Some Topics in Quantum Games
Yshai Avishai
Department of of Physics and Ilse Katz Institute for NanoTechnology
Ben Gurion University of the Negev, Beer Sheva, Israel.
Based on a Thesis Submitted on August 2012 to the
Faculty of Humanities and Social Sciences, Department of Economics
Ben Gurion University of the Negev, Beer Sheva, Israel
in Partial Fulfillment of the Requirements for the Master of Arts Degree
abstract This work concentrates on simultaneous move quantum games of two players. Quantum game theory models the behavior of strategic agents (players) with access to quantum tools for controlling their strategies. The simplest example is to envision a classical (ordinary) twoplayer twostrategies game given in its normal form (a table of payoff functions, think of the prisoner dilemma) in which players communicate with a referee via a specific quantum protocol, and, motivated by this vision, construct a new game with greatly enlarged strategy spaces and a properly designed payoff system. The novel elements in this scheme consist of three axes. First, instead of the four possible positions (CC), (CD), (DC) and (DD) there is an infinitely continuous number of positions represented as different quantum mechanical states. Second, instead of the twopoint strategy space of each player, there is an infinitely continuous number of new strategies (this should not be confused with mixed strategies). Third, the payoff system is entirely different, since it is based on extracting real numbers from a quantum states that is generically a vector of complex number. The fourth difference is apparently the most difficult to grasp, since it is a conceptually different structure that is peculiar to quantum mechanics and has no analog in standard (classical) game theory. This very subtle notion is called quantum entanglement. Its significance in game theory requires a nontrivial modification of one’s mind and attitude toward game theory and choice of strategies. Quantum entanglement is not always easy to define and estimate, but in this work where the classical game is simple enough, it can be (and will) be explicitly defined. Moreover, it is possible to define a certain continuous real parameter such that for there is no entanglement, while for entanglement is maximal.
Naturally, a substantial part of this work is devoted to settling of the mathematical and physical grounds for the topic of quantum games, including the definition of the four axes mentioned above, and the way in which a standard (classical) game can be modified to be a quantum game (I call it a quantization of a classical game).
The connection between game theory and information science is briefly explained. While the four positions of the classical game are formulated in terms of bits, the myriad of positions of the quantum game are formulated in terms of quantum bits. While the two strategies of the classical game are represented by a couple of simple matrices, the strategies of a player in the quantum game are represented by an infinite number of complex unitary matrices with unit determinant. The notion of entanglement is explained and exemplified and the parameter controlling it is introduced. The quantum game is formally defined and the notion of pure strategy Nash equilibrium is defined.
With these tools at, it is possible to investigate some important issues like existence of pure strategy Nash equilibrium and its relation with the degree of entanglement. The main achievement of this work are as follows:

Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon twoplayers twodecisions classical games. based on the method of best response functions

Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by the parameter mentioned above). It has been proved (and I prove it here directly) that when the classical game has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game with maximal entanglement () has no pure strategy Nash equilibrium. By studying a nonsymmetric prisoner dilemma game, I find that there is a critical value such that for there is a pure strategy Nash equilibrium and for the is not. The behavior of the two payoffs as function of start at that of the classical ones at and approach the cooperative classical ones at .

Bayesian quantum games are defined, and it is shown that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal.

