\ket{\mathbf{H}}$. In his first move, Q acts on the penny with unitary
operator $\textsf{U}_{Q}^{\star}$ defined as -

$\textsf{U}_{Q}^{\star}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\
1&-1\end{bmatrix}$

After this, the penny is passed to player C who applies an operators
$\textsf{U}_{C}\in\\{\textsf{F},\textsf{N}\\}$ to it. Next the penny is passed
back to player Q, who again applies the operator $\textsf{U}_{Q}^{\star}$ to
it, as defined above. So, the final state of the penny is
$\textsf{U}_{Q}^{\star}\textsf{U}_{C}\textsf{U}_{Q}^{\star}\ket{\mathbf{H}}$.
The judge of the game now performs a measurement on the penny using the
projection operators
$\\{\outerproduct{\mathbf{H}}{\mathbf{H}},\outerproduct{\mathbf{T}}{\mathbf{T}}\\}$.
If the measurement outcome is $\ket{\mathbf{H}}$, Q wins. Otherwise, if it is
$\ket{\mathbf{T}}$, C wins.

The remarkable thing is that, one can easily calculate,

$\textsf{U}_{Q}^{\star}\textsf{F}_{C}\textsf{U}_{Q}^{\star}\ket{\mathbf{H}}=\ket{\mathbf{H}}$
$\textsf{U}_{Q}^{\star}\textsf{N}_{C}\textsf{U}_{Q}^{\star}\ket{\mathbf{H}}=\ket{\mathbf{H}}$

So, no matter what strategy the classical player uses (given that he is
restricted to the classical strategies F and N), if the Quantum Player uses
the operator $\textsf{U}_{Q}^{\star}$ defined above, the final state of the
penny is always $\ket{\mathbf{H}}$, and thus, Q always wins.

This happens because,

$\textsf{U}_{Q}^{\star}\ket{\mathbf{H}}=\frac{1}{\sqrt{2}}\ket{\mathbf{H}}+\frac{1}{\sqrt{2}}\ket{\mathbf{T}}=\ket{+}$

One can check that, the $\ket{+}$ state is an eigenvector of both the
operators F and N, with an eigenvalue of 1, i.e,

$\textsf{F}\ket{+}=\ket{+}\qquad\textsf{N}\ket{+}=\ket{+}$

So, no matter which operation player C chooses in the 2nd move, the $\ket{+}$
state remains unchanged. Then, in the 3rd move, player Q can again use the
operation $\textsf{U}_{Q}^{\star}$ to revert the $\ket{+}$ state back to the
$\ket{\mathbf{H}}$ state (because
$\textsf{U}_{Q}^{\star}\ket{+}=\ket{\mathbf{H}}$), thus always ensuring a win.
This is the underlying mechanism why Quantum Strategies give an edge to the
player Q in this game.

##  5 Quantum vs. Quantum Players

As we saw in the previous section, in a Quantum Game (one in which quantum
systems are manipulated and communication is done utilizing quantum
technologies), if a player restricts himself to only Classical Strategies, he
risks to always losing to a player utilizing Quantum Strategies. So,
rationality dictates that, all the players in a quantum game will try to use
quantum strategies to have a level playing field.

Next, we quantumize another game in way that both players have access to
quantum strategies. We will see if they can utilize their strategies to obtain
a better payoff than in the classical case.

#### The Prisoner’s Dilemma

This is perhaps one of the most famous and extensively studied game in Game
Theory. There are 2 prisoners A and B, who are suspected of committing a
crime. Each prisoner is asked by a judge if the other has committed the crime.
If one prisoner rats the other one out (Defects), while the other one remains
silent (Cooperates), the cooperative player will receive a sentence of
$\alpha$ years and the defective player will be set free. However, if both
players defect, they will each receive a sentence of $\beta$ years
($\beta<\alpha$). And, if both cooperate, they will each receive a sentence of
$\gamma$ years ($\gamma<\beta$).

We can dexcribe this scenario by the following payoff bi-matrix -

|  | B
---|---|---
|  | $C$ | $D$
A | $C$ | $(-\gamma,-\gamma)$ | $(-\alpha,0)$
$D$ | $(0,-\alpha)$ | $(-\beta,-\beta)$

where, the first element of a tuple represents A’s payoff, and the second
element represents B’s.

In this game, we can see that Defection (D) is a dominant strategy for both
players. I.e, no matter what the other player does, defection results in a
greater payoff for a particular player. This results in a dominant solution
$\mathcal{P}^{D}=(D,D)$ for the game. In addition, $(D,D)$ is also a Nash
Equilibrium, i.e., if one player is adamant at defecting, the other player is
worse-off if he chooses to cooperate. However, we can clearly see that,
$(D,D)$ is _suboptimal_ to the Pareto-Optimal play $(C,C)$, where each player
can get a higher payoff if they both choose to cooperate rather than defect.
Except the dominant play, there are no other (pure or mixed) Nash Equilibrium.

#### The Quantum Prisoner’s Dilemma

Now, we move on to quantumize the Prisonner’s Dillema game. We follow the
scheme proposed by Eisert, Wilkens and Lewenstein (EWL) et al. [9, 2] which
will allow us to generalize and quantumize arbitrary Classical Games.

First, we choose 2 two-dimensional Quantum Systems A and B, with basis states
$\ket{{0}}$ and $\ket{{1}}$. A judge prepares the composite system AB in a
state $\ket{\psi}$, which can be entangled in general. Suppose, it starts out
in the state -

$\ket{\eta_{in}}_{AB}=\frac{1}{\sqrt{2}}\ket{00}+i\frac{1}{\sqrt{2}}\ket{11}$

($\ket{00}$ stands for $\ket{0}_{A}\otimes\ket{0}_{B}$. We omit the explicit
tensor products $\otimes$ between the states for simplicity, and the labels A
and B when they are evident from context.) Next, the two systems $A$ and $B$
are sent to the players A and B. Each player then applies operations $U_{A}$
and $U_{B}$ on their sub-systems and returns them to the judge. The composite
state that is returned to the judge is -

$\ket{\eta_{f}}_{AB}=(U_{A}\otimes U_{B})\ket{\eta_{in}}_{AB}$

Then, the judge performs a projective measurement on the entire system $AB$ in
the basis
$\\{\ket{\eta_{CC}},\ket{\eta_{CD}},\ket{\eta_{DC}},\ket{\eta_{DD}}\\}$,
where,

