**Quantum game theory** is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:

- Superposed initial states
- Entangled initial states
- Superposition of strategies to be used on initial states
- Multiplayer games
- Quantum minimax theorems
- See also
- References
- Further reading

- Superposed initial states,
- Quantum entanglement of initial states,
- Superposition of strategies to be used on the initial states.

This theory is based on the physics of information much like quantum computing.

The information transfer that occurs during a game can be viewed as a physical process. In the simplest case of a classical game between two players with two strategies each, both the players can use a bit (a '0' or a '1') to convey their choice of strategy. A popular example of such a game is the prisoners' dilemma, where each of the convicts can either *cooperate* or *defect*: withholding knowledge or revealing that the other committed the crime. In the quantum version of the game, the bit is replaced by the qubit, which is a quantum superposition of two or more base states. In the case of a two-strategy game this can be physically implemented by the use of an entity like the electron which has a superposed spin state, with the base states being +1/2 (plus half) and −1/2 (minus half). Each of the spin states can be used to represent each of the two strategies available to the players. When a measurement is made on the electron, it collapses to one of the base states, thus conveying the strategy used by the player.

The set of qubits which are initially provided to each of the players (to be used to convey their choice of strategy) may be entangled. For instance, an entangled pair of qubits implies that an operation performed on one of the qubits, affects the other qubit as well, thus altering the expected pay-offs of the game.

The job of a player in a game is to choose a strategy. In terms of bits this means that the player has to choose between 'flipping' the bit to its opposite state or leaving its current state untouched. When extended to the quantum domain this implies that the player can *rotate* the qubit to a new state, thus changing the probability amplitudes of each of the base states. Such operations on the qubits are required to be unitary transformations on the initial state of the qubit. This is different from the classical procedure which chooses the strategies with some statistical probabilities.

Introducing quantum information into multiplayer games allows a new type of "equilibrium strategy" which is not found in traditional games. The entanglement of players' choices can have the effect of a * contract * by preventing players from profiting from other player's betrayal.^{ [1] }

The concepts of a quantum player, a zero-sum quantum game and the associated expected payoff were defined by A. Boukas in 1999 (for finite games) and in 2020 by L. Accardi and A. Boukas (for infinite games) within the framework of the spectral theorem for self-adjoint operators on Hilbert spaces. Quantum versions of Von Neumann's minimax theorem were proved.^{ [2] }^{ [3] }

- Quantum tic tac toe: not a quantum game in the sense above, but a pedagogical tool based on metaphors for quantum mechanics
- Quantum pseudo-telepathy
- Quantum refereed game
- Jan Sładkowski
- Jens Eisert

In physics, the **no-cloning theorem** states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist. One should note that the aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.

**Quantum computing** is the use of quantum phenomena such as superposition and entanglement to perform computation. Computers that perform quantum computations are known as **quantum computers**. Quantum computers are believed to be able to solve certain computational problems, such as integer factorization, substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science.

In physics and computer science, **quantum information** is the information of the state of a quantum system. It is the basic entity of study in **quantum information theory**, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.

**Quantum teleportation** is a process in which quantum information can be transmitted from one location to another, with the help of classical communication and previously shared quantum entanglement between the sending and receiving location. Because it depends on classical communication, which can proceed no faster than the speed of light, it cannot be used for faster-than-light transport or communication of classical bits. While it has proven possible to teleport one or more qubits of information between two (entangled) quanta, this has not yet been achieved between anything larger than molecules.

In quantum computing, a **qubit** or **quantum bit** is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include: the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property which is fundamental to quantum mechanics and quantum computing.

In quantum information theory, a **quantum circuit** is a model for quantum computation in which a computation is a sequence of quantum gates, which are reversible transformations on a quantum mechanical analog of an *n*-bit register. This analogous structure is referred to as an *n*-qubit register. The graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation.

The **measurement problem** in quantum mechanics is the problem of how wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.

The **Bell states**, a concept in quantum information science, are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. The Bell states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1: . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will collapse it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will assign one of two possible values to the other qubit instantly, where the value assigned depends on which Bell state the two qubits are in. Bell states can be generalized to represent specific quantum states of multi-qubit systems, such as the GHZ state for 3 subsystems.

In physics, the **no-communication theorem** or **no-signaling principle** is a no-go theorem from quantum information theory which states that, during measurement of an entangled quantum state, it is not possible for one observer, by making a measurement of a subsystem of the total state, to communicate information to another observer. The theorem is important because, in quantum mechanics, quantum entanglement is an effect by which certain widely separated events can be correlated in ways that suggest the possibility of instantaneous communication. The no-communication theorem gives conditions under which such transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics, such as the EPR paradox, or violations of local realism obtained in tests of Bell's theorem. In these experiments, the no-communication theorem shows that failure of local realism does not lead to what could be referred to as "spooky communication at a distance".

