Quantum complex network

In quantum information Qubits are the equivalent to bits in classical systems. A qubit is a property that when measured only can be found to be in one of two states, that is used to transmit information.[4] The polarization of a photon or the nuclear spin are examples of two state systems that can be used as qubits.[4]

Quantum entanglement is a physical phenomenon characterized by a correlation between the quantum states of two or more particles.[4] While entangled particle do not interact in the classical sense, the quantum state of those particle can not be described independently. Particles can be entangle in different degrees, and the maximally entangled state are the ones the maximize the entropy of entanglement.[9][10] In the context of quantum communication, quantum entanglement qubits are used as a quantum channel capable of transmitting information when combined with a classical channel.[4]

Bell measurement is the joint quantum-mechanical measurement of two qubits, so that after the measurement the two qubits will be maximally entangled.[4][10]

Entanglement swapping is a frequent strategy used in quantum networks that allows the connections in the network to change.[1][11] Lets us suppose that we have 4 qubits, A B C and D, C and D belong to the same station, while A and C belong to two different stations. Qubit A is entangled with qubit C and qubit B is entangled with qubit D. By performing a bell measurement in qubits A and B, not only the qubits A and B will be entangled but it is also possible to create an entanglement state between qubit C and qubit D, despite the fact that there was never an interactions between them. Following this process the entanglement between qubits A and C, and qubits B and D will be lost. This strategy can be used to shape the connection on the network.[1][11][12]

Each node in the network is in possession of a set of qubits that can be in different states. The most convenient representation for the quantum state of the qubits is the dirac notation and represent the two state of the qubits as ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$.[1][11] Two particle are entangled if the joint wave function, ${\displaystyle |\psi _{ij}\rangle }$, can not be decomposed as,[4][10]

${\displaystyle |\psi _{ij}\rangle =|\phi \rangle _{i}\otimes |\phi \rangle _{j},}$

where ${\displaystyle |\phi \rangle _{i}}$ represents the quantum state of the qubit at node i and ${\displaystyle |\phi \rangle _{j}}$ represents the quantum state of the qubit at node j. Another important concept is maximally entangled states. The four states (the Bell states) that maximize the entropy of entanglement can be written as[4][10]

${\displaystyle |\Phi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Phi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}-|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}+|1\rangle _{i}\otimes |0\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}-|1\rangle _{i}\otimes |0\rangle _{j}).}$

The quantum random network model proposed by Perseguers et al.[1] can be thought of as a quantum version of the Erdős–Rényi model. Instead of the typical links used to represent other complex networks, in the quantum random network model each pair of nodes is connected through a pair of entangled qubits. In this case each node contains ${\displaystyle N-1}$ quibits, one for each other node. In a quantum random network, the degree of entanglement between a pair of nodes, represented by ${\displaystyle p}$, plays a similar role to the parameter ${\displaystyle p}$ in the Erdős–Rényi model. While in the Erdős–Rényi model two nodes form a connection with probability ${\displaystyle p}$, in the context of quantum random networks ${\displaystyle p}$ means the probability of an entangled pair of qubits being successful converted to a maximally entangled state using only local operations and classical communications, called LOCC operations.[14] We can think of maximally entangled qubits as the true links between nodes.

Using the notation introduced previously, we can represent a pair of entangled qubits connecting the nodes ${\displaystyle i}$ and ${\displaystyle j}$, as

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1-p/2}}|0\rangle _{i}\otimes |0\rangle _{j}+{\sqrt {p/2}}|1\rangle _{i}\otimes |1\rangle _{j},}$

For ${\displaystyle p=0}$ the two qubits are not entangled,

${\displaystyle |\psi _{ij}\rangle =|0\rangle _{i}\otimes |0\rangle _{j},}$

and for ${\displaystyle p=1}$ we obtain the maximally entangled state, given by

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1/2}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j})}$.

For intermediate values of ${\displaystyle p}$, ${\displaystyle 0<p<1}$, any entangled state can be, with probability ${\displaystyle p}$, successfully converted to the maximally entangled state using LOCC operations.[14]

One of the main features that distinguish this model from its classic version is the fact the in quantum random networks links are only truly established after measurements in the networks being made, and it is possible to take advantage of this fact to shape the final state of the network. Considering an initial quantum complex network with an infinite number of nodes, Perseguers et al.[1] showed that, by doing the right measurements and entanglement swapping, it is possible to collapse the initial network to a network containing any finite subgraph, provided that ${\displaystyle p}$ scales with ${\displaystyle N}$ as,

${\displaystyle p\sim N^{Z},}$

were ${\displaystyle Z\geq -2}$. This result is contrary to what we find in classic graph theory where the type of subgraphs contained in a network is bounded by the value of ${\displaystyle z}$.[15]

The goal of entanglement percolation models is to determine if a quantum network is capable of establishing a connection between two arbitrary nodes through entanglement, and to find best the strategies to create those same connections.[11][16] In a model proposed by Cirac et al.[16] and applied to complex networks by Cuquet et al.,[11] nodes are distributed in a lattice,[16] or in a complex network,[11] and each pair of neighbors share two pairs of entangled qubits that can be converted to a maximally entangle qubit pair with probability ${\displaystyle p}$. We can think of maximally entangled qubits as the true links between nodes. According to classic percolation theory, considering a probability ${\displaystyle p}$ of two neighbors being connected, there is a critical ${\displaystyle p}$ designed by ${\displaystyle p_{c}}$, so that if ${\displaystyle p>p_{c}}$ there is a finite probability of existing a path between two random selected node, and for ${\displaystyle p<p_{c}}$ the probability of existing a path between two random selected nodes goes to zero.[17] ${\displaystyle p_{c}}$ depends only on the topology of the network.[17] A similar phenomena was found in the model proposed by Cirac et al.,[16] where the probability of forming a maximally entangled state between two random selected nodes is zero if ${\displaystyle p<p_{c}}$ and finite if ${\displaystyle p>p_{c}}$. The main difference between classic and entangled percolation is that in quantum networks it is possible to change the links in the network, in a way changing the effective topology of the network, as a consequence ${\displaystyle p_{c}}$ will depend on the strategy used to convert partial entangle qubits to maximally connected qubits.[11][16] A naive approach yields that ${\displaystyle p_{c}}$ for a quantum network is equal to ${\displaystyle p_{c}}$ for a classic network with the same topology.[16] Nevertheless, it was shown that is possible to take advantage of quantum swapping to lower that value, both in regular lattices[16] and complex networks.[11]