$\displaystyle\ket{\eta_{CC}}$ $\displaystyle=(\ket{00}+i\ket{11})/\sqrt{2}$
$\displaystyle\ket{\eta_{CD}}$ $\displaystyle=(\ket{01}-i\ket{10})/\sqrt{2}$
$\displaystyle\ket{\eta_{DD}}$ $\displaystyle=(\ket{00}-i\ket{11})/\sqrt{2}$
$\displaystyle\ket{\eta_{DC}}$ $\displaystyle=(\ket{01}+i\ket{10})/\sqrt{2}$

using the orthonormal projectors
$\Pi_{CC}=\outerproduct{\eta_{CC}}{\eta_{CC}}$, etc. Each player then receives
a payoff depending on the outcome of the measurement, where a measurement
outcome of $\ket{\eta_{CC}}$ corresponds to a play of the Classical Strategy
$(C,C)$, and thus, each of the projectors correspond to particular plays of
the pure strategy game.

An important point to note here is that, in Meyer’s scheme [1], the Classical
Strategies are mapped to Unitary Operators on a suitable Hilbert Space
($\mathbf{F}\rightarrow\textsf{F},\mathbf{N}\rightarrow\textsf{N}$). On the
other hand, in the EWL scheme, they are mapped to orthonormal projectors of a
projective measurement ($(C,C)\rightarrow\Pi_{CC}$, etc.) These are two
different quantumization schemes, and we will see that the EWL scheme is more
general and can be used to quantumize any Classical Game into a Quantum
Correlated game, whereas Meyer’s scheme is applicable only to a certain
special class of games where the players manipulate a common object.

Now, we continue with our discussion of the EWL quantumization scheme. Since
the measurement outcome of the Quantum Game is probabilistic, the expected
payoff of a player, suppose A, is calculated as -

$\displaystyle\bar{\pi}_{A}$
$\displaystyle=p(CC)\pi_{A}(CC)+p(CD)\pi_{A}(CD)+p(DC)\pi_{A}(DC)+p(DD)\pi_{A}(DD)$
$\displaystyle=\bra{\eta_{f}}\Pi_{CC}\ket{\eta_{f}}\cdot\pi_{A}(CC)+\bra{\eta_{f}}\Pi_{CD}\ket{\eta_{f}}\cdot\pi_{A}(CD)$
$\displaystyle\quad+\bra{\eta_{f}}\Pi_{DC}\ket{\eta_{f}}\cdot\pi_{A}(DC)+\bra{\eta_{f}}\Pi_{DD}\ket{\eta_{f}}\cdot\pi_{A}(DD)$
$\displaystyle=\bra{\eta_{f}}\Big{(}\underbrace{\Pi_{CC}\pi_{A}(CC)+\Pi_{CC}\pi_{A}(CC)+\Pi_{CC}\pi_{A}(CC)+\Pi_{CC}\pi_{A}(CC)}_{\hat{\pi}_{A}}\Big{)}\ket{\eta_{f}}$
$\displaystyle=\bra{\eta_{f}}\hat{\pi}_{A}\ket{\eta_{f}}$

where, we define $\hat{\pi}_{A}$ above as player A’s _Payoff Operator_.

Similarly, $\bar{\pi}_{B}=\bra{\eta_{f}}\hat{\pi}_{B}\ket{\eta_{f}}$, with
$\hat{\pi}_{B}$ defined similarly as above.

A play of the Quantum Prisoner’s Dilemma thus consists of a tuple
$(U_{A},U_{B})$ of the operations that A and B choose to use on an initial
state $\ket{\eta_{in}}$, with the expected payoffs defined as above. This is
similar to the framework of Correlated Games, where individual players draw
their strategies from a joint probability distribution. In the quantum
version, the operations $U_{A}$ and $U_{B}$ locally modify the probability
distribution that finally results from the judge’s measurement. However,
initial quantum entanglement allows for much more exotic correlations than is
possible to achieve from a classical joint probability distribution.

Now, we analyze what are the equilibria of this game under different
restrictions for $U_{A},U_{B}$.

### One Parameter Strategies

First, we consider $U$’s of the form -

$U(\theta)=\begin{bmatrix}\cos{(\theta/2)}&\sin{(\theta/2)}\\\
-\sin{(\theta/2)}&\cos{(\theta/2)}\end{bmatrix}$

where, $\theta\in[0,\pi]$. Here, if we define

$U_{C}=U(0)\qquad\text{and}\qquad U_{D}=U(\pi)$

then, we can easily calculate that,

$(U_{\alpha}\otimes
U_{\beta})\ket{\eta_{in}}=\ket{\eta_{\alpha\beta}}\qquad\alpha,\beta\in\\{C,D\\}$

In general,

$\displaystyle\ket{\eta_{f}}$ $\displaystyle=(U(\theta_{A})\otimes
U(\theta_{B})\ket{\eta_{in}}$
$\displaystyle=\cos{(\theta_{A}/2)}\cos{(\theta_{B}/2)}\ket{\eta_{CC}}+i\sin{(\theta_{A}/2)}\sin{(\theta_{B}/2)}\ket{\eta_{DD}}$
$\displaystyle\;\;-\cos{(\theta_{A}/2)}\sin{(\theta_{B}/2)}\ket{\eta_{CD}}+i\sin{(\theta_{A}/2)}\cos{(\theta_{B}/2)}\ket{\eta_{DC}}$

where, we have used the following formula to arrive at the above equality -

$U(\theta)\ket{0}=\cos{(\theta/2)}\ket{0}-\sin{(\theta/2)}\ket{1}\qquad
U(\theta)\ket{1}=\sin{(\theta/2)}\ket{0}+\cos{(\theta/2)}\ket{1}$

And, the change of basis formula -

$\displaystyle\ket{00}$
$\displaystyle=(\ket{\eta_{CC}}+\ket{\eta_{DD}})/\sqrt{2}$
$\displaystyle\ket{01}$
$\displaystyle=(\ket{\eta_{DC}}+\ket{\eta_{CD}})/\sqrt{2}$
$\displaystyle\ket{11}$
$\displaystyle=(\ket{\eta_{CC}}-\ket{\eta_{DD}})/\sqrt{2}i$
$\displaystyle\ket{10}$
$\displaystyle=(\ket{\eta_{DC}}-\ket{\eta_{CD}})/\sqrt{2}i$

From the final form of $\ket{\eta_{f}}$, the expected payoffs are -

$\bar{\pi}_{A}=-\gamma\cdot\absolutevalue{\cos{(\theta_{A}/2)}\cos{(\theta_{B}/2)}}^{2}-\beta\cdot\absolutevalue{\sin{(\theta_{A}/2)}\sin{(\theta_{B}/2)}}^{2}-\alpha\absolutevalue{\cos{(\theta_{A}/2)}\sin{(\theta_{B}/2)}}^{2}$

and, similarly for $\bar{\pi}_{B}$.

