Analyzing three-player quantum games in an EPR type setup
James M. Chappell, Azhar Iqbal, Derek Abbott
1 School of Chemistry and Physics, University of Adelaide, South Australia, Australia
2 School of Electrical and Electronic Engineering, University of Adelaide, South Australia, Australia
We use the formalism of Clifford Geometric Algebra (GA) to develop an analysis of quantum versions of three-player non-cooperative games. The quantum games we explore are played in an Einstein-Podolsky-Rosen (EPR) type setting. In this setting, the players’ strategy sets remain identical to the ones in the mixed-strategy version of the classical game that is obtained as a proper subset of the corresponding quantum game. Using GA we investigate the outcome of a realization of the game by players sharing GHZ state, W state, and a mixture of GHZ and W states. As a specific example, we study the game of three-player Prisoners’ Dilemma.
The field of game theory [1, 2] has a long history , but was first formalized in 1944 with the work of von Neumann and Morgenstern , aiming to develop rational analysis of situations that involve strategic interdependence.
Classical game theory has found increasing expression in the field of physics  and its extension to the quantum regime  was proposed by Meyer  and Eisert et al , though its origins can be traced to earlier works [8, 9, 10, 11]. Early studies in the area of quantum games focused on the two-player two-strategy non-cooperative games, with the proposal for a quantum Prisoners’ Dilemma (PD) being well known . A natural further development of this work was its extension to multiplayer quantum games that was explored by Benjamin and Hayden . Du et al. [13, 14] explored the phase transitions in quantum games for the first time that are central in the present article.
The usual approach in three-player quantum games considers players sharing a three-qubit quantum state with each player accessing their respective qubit in order to perform local unitary transformation. Quantum games have been reported  in which players share Greenberger-Horne-Zeilinger (GHZ) states and the W states , while other works have, for instance, investigated the effects of noise [16, 17] and the benefits of players forming coalitions [18, 19].
A suggested approach [20, 21, 22, 23] in constructing quantum games uses an Einstein-Podolsky-Rosen (EPR) type setting [24, 25, 26, 27, 28, 29, 30, 31]. In this approach, quantum games are setup with an EPR type apparatus, with the players’ strategies being local actions related to their qubit, consisting of a linear combination (with real coefficients) of (spin or polarization) measurements performed in two selected directions.
Note that in a standard arrangement for playing a mixed-strategy game, players are faced with the identical situation, in that in each run, a player has to choose one out of two pure strategies. As the players’ strategy sets remain classical, the EPR type setting avoids a well known criticism  of quantum games. This criticism refers to quantization procedures in which players are given access to extended strategy sets, relative to what they are allowed to have in the classical game. Quantum games constructed with an EPR type setting have been studied in situations involving two players  and also three players . The applications of three-player quantum games include describing three-party situations, involving strategic interaction in quantum communication .
In recent works, the formalism of Clifford’s geometric algebra (GA) [34, 35, 36, 37, 38] has been applied to the analysis of two-player quantum games with significant benefits [39, 40], and so is also adopted here in the analysis of three-player quantum games. The use of GA is justified on the grounds that the Pauli spin algebra is a matrix representation of Clifford’s geometric algebra in , and hence we are choosing to work directly with the underlying Clifford algebra. There are also several other documented benefits of GA such as:
a) The unification of the dot and cross product into a single product, has the significant advantage of possessing an inverse. This results in increased mathematical compactness, thereby aiding physical intuition and insight .
b) The use of the Pauli and Dirac matrices also unnecessarily introduces the imaginary scalars, in contrast to GA, which uses exclusively real elements . This fact was also pointed out by Sommerfield in 1931, who commented that ‘Dirac’s use of matrices simply rediscovered Clifford algebra’ .
c) In the density matrix formalism of quantum mechanics, the expectation for an operator is given by , from which we find the isomorphism to GA, Tr, the subscript zero, indicating to take the scalar part of the algebraic product , where and are now constructed from real Clifford elements. This leads to a uniquely compact expression for the overlap probability between two states in the -particle case, given by Eq. (13), which allows straightforward calculations that normally require complex matrices representing operations on three qubits.
d) Pauli wave functions are isomorphic to the quaternions, and hence represent rotations of particle states . This fact paves the way to describe general unitary transformations on qubits, in a simplified algebraic form, as rotors. In regard to Hestenes’ analysis of the Dirac equation using GA, Boudet  notes that, ‘the use of the pure real formalism of Hestenes brings noticeable simplifications and above all the entire geometrical clarification of the theory of the electron.’
e) Recent works [39, 40, 6] show that GA provides a better intuitive understanding of Meyer’s quantum penny flip game , using operations in -space with real coordinates, permitting helpful visualizations in determining the quantum player’s winning strategy. Also, Christian [45, 46] has recently used GA to produce thought provoking investigations into some of the foundational questions in quantum mechanics.
