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Quantum complex networks are complex networks whose nodes are quantum computing devices. [1] [2] Quantum mechanics has been used to create secure quantum communications channels that are protected from hacking. [3] [4] Quantum communications offer the potential for secure enterprise-scale solutions. [5] [2] [6]
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 [3] Quantum teleportation can transfer data at a higher rate than classical channels. [4] [ relevant? ]
Successful quantum teleportation experiments in 1998. [7] Prototypical quantum communication networks arrived in 2004. [8] Large scale communication networks tend to have non-trivial topologies and characteristics, such as small world effect, community structure, or scale-free. [6]
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. [3] Photon polarization or nuclear spin are examples of binary phenomena that can be used as qubits. [3]
Quantum entanglement is a physical phenomenon characterized by correlation between the quantum states of two or more physically separate qubits. [3] Maximally entangled states are those that maximize the entropy of entanglement. [9] [10] In the context of quantum communication, entangled qubits are used as a quantum channel. [3]
Bell measurement is a kind of joint quantum-mechanical measurement of two qubits such that, after the measurement, the two qubits are maximally entangled. [3] [10]
Entanglement swapping is a strategy used in the study of quantum networks that allows connections in the network to change. [1] [11] For example, given 4 qubits, A, B, C and D, such that qubits C and D belong to the same station[ clarification needed ], while A and C belong to two different stations[ clarification needed ], 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. [1] [11] [12]
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 and means that a qubit in node is entangled to a qubit in node , although those two qubits are in different places and so cannot physically interact. [1] [11] Quantum networks where the links are interaction terms[ clarification needed ] instead of entanglement are also of interest [13] [ which? ]
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 and . [1] [11] In this notation, two particles are entangled if the joint wave function, , cannot be decomposed as [3] [10]
where represents the quantum state of the qubit at node i and 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: [3] [10]
The quantum random network model proposed by Perseguers et al. (2009) [1] can be thought of as a quantum version of the Erdős–Rényi model. In this model, each node contains qubits, one for each other node. The degree of entanglement between a pair of nodes, represented by , plays a similar role to the parameter in the Erdős–Rényi model in which two nodes form a connection with probability , whereas in the context of quantum random networks, refers to the probability of converting an entangled pair of qubits to a maximally entangled state using only local operations and classical communication. [14]
Using Dirac notation, a pair of entangled qubits connecting the nodes and is represented as
For , the two qubits are not entangled:
and for , we obtain the maximally entangled state:
For intermediate values of , , any entangled state is, with probability , successfully converted to the maximally entangled state using LOCC operations. [14]
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.[ relevant? ] For an initial quantum complex network with an infinite number of nodes, Perseguers et al. [1] showed that, the right measurements and entanglement swapping, make it possible[ how? ] to collapse the initial network to a network containing any finite subgraph, provided that scales with as , where . This result is contrary to classical graph theory, where the type of subgraphs contained in a network is bounded by the value of . [15] [ why? ]
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. [11] [16]
Cirac et al. (2007) [16] applied a model to complex networks by Cuquet et al. (2009), [11] in which 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 entangled qubit pair with probability . We can think of maximally entangled qubits as the true links between nodes. In classical percolation theory, with a probability that two nodes are connected, has a critical value (denoted by ), so that if a path between two randomly selected nodes exists with a finite probability, and for the probability of such a path existing is asymptotically zero. [17] depends only on the network topology. [17]
A similar phenomenon was found in the model proposed by Cirac et al. (2007), [16] where the probability of forming a maximally entangled state between two randomly selected nodes is zero if and finite if . The main difference between classical 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 result, depends on the strategy used to convert partially entangled qubits to maximally connected[ clarification needed ] qubits. [11] [16] With a naïve approach, for a quantum network is equal to for a classic network with the same topology. [16] Nevertheless, it was shown that is possible to take advantage of quantum swapping to lower both in regular lattices [16] and complex networks. [11]
Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light.
In quantum computing, a qubit or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault-tolerant quantum computation that can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum preparation, and faulty measurements.
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1: . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will "collapse" it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will "collapse" the other qubit to a state whose measurement will yield one of two possible values, where the value depends on which Bell state the two qubits are in initially. Bell states can be generalized to certain quantum states of multi-qubit systems, such as the GHZ state for 3 or more subsystems.
The Peres–Horodecki criterion is a necessary condition, for the joint density matrix of two quantum mechanical systems and , to be separable. It is also called the PPT criterion, for positive partial transpose. In the 2×2 and 2×3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply. The theorem was discovered in 1996 by Asher Peres and the Horodecki family
In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determining if a state is separable is not straightforward and the problem is classed as NP-hard.
In quantum information theory, superdense coding is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assumption of sender and receiver pre-sharing an entangled resource. In its simplest form, the protocol involves two parties, often referred to as Alice and Bob in this context, which share a pair of maximally entangled qubits, and allows Alice to transmit two bits to Bob by sending only one qubit. This protocol was first proposed by Charles H. Bennett and Stephen Wiesner in 1970 and experimentally actualized in 1996 by Klaus Mattle, Harald Weinfurter, Paul G. Kwiat and Anton Zeilinger using entangled photon pairs. Superdense coding can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair.
LOCC, or local operations and classical communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed conditioned on the information received.
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state that involves at least three subsystems. It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989. Extremely non-classical properties of the state have been observed.
The W state is an entangled quantum state of three qubits which in the bra-ket notation has the following shape
In quantum computing, a graph state is a special type of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. In particular, they are a convenient way of representing certain types of entangled states.
The one-way or measurement-based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.
In quantum information and quantum computing, a cluster state is a type of highly entangled state of multiple qubits. Cluster states are generated in lattices of qubits with Ising type interactions. A cluster C is a connected subset of a d-dimensional lattice, and a cluster state is a pure state of the qubits located on C. They are different from other types of entangled states such as GHZ states or W states in that it is more difficult to eliminate quantum entanglement in the case of cluster states. Another way of thinking of cluster states is as a particular instance of graph states, where the underlying graph is a connected subset of a d-dimensional lattice. Cluster states are especially useful in the context of the one-way quantum computer. For a comprehensible introduction to the topic see.
Entanglement distillation is the transformation of N copies of an arbitrary entangled state into some number of approximately pure Bell pairs, using only local operations and classical communication.
Optical cluster states are a proposed tool to achieve quantum computational universality in linear optical quantum computing (LOQC). As direct entangling operations with photons often require nonlinear effects, probabilistic generation of entangled resource states has been proposed as an alternative path to the direct approach.
In quantum physics, the "monogamy" of quantum entanglement refers to the fundamental property that it cannot be freely shared between arbitrarily many parties.
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