The payoff functions are exactly equivalent to a classical mixed strategy
Prisoner’s Dilemma game with probability $p=\cos^{2}(\theta/2)$ to Cooperate.
The players $A$ and $B$ each try to convert the state $\ket{\eta_{in}}$ to the
state $\eta_{DC}$ and $\eta_{CD}$ respectively, to obtain maximum payoff. So,
just like the classical game, each player has a dominant strategy which is to
defect (i.e, play $U_{D}$). However, once a player plays $U_{D}$, the best
option for the other player is to play $U_{D}$ as well, which modifies the
initial state to $\ket{\eta_{DD}}$. This results in a measurement outcome of
$(D,D)$, and thus the players settle in the dominant but Pareto-suboptimal
Nash Equilibrium. This case doesn’t provide any advantage over the classical
case.

### Two Parameter Strategies

Next, we extend the set of allowed strategies to the 2-parameter Unitary
Operators described below:

$U(\theta,\phi)=\begin{bmatrix}e^{i\phi}\cos{(\theta/2)}&\sin{(\theta/2)}\\\
-\sin{(\theta/2)}&e^{-i\phi}\cos{(\theta/2)}\end{bmatrix}$

with $\theta\in[0,\pi]$ and $\phi\in[0,\pi/2]$.

As before, the classical pure strategies correspond to the operators -

$U_{C}=U(0,0)\qquad\text{and}\qquad U_{D}=U(\pi,0)$

As it stands, the strategy $U_{D}$ is no longer a dominant strategy. If one
player plays $U_{D}$, the other player can maximize his/her payoff by
deviating from $U_{D}$. For example, if player B plays $U_{D}$, we have,

$(I\otimes U_{D})\ket{\eta_{in}}=\ket{\eta_{CD}}$

However, in the space of 2-parameter strategies, $A$ can now use the operation

$U_{Q}=U(0,\pi/2)=\begin{bmatrix}i&0\\\ 0&-i\end{bmatrix}$

which results in the state (after B’s play of $U_{D}$),

$(U_{Q}\otimes U_{D})\ket{\eta_{in}}=(U_{Q}\otimes
I)\ket{\eta_{CD}}=\ket{\eta_{DC}}$

Since, $\ket{\eta_{DC}}$ results in the maximum payoff for player $A$, $U_{Q}$
is the best response to $U_{D}$. So, $(D,D)$ is no longer a Nash Equilibrium.

To find a new Nash Equilirbium, we write the general state after $A$ and $B$’s
operations $U(\theta_{A},\phi_{A})$ and $U(\theta_{B},\phi_{B})$,
$\ket{\eta_{f}}=U(\theta_{A},\phi_{A})\otimes
U(\theta_{B},\phi_{B})\ket{\eta_{in}}$, which gives after some careful
calculation -

$\displaystyle\ket{\eta_{f}}$
$\displaystyle=\cos{(\phi_{A}+\phi_{B})}\cos{(\theta_{A}/2)}\cos{(\theta_{B}/2)}\ket{\eta_{CC}}$
$\displaystyle+i\left[\sin{(\theta_{A}/2)}\sin{(\theta_{B}/2)}+\sin(\phi_{A}+\phi_{B})\cos{(\theta_{A}/2)}\cos{(\theta_{B}/2)}\right]\ket{\eta_{DD}}$
$\displaystyle+i\left[\cos{(\phi_{B})}\sin{(\theta_{A}/2)}\cos{(\theta_{B}/2)}-\sin{(\phi_{A})}\cos{(\theta_{A}/2)}\sin{(\theta_{B}/2)}\right]\ket{\eta_{DC}}$
$\displaystyle+\left[\sin{(\phi_{B})}\sin{(\theta_{A}/2)}\cos{(\theta_{B}/2)}-\cos{(\phi_{A})}\cos{(\theta_{A}/2)}\sin{(\theta_{B}/2)}\right]\ket{\eta_{CD}}$

From the expression of $\ket{\eta_{f}}$, we can calculate the expected Payoff
of $A$ as -

$\displaystyle\bar{\pi}_{A}=$
$\displaystyle-\alpha\cdot\absolutevalue{\sin{(\phi_{B})}\sin{(\theta_{A}/2)}\cos{(\theta_{B}/2)}-\cos{(\phi_{A})}\cos{(\theta_{A}/2)}\sin{(\theta_{B}/2)}}^{2}$
$\displaystyle-\beta\cdot\absolutevalue{\cos{(\phi_{A}+\phi_{B})}\cos{(\theta_{A}/2)}\cos{(\theta_{B}/2)}}^{2}$
$\displaystyle-\gamma\cdot\absolutevalue{\sin{(\theta_{A}/2)}\sin{(\theta_{B}/2)}+\sin(\phi_{A}+\phi_{B})\cos{(\theta_{A}/2)}\cos{(\theta_{B}/2)}}^{2}$

And, similarly for $\bar{\pi}_{B}$. For any play $(\theta_{B},\phi_{B})$ of
$B$, we can optimize this expression by setting -

$\partialderivative{\bar{\pi}_{A}}{\theta_{A}}=0\qquad\text{and}\qquad\partialderivative{\bar{\pi}_{A}}{\phi_{A}}=0$

From this, we get,

$\displaystyle\tan(\theta_{A}/2)\cos{(\phi_{B})}$
$\displaystyle+\tan(\theta_{B}/2)\cos{(\phi_{A})}=0$ (5)
$\displaystyle\tan(\theta_{A}/2)\sin{(\phi_{B})}$
$\displaystyle+\cot(\theta_{B}/2)\cos{(\phi_{A})}=0$ (6)

which can be solved for a fixed $(\theta_{B},\phi_{B})$ to find the optimal
$(\theta_{A},\phi_{A})$. For example, if $(\theta_{B},\phi_{B})=(\pi,0)$, the
optimal move for $A$ is $(\theta_{A},\phi_{A})=(0,\pi/2)$ which results in the
operator $U_{Q}$ defined above.