Our quantum games use an EPR type setting and players have access to general pure quantum states. We determine constraints that ensure a faithful embedding of the mixed-strategy version of the original classical game within the corresponding quantum game. We find how a Pareto-optimal quantum outcome emerges in three-player quantum PD game at high entanglement. We also report phase transitions taking place with increasing entanglement when players share a mixture of GHZ and W type states in superposition.
In an earlier paper , two of the three authors contributed to developing an entirely probabilistic framework for the analysis of three-player quantum games that are also played using an EPR type setting, whereas the present paper, though using an EPR type setting, provides an analysis from the perspective of quantum mechanics, with the mathematical formalism of GA. The previous work analyzed quantum games from the non-factorizable property of a joint probability distribution relevant to a physical system that the players shared in order to implement the game. For the game of three-player Prisoners’ Dilemma, our probabilistic analysis showed that non-factorizability of a joint probability distribution indeed can lead to a new equilibrium in the game. The three-player quantum Prisoners’ Dilemma, in the present analysis, however, moves to the next step and explores the phase structure relating players’ payoffs with shared entanglement and also the impact of players sharing GHZ and W states and their mixture. We believe that without using the powerful formalism of GA, a similar analysis will nearly be impossible to perform using an entirely probabilistic approach as developed in .
EPR setting for playing quantum games
The EPR setting [20, 22, 23] two player quantum games involves a large number of runs when, in a run, two halves of an EPR pair originate from the same source and move in the opposite directions. Player Alice receives one half whereas player Bob receives the other half. To keep the non-cooperative feature of the game, it is assumed that players Alice and Bob are located at some distance from each other and are not unable to communicate between themselves. The players, however, can communicate about their actions, which they perform on their received halves, to a referee who organizes the game and ensures that the rules of the game are followed. The referee makes available two directions to each player. In a run, each player has to choose one of two available directions. The referee rotates Stern-Gerlach type detectors  along the two chosen directions and performs quantum measurement. The outcome of the quantum measurement, on Alice’s side, and on Bob’s side of the Stern-Gerlach detectors, is either or . Runs are repeated as the players receive a large number of halves in pairs, when each pair comes from the same source and the measurement outcomes are recorded for all runs. A player’s strategy, defined over a large number of runs, is a linear combination (with normalized and real coefficients) of the two directions along which the measurement is performed. The referee makes public the payoff relations at the start of the game and announces rewards to the players after the completion of runs. The payoff relations are constructed in view of a) the matrix of the game, b) the list of players’ choices of directions over a large number of runs, and c) the list of measurement outcomes that the referee prepares using his/her Stern-Gerlach apparatus.
For a three-player quantum game, this setting is extended to consider three players Alice, Bob and Chris who are located at the three arms of an EPR system . In the following they will be denoted by and , respectively. As it is the case with two-player EPR setting, in a run of the experiment, each player chooses one out of two directions.
We have used the EPR setting in view of the well known Enk and Pike’s criticism  of quantum games that are played using Eisert et al’s setting . Essentially this criticism attempts to equate a quantum game to a classical game in which the players are given access to an extended set of classical strategies. The present paper uses an EPR setting in which each player has two classical strategies consisting of the two choices he/she can make between two directions along which a quantum measurement can be performed. That is, the player’s pure strategy, in a run, consists of choosing one direction out of the two. As the sets of strategies remain exactly identical in both the classical and the quantum forms of the game, it is difficult to construct an Enk and Pike type argument for a quantum game that is played with an EPR setting.
As Fig. 1 shows, we represent Alice’s two directions as . Similarly, Bob’s directions are and Chris’ are . The players measurement directions form a triplet out of eight possible cases and measurement is performed along the chosen directional triplet. The measurement outcome for each player along their chosen direction is or .
