Improving documentation for QuantumNetwork class + bug fix

doc/source/api-reference/qibo.rst

@@ -1781,6 +1781,13 @@ Frame Potential
 Quantum Networks
 ^^^^^^^^^^^^^^^^
 
+Quantum network is an object that unifies the representation of quantum states, channels,
+observables, and higher-order quantum operators.
+
+For more details, see G. Chiribella *et al.*, *Theoretical framework for quantum networks*,
+`Physical Review A 80.2 (2009): 022339
+<https://journals.aps.org/pra/abstract/10.1103/PhysRevA.80.022339>`_.
+
 .. autoclass:: qibo.quantum_info.quantum_networks.QuantumNetwork
     :members:
     :member-order: bysource

doc/source/code-examples/tutorials/quantum_networks/README.md

@@ -0,0 +1,122 @@
+# Quantum Networks
+
+## The Quantum Network Model
+
+The quantum network model is a mathematical framework that allows us to uniquely describe quantum information processing that involves multiple points in time and space.
+Each distinguished point in time and space is treated as a linear system $\mathcal{H}_ i$.
+A quantum network involving $n$ points in time and space is a Hermitian operator $\mathcal{N}$ that acts on the tensor product of the linear systems $\mathcal{H}_ 0 \otimes \mathcal{H}_ 1 \otimes \cdots \otimes \mathcal{H}_ {n-1}$.
+Each system $\mathcal{H}_ {i}$ is either an input or an output of the network.
+
+A physically implementable quantum network is described by a semi-positive definite operator $\mathcal{N}$ that satisfies the causal constraints.
+
+A simple example is a quantum channel $\Gamma: \mathcal{H}_ 0 \to \mathcal{H}_ 1$, where $\mathcal{H}_ 0$ is the input system and $\mathcal{H}_ 1$ is the output system.
+The quantum channel is a linear map, such that it maps any input quantum state to an output quantum state, which is a sufficient and necessary condition for the map to be physical.
+A Hermitian operator $J^\Gamma$ acting on $\mathcal{H}_ 0\otimes \mathcal{H}_ 1$ is associated with a quantum channel $\Gamma$, if $J^\Gamma$ satisfies the following conditions:
+$$J^\Gamma \geq 0, \quad \text{and} \quad \text{Tr}_ {\mathcal{H}_ 1} J^\Gamma = \mathbb{I}_ {\mathcal{H} _0} .$$
+
+The first condition is called *complete positivity*, and the second condition is called *trace-preserving*.
+In particular, the second condition ensures that the information of the input system is only accessible through the output system.
+
+In particular, a quantum state $\rho$ may be also considered as a quantum network, where the input system is the trivial system $\mathbb{C}$, and the output system is the quantum system $\mathcal{H}$.
+The constraints on the quantum channels are then equivalent to the constraints on the quantum states:
+$$\rho \geq 0, \quad \text{and} \quad \text{Tr} \rho = \mathbb{I}_ \mathbb{C} = 1\ .$$
+
+> For more details, see G. Chiribella *et al.*, *Theoretical framework for quantum networks*,
+> [Physical Review A 80.2 (2009): 022339](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.80.022339).
+
+## Quantum Network in `qibo`
+
+The manipulation of quantum networks in `qibo` is done through the `QuantumNetwork` class.
+
+```python
+from qibo.quantum_info.quantum_networks import QuantumNetwork
+```
+
+A quantum state is a quantum network with a single input system and a single output system, where the input system is the trivial 1-dimensional system.
+We need to specify the dimensions of each system in the `partition` argument.
+
+```python
+from qibo.quantum_info import random_density_matrix, random_unitary
+
+state = random_density_matrix(2)
+state_choi = QuantumNetwork(state, (1,2))
+print(f'A quantum state is a quantum netowrk of the form {state_choi}')
+```
+
+```
+>>> A quantum state is a quantum netowrk of the form J[1 -> 2]
+```
+
+A general quantum channel can be created in a similar way.
+
+```python
+from qibo.gates import DepolarizingChannel
+
+test_ch = DepolarizingChannel(0,0.5)
+N = len(test_ch.target_qubits)
+partition = (2**N, 2**N)
+
+depolar_choi = QuantumNetwork(test_ch.to_choi(), partition)
+print(f'A quantum channel is a quantum netowrk of the form {depolar_choi}')
+```
+
+```
+>>> A quantum channel is a quantum netowrk of the form J[2 -> 2]
+```
+
+One may apply a quantum channel to a quantum state, or compose two quantum channels, using the `@` operator.
+
+```python
+new_state = depolar_choi @ depolar_choi @ state_choi
+```
+
+## Example
+
+For 3-dimensional systems, an unital channel may not be a mixed unitary channel.
+
+> Example 4.3 in (Watrous, John. The theory of quantum information. Cambridge university press, 2018.)
+
+```python
+A1 = np.array([
+    [0,0,0],
+    [0,0,1/np.sqrt(2)],
+    [0,-1/np.sqrt(2),0],
+])
+A2 = np.array([
+    [0,0,1/np.sqrt(2)],
+    [0,0,0],
+    [-1/np.sqrt(2),0,0],
+])
+A3 = np.array([
+    [0,1/np.sqrt(2),0],
+    [-1/np.sqrt(2),0,0],
+    [0,0,0],
+])
+
+Choi1 = QuantumNetwork(A1, (3,3), pure=True) * 3
+Choi2 = QuantumNetwork(A2, (3,3), pure=True)*3
+Choi3 = QuantumNetwork(A3, (3,3), pure=True)*3
+```
+
+The three channels are pure but not unital. Which means they are not unitary.
+
+```python
+print(f"Choi1 is unital: {Choi1.unital()}")
+print(f"Choi2 is unital: {Choi2.unital()}")
+print(f"Choi3 is unital: {Choi3.unital()}")
+```
+
+```
+>>> Choi1 is unital: False
+Choi2 is unital: False
+Choi3 is unital: False
+```
+
+However, the mixture of the three operators is unital.
+As the matrices are orthogonal, they are the extreme points of the convex set of the unital channels.
+Therefore, this mixed channel is not a mixed unitary channel.
+
+```python
+Choi = Choi1/3 + Choi2/3 + Choi3/3
+print(f"The mixed channel is unital: {Choi.unital()}")
+```