Particularly, we can calculate the best response for the move
$(\theta_{B},\phi_{B})=(0,\pi/2)$ (i.e, $U_{Q}$) to be -

$\mathcal{BR}_{A}(0,\pi/2)=\\{(\theta_{A},\phi_{A})\;\big{|}\;\theta_{A}\in[0,\pi];\;\phi_{A}=\pi/2\\}$

So, $(0,\pi/2)\in\mathcal{BR}_{A}(0,\pi/2)$, and so, $(U_{Q},U_{Q})$ is a
Quantum Nash Equilibrium for the Quantum Prisoner’s Dilemma in the set of
2-parameter of strategies.

We can easily calculate that,
$\bar{\pi}_{A}(U_{Q},U_{Q})=\bar{\pi}_{B}(U_{Q},U_{Q})=-\gamma$, which is the
Pareto-Optimal solution. So, in this case, Quantum Strategies allow players to
realize the Pareto-Optimal play as a Nash Equilibrium, which is not possible
in the classical game.

So far, we have considered Pure Strategy Quantum Games where the players only
apply _one_ of the Unitary Operators they have access to. We can extend this
notion further to _Mixed Strategy Quantum Games_ , where the initial state of
the Quantum System can be mixed, and the players can apply their strategies
probabilistically. To do that, we have to extend our description of Quantum
Mechanics to account for this stochasticity.

### Three Parameter Strategies

Now, let’s analyze the most general case, where $U_{A},U_{B}$ can be any
allowable Unitary Operator. In this case, the operators can be described by 3
parameters as follows -

$U(\theta,\phi,\lambda)=\begin{bmatrix}e^{i\phi}\cos{(\theta/2)}&e^{-i\lambda}\sin{(\theta/2)}\\\
-e^{i\lambda}\sin{(\theta/2)}&e^{-i\phi}\cos{(\theta/2)}\end{bmatrix}$

As it follows, there exists no Nash Equilibria in this set of strategies.
Because, for any strategy $U_{A}=U(\theta_{A},\phi_{A},\lambda_{A})$, B can
choose a strategy $U_{B}$ such that,

$(U_{A}\otimes U_{B})\ket{\eta_{i}}=\ket{\eta_{CD}}$

i.e, B can manipulate the state so that it gives him the maximum payoff at the
expense of A. Particularly, B can choose the operator -

$U_{B}=U_{D}U_{A}^{\dagger}=U(\pi,0,0)\cdot
U(-\theta_{A},-\phi_{A},-\lambda_{A})$

Since, by symmetry, A can do the same thing, there exists no Nash Equilibrium
for this set of _pure_ Quantum Strategies. However, a Nash Equilibrium _can_
be found, if we allow for _Mixed_ Quantum Strategies. To do that, we have to
extend our description of Quantum Mechanics using another mathematical tool.

##  6 The Density Matrix Formalism

To allow for Mixed Quantum Strategies, we have to find a way to describe
probabilistic Quantum States and and probabilistic Quantum Operations. In
Quantum Mechanics, the Measurement Postulate is the main vehicle which
connects the state-vector description of a system to results observed in real
world experiments, by providing a means to calculate the probabilities for
different outcomes. Now, we need another quantity that will enable us to
calculate measurement probabilities from an ensemble of states. This
description is due to von Neumann [13] and Landau [14].

Suppose, we have an ensemble of states
$\mathcal{E}:=\\{(q_{1},\ket{\psi_{1}}),(q_{2},\ket{\psi_{2}}),\ldots(q_{n},\ket{\psi_{n}})\\}$,
where the state $\ket{\psi_{i}}$ occurs with the probability $q_{i}$. If we
are performing a measurement of the observable $O$ on this ensemble, then the
probability that the outcome $o$ will occur (which has a corresponding
projector $\Pi_{o}$) is -

$\displaystyle p(o)$ $\displaystyle=q_{1}\cdot
p(o\,\big{|}\ket{\psi_{1}})+q_{2}\cdot
p(o\,\big{|}\ket{\psi_{2}})+\ldots+q_{n}\cdot p(o\,\big{|}\ket{\psi_{n}})$
$\displaystyle=q_{1}\cdot\bra{\psi_{1}}\Pi_{o}\ket{\psi_{1}}+q_{2}\cdot\bra{\psi_{2}}\Pi_{o}\ket{\psi_{2}}+\ldots+q_{n}\cdot\bra{\psi_{n}}\Pi_{o}\ket{\psi_{n}}$
$\displaystyle=q_{1}\cdot\Tr\left[\Pi_{o}\outerproduct{\psi_{1}}{\psi_{1}}\right]+q_{2}\cdot\Tr\left[\Pi_{o}\outerproduct{\psi_{2}}{\psi_{2}}\right]+\ldots+q_{n}\cdot\Tr\left[\Pi_{o}\outerproduct{\psi_{n}}{\psi_{n}}\right]$
$\displaystyle=\Tr[\Pi_{o}\bigg{(}\underbrace{q_{1}\outerproduct{\psi_{1}}{\psi_{1}}+q_{2}\outerproduct{\psi_{2}}{\psi_{2}}+\ldots+q_{n}\outerproduct{\psi_{n}}{\psi_{n}}}_{\rho}\bigg{)}\Big{]}$

where, $\Tr$ is the Trace Operator.

The quantity in the parentheses is the desired quantity that will allow us to
describe the ensemble of states and extract measurement probabilities from it.
This quantity $\rho$ is called the _Density Matrix_ of the ensemble. So, the
density matrix corresponding to the ensemble $\mathcal{E}$ is -

$\rho_{\mathcal{E}}=q_{1}\outerproduct{\psi_{1}}{\psi_{1}}+q_{2}\outerproduct{\psi_{2}}{\psi_{2}}+\ldots+q_{n}\outerproduct{\psi_{n}}{\psi_{n}}=\sum_{i}q_{i}\outerproduct{\psi_{i}}{\psi_{i}}$

Such states $\rho_{\mathcal{E}}$ are called _Mixed_ states as opposed to
_Pure_ states which are of the form $\rho=\outerproduct{\psi}{\psi}$.

