quantum complex network

In theory, it is possible to take advantage of quantum mechanics to create secure communications using features such as quantum key distribution is an application of quantum cryptography that enables secure communications Quantum teleportation can transfer data at a higher rate than classical channels.{{Relevance inline|date=December 2021}}

Successful quantum teleportation experiments in 1998.{{cite journal|last1=Boschi|first1=D.|last2=Branca|first2=S.|last3=De Martini|first3=F.|last4=Hardy|first4=L.|last5=Popescu|first5=S.|title=Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels|journal=Physical Review Letters|volume=80|issue=6|pages=1121–1125|doi=10.1103/physrevlett.80.1121|bibcode=1998PhRvL..80.1121B|arxiv=quant-ph/9710013|year=1998|s2cid=15020942}} Prototypical quantum communication networks arrived in 2004.{{cite conference

| last1 = Elliott | first1 = Chip

| last2 = Colvin | first2 = Alexander

| last3 = Pearson | first3 = David

| last4 = Pikalo | first4 = Oleksiy

| last5 = Schlafer | first5 = John

| last6 = Yeh | first6 = Henry

| editor1-last = Donkor | editor1-first = Eric J.

| editor2-last = Pirich | editor2-first = Andrew R.

| editor3-last = Brandt | editor3-first = Howard E.

| arxiv = quant-ph/0503058

| contribution = Current status of the DARPA quantum network (Invited Paper)

| date = May 2005

| doi = 10.1117/12.606489

| publisher = SPIE

| title = Quantum Information and Computation III}} Large scale communication networks tend to have non-trivial topologies and characteristics, such as small world effect, community structure, or scale-free.

In quantum information theory, qubits are analogous to bits in classical systems. A qubit is a quantum object that, when measured, can be found to be in one of only two states, and that is used to transmit information. Photon polarization or nuclear spin are examples of binary phenomena that can be used as qubits.

Quantum entanglement is a physical phenomenon characterized by correlation between the quantum states of two or more physically separate qubits. Maximally entangled states are those that maximize the entropy of entanglement.{{cite journal|last1=Eisert|first1=J.|last2=Cramer|first2=M.|last3=Plenio|first3=M. B.|title=Colloquium: Area laws for the entanglement entropy|journal=Reviews of Modern Physics|date=February 2010|volume=82|issue=1|pages=277–306|doi=10.1103/RevModPhys.82.277|bibcode=2010RvMP...82..277E|arxiv=0808.3773}}{{cite book | title = Quantum Entanglement in Electron Optics: Generation, Characterization, and Applications | first1 = Naresh | last1 = Chandra | first2 = Rama | last2 = Ghosh | publisher = Springer | date = 2013 | isbn = 978-3642240706 | page = 43 | url = {{google books|id=be6BCYFzqFQC|page=43|plainurl=yes}} | series = Springer Series on Atomic, Optical, and Plasma Physics | volume = 67 }} In the context of quantum communication, entangled qubits are used as a quantum channel.

Bell measurement is a kind of joint quantum-mechanical measurement of two qubits such that, after the measurement, the two qubits are maximally entangled.

Entanglement swapping is a strategy used in the study of quantum networks that allows connections in the network to change. For example, given 4 qubits, A, B, C and D, such that qubits C and D belong to the same station{{Clarify|date=December 2021}}, while A and C belong to two different stations{{Clarify|date=December 2021}}, and where qubit A is entangled with qubit C and qubit B is entangled with qubit D. Performing a Bell measurement for qubits A and B, entangles qubits A and B. It is also possible to entangle qubits C and D, despite the fact that these two qubits never interact directly with each other. Following this process, the entanglement between qubits A and C, and qubits B and D are lost. This strategy can be used to define network topology.{{cite report | first = Bob | last = Coecke | title = The logic of entanglement | publisher = Department of Computer Science, University of Oxford| date = 2003 | volume = RR-03-12 | arxiv = quant-ph/0402014 | url = http://www.cs.ox.ac.uk/publications/publication2580-abstract.html }}

While models for quantum complex networks are not of identical structure, usually a node represents a set of qubits in the same station (where operations like Bell measurements and entanglement swapping can be applied) and an edge between node $i$ and $j$ means that a qubit in node $i$ is entangled to a qubit in node $j$, although those two qubits are in different places and so cannot physically interact.{{cite journal | last1 = Cuquet | first1 = M. | last2 = Calsamiglia | first2 = J. | title = Entanglement percolation in quantum complex networks|journal=Physical Review Letters | date = 10 December 2009 | origyear = 6 June 2009 | volume = 103 | issue = 24 | pages = 240503 | doi = 10.1103/physrevlett.103.240503 | pmid = 20366190 | arxiv = 0906.2977 | bibcode = 2009PhRvL.103x0503C | s2cid = 19441960 }} Quantum networks where the links are interaction terms{{Clarify|date=December 2021}} instead of entanglement are also of interest.{{Cite thesis |last=Nokkala |first=Johannes |year=2018 |title=Quantum complex networks|type=Doctoral dissertation |url=https://www.utupub.fi/handle/10024/146194 |publisher=University of Turku|series=Turun Yliopiston Julkaisuja – Annales Universitatis Turkuensis}}{{Which|date=December 2021}}

Each node in the network contains a set of qubits in different states. To represent the quantum state of these qubits, it is convenient to use Dirac notation and represent the two possible states of each qubit as $|0\backslash rangle$ and $|1\backslash rangle$. In this notation, two particles are entangled if the joint wave function, $|\backslash psi\_\{ij\}\backslash rangle$, cannot be decomposed as

