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A multiple round interactive quantum games is a subset of games that falls under the general theory of quantum games.

Background
Game theory had been studied extensively by mathematicians using classical (non-quantum mechanical) models. Around the year 2000, many researchers {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  because the results of such games lack the generality 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, 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.

Definitions
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, $$\mathcal X_1,\cdots, \mathcal X_n$$ are represented by the tensor product notation $$\mathcal X_1\otimes\cdots\otimes\mathcal X_n$$.

An operator between complex Euclidean spaces $$\mathcal X$$ and $$\mathcal Y$$ is a linear mapping denoted by $$L(\mathcal X,\mathcal Y)$$. $$L(\mathcal X)$$ is a short form for $$L(\mathcal X,\mathcal X)$$.

A super-operator is a linear mapping of the form $$\Phi:L(\mathcal X)\rightarrow L(\mathcal Y)$$.

A $$n$$-turn quantum interactive strategy is a combination of $$n$$-turn strategy and co-strategy, which correspond to the strategies players Alice and Bob can choose from.

Let $$\mathcal{X}_1,\cdots,\mathcal X_n, \mathcal Y_1,\cdots, \mathcal Y_n, \mathcal Z_1,\cdots,\mathcal Z_n, \mathcal W_1,\cdots, \mathcal W_n$$ be complex Euclidean spaces.

n-turn strategy
An n-turn non-measuring strategy with the input spaces $$\mathcal{X}_1,\cdots,\mathcal X_n,$$ and output spaces $$ \mathcal Y_1,\cdots, \mathcal Y_n$$ consist of

1. a memory space $$\mathcal Z_1,\cdots, \mathcal Z_n$$

2. an n-tuple of admissible mappings $$(\Phi_1, \cdots, \Phi_n)$$ of the form
 * $$\Phi_1: L(\mathcal X_1)\rightarrow L(\mathcal Y_1\otimes \mathcal Z_1)$$
 * $$\Phi_k: L(\mathcal X_k\otimes Z_{k-1})\rightarrow L(\mathcal Y_k\otimes\mathcal Z_k),\ (2\leq k\leq n)$$

An $$n$$-turn measuring strategy consists of 1 and 2 above and

3. a measurement $$\{ P_a:a\in\Sigma\} $$

n-turn co-strategy
Similar to $$n$$-turn strategy, an $$n$$-turn co-strategy consists of input spaces $$ \mathcal Y_1,\cdots, \mathcal Y_n$$, output spaces $$\mathcal{X}_1,\cdots,\mathcal X_n,$$, as well as the following:

1. a memory space $$ \mathcal W_0,\cdots,\mathcal W_n$$

2. a density operator $$\rho_0 =D(\mathcal X_1\otimes \mathcal W_0)$$

3. an n-tuple of admissible mappings $$(\Psi_1, \cdots, \Psi_n)$$ of the form


 * $$\Psi_k: L(\mathcal Y_k\otimes \mathcal W_{k-1})\rightarrow L(\mathcal X_{k+1}\otimes \mathcal W_k)$$


 * $$ \Psi_n:L(\mathcal Y_n\otimes\mathcal W_{n-1})\rightarrow L(\mathcal W_n)$$

An n-turn measuring co-strategy consists of 1, 2 and 3 above and

4. a measurement $$\{Q_b:b\in\Gamma\}$$

Example
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 $$\rho_0\in D(\mathcal X_1\otimes\mathcal W_0) $$, and sends part of the state in $$L(\mathcal X_1)$$ to Alice, while keeping $$L(\mathcal W_1)$$.


 * Alice (with 2-turn strategy) applies operator $$ \Phi_1$$ on $$L(\mathcal X_1)$$ he receives, which outputs some state in $$L(\mathcal Y_1\otimes Z_1)$$. Alice sends $$L(\mathcal Y_1)$$ to Bob.