The recipe to get the measurement probability for an outcome $o$ with
projector $\Pi_{o}$ is -

$p(o)=\Tr[\Pi_{o}\rho]=\Tr[\rho\Pi_{o}]$

where, in the last equality, we have used the cyclic property of Trace.

Evolution of states can be described in the Density Matrix formalism as
follows. Suppose, a Unitary Operator $U$ acts on the ensemble $\mathcal{E}$.
This means that, each state of the ensemble has been transformed to -

$\ket{\psi_{i}}\rightarrow U\ket{\psi_{i}}$

Hence, the ensemble $\mathcal{E}$ is transformed to -

$\mathcal{E}\rightarrow\mathcal{E^{\prime}}:=\\{(q_{1},U\ket{\psi_{1}}),(q_{2},U\ket{\psi_{2}}),\ldots(q_{n},U\ket{\psi_{n}})\\}$

So, the density matrix of the evolved ensemble is -

$\rho_{\mathcal{E^{\prime}}}=\sum_{i}q_{i}U\outerproduct{\psi_{i}}{\psi_{i}}U^{\dagger}=U\left(\sum_{i}q_{i}\outerproduct{\psi_{i}}{\psi_{i}}\right)U^{\dagger}=U\rho_{\mathcal{E}}U^{\dagger}$

Simply put, under a Unitary Transformation, a Density Matrix evolves as -

$\rho\longrightarrow\rho^{\prime}=U\rho U^{\dagger}$

Using this machinery, we can describe even more general scenarios. For
example, if a unitary transformation $U_{i}$ acts on a density matrix $\rho$
with probability $s_{i}$, then the resulting Density Matrix is -

$\rho\longrightarrow\rho^{\prime}=\sum_{i}s_{i}\cdot U_{i}\rho
U_{i}^{\dagger}=\sum_{i}E_{i}\rho E_{i}^{\dagger}$

where,

$E_{i}=\sqrt{s_{i}}U_{i}$

are called the ‘Kraus Operators’ of the generalized evolution.

We can use the language of Desnity Matrices to formulate Mixed Strategy
Quantum Games. However, before doing that, we generalize the approaches of
Meyer [1] and EWL [2] to formulate a protocol to quantumize any Classical Game
in this language. We will then use this protocol to formulate a Mixed Strategy
Quantum Game due to Marinatto and Weber [3].

##  7 Game Quantumization Protocol

Following the description of EWL, we now describe a general protocol to
quantumize any $n$-player finite classical pure-strategy game
$\mathbf{G}(\mathcal{S},\pi)$ (as desribed in eq. 1-3) to the pure strategy
quantum game $\mathbf{G}^{Q}$. The steps go as follows -

  1. 1.

First, we choose $n$ quantum systems $Q_{i}$ with dimensions
$\absolutevalue{\mathcal{S}_{i}}$ respectively.

  2. 2.

A judge prepares the composite system $\mathcal{Q}$,

$\mathcal{Q}=Q_{1}\otimes Q_{2}\otimes\ldots\otimes Q_{n}$

in an initial state $\rho$, which might in general be entangled and mixed.

  3. 3.

The judge then passes each of the subsystems $Q_{i}$ to the player $i$.

  4. 4.

Each player $i$ then operates on $Q_{i}$ with a Unitary operator $U_{i}$, and
returns their respective subsystem to the judge. The combined unitary
operation on the entire system is $\mathcal{U}=U_{1}\otimes
U_{2}\otimes\ldots\otimes U_{n}$. So, the state returned to the judge is -

$\rho_{f}=\mathcal{U}\rho_{in}\mathcal{U}^{\dagger}$

The set of operators $U_{i}$ now represent the strategy-space of the player
$i$ in the Quantum Game $\mathbf{G}^{Q}$, and the operator $\mathcal{U}$ now
represents a play $\mathcal{P}_{\mathbf{G}^{Q}}$ of $\mathbf{G}^{Q}$.

  5. 5.

The judge then performs a measurement on the state $\rho_{f}$ of the combined
system $\mathcal{Q}$, and gets an outcome corresponding to a play
$\mathcal{P}_{\mathbf{G}}$ of the pure-strategy game $\mathbf{G}$. The
projection operators that the judge uses for measurement can be represented as
-

$\Pi_{\mathbf{G}}=\\{\Pi_{\mathcal{P}_{\mathbf{G}}}\;\big{|}\;\mathcal{P}_{\mathbf{G}}:=(s_{1}^{\alpha_{1}},s_{2}^{\alpha_{2}},\ldots,s_{n}^{\alpha_{n}}),\forall
s_{i}^{\alpha_{i}}\in\mathcal{S}_{i}\\}$

That is, there is a projector corresponding to each play of pure strategies
$\mathcal{P}_{\mathbf{G}}$. This is consistent with our description of
$\mathcal{Q}$, because, the dimension of $\mathcal{Q}$ is,

$\dim(\mathcal{Q})=\dim(Q_{1})\cdot\dim(Q_{2})\cdot\ldots\cdot\dim(Q_{n})=\absolutevalue{\mathcal{S}_{1}}\cdot\absolutevalue{\mathcal{S}_{2}}\cdot\ldots\cdot\absolutevalue{\mathcal{S}_{n}}$

which is exactly equal to the number of plays of the pure-strategy game
$\mathbf{G}$, and thus, we can always find a complete set of orthonormal
projectors equal to this number.

  6. 6.

