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One of the major and important developments in quantum computing is the quantum game theory. Be it classical game theory or quantum mechanics; quantum game theory has implications in both of them. However, there is no exact similarity between the classical game theory and the quantum one, as there are for sure some traditional results that quantum game theory doesn’t take into account. So, I would try to provide proper and somewhat complete information about quantum games.
Let us start with some history of quantum games and how did it start from the beginning. Well, the first stone was laid in 1900, when Max Planck proposed the energy units or what he called as “quanta” regarding the electromagnetic radiation energy, that they could be emitted or absorbed only in multiples of h: hv, 2hv, 3hv …, where h is the Planck’s constant and v is the frequency. After that Albert Einstein’s major contribution came when he gave the concept of the photoelectric effect using Planck’s quantum, that in metals, there should be a threshold frequency of light, above which it would release electrons, no matter how much the intensity of light would be. He called each collection of particles as “photons” having energy E = hv. After that, Niels Bohr proposed a model of the atom where the nucleus is surrounded by electrons having orbits attaining only some discrete values which are related to the energy of Planck’s 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 energy level (quantized). For the state ψ, there is an N-eigenvector basis which is associated with En eigenvalues. After that, in the next year, Erwin Schrodinger gave his famous equation where, and V is the potential energy. After some years John von Neumann proposed the concept of Hilbert space. During these periods the game theory begins to develop. Von Neumann firstly solved the two-person zero-sum games and then formed a conjecture about the N-person games, the complexity of whom was very high as there was the possibility of the formation of coalitions, which could provide an advantage to some people over others.
Today, we could say that quantum computation has a subbranch that we could call as quantum game theory. We now see that even if there is the entanglement of superposed states or not, the superposition makes the quantum games dissimilar with the classical games. (Superposition is defined as if two quantum states are added together, then the net result would also be a quantum state.)
Von Neumann and Morgenstern stated that for clarification of some of the concepts of economics, the concepts of physics need to be used and there were a lot of social scientists who clearly opposed such parallelism as they think that physics doesn’t have the power to explain the economic theory which is based on human psychology and human behavior, but Neumann and Morgenstern termed these statements as premature. We could also think along the same lines as it is always said that the human brain in itself is also a quantum computer and quantum games definitely have something for both of quantum mechanics as well as economics.
We will first define some of the concepts that we are going to use later involving the vectors and matrices. The following vectors are pretty useful in our case. Now, we could represent any point in the complex space by using these two vectors like au + bd, and here a and b are complex scalars. The moves in a game of a particular player could be clearly represented by using u or d choice and the sequence of the moves by bits of binary number or the qubits, which are the quantum equivalents of the bits. A vector in the Hilbert space is given by qubits while a bit is a single number. Now, for the transformation of one state to another we need some tools which could be done by using the Pauli spin matrices, which are the following.
All of these matrices have some effect on our base states, i.e., u and d, like,
1u = u, 1d = d
As we know that the electrons have two states of spin, which are spin down and spin up. Now, let us 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 that Surya could either apply 1 or matrix to u which could result in
1u = u or
After that Mayank again takes his turn, but he doesn’t know the action of the Bob and also about the state of the electron, by applying either 1 or to the spin of the electron. Then finally Surya takes his final turn without knowing the action of Mayank and the state of the electron whether it’s in spin up or spin down. Then the state of the electron is measured. If the state of the electron is in the u state, then Surya would win $1 and Mayank would lose $1. Similarly, if the state of the electron is in the d state, then Mayank would win $1 and Surya would then lose out on $1.
Now, this is the whole game and we would consider both the probabilistic moves as well as the quantum superposition. As it is a zero-sum game, the payoff would be exactly opposite to the Surya as that of Mayank.