Over a large number of runs the players sequentially receive three-particle systems emitted from a source and a record is maintained of the players’ choices of directions over all runs. One of the eight possible outcomes emerges out of the measurement in an individual run, with the first entry for Alice’s outcome, the second entry for Bob’s outcome and the third entry for Chris’ outcome.
In the following we express the players’ payoff relations in terms of the outcomes of these measurements. These payoffs depend on the triplets of the players’ strategic choices made over a large number of runs and on the dichotomic outcomes of the measurements performed along those directions.
Players’ sharing a symmetric initial state
We consider the situation in which an initial quantum state of three qubits is shared among three players. To obtain a fair game, we assume this state is symmetric with regard to the interchange of the three players. The GHZ state is a natural candidate given by
where we have an entanglement angle , which has been shown  to be capable of producing the maximally entangled three qubit state. Alternatively we could start with the W entangled state
The other symmetric state would be an inverted W state
After the measurement along three directions selected by the players, each player is rewarded according to a payoff matrix , for each player . Thus the expected payoffs for a player is given by
where is the probability the state is obtained after measurement, with , along the three directions chosen by Alice, Bob and Chris respectively. In the EPR setting, can be either of Alice’s two directions i.e. or and similarly for Bob and Chris.
Clifford’s geometric algebra
To set up the GA framework for representing quantum states, we begin by defining as a right-handed set of orthonormal basis vectors, with
where is Kronecker delta. Multiplication between algebraic elements is defined to be the geometric product, which for two vectors and is given by
where is the conventional symmetric dot product and is the anti-symmetric outer product related to the Gibb’s cross product by , where . For distinct basis vectors we find
This can be summarized by
where is the Levi-Civita symbol. We can therefore see that squares to minus one, that is and commutes with all other elements and so has identical properties to the unit imaginary . Thus we have an isomorphism between the basis vectors and the Pauli matrices through the use of the geometric product.
where are real scalars.
For a single particle we then have the basis vectors
and so for three particles we can use as a basis
|where to reduce the number of superscripts representing particle number we write as . General unitary operations are equivalent to rotors in GA , represented as|
which is in Euler angle form and can completely explore the available space of a single qubit. Using the definition of unitary operations given by Eq. (12) we define for general unitary transformations acting locally on each of the three players qubit in order to generalize the starting state, that is the GHZ or W states, as far as possible.
We define a separable state , where and are single particle rotors, which allow the players’ measurement directions to be specified on the first, second and third qubit respectively. The state to be measured is now projected onto this separable state . The overlap probability between two states and in the -particle case is given in Ref.  as
where the angle brackets mean to retain only the scalar part of the expression and and are defined for 3 particles in Ref.  as
|The operator acts the same as complex conjugation: flipping the sign of and inverting the order of the terms.|
where and represent the referee’s local unitary actions, written as rotors and in GA, on the respective player’s qubits, in order to generalize the starting state. Referring to Eq. (13), we firstly calculate
where . We also calculate
For measurement defined with , and allowing a rotation of the detectors by an angle , where we have written as , we find
|From Eq. (13) we find|
where refers to measuring a or state, respectively, and using the standard results listed in the Appendix, we have
|with , representing the two measurement directions available to each player. Also from Eq. (13) we have|
So we find from Eq. (13) the probability to observe a particular state after measurement as
For instance, at we obtain
which shows a product state, as expected. Alternatively with general entanglement, but no operation on the third qubit, that is , we have
which shows that for the GHZ type entanglement each pair of qubits is mutually unentangled.
Obtaining the payoff relations
We extend the approach of Ichikawa and Tsutsui  to three qubits and represent the permutation of signs introduced by the measurement process. For Alice we define
|Using Eq. (4), we then can find the payoff for each player|
|where, as Eqs. (20) show, the three measurement directions are held in . Alternatively, in order to produce other quantum game frameworks [7, 48], we can interpret the rotors , held in , as the unitary operations which can be applied by each player to their qubit, where in this case, the measurement directions will be set by the referee.|
Payoff relations for a symmetric game
For a symmetric game we have . This requires and . The payoff relations (29) are then reduced to