Finally, the judge calculates the classical payoffs
$\pi_{i}(\mathcal{P}_{\mathbf{G}})$ for the play resulting from the
measurement, and relays them to the player $i$. Since the outcome of the
measurement is probabilistic, the expected payoff of each player is -

$\bar{\pi}_{i}(\mathcal{U})=\sum_{\mathcal{P}_{\mathbf{G}}}p(\mathcal{P}_{\mathbf{G}}\,\big{|}\,\mathcal{U}\rho_{in}\mathcal{U}^{\dagger})\cdot\pi_{i}(\mathcal{P}_{\mathbf{G}})$

Steps 5 and 6 can also be described succinctly using the language of _Payoff
Operators_. We can define the payoff operator $\hat{\pi}_{i}$ for the $i$-th
player as -

$\hat{\pi}_{i}=\sum_{\mathcal{P}_{\mathbf{G}}}\pi_{i}(\mathcal{P}_{\mathbf{G}})\cdot\Pi_{\mathcal{P}_{\mathbf{G}}}$

As defined, the summation given in the definition is actually the Spectral
Decomposition of the operator $\hat{\pi}_{i}$ (the
$\Pi_{\mathcal{P}_{\mathbf{G}}}$ being orthonormal), and each of the operators
$\hat{\pi}_{i}$ are Hermitian (because the $\pi_{i}(\mathcal{P}_{\mathbf{G}})$
are real). As a result, the $\hat{\pi}_{i}$’s are valid Quantum Observables.
In addition, since they share a common eigenbasis, all the operators
$\hat{\pi}_{i}$ commute -

$[\hat{\pi}_{i},\hat{\pi}_{j}]=0\quad\forall\;i,j$

Using these commutation relations, we can flip the argument for measurement.
Given a set of mutually commuting Payoff Operators $\\{\hat{\pi}_{i}\\}$,
there exists a common eigenbasis for the set of operators. We will measure the
state $\mathcal{U}\rho_{in}\mathcal{U}^{\dagger}$ in this common eigenbasis of
the $\hat{\pi}_{i}$’s and calculate the final payoffs from the measurement
results. This gives us a way to describe the quantum game, without explicitly
mentioning the measurement basis.

Also, the expected payoff of each player is more concisely expressed using the
payoff operators. We simply have,

$\bar{\pi}_{i}(\mathcal{U})=\Tr\left[\mathcal{U}\rho\,\mathcal{U}^{\dagger}\hat{\pi}_{i}\right]=\Tr\left[\rho\,\mathcal{U}^{\dagger}\hat{\pi}_{i}\mathcal{U}\right]$
(7)

So, the quantum game can be described as
$\mathbf{G}^{Q}(\rho_{in},\\{\mathcal{U}\\},\\{\hat{\pi}_{i}\\})$ with
$[\hat{\pi}_{i},\hat{\pi}_{j}]=0$ for all pairs $i,j$. This description can be
generalized to _Mixed Quantum Games_
$\mathbf{G}^{Q}_{M}(\rho_{in},\\{\mathcal{E}\\},\\{\hat{\pi}_{i}\\})$ where
the set of Unitary Operators $\\{\mathcal{U}\\}$ is replaced by the set

$\\{\mathcal{E}\\}=\\{\mathcal{E}_{1}\otimes\mathcal{E}_{2}\otimes\ldots\otimes\mathcal{E}_{n}\\}$

where, each $\mathcal{E}_{i}$ is a generalized Quantum Evolution Operator
(which is a Completely Positive Trace Preserving (CPTP) Map on the Reduced
Density Matrix $\rho_{i}=\Tr_{1,2,\ldots,i-1,i+1,\ldots,n}[\rho]$). In
general, $\mathcal{E}_{i}$ include strategies where the player $i$ selects an
operator $\\{U_{i}\\}$ with some probability measure $\mu(U_{i})$, as well as
other more general situations.

##  8 A Mixed Strategy Quantum Game

We end our discussion by using the protocol described in the previous section
to quantumize another game. This quantumization was done by Marinatto and
Weber [3], but our presentation is much more succinct using the protocol we
have developed.

#### The Battle of Sexes

In this game, there are two players Alice (A) and Bob (B) who are husband and
wife. They want to decide on an activity to do on a certain evening. Alice
wants to go outside to watch an Opera (O), while Bob wants to stay at home and
watch TV (T). However, they would rather spend time together than alone. This
scenario can be described by the following payoff bi-matrix -

|  | B
---|---|---
|  | O | T
A | O | $(\alpha,\beta)$ | $(\gamma,\gamma)$
T | $(\gamma,\gamma)$ | $(\beta,\alpha)$

where, the first element of a tuple represents Alice’s payoff, and the second
element represents Bob’s. To reflect the preferences of the players in this
game, we have $\alpha>\beta>\gamma$.

This game has 2 pure-strategy Nash Equilibria $\mathcal{P}_{1}=(O,O)$ and
$\mathcal{P}_{2}=(T,T)$ with payoffs $(\alpha,\beta)$ and $(\beta,\alpha)$
respectively. In addition, it also has a mixed strategy Nash Equilibrium
$\mathcal{P}_{M}=(\mathbf{p}_{A},\mathbf{q}_{B})$ with,

$\mathbf{p}_{A}=\left[\frac{\alpha-\gamma}{\alpha+\beta-\gamma}\quad\frac{\beta-\gamma}{\alpha+\beta-\gamma}\right]$
$\mathbf{q}_{B}=\left[\frac{\beta-\gamma}{\alpha+\beta-\gamma}\quad\frac{\alpha-\gamma}{\alpha+\beta-\gamma}\right]$

where the first and second elements of the vectors are the probabilities of
selecting $O$ and $T$ for the corresponding players.

This results in an expected payoff of -

$\pi_{A}(\mathcal{P}_{M})=\pi_{B}(\mathcal{P}_{M})=\frac{\alpha\beta-\gamma^{2}}{\alpha+\beta-2\gamma}$

To quantumize this game, we choose,

$\mathcal{Q}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$

with $\mathcal{H}_{A},\mathcal{H}_{B}$ two-dimensional quantum systems
(qubits). For demonstration purposes, we choose
$\rho_{in}=\outerproduct{\Phi^{+}}{\Phi^{+}}$ (according to [3]), where,

$\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}\ket{00}+\frac{1}{\sqrt{2}}\ket{11}$

is a maximally entangled state.
We restrict the set of operations of $A$ and $B$ to

$\\{U_{A}\\}=\\{U_{B}\\}=\\{\textsf{X},\textsf{I}\\}$

where,

$\textsf{X}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\qquad\textsf{I}=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$

