IndexHistoryMathematical ContextConclusionReferencesOne of the most important and important developments in quantum computing is quantum game theory. Be it classical game theory or quantum mechanics; quantum game theory has implications for both. However, there is no exact similarity between classical and quantum game theory, as there are definitely some traditional outcomes that quantum game theory does not take into account. So, I would try to provide adequate and somewhat complete information about quantum games. Say no to plagiarism. Get a tailor-made essay on "Why Violent Video Games Shouldn't Be Banned"? Get an original essayHistoryLet's start with some history of quantum games and how they started from the beginning. Well, the first stone was laid in 1900, when Max Planck proposed energy units or what he called “quanta” regarding the energy of electromagnetic radiation, which could only be emitted or absorbed in multiples of h: hv, 2hv, 3hv … , where h is Planck's constant and v is the frequency. Later Albert Einstein's major contribution came when he provided the concept of the photoelectric effect using the Planck quantum, according to which in metals there should be a threshold frequency of light, above which it would release electrons, no matter how intense the light would be. . He called every set of particles “photons” with energy E = hv. Subsequently, Niels Bohr proposed a model of the atom in which the nucleus is surrounded by electrons whose orbits reach only some discrete values ​​related to the energy of the Planck quantum. Then came Werner Heisenberg with the new quantum theory in 1925, where he introduced the concept of eigenvector and eigenvalue system where E represents the (quantized) energy level. For the state ψ there exists an N-eigenvector basis associated with the eigenvalues ​​En. Subsequently, the following year, Erwin Schrodinger provided his famous equation where e V is the potential energy. After a few years John von Neumann proposed the concept of Hilbert space. During these periods game theory began to develop. Von Neumann first solved two-person zero-sum games and then formulated a conjecture about N-person games, the complexity of which was very high since there was the possibility of the formation of coalitions, which could provide an advantage to some people over others. Today, we might say that quantum computing has a sub-branch that we might call quantum game theory. We now see that whether or not there is entanglement of superimposed states, the superposition makes quantum games dissimilar to classical games. (Superposition is defined as if two quantum states were added together, then the net result would also be a quantum state.) Von Neumann and Morgenstern stated that to clarify some concepts from economics, it is necessary to use concepts from physics, and there were many social scientists who clearly objected to such parallelism because they thought that physics did not have the power to explain economic theory that is based on human psychology and human behavior, but Neumann and Morgenstern called these claims premature. We could also think along the same lines since it is always said that the human brain itself is also a quantum computer and that quantum games definitely have something for both quantum mechanics and economics. Mathematical Background We will first define some of the concepts we will use later involving vectors and matrices. The following vectors are quite useful in our case. Now, we could represent any point in complex space using these twovectors like au + bd, and here a and b are complex scalars. The moves in a game of a particular player could be clearly represented using the choice uode sequence of moves via binary number bits or qubits, which are the quantum equivalents of bits. A vector in Hilbert space is given by qubits while a bit is a single number. Now, for the transformation of one state into another we need some tools which could be made using Pauli spin matrices, which are as follows. All these matrices have some effect on our basic states, i.e. u and d, like, 1u = u, 1d = dA since we know that electrons have two spin states, spin down and spin up. Now, let's think about a simple electron spin-flip game played between Mayank and Surya. First of all, Mayank keeps the electron in the up spin state, after which Surya could apply 1 or an au matrix which could result in 1u = uo After which Mayank takes his turn again, but he doesn't know the action of Bob and also on the state of the electron, applying 1 or to the spin of the electron. Then finally Surya takes his final turn without knowing Mayank's action and the state of the electron whether it is spin up or spin down. Then the state of the electron is measured. If the state of the electron was in state u, then Surya would win $1 and Mayank would lose $1. Similarly, if the state of electron is in state d, then Mayank would win $1 and Surya would lose $1. Now, this is the whole game and we will consider both probabilistic moves and quantum superposition. Since it is a zero-sum game, the profit would be exactly opposite to Surya's as to Mayank's. In the game, we said that both players did not know the other players' moves, that is, what the other person will do. Now if we remove this assumption, Mayank would