The basic ingredients of a quantum game based on a twoplayers three decisions classical games. This requires the definition of trits (instead of bits) and quantum trits (instead of quantum bits). It is shown that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
1 Intoduction
This introductory Section contains the following parts: 1) A prolog that specifies the arena of the thesis and sets the relevant scientific framework in which the research it is carried out. 2) An acknowledgment expressing my gratitudes to my supervisor and for all those who helped me passing an enjoyable period in the department of Economics at BGU. 3) An abstract with a list of novel results achieved in this work. 4) A background that surveys the history and prospects of the topics discussed in this work. 5) Content of the following Sections of the thesis.
1.1 Prolog
This manuscript is based on the MA thesis written by the author under the supervision of professor Oscar Volij, as partial fulfillment of academic duties toward achieving second degree in Economics in the Department of Economics at Ben Gurion University. The subject matter is focused on the topic of quantum games, an emergent subdiscipline of physics and mathematics. It has been developed rapidly during the last fifteen years, together with other similar fields, in particular quantum information to which it is intimately related. Even before being acquainted with the topic of quantum games the reader might wonder (and justly so) what is the relation between quantum games and Economics . This research will not touch upon this interface, but numerous references relating quantum games and Economics will be mentioned. Similar questions arose in relation to the amalgamation of quantum mechanics and information science. If information is stored in our hard disk in bits, what has quantum mechanics to do with that? But in 1997 it was shown by Shor that by using quantum bits instead of bits, some problems that require a huge amount of time to be solved on ordinary computers could be solved in much shorter time using quantum computers. It was also shown that quantum computers can break secret codes in a shorter time than ordinary computers do, and that might affect our everyday life as for example, breaking our credit card security codes or affecting the crime of counterfeit money. Game theory is closely related with information science because taking a decision (like confess or don’t confess in the prisoner dilemma game) is exactly like determining the state of a bit, 0 or 1. Following the crucial role of game theory in Economics, and the intimate relation between game theory and information science, it is then reasonable to speculate that the dramatic impetus achieved in information science due to its combination with quantum mechanics might repeat itself in the application of quantum game theory in Economics.
As I stressed at the onset, the present work focuses on some aspects of quantum game theory, especially, quantum games based on simultaneous games with two players and two or three point strategic space for each player. The main effort is directed on the elucidation of pure strategy Nash equilibria in quantum games with full information and in games with incomplete information (Bayesian games). I do not touch the topic of the interface between quantum games and economics, since this aspect is still in a very preliminary stage.
Understanding the topics covered in this work requires a modest knowledge of mathematics and the basic ingredients of quantum mechanics. Yet, the writing style is not mathematically oriented. Bearing in mind that the target audience is mathematically oriented economists, I tried my best to explain and clarify every topic that appears to be unfamiliar to nonexperts. It seems to me that mathematically oriented economists will encounter no problem in handling this material. The new themes required beyond the central topics of mathematics used in economic science include complex numbers, vector fields, matrix algebra, group theory, finite dimensional Hilbert space and a tip of the iceberg of quantum mechanics. But all these topics are required on an elementary level, and are covered in the pertinent appendices.
1.2 Background
There are four scientific disciplines that seem to be intimately related. Economics, Quantum Mechanics, Information Science and Game Theory. The order of appearance in the above list is chronological. The birth of Economics as an established scientific discipline is about two hundred years old. Quantum mechanics has been initiated more than hundred years ago by Erwin Schrödinger, Werner Heisenberg, Niels Bohr, Max Born, Wolfgang Pauli, Paul Dirac and others. It has been established as the ultimate physical theory of Nature. The Theory of Information has been developed by Claude Elwood Shanon in 1949 [1], and Game Theory has been developed by John Nash in 1951[2].
The first connection between two of these four disciplines has been discovered in 1953 when the science of game theory and its role in Economics has been established by von Newmann and Morgenstern [3] (Incidentally, von Newmann laid the mathematical foundations of quantum mechanics in the early fifties). Almost half a century later, in 1997, the relevance of quantum mechanics for information was established[4] and that marked the birth of a new science, called quantum information.
These facts invite two fundamental questions: 1) Is quantum mechanics relevant for game theory? That is, can one speak of quantum games where the players use the concepts of quantum mechanics in order to design their strategies and payoff schemes? 2) If the answer is positive, is the concept of quantum game relevant for Economics?
The answer to the first question is evidently positive. In the last two and a half decades, the theory of quantum games has emerged as a new discipline in mathematics and physics and attracts the attention of many scientists. Pioneering works before the turn of the century include Refs. [5, 6, 7, 8]. The present work is inspired by some works published after the turn of the century that developed the concept of quantum games that are based on standard (classical) games albeit with quantum strategies and a referee that imposes an entanglement[9, 10, 11, 12, 13, 14] and others. Quantum game theory combines game theory, that is, the mathematical formulation of competitions and conflicts, with the physical nature of quantum information.
The question why game theory can be interesting and what it adds to classical game theory was addressed in some of the references listed above. Some of the reasons are:

The role of probability in quantum mechanics is rather fundamental. Since classical games also use the concept of probability, the interface between classical and quantum game theory promises to be conceptually rich.

Since quantum mechanics is the theory of Nature, it must show up also in people mind when they communicate with each other.

Searching for quantum strategies in quantum game may lead to new quantum algorithms designed to solve complicated problems in polynomial time.
The answer to the second question, the relevance of quantum game to economics is less deterministic. Numerous works were published on this interface[15] and they give stimulus for further investigations. I feel however that this topics is still at a very early stage and requires a lot of new ideas and breakthroughs before it can be established as a sound scientific discipline.
As I have already indicated, the present thesis rests within the arena of quantum games and does not touch the interface between quantum games and economics. Its main achievement is the suggestion and the testing of a numerical method based on best response functions in the quantum game for searching pure strategy Nash equilibria.
1.3 Content of Sections

In Section 2 we cast the classical 2player 2strategies game in the language of classical information. Using the prisoner dilemma game as a guiding example we present the four positions on the game table (C,C),(C,D),(D,C) and (D,D) as two bit states (0,0),(0,1),(1,0) and (1,1) and define the classical strategies as operations on bits, that is known in the theory of information as classical gates. At the end of this Section we briefly discuss the information theory representation of 2player three strategies classical games.