Finally, we choose the projective measurement operators

$(\Pi_{OO},\Pi_{OT},\Pi_{TO},\Pi_{TT})=(\outerproduct{\Phi^{+}}{\Phi^{+}},\outerproduct{\Psi^{+}}{\Psi^{+}},\outerproduct{\Psi^{-}}{\Psi^{-}},\outerproduct{\Phi^{-}}{\Phi^{-}})$

where,

$\displaystyle\ket{\Phi^{+}}=(\ket{00}+\ket{11})/\sqrt{2}$
$\displaystyle\ket{\Psi^{+}}=(\ket{01}+\ket{10})/\sqrt{2}$
$\displaystyle\ket{\Phi^{-}}=(\ket{00}-\ket{11})/\sqrt{2}$
$\displaystyle\ket{\Psi^{-}}=(\ket{01}-\ket{10})/\sqrt{2}$

In this set of strategies, there are 2 pure Quantum Nash Equilibria -

$\mathcal{P}^{Q}_{1}=(\textsf{I},\textsf{I})\qquad\mathcal{P}^{Q}_{2}=(\textsf{X},\textsf{X})$

Each with expected payoff -

$\bar{\pi}_{A}=\bar{\pi}_{B}=\frac{\alpha+\beta}{2}$

Now, we analyze this game for mixed strategies. In this case, player $A$
applies I and X with probabilities $p_{A1}=p_{A}$ and $p_{A2}=(1-p_{A})$, and
similarly for $B$, with probabilities $p_{B1}=p_{B}$ and $p_{B2}=(1-p_{B})$.
In the mixed strategy game, the final state of $\mathcal{Q}$ is -

$\rho_{f}=\sum_{i,j}p_{Ai}p_{Bj}\cdot(U_{Ai}\otimes
U_{Bj})\rho(U_{Ai}^{\dagger}\otimes U_{Bj}^{\dagger})$

This results in the state,

$\rho_{f}=[p_{A}p_{B}+(1-p_{A})(1-p_{B})]\outerproduct{\Phi^{+}}{\Phi^{+}}+[p_{A}(1-p_{B})+(1-p_{A})p_{B}]\outerproduct{\Psi^{+}}{\Psi^{+}}$

with expected payoff,

$\displaystyle\bar{\pi}_{A}$ $\displaystyle=\Tr[\rho_{f}\hat{\pi}_{A}]$
$\displaystyle=[p_{A}p_{B}+(1-p_{A})(1-p_{B})]\cdot\frac{\alpha+\beta}{2}+[p_{A}(1-p_{B})+(1-p_{A})p_{B}]\cdot\gamma$

So, the Best Response of $B$ for a particular $p_{A}$ is -

$p_{B}^{\star}=\mathcal{BR}(p_{A})=\begin{cases}0&p_{A}>1/2\\\
1&p_{A}<1/2\end{cases}$

However, for $p_{A}=1/2$, any $p_{B}$ will give the same payoff. Particularly,
$(p_{B}=1/2)\in\mathcal{BR}(p_{A}=1/2)$. Hence, there exists a Mixed Strategy
Quantum Nash Equilibrium,

$\mathcal{P}^{Q}_{M}=(\left[\frac{1}{2}\;\;\frac{1}{2}\right]\,,\,\left[\frac{1}{2}\;\;\frac{1}{2}\right])$

This gives an expected payoff of

$\bar{\pi}_{A}(\mathcal{P}^{Q}_{M})=\bar{\pi}_{B}(\mathcal{P}^{Q}_{M})=\frac{\alpha+\beta+2\gamma}{4}$

We see that, among the plays $\mathcal{P}_{1,2,M}$ and
$\mathcal{P}^{Q}_{1,2,M}$, the plays $\mathcal{P}^{Q}_{1,2}$ are Pareto-
Optimal. Thus the quantum version of the game allows players to achieve a
pareto-optimal solution as a Nash Equilibrium, which is not possible in the
classical case.

##  9 Alternate Quantumization Protocol

In certain scenarios, a special class of Classical Games can be quantumized in
a different manner. This approach was followed by Meyer [1], which we
generalize in this section. We describe this class of Games as
$\mathbf{G}(\mathcal{C}^{(k)},\mathcal{M},\pi)$. In this class of games, the
players manipulate a common classical system $\mathcal{C}$ which can be in be
one of a finite set of classical states
$\mathcal{C}^{(k)}=\\{\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}\\}$,
and the players can select from a set of _moves_
$\mathcal{M}=\\{M_{1},M_{2},\ldots,\\}$ which transform the system from one
state to another, i.e,

$M_{i}(\mathcal{C}_{j})=\mathcal{C}_{ij}\qquad\text{where},\,\mathcal{C}_{ij}\in\mathcal{C}^{(k)}$

Here, $\mathcal{C}_{ij}$ is called the Transition Matrix of the game.
Depending on the structure of the game, a player can play a move multiple
times.

In these games, only the initial state of the system is known to the players,
and the running state remains hidden as the game progresses. The final state
is revealed at the end of the game only after all of the players have played
their moves. The payoff of each player depends on this final state.

For example, in the Penny-Flip game, the common Classical System is a penny
with $\mathcal{C}^{(2)}$ = {Heads (H), Tails (T)}, the players moves are
$\mathcal{M}$ = {Flip (F), No-Flip (N)}, and the payoff of the players depends
on the final state of the penny being H or T, with
$\pi_{Q}(H)=-\pi_{Q}(T)=\pi_{C}(T)=-\pi_{C}(H)=1$.

As an alternate to the EWL protocol, this class of games can also be
quantumized as follows -

  1. 1.

The Classical System $\mathcal{C}^{(k)}$ is replaced by a quantum system
$\mathcal{Q}$ with

$\dim(\mathcal{Q})=k$

which has orthonormal basis states
$\\{\ket{\mathcal{C}_{1}},\ket{\mathcal{C}_{2}},\ldots,\ket{\mathcal{C}_{k}}\\}$

  2. 2.

The set of classical moves $\mathcal{M}$ is now replaced by the set of _all_
Unitary operators $U$ which can act on $\mathcal{Q}$. This set also contains
the operators $U_{M_{i}}$ such that,

$U_{M_{i}}\ket{\mathcal{C}_{j}}=\ket{\mathcal{C}_{ij}}$

These $U_{M_{i}}$’s are the so-called “classical strategies” in this quantum
game. However, players may use quantum strategies or quantum moves other than
these classical moves as well. A play of this quantum game $\mathcal{P}^{Q}$
consists of a sequence $\mathcal{U}=\ldots U_{i}\ldots U_{2}U_{1}$ of unitary
operators played by the players.