A **qutrit** is a unit of quantum information that is realized by a quantum system described by a superposition of three mutually orthogonal quantum states.

A **trapped ion quantum computer** is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic states of each ion, and quantum information can be transferred through the collective quantized motion of the ions in a shared trap. Lasers are applied to induce coupling between the qubit states or coupling between the internal qubit states and the external motional states.

**Quantum networks** form an important element of quantum computing and quantum communication systems. Quantum networks facilitate the transmission of information in the form of quantum bits, also called qubits, between physically separated quantum processors. A quantum processor is a small quantum computer being able to perform quantum logic gates on a certain number of qubits. Quantum networks work in a similar way to classical networks. The main difference is that quantum networking, like quantum computing, is better at solving certain problems, such as modeling quantum systems.

In physics, in the area of quantum information theory, a **Greenberger–Horne–Zeilinger state** is a certain type of entangled quantum state that involves at least three subsystems. It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989. Extremely non-classical properties of the state have been observed.

The **W state** is an entangled quantum state of three qubits which has the following shape

In quantum computing, the **Gottesman–Knill theorem** is a theoretical result by Daniel Gottesman and Emanuel Knill that states that stabilizer circuits, circuits that only consist of gates from the normalizer of the qubit Pauli group, also called Clifford group, can be perfectly simulated in polynomial time on a probabilistic classical computer. The Clifford group can be generated solely by using CNOT, Hadamard, and phase gates; and therefore stabilizer circuits can be constructed using only these gates.

**Quantum pseudo-telepathy** is the fact that in certain Bayesian games with asymmetric information, players who have access to a shared physical system in an entangled quantum state, and who are able to execute strategies that are contingent upon measurements performed on the entangled physical system, are able to achieve higher expected payoffs in equilibrium than can be achieved in any mixed-strategy Nash equilibrium of the same game by players without access to the entangled quantum system.

**Entanglement distillation** is the transformation of *N* copies of an arbitrary entangled state into some number of approximately pure Bell pairs, using only *local operations and classical communication* (LOCC).

In quantum mechanics, the **cat state**, named after Schrödinger's cat, is a quantum state that is composed of two diametrically opposed conditions *at the same time*, such as the possibilities that a cat be alive and dead at the same time. Schrödinger's cat is sometimes connected to the many worlds hypothesis by its proponents.

Being a component part of network science the study of quantum complex networks aims to explore the impact of complexity science and network architectures in quantum systems. According to quantum information theory it is possible to improve communication security and data transfer rates by taking advantage of quantum mechanics. In this context the study of quantum complex networks is motivated by the possibility of quantum communications being used on a massive scale in the future. In such case it is likely that quantum communication networks will acquire non trivial features as is common in existing communication networks today.

In quantum computing, a *qubit* is a unit of information analogous to a bit in classical computing, but it is affected by quantum mechanical properties such as superposition and entanglement which allow qubits to be in some ways more powerful than classical bits for some tasks. Qubits are used in quantum circuits and quantum algorithms composed of quantum logic gates to solve computational problems, where they are used for input/output and intermediate computations.

- ↑ Simon C. Benjamin and Patrick M. Hayden (13 August 2001), "Multiplayer quantum games",
*Physical Review A*,**64**(3): 030301, arXiv: quant-ph/0007038 , Bibcode:2001PhRvA..64c0301B, doi:10.1103/PhysRevA.64.030301 , arXiv:quant-ph/0007038 - ↑ Boukas, A. (2000). "Quantum Formulation of Classical Two Person Zero-Sum Games".
*Open Systems & Information Dynamics*.**7**: 19–32. doi:10.1023/A:1009699300776. - ↑ Accardi, Luigi; Boukas, Andreas (2020). "Von Neumann's Minimax Theorem for Continuous Quantum Games".
*Journal of Stochastic Analysis*.**1**(2). Article 5. doi: 10.31390/josa.1.2.05 .

- Ball, Philip (18 Oct 1999). "Everyone wins in quantum games".
*Nature*. doi:10.1038/news991021-3. ISSN 0028-0836. Archived from the original on 29 April 2005. - Piotrowski, E. W.; Sładkowski, J. (2003). "An Invitation to Quantum Game Theory" (PDF).
*International Journal of Theoretical Physics*. Springer Nature.**42**(5): 1089–1099. doi:10.1023/a:1025443111388. ISSN 0020-7748.

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