In Section 3 All the quantum mechanical tools necessary for the conduction of a quantum game are introduced. These include a very short introduction to the concept of Hilbert space (discussed in more details in Section 7), followed by the definition of quantum bits, that is the fundamental unit of quantum information. Then the quantum strategies of the players are defined as unitary complex matrices with unit determinant. The quantum states of a two players in a quantum game are then defined, and their relation to the two qubit states is clarified. This leads us to the basic concept of entanglement and entanglement operators that play a crucial role in the protocol of the quantum game. In addition, the concept of partial entanglement is explained (as it will be used in Section 5).

In Section 5 we introduce our numerical formalism to construct the best response functions and to search for pure strategy Nash equilibrium by identifying the intersections of the best response functions. The method is then used on a specific game and the relation between the payoffs and the degree of entanglement is clarified.

In Section 6 we briefly discuss more advanced topics such as Bayesian quantum games, mixed strategies, quaternionic formulation of quantum games and quantum games based on twoplayers three decision classical games. These requires the introduction of quantum trits (qutrits) and the definition of strategies as complex unitary matrices with unit determinant.

Finally, in Section 7 we collect the minimum necessary mathematical apparatus in a few appendices, including complex numbers, linear vector spaces, matrices, elements of group theory, introduction to Hilbert soace and, eventually, the basic concepts of quantum mechanics.
2 Information Theoretic Language for Classical Games
The standard notion of games as appears in the literature will be referred to as a classical games, to distinguish it from the notion of quantum games that is the subject of this work. In the present Section we will use the language of information theory in the description of simultaneous classical games. Usually these games will be represented in their normal form (a payoff table). Except for the language used, nothing is new here.
2.1 Two Players  Two Decisions Games: Bits
Consider a two player game with pure strategy such as the prisoner dilemma, given below in Eq. (8). The formal definition is,
(1) 
Each player can choose between two strategies and for Confess or Don’t Confess. Let us modify the presentation of the game just a little bit in order to adapt it to the nomenclature of quantum games. When the two prisoners appear before the judge, he tells them that he assumes that they both confess and let them decide whether to change their position or leave it at C. This modification does not affect the conduction of the game. The only change is that instead of choosing C or D as strategy, the strategy to be chosen by each player is either to replace C by D or leave it C as it is. Of course, if the judge would tell the prisoner that he assumes that prisoner 1 confesses and prisoner 2 does not, then the strategies will be different, but again, each one’s strategy space has the two points { Don’t replace, Replace }.
Now let us use different notations than C and D say 0 and 1. This has nothing to do with the numbers 0 and 1, they just
stand for the two different symbols. We can equally consider two colors, red and blue.
Such two symbols form a bit. We thus have:
Definition: A bit is an object that can have two different states.
A bit is the basic ingredient of information science and is used ubiquitously in numerous information devices such as hard disks, transmission lines and other information storage devices. There are several notations used in information theory to denote the two states of a bit. The simplest one is just to say that the bit state is 0 or 1. But this notation is inconvenient when it is required to perform some operation on bits like replace or don’t replace. A more informative description is to consider bit states as two dimensional vectors (see below). Yet a third notation that anticipates the formulation of quantum games is to denote the two states of a bit as and . This ket notation might look strange at first glance but it proves very useful in analyzing quantum games. In summary we have,
(2) 
2.1.1 Two Bit States
Looking at the game table in Eq. (8), the prisoner dilemma game table has four squares marked by (C,C), (C,D),(D,C), and (D,D). In our modified language, any square in the game table is called a twobit state, because each player knows what is his bit value in this square. The corresponding four twobit states are denoted as (0,0),(0,1),(1,0), (1,1). In this notation (exactly as in the former notation with C and D) it is understood that the first symbol (from the left) belongs to player 1 and the second belongs to player 2.
Thus, in our language, when the prisoners appear before the judge he tells them ”your twobit state at the moment is (0,0) and now I ask anyone to decide whether to replace his bit value from 0 to 1 or leave it as it is”. As for the single bit states that have several equivalent notations specified in Eq. (2), two bit states have also several different notations. In the vector notation of Eq. (2) the four twobit states listed above are obtained as outer products of the two bits
(3) 
Again, it is understood that the bit composing the left factor in the outer product belongs to player 1 (the column player) and the the right factor in the outer product belongs to player 2 (the row player).
Generalization to players twodecision games is straightforward. A set of bits can exist in one of different configurations and
described by a vector of length where only one component is 1, all the others being 0.
Ket notation for two bit states: The vector notation of Eq. (3) requires a great deal of page space,
a problem that can be avoided by using the ket notation. In this framework, the four twobit states are respectively denoted as
(see the comment after after Eq. (3)),
(4) 
For example, in the prisoner dilemma game, these four states correspond respectively to
.
2.1.2 Classical Strategy as an Operation on Bits
Now we come to the description of the classical strategies (replace or do not replace) using our information theoretic language. Since we have agreed to represent bits as two components vectors, execution of operation of each player on his own bit (replace or do not replace) is represented by a real matrix. In classical information theory, operations on bits are referred to as gates. Here we will be concerned with the two simplest operations performed on bits changing them from one configuration to another. An operation on a bit state that results in the same bit state is accomplished by the unit matrices . An operation on a bit state that results in the other bit state is accomplished by a matrix denoted as .
An important notational comment: The 1 in the matrix is designed to guarantee that det=1, in analogy with the strategies of the quantum game to be defined in the following Sections. As far as the classical game is concerned, this sign has no meaning, because a bit state or is not a number, it is just a symbol. So that we can agree that for classical games, the vectors and represent the same bit, and the vectors and represent the same bit, 
(5) 
Written in ket notation we have,
(6) 
In the present language, the two strategies of each player are the two matrices and and the four elements of are the four matrices, (7) 
In this notation, following the comment after Eq. (3), the left factor in the outer product is executed by player 1 (the column player) on his bit, while the right factor in the outer product is executed by player 2 (the row player). In matrix notation each operator listed in Eq. (7) acts on a four component vector as listed in Eq. (3).
Example: Consider the classical prisoner dilemma with the normal form,
Prisoner 1
Prisoner 2
1 (C) Y (D) 1 (C) 4,4 6,2 Y (D) 2,6 5,5 
(8) 
The entries stand for the number of years in prison.
2.1.3 Formal Definition of a Classical Game in the Language of Bits
The formal definition is,
(9) 
The two differences between this definition and the standard definition of Eq. (1) is that the players face an initial twobit state presumed by the judge (usually and the twopoint strategy space of each players contains the two gates instead of . The conduction of a pure strategy classical twoplayerstwo strategies simultaneous game given in its normal form (a payoff matrix) follows the following steps:

A referee declares that the initial configuration is some fixed 2 bit state. This initial state is one of the four 2bit states listed in Eq. (4). The referee’s choice does not, in any way, affect the final outcome of the game, it just serves as a starting point. For definiteness assume that the referee suggests the state as the initial state of the game. We already gave an example: In the story of the prisoner dilemma it is like the judge telling them that he assumes that they both confess.

In the next step, each player decides upon his strategy ( or ) to be applied on his respective bit. For example, if each player choses the strategy we note from Eq. (5) that
(10) Thus, a player can choose either to leave his bit as suggested by the referee or to change it to the second possible state. As a result of the two operations, the two bit state assumes it final form.

The referee then “rewards” each players according to sums appearing in the corresponding payoff matrix. Explicitly,
The procedure described above is schematically shown in Fig. 1.
A pure strategy Nash equilibrium (PSNE) is a pair of strategies such that
(11) 
In the present example, it is easy to check that, given the initial state from the referee, the pair of strategies leading to NE is . However, this equilibrium is not Pareto efficient, namely there is a strategy set such that for . In the present example
the strategy set leaves the system in the state and
==.
2.1.4 Mixed Strategy in the Language of Bits
This technique of operation on bits is naturally extended to treat, mixed strategy games. Then by operating on the bit state by the matrix with , we get the vector,
(12) 
that can be interpreted as a mixed strategy of choosing pure strategy with probability and pure strategy with probability . Following our example, assuming
player 1 choses with probability and with probability
and player 2 choses with probability and with probability the
combined operation on the initial state is,
.
3 The Quantum Structure: Qubits
In quantum mechanics, the analog of a bit is a quantum bit, briefly referred to as qubit. Physically, this is a two level system. The most simple example is the two spin states of an electron. In order to explain this concept we need to carry out some preparatory work. ^{1}^{1}1For understanding this section, the reader is assumed to have gone through the Appendix on Quantum Mechanics.
3.1 Two Dimensional Hilbert Space
As discussed in the Appendix 7.5, a Hilbert space is a linear vector space above the field of complex numbers, see Appendix 7.1. The dimension of a Hilbert space is the maximal number of linearly independent vectors belonging to . A Hilbert space might have any dimension, including infinite. In quantum information we mainly encounter finite dimensional Hilbert spaces. In quantum games the dimension of Hilbert space pertaining to a given player is equal to the number of his classical strategies. One of the simplest cases relevant to game theory is a classical game with two playerstwo decisions game. Therefore, for the time being we will be concerned with twodimensional Hilbert space, denoted as . As we learn from Appendix 7.5, we can define a set of two linearly independent orthogonal vectors (kets) in denoted as . The fact that the notation of basis states is the same as that used for bits is of course not accidental.
An arbitrary state (or vector) is written as . As we also recall from Appendix 7.5 the Hilbert space is endowed with an inner product, that is, a mapping written as . The basis states have the following properties,

Orthogonality and normalization: .