  3. 3.

$\mathcal{Q}$ is prepared in an initial state $\ket{\psi}$ known to the
players. After all the players have played their respective Quantum Moves, the
final state of $\mathcal{Q}$, $\ket{\psi_{f}}=\mathcal{U}\ket{\psi}$ is
returned to a neutral judge. The judge measures $\mathcal{Q}$ in the
$\\{\ket{\mathcal{C}_{i}}\\}$ basis. Depending on the measurement result, each
player receives a payoff. The expected payoff of the $i$-th player is
calculated as -

$\displaystyle\bar{\pi}_{i}(\mathcal{U})$
$\displaystyle=\sum_{j}p(\mathcal{C}_{j})\cdot\pi_{i}(\mathcal{C}_{j})$
$\displaystyle=\sum_{j}\bra{\psi_{f}}(\outerproduct{\mathcal{C}_{i}}{\mathcal{C}_{i}})\ket{\psi_{f}}\cdot\pi_{i}(\mathcal{C}_{j})$
$\displaystyle=\bra{\psi_{f}}\left(\sum_{j}\pi_{i}(\mathcal{C}_{j})\outerproduct{\mathcal{C}_{i}}{\mathcal{C}_{i}}\right)\ket{\psi_{f}}$
$\displaystyle=\bra{\psi_{f}}\hat{\pi}_{i}\ket{\psi_{f}}$
$\displaystyle=\bra{\psi}\mathcal{U}^{\dagger}\hat{\pi}_{i}\mathcal{U}\ket{\psi}$
$\displaystyle=\Tr[\outerproduct{\psi}{\psi}\mathcal{U}^{\dagger}\hat{\pi}_{i}\mathcal{U}]$
$\displaystyle=\Tr[\rho\,\mathcal{U}^{\dagger}\hat{\pi}_{i}\mathcal{U}]$ (8)

where,

$\hat{\pi}_{i}=\sum_{j}\pi_{i}(\mathcal{C}_{j})\outerproduct{\mathcal{C}_{i}}{\mathcal{C}_{i}}$

is the Payoff Operator for player $i$, and

$\rho=\outerproduct{\psi}{\psi}$

is the initial state of $\mathcal{Q}$.

We can see that eq. 8 is the same as eq. 7, and thus the formula using Payoff
Operators also holds in this case. This protocol can be easily generalized to
the case where the starting state is a mixed state. The difference is in
$\mathcal{U}$, where in eq. 7 the Unitary Operators are applied in ’
_parallel_ ’, whereas in eq. 8 they are applied in ‘ _sequence_ ’.

In Meyer’s [1] quantumization of the Penny Flip game,
$\mathcal{Q}=\mathcal{H}^{(2)}$ is a 2-dimensional quantum system, i.e, a
qubit. The initial state is $\rho=\outerproduct{\mathbf{H}}{\mathbf{H}}$, and
the sequence of operations the players use is

$\mathcal{U}=\textsf{U}_{Q}^{\star}\textsf{U}_{C}\textsf{U}_{Q}^{\star}$

with $\textsf{U}_{C}\in\\{\textsf{F},\textsf{N}\\}$ defined in eq. 4. The
Payoff Operators are -

$\hat{\pi}_{Q}=\outerproduct{\mathbf{H}}{\mathbf{H}}-\outerproduct{\mathbf{T}}{\mathbf{T}}\qquad\hat{\pi}_{C}=\outerproduct{\mathbf{T}}{\mathbf{T}}-\outerproduct{\mathbf{H}}{\mathbf{H}}$

In this language,

$\rho_{f}=\mathcal{U}\rho\mathcal{U}^{\dagger}=\outerproduct{\mathbf{H}}{\mathbf{H}}$

So, the payoffs are -

$\bar{\pi}_{Q}=\Tr[\hat{\pi}_{Q}\outerproduct{\mathbf{H}}{\mathbf{H}}]=1\qquad\bar{\pi}_{C}=\Tr[\hat{\pi}_{C}\outerproduct{\mathbf{H}}{\mathbf{H}}]=-1$

As is evident from the payoffs, player Q always wins.

##  10 Discussions and Conclusion

In this exposition, we have reviewed the the foundational works of Meyer [1],
Eisert et al. [9, 2], Marinatto et al. [3] and Landsburg [4, 11] in
formulating Pure and Mixed strategy Quantum Games. We have generalized their
works into two _Quantumization Protocols_ , the first of which is general and
follows the approach of EWL, while the second is applicable to some special
games and follows the approach of Meyer. In all of the cases, we have shown
that the corresponding quantum games allow the players to achieve additional
Quantum Nash Equilibria that Pareto-dominate the Classical Nash Equilibria.

Existence of of Quantum Nash Equilibria are shown for a finite (and
restricted) set of Quantum Strategies. This restriction allows us to use the
mathematical techniques of finite games, and particularly Nash’s theorem [12]
to prove the existence of Quantum Nash Equilibria. The case of the existence
of Mixed Strategy Quantum Nash Equilibria for the entire set of Quantum
Strategies ($U\in SU(2)$) requires more sophisticated techniques, since the
strategy space then becomes infinite. To do this, we need to define a
probability measure on $SU(2)$ or $SU(n)$, and start from there. Nevertheless,
the framework that we have introduced is very general and can be used to
perform such an analysis. Another avenue of investigation is what effect the
initial state has on the achievable equilibria, as clearly, the final payoff
depends on the initial state of the Quantum System.

In addition, the description of the dynamics of Quantum Games have been barely
touched upon. This is because, the Payoffs of these games are calculated
through Projective Quantum Measurements, which reveal the probabilities of the
corresponding projectors and _not_ the probability ‘amplitudes’ of the Quantum
State. It is these amplitudes which are modified by Quantum Strategies and
which control the evolution of the state. So, the dynamics of these Quantum
Games have to be treated in a framework in which the players have some sort of
_incompleteness_ in their knowledge of the strategies other players are using.
Interestingly, there is actually very little work treating this dynamics, and
most of the body of literature on Quantum Games deals with Static Quantum
Games. We leave this exploration for future investigation.

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