 * Bob applies $$ \Psi_1$$ to the state $$L(\mathcal W_1)$$ he kept and $$L(\mathcal Y_1)$$ he received to get $$L(\mathcal X_2\otimes\mathcal W_2)$$, and sends $$L(\mathcal X_2)$$ to Alice.


 * Alice applies $$ \Phi_2$$ on state in $$L(\mathcal X_2\otimes\mathcal Z_2)$$, and sends $$L(\mathcal Y_2)$$ to Bob. For a 2-turn measuring strategy, Alice measures her state in $$L(\mathcal Z_2)$$ with $$\{P_a\}$$ to get value $$a$$.


 * Bob applies $$ \Psi_2$$ on state in $$L(\mathcal Y_2\otimes\mathcal W_2)$$ to get a final state $$L(\mathcal W_2)$$. For a 2-turn measuring co-strategy, Bob measures his state in $$L(\mathcal W_2)$$ with $$\{Q_b\}$$ to get $$b$$.

$$ (a, b)$$ 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 $$(a,b)$$ is


 * $$\langle P_a\otimes Q_b,(I_{L_{\mathcal Z_2}}\otimes\Psi_2)(\Phi_2\otimes I_{L_{\mathcal Z_2}})(I_{L_{\mathcal Z_1}}\otimes\Psi_1)(\Phi_1\otimes I_{L_{\mathcal Z_1}})\rho_0\rangle$$.

In general, for an n-turn quantum interactive strategy with measuring strategy and co-strategy, the probability of arriving at output $$(a,b)\in\Sigma\times\Gamma$$ is


 * $$\langle P_a\otimes Q_b,(I_{L_{\mathcal Z_n}}\otimes\Psi_n)(\Phi_n\otimes I_{L_{\mathcal Z_n}})\cdots(I_{L_{\mathcal Z_1}}\otimes\Psi_1)(\Phi_1\otimes I_{L_{\mathcal Z_1}})\rho_0\rangle$$

Representation of Strategy
Although $$n$$-turn strategy and co-strategy can be described by a sequence of super-operators, this representation can be inconvenient in some situations. It is possible to describe a sequence of super-operators $$(\Phi_1,\cdots,\Phi_n)$$ for a strategy using a single super-operator


 * $$\Xi: L(\mathcal X_{1\cdots n})\rightarrow L(\mathcal Y_{1\cdots n})$$.

The super operator $$\Xi$$ takes input state $$\xi$$ from spaces $$\mathcal X_1,\cdots,\mathcal X_n$$ one piece at a time, applying the corresponding $$\Phi_k$$ super operator, and trace out space $$\mathcal Z_n$$ at the end to give $$\Xi(\xi)\in D(\mathcal Y_{1\cdots n})$$. The Choi-Jamiolkowski representation of the strategy $$(\Phi_1,\cdots,\Phi_n)$$ is defined as the Choi-Jamiolkowski representation $$J(\Xi)$$ of the super-operator $$\Xi$$, where
 * $$J(\Phi) = \sum_{i\leq i, j\leq N}\Phi(|i\rangle\langle j|)\otimes |i\rangle\langle j|\in L(\mathcal Y\otimes\mathcal X)$$.

Although $$n$$-turn strategy and co-strategy can be described by a sequence of super-operators, this representation can be inconvenient in some situations. It is possible to describe a sequence of super-operators $$(\Phi_1,\cdots,\Phi_n)$$ for a strategy using a single super-operator

A measuring strategy with measurement $$\{P_a:a\in\Sigma\}$$ can be described by the collection of super-operators $$\{\Xi_a:a\in\Sigma\}$$. Each of them has the form $$\Xi_a : L(\mathcal X_{1\cdots n})\rightarrow L(\mathcal Y_{1\cdots n})$$, and are defined almost exactly the same way as $$\Xi$$, except the partial trace on $$\mathcal Z_n$$ is replaced by the mapping


 * $$X\mapsto \operatorname{Tr}_{\mathcal Z_n}((P_a\otimes I_{\mathcal Y_{1\cdots n}}) X)$$.

