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InTech eBooks

Entanglement in Two and Three Quantum Mechanical Systems

L. M. Arvalo Aguilar, M. M. Mndez Otero, P. Mastranzo

Published Feb 22, 2012• 0 citations
Publisher

Abstract

Entanglement is one of most peculiar characteristic of Quantum Mechanics (probably the other most mentioned is the particle tunneling through a barrier) because it entails correlations that supersedes classical correlations. It is conceived as the genuine quantum characteristic by many researchers and quite recently it has emerged as a physical resource to produce non-classical task, like quantum teleportation, quantum cryptography and quantum information. Entanglement could be distributed (efficient quantum communication is equivalent to efficient entanglement distribution (Plenio & Virmani, 2007)), concentrated (given an amount of entanglement, equal to the full content of entanglement in n pairs of identical entangled pure state systems, it is possible, using local operations on each system, to concentrate the total amount of entanglement into a smaller number m, m < n, of maximally pure entangled state system (Bennett, et al.; 1996)) and used to perform many quantum information tasks useful to overcome technical restrictions present on classical communication. In this chapter, we review the entanglement of quantum mechanical systems. First, in Section 2, we give a characterization of entanglement in terms of its special features as resource and its mathematical structure. Then, in section 3, we review some of the Bell inequalities. In Sections 4 and 5, we review some of the most important entanglement measures published for two and three entangled systems. After that, in Section 6, we give a characterization of quantum gates and operators by its entanglement power, i. e. the amount of entanglement that they can produce when acting on an arbitrary state. Finally, in Section 7, we briefly review the experimental detection of entanglement.