Linear independence If (namely, they are complex numbers, see Appendix 7.1), then .

Expanding vectors: Every vector (state) can be expressed as a linear combination
, with .
The last equality is obtained by performing the inner products and and using the orthogonality of the bases states discussed in item 1. A more concrete way to say it is that we “multiply” the two sides of the expression on the left once by and once by . This show the power of the Dirac notation.
3.2 Qubits
The quantum bit (shortly qubit) is the basic unit of quantum information, in the same token that bit is the basic unit of classical information. While the notion of bit is familiar to anyone who has a basic knowledge in information storage (on a hard disk for example) and information transfer, the notion of qubit is much less familiar. Until a few years ago it could be argued that qubit are simple quantum system that cannot be used in such discipline as information science, economics, computational resources and cryptography. This is definitely not the case nowadays as the fields of quantum information and quantum computation become closer and closer to reality. For economists, in general, and for game theorists in particular, the concept of qubit requires some change of mind in the sense that a decision (a strategy) is not simple yes or no (for pure strategy) or simple yes with probability and no with probability . Similar to the classical game, where a decision is an operation on bits (see Eq. (6) a strategy is an operation on qubit. However, since a qubit has a much richer structure than a bit, a quantum strategy is much richer than a classical one. But before speaking of quantum games and quantum strategy we need to define the basic unit (like the hydrogen atom in chemistry).
3.3 Definition and Manipulation of Qubits
Now we come to the central definition:
Definition A qubit is a vector such that . The collection of all qubits is a set and not a space (the vector sum of two qubits is, in general, not a qubit, and hence it has no meaning in what follows). The cardinality of the set of qubits is hence (recall that there are only two bits). Two qubits and that differ by a unimodular factor (see appendix 7.1) are considered identical. This is called phase freedom. 
A convenient way to underline the difference between bits and qubits is to write them as vectors,
(13) 
Another standard notation is to write the basis states in terms of arrows. The three notations
(14) 
are in use. The arrow notation is borrowed from physics where the two directions represents the two orientations of an electron’s spin. Thus, all the definitions used below to denote a qubit are equivalent,
(15) 
where means, literally, can also be written as.
The number of degrees of freedom (parameters) of a qubit is 2 (two complex numbers with one constraint combined with the phase freedom). The phase freedom allows us to chose to be real and positive. An elegant way to represent a qubit is by choosing two angles and such that ==:
(16) 
The two angles and determine a point on the unit sphere (globe) with Cartesian coordinates,
(17) 
Therefore, every point on the unit sphere with spherical angles uniquely define a qubit according to Eq. (16). In physics this construction is referred to as Bloch Sphere, as displayed in Fig. 2. In particular, the north pole corresponds to and the south pole, corresponds to .
3.4 Operations on a Single Qubit: Quantum Strategies
In Eq. (6) and (7) we defined two classical strategies, and as operations on bits. According to Eq. (5) they are realized by matrices and act on the bit vectors and . In this subsection we develop the quantum analogs: We are interested in operations on qubits, (also referred to as single qubit quantum gates) that transform a qubit into another qubit .
There are some restrictions on the allowed operations on qubits. First, a qubit is a vector in two dimensional Hilbert space and therefore, operations on a single qubit must be realized by complex matrices. Second, we have seen in Fig. 2 that a qubit is a point on a point on the Bloch sphere and therefore, the new qubit must have the same unit length (the radius of the Bloch sphere). In other words the unit length of a qubit must be conserved under any operation. From what we learn from Appendix 7.3, this means that any allowed operation on a qubit is defined by a unitary matrix . In the notation of Eq. (13) a unitary operation on a qubit represented as a two component vector is defined as,
(18) 
For reasons to become clear later on we will restrict ourselves to unitary transformations with unit determinant, Det[U]=1. The collection of all unitary matrices with unit determinant, form a group under the usual rule of matrix multiplication. This is the group (see Appendix 7.4 on group theory), that plays a central role in physics as well as in abstract group theory. The most general form of a matrix is,
(19) 
Although we have not yet defined the notion of quantum game, we assert that, in analogy with Eq. (6 (that defines player’s classical strategies as operations on bits), the operation on qubits (such that each player acts with his matrix on his qubit), is an implementation of each player’s quantum strategy. Thus,