A measurement must satisfy $$\sum_a P_a = I$$ so $$\sum_a\Xi_a = \Xi$$.

Linear isometry representation
An alternative way of expressing the Choi-Jamiolkowski representation of a strategy is based on its linear isometry descriptions $$(A_1,\cdots, A_n)$$. Define $$A\in L(\mathcal X_{1\cdots n},\mathcal Y_{1\cdots, n}\otimes \mathcal Z_n)$$ to be
 * $$A = (I_{\mathcal Y_{1\cdots n-1}}\otimes A_n)((I_{\mathcal Y_{1\cdots n-2}}\otimes A_{n-1}\otimes I_{\mathcal X_n})\cdots (A_1\otimes I_{\mathcal X_{1\cdots n-1}})$$.

Then the Choi-Jamiolkowski representation is $$Q = \operatorname{Tr}_{\mathcal Z_n}(\operatorname{vec}(A)\operatorname{vec}(A)^*)$$ where $$\operatorname{vec}$$ is the vectorization operation.

The representation for a measuring strategy measurement $$\{P_a:a\in\Sigma\}$$ is then $$\{Q_a:a\in\Sigma\}$$ defined by


 * $$Q_a = \operatorname{Tr}_{\mathcal Z_n}(\operatorname{vec}(B_a)\operatorname{vec}(B_a)^*)$$ where $$ B_a = (\sqrt{P_a}\otimes I_{\mathcal Y_{1\cdots n}})A$$.

The Choi-Jamiolkowski representation of measuring and non-measuring co-strategies are defined similarly, with the resulting operators transposed due to technical reasons.

Properties
Representing n-turn strategy and co-strategy in Choi-Jamilkowski representation lead to the discovery of the following three properties of the representations.

Define $$\mathcal S_n(\mathcal X_{1\cdots n},\mathcal Y_{1\cdots n})$$ to be the set of all representation of n-turn strategies, and $$\text{co-}\mathcal S_n(\mathcal X_{1\cdots n},\mathcal Y_{1\cdots n})$$ to be the set of all representation of n-turn co-strategies.

Interaction output probabilities
The probability of getting output $$(a,b)$$ at the end of an n-turn interaction with a measuring strategy $$\{Q_b:b\in\Gamma\}$$ and a measuring co-strategy $$\{R_a:a\in\Sigma\}$$ is
 * $$\langle Q_a,R_b\rangle $$

where $$\langle A,B\rangle = \text{Tr}(A^*B)$$ is the Hilbert-Schmidt inner product.

Characterization of strategy representations
Let $$\operatorname{Pos}(\mathcal X)$$ denote the set of all positive-semidefinite operators acting on space $$\mathcal X$$. Given $$Q\in\operatorname{Pos}(\mathcal Y_{1\cdots n}\otimes \mathcal X_{1\cdots n})$$, $$Q\in \mathcal S_n(\mathcal X_{1\cdots n},\mathcal Y_{1\cdots n})$$, i.e. $$Q$$ is a representation for a strategy if and only if


 * $$\operatorname{Tr}_{\mathcal Y_n}(Q) = R\otimes I_{\mathcal X_n}$$

for $$R\in\mathcal S_{n-1}(\mathcal X_{1\cdots n-1},\mathcal Y_{1\cdots n-1})$$

Maximum Output Probabilities
Given an n-turn measuring strategy $$\{Q_a:a\in\Sigma\}$$, the maximum probability for this strategy to be forced to output $$a$$ across all compatible co-strategies is
 * $$\min\{p\in[0,1]: Q_a\leq pR\text{ for some }R\in\mathcal S_n(\mathcal X_{1\cdots n},\mathcal Y_{1\cdots n})\}$$

Applications
Strong coin flipping

Quantum refereed game