:$|\backslash psi\_\{ij\}\backslash rangle=|\backslash phi\backslash rangle\_i\backslash otimes\; |\backslash phi\backslash rangle\_j,$

where $|\backslash phi\backslash rangle\_i$ represents the quantum state of the qubit at node i and $|\backslash phi\backslash rangle\_j$ 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 between two qubits can be written as follows:

:$|\backslash Phi\_\{ij\}^+\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_i\; \backslash otimes\; |0\backslash rangle\_j\; +\; |1\backslash rangle\_i\; \backslash otimes\; |1\backslash rangle\_j),$

:$|\backslash Phi\_\{ij\}^-\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_i\; \backslash otimes\; |0\backslash rangle\_j\; -\; |1\backslash rangle\_i\; \backslash otimes\; |1\backslash rangle\_j),$

:$|\backslash Psi\_\{ij\}^+\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_i\; \backslash otimes\; |1\backslash rangle\_j\; +\; |1\backslash rangle\_i\; \backslash otimes\; |0\backslash rangle\_j),$

:$|\backslash Psi\_\{ij\}^-\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_i\; \backslash otimes\; |1\backslash rangle\_j\; -\; |1\backslash rangle\_i\; \backslash otimes\; |0\backslash rangle\_j).$

The quantum random network model proposed by Perseguers et al. (2009) can be thought of as a quantum version of the Erdős–Rényi model. In this model, each node contains $N-1$ qubits, one for each other node. The degree of entanglement between a pair of nodes, represented by $p$, plays a similar role to the parameter $p$ in the Erdős–Rényi model in which two nodes form a connection with probability $p$, whereas in the context of quantum random networks, $p$ refers to the probability of converting an entangled pair of qubits to a maximally entangled state using only local operations and classical communication.{{cite journal|last1=Werner|first1=Reinhard F.|title=Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model|journal=Physical Review A|date=15 Oct 1989|volume=40|issue=8|pages=4277–4281|doi=10.1103/physreva.40.4277|pmid=9902666|bibcode=1989PhRvA..40.4277W}}

Using Dirac notation, a pair of entangled qubits connecting the nodes $i$ and $j$ is represented as

:$|\backslash psi\_\{ij\}\backslash rangle=\backslash sqrt\{1-p/2\}|0\backslash rangle\_i\; \backslash otimes\; |0\backslash rangle\_j\; +\; \backslash sqrt\{p/2\}\; |1\backslash rangle\_i\backslash otimes|1\backslash rangle\_j,$

For $p=0$, the two qubits are not entangled:

:$|\backslash psi\_\{ij\}\backslash rangle=|0\backslash rangle\_i\; \backslash otimes\; |0\backslash rangle\_j,$

and for $p=1$, we obtain the maximally entangled state:

:$|\backslash psi\_\{ij\}\backslash rangle=\backslash sqrt\{1/2\}(|0\backslash rangle\_i\; \backslash otimes\; |0\backslash rangle\_j\; +\; |1\backslash rangle\_i\backslash otimes|1\backslash rangle\_j)$.

For intermediate values of $p$,

One feature that distinguishes this model from its classical analogue is the fact that, in quantum random networks, links are only truly established after they are measured, and it is possible to exploit this fact to shape the final state of the network.{{Relevance inline|date=December 2021|reason=This would fit better in the section "Important concepts".}} For an initial quantum complex network with an infinite number of nodes, Perseguers et al. showed that, the right measurements and entanglement swapping, make it possible{{How|date=December 2021}} to collapse the initial network to a network containing any finite subgraph, provided that $p$ scales with $N$ as $p\backslash sim\; N^Z$, where $Z\backslash geq-2$. This result is contrary to classical graph theory, where the type of subgraphs contained in a network is bounded by the value of $z$.{{cite journal|last1=Albert|first1=Réka|last2=Barabási|first2=Albert L.|title=Statistical mechanics of complex networks|journal=Reviews of Modern Physics|date=Jan 2002|volume=74|issue=1|pages=47–97|doi=10.1103/revmodphys.74.47|bibcode=2002RvMP...74...47A|arxiv=cond-mat/0106096|s2cid=60545}}{{Why|date=December 2021}}

Entanglement percolation models attempt to determine whether a quantum network is capable of establishing a connection between two arbitrary nodes through entanglement, and to find the best strategies to create such connections.{{cite journal|last1=Acin|first1=Antonio|last2=Cirac|first2=J. Ignacio|last3=Lewenstein|first3=Maciej|title=Entanglement percolation in quantum networks|journal=Nature Physics|date=25 February 2007|volume=3|issue=4|pages=256–259|doi=10.1038/nphys549|arxiv=quant-ph/0612167|bibcode=2007NatPh...3..256A|s2cid=118987352}}

Cirac et al. (2007) applied a model to complex networks by Cuquet et al. (2009), in which nodes are distributed in a lattice or in a complex network, and each pair of neighbors share two pairs of entangled qubits that can be converted to a maximally entangled qubit pair with probability $p$. We can think of maximally entangled qubits as the true links between nodes. In classical percolation theory, with a probability $p$ that two nodes are connected, $p$ has a critical value (denoted by $p\_c$), so that if $p>p\_c$ a path between two randomly selected nodes exists with a finite probability, and for

A similar phenomenon was found in the model proposed by Cirac et al. (2007), where the probability of forming a maximally entangled state between two randomly selected nodes is zero if

• Erdős–Rényi model
• Gradient network
• Network dynamics
• Network topology
• Quantum key distribution
• Quantum teleportation

{{Reflist}}

• [http://www.quantiki.org/wiki/LOCC_operations LOCC operations]

{{Quantum information}}

Category:Network theory

Category:Quantum information theory