Definition In quantum games, the (infinite number of) quantum strategies of each player is the infinite set of his matrices as defined in Eq. (19). The infinite collection of these matrices form the group SU(2) of unitary matrices with unit determinat. Since the functional form of the matrix is given by Eq. (19), the strategy of player is determined by his choice of the three angles . Here is just a short notation for the three angles. The three angles are referred to as the Euler angles. 
The quantum strategy specified by the matrix as specified above has a geometrical interpretation. This is similar to the geometrical interpretation given to qubit as a point on the Bloch sphere in Fig. 1, where the two angles determine a point on the boundary of a sphere of unit radius in three dimensions. Such a (Bloch) sphere, is a two dimensional surface denoted by . On the other hand, the three angles defining a quantum strategy determine a point on the surface of the unit sphere in four dimensional space, (the 4 dimensional Euclidean space). The unit sphere is in this space is defined as the collection of points with Cartesian coordinates restricted by the equation . This equality defines the surface of a three dimensional sphere denoted by (impossible to draw a figure). The equality is satisfied by writing the four Cartesian coordinates as,
(20) 
An alternative definition of a player’s strategy is therefore as follows:
Definition A strategy of player in a quantum analog of a twoplayers twostrategies classical game is a point 
Thus, instead of a single number or as a strategy of the classical game, the set of quantum strategies has a cardinality .
3.4.1 Classical Strategies as Special Cases of Quantum Strategies
A desirable property from a quantum game is that the players can reach also their classical strategies. Of course, the interesting case is that reaching the classical strategies does not lead to Nash equilibrium, but the payoff awarded to players in a quantum game that use their classical strategies serve as a useful reference point. Therefore, we ask the question whether, by an appropriate choice of the three angles the quantum strategy is reduced to one of the two classical strategies or . First, it is trivially seen that . It is now clear why we chosen the classical strategy that flips the state of a bit as and not as , because Det[]=1 whereas Det[]=1. On the other hand, we notes that . The quantum game procedure to be described in the next Section is such that the difference between and does not affects the payoff at all, and therefore, we may conclude that the classical strategies are indeed, obtained as special cases of the quantum strategies,
(21) 
3.5 Two qubit States
In Eqs. (3) and (4) we represented twobit states as tensor products
of two onebit states. Equivalently, a twobit state is represented by a four dimensional
vector, three of whose components are 0 and one component is 1 see Eq. (3).
Since each bit can be found in one of two states or there are exactly four twobit states. With twoqubit states, the situation is dramatically different in two respects. First, as noted in connection with Eq. (15), each qubit with
can be found in an infinite number of states.
This is easily understood by noting that, according to Eq. (16)
and Fig. 2, each qubit is a point on the twodimensional (Bloch) sphere.
Accordingly, once we construct twoqubit states by tensor products of two onequbit states we expect a twoqubit state to be represented by a four dimensional vector of complex numbers. Second, and much more profound, there are four dimensional vectors that are not represented as a tensor product of two
twodimensional vectors. Namely, in contrast with the classical twobit states, there are twoqubit states that are not represented as a tensor product of two onequbit states. This is referred to as entanglement and will be explained further below.
In a twoplayers twostrategies classical game,
each player has its own bit upon which he can operate (namely, chose his strategy). Below we shall define a quantum game that is
based on twoplayer twostrategies classical game. In such game,
each player has its own qubit upon which he can operate by an matrix (namely, chose his quantum strategy).
3.5.1 Outer (tensor) product of two qubits
In analogy with Eq. (3) that defines the 4 twobit states we define an outer (or tensor) product of two qubits using the notation of Eq. (13) as follows: Let and be two qubits numbered 1 and 2. We define their outer (or tensor) product as,
(22) 
In terms of 4 component vectors, the tensor products of the elements such as are the same as the twobit states defined in Eq. (3), and therefore, in this notation we have,
(23) 
A tensor product of two qubits as defined above is an example of a two qubit state, briefly referred to as 2qubits. The coefficients of the four products in Eq. (22) (or, equivalently, the four vectors in Eq. (23)), are complex numbers referred to as amplitudes. Thus, we say that the amplitude of in the 2qubits is and so on. Using simple trigonometric identities it is easily verified that the sum of the coefficients is 1, namely,
(24) 
2qubits can also be related to a Bloch sphere (but we will not do it here).
We have seen in Eq. (23) that a tensor product of two qubits is a 2qubits that is written as a linear combination of the four basic 2qubit states
(25) 
From the theory of Hilbert spaces we know the the 2qubits defined in Eq. (25 form a basis in .
This bring as to the following
Definition: A general 2qubits has the form, (26) 
