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A multiple round interactive quantum games is a subset of games that falls under the general theory of quantum games.[1]
Background
[edit]Game theory had been studied extensively by mathematicians using classical (non-quantum mechanical) models. Around the year 2000, many researchers[2] [3] {references} explored ways to extend games well studied in classical game theory to allow the players to use quantum strategies, and showed that the quantum version would give significant advantages over the classical counterparts. Even though classical games can be considered as a special case of quantum games, some controversy were drawn [4][5] because the results of such games lack the generality [1] to be applied to any other areas of quantum information and computation. A more general theory of quantum game was proposed in 2007 by Gutoski[1], where a quantum game can be a general set of rules that allows the players to choose their strategies with no restrictions beyond the ones imposed by the theory of quantum information.
Multiple round interaction competitive quantum games fall under the general theory of quantum games, and have applications a variety of fields such as quantum cryptography, computational complexity, communication complexity, and distributed computation. [1]
Definitions
[edit]The information used for quantum strategies are represented by vectors in complex Euclidean space, or finite dimensional inner product space over the complex numbers. A (finite) sequence of complex Euclidean spaces, are represented by the tensor product notation .
An operator between complex Euclidean spaces and is a linear mapping denoted by . is a short form for .
A super-operator is a linear mapping of the form .
A -turn quantum interactive strategy is a combination of -turn strategy and co-strategy, which correspond to the strategies players Alice and Bob can choose from.
Let be complex Euclidean spaces.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/QuantumStrategy.svg/512px-QuantumStrategy.svg.png)
n-turn strategy
[edit]An n-turn non-measuring strategy with the input spaces and output spaces consist of
1. a memory space
2. an n-tuple of admissible mappings of the form
An -turn measuring strategy consists of 1 and 2 above and
3. a measurement
n-turn co-strategy
[edit]Similar to -turn strategy, an -turn co-strategy consists of input spaces , output spaces , as well as the following:
1. a memory space
2. a density operator
3. an n-tuple of admissible mappings of the form
An n-turn measuring co-strategy consists of 1, 2 and 3 above and
4. a measurement
Example
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/ea/2_turn_interactive_quantum_game.svg/512px-2_turn_interactive_quantum_game.svg.png)
Consider the interaction between a 2-turn measuring strategy and co-strategy. Assume Alice and Bob are players of a 2-turn interactive quantum game shown in the figure on the left.
- Bob (with 2-turn co-strategy) starts with some state , and sends part of the state in to Alice, while keeping .
- Alice (with 2-turn strategy) applies operator on he receives, which outputs some state in . Alice sends to Bob.
- Bob applies to the state he kept and he received to get , and sends to Alice.
- Alice applies on state in , and sends to Bob. For a 2-turn measuring strategy, Alice measures her state in with to get value .
- Bob applies on state in to get a final state . For a 2-turn measuring co-strategy, Bob measures his state in with to get .
are the outcome of the quantum strategies Bob and Alice chose, which can be used to determine the pay-off of the game. The probability of getting the outcome is
- .
In general, for an n-turn quantum interactive strategy with measuring strategy and co-strategy, the probability of arriving at output is
Representation of Strategy
[edit]Although -turn strategy and co-strategy can be described by a sequence of super-operators, this representation can be inconvenient in some situations.[1] It is possible to describe a sequence of super-operators for a strategy using a single super-operator
- .
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Quantum_strategy_representation.svg/512px-Quantum_strategy_representation.svg.png)
The super operator takes input state from spaces one piece at a time, applying the corresponding super operator, and trace out space at the end to give . The Choi-Jamiolkowski representation of the strategy is defined as the Choi-Jamiolkowski representation of the super-operator , where
- .
Although -turn strategy and co-strategy can be described by a sequence of super-operators, this representation can be inconvenient in some situations.[1] It is possible to describe a sequence of super-operators for a strategy using a single super-operator
A measuring strategy with measurement can be described by the collection of super-operators . Each of them has the form , and are defined almost exactly the same way as , except the partial trace on is replaced by the mapping
- .
A measurement must satisfy so .
Linear isometry representation
[edit]An alternative way of expressing the Choi-Jamiolkowski representation of a strategy is based on its linear isometry descriptions . Define to be
- .
Then the Choi-Jamiolkowski representation is where is the vectorization operation.
The representation for a measuring strategy measurement is then defined by
- where .
The Choi-Jamiolkowski representation of measuring and non-measuring co-strategies are defined similarly, with the resulting operators transposed due to technical reasons.
Properties
[edit]Representing n-turn strategy and co-strategy in Choi-Jamilkowski representation lead to the discovery of the following three properties of the representations. [1]
Define to be the set of all representation of n-turn strategies, and to be the set of all representation of n-turn co-strategies.
Interaction output probabilities
[edit]The probability of getting output at the end of an n-turn interaction with a measuring strategy and a measuring co-strategy is
where is the Hilbert-Schmidt inner product.
Characterization of strategy representations
[edit]Let denote the set of all positive-semidefinite operators acting on space . Given , , i.e. is a representation for a strategy if and only if
for
Maximum Output Probabilities
[edit]Given an n-turn measuring strategy , the maximum probability for this strategy to be forced to output across all compatible co-strategies is
Applications
[edit]See Also
[edit]References
[edit]- ^ a b c d e f g Gutoski, G (2007). "Toward a general theory of quantum games". Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing: 565–574. arXiv:quant-ph/0611234. doi:10.1145/1250790.1250873. ISBN 9781595936318. S2CID 2329605.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ Eisert, J (1999). "Quantum games and quantum strategies". Physical Review Letters. 83 (15): 3077–3080. arXiv:quant-ph/9806088. doi:10.1103/PhysRevLett.83.3077. S2CID 30550760.
{{cite journal}}
: Unknown parameter|coauthors=
ignored (|author=
suggested) (help) - ^ Myer, D (1999). "Quantum Strategies". Physical Review Letters. 82 (5): 1502–1555. arXiv:quant-ph/9804010. doi:10.1103/PhysRevLett.82.1052. S2CID 7361611.
- ^ Enk, S.V. (2002). "Classical rules in quantum games". Physical Review A. 66 (2): 024306. arXiv:quant-ph/0203133. doi:10.1103/PhysRevA.66.024306. S2CID 119509677.
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ignored (|author=
suggested) (help) - ^ Benjamin, S (2001). "Comment on "Quantum games and quantum strategeis"". Physical Review Letters. 87 (6): 069801. arXiv:quant-ph/0003036. doi:10.1103/PhysRevLett.87.069801. S2CID 17148657.
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