definitely know about Surya's first move and would therefore choose his move accordingly, but this wouldn't make such a difference since Surya would have the last move and could clearly choose his move which causes l electron is in the spin-up state and this would make it win. In this way Surya always won and therefore it could not be called a game. So, we would have to provide limited information to both players in order to call it a game. Now we would have two strategies, and for Mayank and Surya respectively. We would call them mixed strategies since there is some probability involved in a particular move. playing 1 with probability and playing with probability Now, the expected payoffs for Mayank would be, whatever Surya does, they would always be: And for Surya these would be :Let's take a quantum state as the following form:We call it qubits in quantum computing. A and b here are the amplitudes, and by measuring them, we would get the probability of the basic states of u and d as e, respectively, and. Now, let's go back to our basic spin-flip game where Mayank sets the state of the electron initially to u, but also follows a mixed strategy by playing 1 and with equal probability. But now there is a twist in this game as we would allow Surya to cheat, what Surya could do is know some other Pauli spin matrices which are and also their linear combinations. Apart from this, Surya also has the finishing move to play. Now, suppose Surya plays the operator: As we saw before, Mayank's mixed strategies will not change this state in any way and so Surya will play H again to get: This clearly shows that Surya would always win and he could only do so because he could use state superposition. This clearly shows us that quantum strategies could increase a player's expected profit and that they are at least as good as their classical counterparts. Game theoryQuantum games has attracted much attention since its inception and the quantization of games was shown by Eisert, Lewenstein, and Wilkens which was later generalized by Weber and Marinatto. They showed that quantum game theory can be applied to any game that is of the form 2xn, in which each player has n number of strategies. This has been used to quantize many classic games such as Prisoner's Dilemma, or Battle of Sexes, for evolutionary game theory in the game Stag Hunt, in the Monty Hall problem, etc. All results clearly show that the quantization process and the relationship of classical problems to their background are not unique. In these cases, we may find Nash equilibria but, like classical problems, in most of these cases, they are not Pareto Optimal. On the other hand, they show that if games were to be played with quantum mechanics, then this would be more efficient and would also provide an upper bound on the efficiency. There are still many connections between scientific models and quantum information theory that have not yet been explored. Quantum game theory basically provides tools for phenomena that are not physical processes. We all know that as technological developments occur, we will definitely be able to build quantum computers sooner or later. We know that there is a defined set of acts involved in quantum games, from preparing a system to measuring it and then also getting a reward for it that would not depend on the result of the measurement, which shows that quantum games have much bigger consequences. Complexity theory to solve a certain computational problem applies lower bounds to different resources and from the cryptography point of view, the main problem is that we might be able to break cryptographic systems by proving non-trivial lower bounds on their complexity. Our main goal is to get some information by asking some questions. There are many games where there is no adequate description in terms of probability theory and logic regarding the strategies of the agents, these games could be clearly analyzed using the rules of quantum game theory and we see that the results that arrive are very interesting and promising. At the moment we might even be talking about technological developments to a level where opening a quantum casino would actually be feasible, which might cost us a lot of money but could prove to be totally worthwhile. Please note: this is just an example. Get a custom paper now from our expert writers. Get a Custom EssayConclusion Well, we could definitely establish this fact that quantum game theory has a subset that we could call traditional game theory and quantum game theory has a much larger and richer structure and also a very large set of results . This justification is quite suitable for defining quantum game theory. We could lose nothing and, indeed, gain much more than by moving to quantum game theory. And therefore, using the traditional sense of game theory should be discarded since it is neither a Nash equilibrium nor an evolutionarily stable strategy. In the example shown above of Spin reversal of the electronic states game, we clearly showed that Surya was clearly able to exploit the quantum superposition using the H transform (linear combination of ) so that he could win the game at all times (but depended on the sequence in which they had to take the risk). Markets were governed by classical laws in earlier stages, but since,39