Note the difference between this expression and the outer product of two qubits as defined in Eq. (22), in which the coefficients are certain products of the coefficients of the qubit factors. In the expression (26) the coefficients are arbitrary as long as they satisfy the normalization condition. Therefore, Eq. (22) is a special case of (26) but not viceversa. This observation leads us naturally to the next topic, that is, entanglement.
3.6 Entanglement
Entanglement is one of the most fundamental concepts in quantum information and in quantum game theory. In order to introduce it we ask the following question: Let
(27) 
as already defined in Eq. (26) denote a general 2qubits. Is it always possible to represent it as a tensor product of two single qubit states as in Eqs. (22) or (23) ?? The answer is NO. Few counter examples with two out of the four coefficients set equal to 0 are,
(28) 
where the notations T=triplet and S=singlet are borrowed from physics. These four 2qubits are referred to as maximally entangles Bell states. We now have,
Definition A 2qubits as defined in Eq. (27) is said to be entangled iff it cannot be represented as a tensor product of two single qubit states as in Eqs. (22) or (23). 
Entanglement is a pure quantum mechanical effect that appears in manipulating 2qubits. It does not occur in manipulations of bits. There are only four 2bit states as defined in Eq. (3), all of them
are obtained as tensor products of single bit states, so that by definition they are not entangled.
The concept of entanglement is of utmost importance in many aspects of quantum mechanics. It led
to a very long debate initiated by a paper written in 1935 by Albert Einstein, Boris Podolsky and Nathan Rosen referred to as the EPR paradox that questioned the completeness of quantum mechanics.
The answer to this paradox was given by John Bell in 1964. Entanglement plays a central role in
quantum information. Here we will see that it also plays a central role in quantum game theory.
Strictly speaking, without entanglement, quantum game theory reduces to the classical one.
3.7 Operations on 2qubits (2qubits Gates)
An important tool in manipulating 2qubits are operations transforming one 2qubits to another. Borrowing from the theory of quantum information these are called twoqubit gates. Writing a general 2qubits as defined in Eq. (27) in terms of its 4 vector of coefficients,
(29) 
a 2qubit gate is a unitary matrix (with unit determinant) acting on the 4 vector of coefficients, in analogy with Eq. (18),
(30) 
In the same token as we required the matrices operating on a single qubit state to have unit determinant, that is , we require also to have a unit determinant, that is, , the group of unitary complex matrices with unit determinant.
3.7.1 2qubit Gates Defined as Outer Product of Two 1qubit Gates
Let us recall that the twoplayer strategies in a classical game are defined as outer product of each single player strategy ( or ), defined in Eq. (7) that operate on two bit states as exemplified in Eq. (10). Let us also recall that each player in a quantum game has a strategy that is a matrix as defined in Eq. (19). Therefore, we anticipate that the twoplayer strategies in a quantum game are defined as outer product of the two single player strategies. Thus, a 2qubit gate of special importance is the outer product operation where each player acts on his own qubit. Explicitly, the operation of on given in (29) is,
(31) 
Again, before defining the notion of quantum game, we assert that this operation defines the set of combined quantum strategies in analogy with the classical game set of combined strategies defined in Eq. (7). Thus,
The (infinite numbers of) elements in the set of combined (quantum) strategies are matrices, . These matrices act on two qubit states defined above, e.g Eq. (29). The single qubit operations are defined in Eq. (18). 
3.7.2 Entanglement Operators (Entanglers)
We have already underlined the crucial importance of the concept of entanglement in quantum games. Therefore, of crucial importance for quantum game is an operation executed by an entanglement operator that acts on a nonentangled 2qubits and turns it into an entangled 2qubits. Anticipating the importance and relevance of Bell’s states introduced in Eq. (28) for quantum games, we search entanglement operators that operate on the nonentangled state and create the maximally entangled Bell states such as or as defined in Eq. (ref17). For reason that will become clear later we should require that is unitary, that is, (see Appendix 7.3). With a little effort we find,
(32) 
(33) 
It is straight forward to check that and as defined above are unitary and that application of instead of on the initial state in Eq. (32) yields the second Bell’s state also defined in Eq. (28), while . There is, however, some subtle difference between and that will surface later on.
3.7.3 Partial Entanglement Operators
Intuitively, the Bell’s states defined in Eq. (28) are Maximally entangled because the two coefficients before the two bit states (say, and ) have the same absolute value, . We may think of an entangled state where the weights of the two 2bit states are unequal, in that case we speak of partially entangled state. Thus, instead of the maximally entangled Bell states and defined in Eqs. (28), (32) and (33) we may consider the partially entangled state and that depend on a continuous parameter (an angle) defined as,
(34) 
(35) 