DOI
10.5772/34900

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Entanglement is one of most peculiar characteristic of Quantum Mechanics (probably the other most mentioned is the particle tunneling through a barrier) because it entails correlations that supersedes classical correlations. It is conceived as the genuine quantum characteristic by many researchers and quite recently it has emerged as a physical resource to produce non-classical task, like quantum teleportation, quantum cryptography and quantum information. Entanglement could be distributed (efficient quantum communication is equivalent to efficient entanglement distribution (Plenio & Virmani, 2007)), concentrated (given an amount of entanglement, equal to the full content of entanglement in n pairs of identical entangled pure state systems, it is possible, using local operations on each system, to concentrate the total amount of entanglement into a smaller number m, m < n, of maximally pure entangled state system (Bennett, et al.; 1996)) and used to perform many quantum information tasks useful to overcome technical restrictions present on classical communication. In this chapter, we review the entanglement of quantum mechanical systems. First, in Section 2, we give a characterization of entanglement in terms of its special features as resource and its mathematical structure. Then, in section 3, we review some of the Bell inequalities. In Sections 4 and 5, we review some of the most important entanglement measures published for two and three entangled systems. After that, in Section 6, we give a characterization of quantum gates and operators by its entanglement power, i. e. the amount of entanglement that they can produce when acting on an arbitrary state. Finally, in Section 7, we briefly review the experimental detection of entanglement.
Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation. The tensor product of two vector spaces Definition and properties of the tensor product We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌ N 1 and ℌ M 2 be the spaces of states of the two systems, of dimension N and M , respectively.
All our former experience with application of quantum theory seems to say that what is predicted by quantum formalism must occur in the laboratory. But the essence of quantum formalism---entanglement, recognized by Einstein, Podolsky, Rosen, and Schr\"odinger---waited over 70years70\phantom{\rule{0.3em}{0ex}}\text{years}70years to enter laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, has potential for many quantum processes, including canonical ones: quantum cryptography, quantum teleportation, and dense coding. However, it appears that this new resource is complex and difficult to detect. Although it is usually fragile to the environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure. This article reviews basic aspects of entanglement including its characterization, detection, distillation, and quantification. In particular, various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, and quantum cryptography are discussed, and some interrelations are pointed out. The basic role of entanglement in quantum communication within a distant laboratory paradigm is stressed, and some peculiarities such as the irreversibility of entanglement manipulations are also discussed including its extremal form---the bound entanglement phenomenon. The basic role of entanglement witnesses in detection of entanglement is emphasized.
Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information. In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence. Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps. We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.
Nonlocality and quantum entanglement constitute two special aspects of the quantum correlations existing in quantum systems, which are of paramount importance in quantum-information theory. Traditionally, they have been regarded as identical (equivalent, in fact, for pure two qubit states, that is, Gisin's Theorem ), yet they constitute different resources. Describing nonlocality by means of the violation of several Bell inequalities, we obtain by direct optimization those states of two qubits that maximally violate a Bell inequality, in terms of their degree of mixture as measured by either their participation ratio R = 1/Tr(ρ 2 ) or their maximum eigenvalue λ max . This optimum value is obtained as well, which coincides with previous results. Comparison with entanglement is performed too. An example of an application is given in the XY model. In this novel approximation, we also concentrate on the nonlocality for linear combinations of pure states of two qubits, providing a closed form for their maximal nonlocality measure. The case of Bell diagonal mixed states of two qubits is also extensively studied. Special attention concerning the connection between nonlocality and entanglement for mixed states of two qubits is paid to the so-called maximally entangled mixed states. Additional aspects for the case of two qubits are also described in detail. Since we deal with qubit systems, we will perform an analogous study for three qubits, employing similar tools. Relation between distillability and nonlocality is explored quantitatively for the whole space of states of three qubits. We finally extend our analysis to four-qubit systems, where nonlocality for generalized Greenberger–Horne–Zeilinger states of arbitrary number of parties is computed.
In the case of mixed states ρAB of the composite system A ⊗ B, the previous relation generalizes to ρAB = ∑ k pkρ A k ⊗ρk , where pk are probabilities and ρk ’s (ρk ’s) are generic density matrix of the system A (B). The entanglement derives from “subtle” nonlocal correlations between the parts of a quantum system and combines three basic structural elements of quantum theory: the superposition principle, the quantum non-separability property and the exponential scaling of the state space with the number of partitions. This unique resource can be used to perform computational and cryptographic tasks otherwise impossible with classical systems. An entangled state shared by two or more separated parties is a valuable resource for fundamental quantum protocols, such as quantum teleportation [2], quantum computing [3] and quantum cryptography. By using entangled states we can deeply investigate the nonlocal properties of the quantum world. In fact, an entangled system exhibits correlations between two (or more) parties, which cannot be reproduced by a classical theory or, in general, by a Local Hidden Variables Theory (LHVT). Quantum optics represents an excellent experimental test bench for various novel concepts introduced within the framework of quantum information theory. Quantum states of photons, generally produced by a probabilistic process, namely spontaneous parametric down conversion (SPDC), may be easily and accurately manipulated using linear and nonlinear optical devices and measured by efficient single-photon detectors. Photonic qubits represent the typical carrier of quantum information and can be distributed over long distances, either in free-space or in low-loss optical fibers. Two-particle experiments correspond to operate with qubits belonging to a (2 × 2) Hilbert space. Many quantum information tasks and fundamental tests of quantum mechanics, such as the simulation of properties of quantum systems, the realization of quantum algorithms of increasing complexity and the investigation of the quantum world at a mesoscopic level, deal with a large number of qubits. In order to take full advantage of the possibilities offered by quantum mechanics, more qubits must be added to a quantum state. Two approaches may be followed to increase the number of qubits. The first one consists of increasing the number of entangled particles. In this way, multi-qubit entangled states are created by distributing the qubits between the particles so that each particle carries one qubit. A second strategy consists of encoding more than one qubit in each particle, by exploiting different degrees of freedom (DOFs) of the photon such as the polarization, the path, the emission time, the orbital angular momentum. This has been used to create two-photon fourand six-qubit states and up to five-photon ten-qubit graph states. The entanglement of two particles in different DOFs corresponds to socalled hyperentanglement. It is worth noting that multi-qubit hyperentanglement allows to achieve higher
Quantum entanglement is an indispensable resource for many significant quantum information processing tasks. However, in practice, it is difficult to distribute quantum entanglement over a long distance, due to the absorption and noise in quantum channels. A solution to this challenge is a quantum repeater, which can extend the distance of entanglement distribution. In this scheme, the time consumption of classical communication and local operations takes an important place with respect to time efficiency. Motivated by this observation, we consider a basic quantum repeater scheme that focuses on not only the optimal rate of entanglement concentration but also the complexity of local operations and classical communication. First, we consider the case where two different two-qubit pure states are initially distributed in the scenario. We construct a protocol with the optimal entanglement-concentration rate and less consumption of local operations and classical communication. We also find a criterion for the projective measurements to achieve the optimal probability of creating a maximally entangled state between the two ends. Second, we consider the case in which two general pure states are prepared and general measurements are allowed. We get an upper bound on the probability for a successful measurement operation to produce a maximally entangled state without any further local operations.