In the game, we said that both the players didn’t know about the moves of other players, i.e., what the other person is going to do. Now, if we remove this assumption then Mayank would definitely know about Surya’s first move and then choose his move accordingly but this would not make such a difference as then Surya has the last move and he could clearly choose his move that makes the electron to be in the spin-up state and that would make him win. Like this Surya would win all the time and then it couldn’t be called a game. So, we would need to provide limited information to both the players to call it a game. Now, we would have two strategies, and, for Mayank and Surya respectively. We would call them mixed strategies as there is some probability involved with a particular move.
playing 1 with probability and playing with probability
Now, the expected payoffs for Mayank would be, no matter what Surya does, would be always:
And for Surya these would be:
Let us take a quantum state like the following form:
We call this a qubit in quantum computation. The a and b here are the amplitudes and on measurement, we would obtain the probability of base states of u and d as and, respectively, and.
Now, let us return to our basic game of spin-flip where Mayank sets the electron state initially to u, but also, he follows a mixed strategy by playing 1 and with equal probability. But there is a twist in this game now as we would allow Surya to cheat, what Surya could do is that he knows some other Pauli spin matrices which are and also their linear combinations. Apart from that Surya also has the final move to play. Now, let us suppose that Surya would play the operator :
As we saw earlier that the mixed strategies of Mayank will in no way change this state and then Surya will play H again to get:
This clearly shows that Surya would always win and he is able to do it only because he could use the superposition of states. This clearly shows us that quantum strategies could increase a player’s expected payoff and that they are good at least as their classical counterparts.
The Quantum game theory garnered a lot of attention since its beginning and the quantization of the games has been shown by Eisert, Lewenstein, and Wilkens which was later generalized by Weber and Marinatto. They showed that the quantum game theory could be applied to any game which is of the form 2xn, where each player has n number of strategies. This has been used for quantizing a lot of classic games like Prisoner’s Dilemma, or be it Battle of Sexes, for evolutionary game theory in Stag Hunt game, The Monty Hall problem, etc. All the results clearly show that the quantization process and the relation of the classical problems to their background are not unique. In these cases, we could find the Nash equilibria but like the classical problems, in the majority of these cases, they are not Pareto Optimal. On the other hand, they show that if the games are to be played quantum mechanically, then that would be more efficient and it also provides an upper bound on the efficiency. There are still a lot of 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 are taking place, sooner or later, we would definitely be able to construct quantum computers. We know that there is a definite set of acts that are involved in quantum games as from the preparation of a system to measuring it and then also getting some reward for it which would not depend upon the outcome of the measurement, which shows that the quantum games have much bigger consequences. The complexity theory for solving a certain computational problem applies lower bounds on different resources and from the viewpoint of cryptography, the main problem is that we could be able to break cryptographic systems by proving the nontrivial lower bounds on their complexity. Our main goal is to get some information by asking some questions. There are a lot of games where there is no adequate description in terms of theory and logic of probability about the strategies of the agents, these games could be analyzed clearly by using the rules of the quantum game theory and we see that the results that come out are very interesting and promising. Presently we could also talk about the technological developments up to such a level that opening a quantum casino would be really feasible, that could cost us big bucks but it could totally prove its worth.
Well, we could establish this fact for sure that the quantum game theory has a subset that we could call as the traditional game theory and the quantum game theory has a much bigger and richer structure and also a very broad set of outcomes. That justification is quite apt for defining quantum game theory. We could lose nothing and in fact, gain much more than switching to the quantum game theory. And so, using the traditional sense of game theory should be discarded as it is neither a Nash equilibrium nor an evolutionarily stable strategy. In the example shown above of the Spin flip of the electron states game, we clearly showed that Surya was clearly able to exploit the quantum superposition by using the H transform (linear combination of ) so that he could win the game all the time (But it did depend upon the sequence in which they have to take their chances). The markets were governed by classical laws in earlier stages but as the efficiency of quantum algorithms is proved to be incomparable with them, the classical behavior will fall much behind than those of the quantum behaviors. As nature plays the quantum games already, it would clearly appear that we humans also do by using our personal quantum computers, i.e., our brains. One could say that nature has for sure taken advantage of quantum computing in the evolvement of our complex brains if the decision of humans could be traced to microscopic quantum events.
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