The notion of partial entanglement can be put on a more rigorous basis once we have a tool to determine the degree of entanglement. Such a tool does exists, called Entanglement Entropy but it will not be detailed here. The reason for introducing partial entanglement is that it is intimately related with the existence (or the absence) of pure strategy Nash equilibrium in quantum games as will be demonstrated below.
In the same way that we designed the entanglement operators and that, upon acting on the twobit state yield the maximally entangled Bell’s states and , we need to design analogous partial entanglement operators and that, upon acting on the twobit state yield the partilly entangled states and . With a little effort we find,
(36) 
4 Quantum Games
We come now to the heart of our work, that is, description and search for pure strategy Nash equilibrium in these games. Quantum games have different structures and different rules than classical games. The skeptical reader might justly argue that introducing a quantum game with an attempt to confront it with its classical analogue is meaningless. It is just like inventing a new chess game by using a chessboard (instead of the usual one) and adding four more pieces to each player.
There is, however two points that connect a classical game with its quantum analog. First, the quantum game is based on a classical game and the payoffs in the quantum game are determined by the payoff function of the classical game. Second, the classical strategies are obtained as a special case of the quantum strategies. Depending on the entanglement operators defined in Eq. (36), the players may even reach the classical square in the game table. In most cases, however, this will not lead to a Nash equilibrium.
4.1 How to Quantize a Classical Game?
With all these complex numbers running around, it must be quite hard to imagine how this formalism can be connected to a game in which people have to take decisions and get tangible rewards that depend on their opponent’s decisions, especially when these rewards are expressed in real numbers (dollars or years in prison). Whatever we do, at the end of the day, a passage to real numbers must take place. To show how it works, we start with an old faithful classical game (e.g the prisoner dilemma) and show how to turn into into a quantum game that still ends with rewarding its players with tangible rewards. This procedure is referred as quantization of a classical game. We will carry out this task in two steps. In the first step we will consider a classical game and endow each player with a quantum strategy (The matrix defined in Eq (19). At the same time, we will also design a new payoff system that translates the complex numbers appearing in the state of the system into a real reward. This first step leads us to a reasonable description of a game, but proves to be inadequate if we want to achieve a really new game, not just the classical game from which we started our journey. This task will be achieved in the second step.
Suppose we start with the same classical game as described in Section 1, that is given in its normal form with specified payoff functions as,
Player 2
Player 1 u(I,I),u(I,I) u(I,Y),u(I,Y,1) u(Y,I),u(Y,I) u(Y,Y),u(Y,Y)
It is assumed that the referee already decreed that the initial state is , and asks the players to choose their strategies. There his, however, one difference: Instead of using the classical strategies of either leaving a bit untouched (the strategy ) or operating on it with the second strategy , the referee allows each player to use his quantum strategy defined in Eq. (19). Before we find out how all this will help the players, let us find out what will happen with the state of the system after such an operation. For that purpose it is convenient to use the vector notations specified in Eq. (2) or (13), (14), (15) and let each player act on his own qubit with his own as strategy as explained through Eq. (31), thereby leading the system from its initial state to its final state given by,
(37) 
With the help of Eq. (29) we may then write,
(38) 
From Eq. (19) it is easy to determine the dependence of the coefficients on the angles (that is the strategies of the two players) , for example and so on. Since is a 2qubits then, as we have stressed all around, in Eqs. 26 or (30) we have . This leads us naturally to suggest the following payoff system.
The payoff of player is calculated similar to the calculation of payoffs in correlated equilibrium classical games, with the absolute value squared of the amplitudes (themselves are complex numbers) as the corresponding probabilities, (39) 
For example, prisoner’s 1 and 2 years in prison in the prisoner dilemma game table, Eq. (8) are,
(40) 
The alert reader must have noticed that this procedure ends up in a classical game with mixed strategies. First, once absolute values are taken, the role of the two angles and is void because
(41) 
What is more disturbing is that we arrive at an old format of classical games with mixed strategies. Since , we immediately identify the payoffs in Eq. (39) as those resulting from mixed strategy classical game where a prisoner chooses to confess with probability and to don’t confess with probability . In particular, the pure strategies are obtained as specified in Eq. (21). Thus while the analysis of the first step taught us how to use quantum strategies and how to design a payoff system applicable for a complex state of the system </