# Nash Embedding and Equilibrium in Pure Quantum States

## Abstract

With respect to probabilistic mixtures of the strategies in non-cooperative games, quantum game theory provides guarantee of fixed-point stability, the so-called Nash equilibrium. This permits players to choose mixed quantum strategies that prepare mixed quantum states optimally under constraints. Here, we show here that fixed-point stability of Nash equilibrium can also be guaranteed for pure quantum strategies via an application of the Nash embedding theorem, permitting players to prepare pure quantum states optimally under constraints.

- Authors:

- Khalifa Univ., Abu Dhabi (United Arab Emirates)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1513384

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Quantum Technology and Optimization Problems

- Additional Journal Information:
- Journal Volume: 11413; Journal Issue: 0; Conference: 1. Quantum Technology and Optimization Problems International Workshop, Munich (Germany), 18 Mar 2019

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Nash embedding; Fixed-point stability; Nash equilibrium

### Citation Formats

```
Khan, Faisal Shah, and Humble, Travis S. Nash Embedding and Equilibrium in Pure Quantum States. United States: N. p., 2019.
Web. doi:10.1007/978-3-030-14082-3_5.
```

```
Khan, Faisal Shah, & Humble, Travis S. Nash Embedding and Equilibrium in Pure Quantum States. United States. doi:10.1007/978-3-030-14082-3_5.
```

```
Khan, Faisal Shah, and Humble, Travis S. Tue .
"Nash Embedding and Equilibrium in Pure Quantum States". United States. doi:10.1007/978-3-030-14082-3_5. https://www.osti.gov/servlets/purl/1513384.
```

```
@article{osti_1513384,
```

title = {Nash Embedding and Equilibrium in Pure Quantum States},

author = {Khan, Faisal Shah and Humble, Travis S.},

abstractNote = {With respect to probabilistic mixtures of the strategies in non-cooperative games, quantum game theory provides guarantee of fixed-point stability, the so-called Nash equilibrium. This permits players to choose mixed quantum strategies that prepare mixed quantum states optimally under constraints. Here, we show here that fixed-point stability of Nash equilibrium can also be guaranteed for pure quantum strategies via an application of the Nash embedding theorem, permitting players to prepare pure quantum states optimally under constraints.},

doi = {10.1007/978-3-030-14082-3_5},

journal = {Quantum Technology and Optimization Problems},

number = 0,

volume = 11413,

place = {United States},

year = {2019},

month = {2}

}

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