Physics
Quantum Entanglement
Quantum entanglement is a phenomenon in quantum physics where two or more particles become interconnected in such a way that the state of one particle instantly influences the state of the other, regardless of the distance between them. This interconnectedness challenges classical notions of locality and suggests a fundamental interconnectedness at the quantum level.
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11 Key excerpts on "Quantum Entanglement"
eBook - PDFHandbook of Nanophysics
Nanoparticles and Quantum Dots
- Klaus D. Sattler(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
15–22 Generally speaking, if two particles are in an entangled state then, even if the particles are physically separated by a great distance, they behave in some respects as a single entity rather than as two separate entities. There is no doubt that entanglement has been lying in the heart of the foundation of quantum mechanics. 23 Besides quantum computations, entanglement has also been the core of many other active research such as quantum tele-portation, 6,24 dense coding, 25,26 quantum communication, 27 and quantum cryptography. 28 It is believed that the conceptual puz-zles posed by entanglement—and discussed more than 50 years ago—have now become a physical source to brew completely novel ideas that might result in useful applications. A big challenge faced by all the above-mentioned applications is to prepare the entangled states, which is much more subtle than classically correlated states. To prepare an entangled state of good quality is a preliminary condition for any successful experiment. In fact, this is not only a problem involved in experiments, but this also poses an obstacle to theories since the issue of how to quan-tify entanglement is still unsettled, which is now becoming one of the central topics in quantum information theory. Any func-tion that quantifies entanglement is called an entanglement mea-sure. It should tell us how much entanglement there is in a given multipartite state. Unfortunately there is currently no consen-sus as to the best method to define entanglement for all possible multipartite states. The theory of entanglement is only partially developed 23,29–32 and can only be applied in a limited number of scenarios, where there is unambiguous way to construct suitable measures. Two important scenarios are (1) the case of a pure state of a bipartite system, that is, a system consisting of only two com-ponents and (2) a mixed state of two spin- 1/2 particles.
eBook - PDFEntangled World
The Fascination of Quantum Information and Computation
- Jürgen Audretsch(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
The entanglement of states means, generally formulated, that none of the systems by itself has a value for the state-dependent attributes in question – neither has it a value with an uncertainty that depends on other attributes of the same system. Quantum theory, instead, gives us correlations between the relevant systems with respect to their state-dependent attributes. These correlations are independent of the spatial or spatial-temporal distance of the systems in question. These correlations are not a case of causality; because the condition for a causal connection between the changes of the states of physical systems is the existence of a state for each one of the systems in the sense of the principle of separability. The correlations of the entan-glement of states, on the other hand, break this principle. Moreover, the entanglement of states means a reversal of the principle of sepa-rability. Instead of each of the relevant systems having a state that is independent of all the other systems, only the systems taken together can have a well-defined state: only the entire system is in a pure state. From the state of the entire system – and only from this one – it is 4) Compare to Section 2.4. 276 Quantum theory: a challenge for philosophy! defined whatever is true about the state-dependent attributes of the subsystems, namely that only relations between the subsystems ex-ist in the sense of the Einstein–Podolsky–Rosen correlations we have mentioned. Starting from correlations between quantum systems and measur-ing instruments over Einstein–Podolsky–Rosen correlations in be-tween quantum systems, we can extend the discussion of the latter correlations so far that we finally get to a whole network of such cor-relations. Entanglements of states like the examples mentioned here are from the point of view of the formalism of quantum theory not exceptional cases.
eBook - PDF- Franck Laloë(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
6 Quantum Entanglement In this chapter, we study the properties of Quantum Entanglement, and more gener- ally the way correlations can appear in quantum mechanics. Quantum Entanglement is an important notion that we have already discussed, for instance in the context of the Von Neumann chain or of the Schrödinger cat, but here we give more details on its properties. In classical physics, the notion of correlation is well known. It hinges on the cal- culation of probabilities and on linear averages over a number of possibilities. A distribution gives the probability of having the first system in a some given state and the second in another state. If this distribution is not a product, the two systems are correlated. If it is a product, they are uncorrelated; measuring the properties of one system does not bring any information on the other. This is in particular the case if the state of each of the two systems is perfectly defined (which also defines the state of the whole system perfectly well). The notion of correlation between sub-systems therefore stems from the multiplicity of possible states of the whole system; fluctuations of this state are necessary to give its full meaning to the classical notion of correlation. In quantum mechanics, the situation is different: as we have seen (in particular in Chapter 4), even a physical system that is perfectly defined by a given state vector already contains fluctuations. This leads naturally to another notion of correlation, independent of any fluctuation of the state. For instance, the components of two 1/2 spins are strongly correlated in a singlet state (§4.1.1), which is a pure state. This is because the principle of linear superposition of quantum mechanics allows one to introduce superpositions directly inside the state vector; this is very different from superpositions of probabilities, which are quadratic functions of this state vector.
eBook - PDF- Gonzalo J. Olmo(Author)
- 2012(Publication Date)
- IntechOpen(Publisher)
3. Quantum Entanglement 3.1. Introduction Back to the early years of the quantum development, in 1935, Schrödinger [64, 65] coined the word ’entanglement’ to describe a puzzling feature of the quantum theory that was formerly Inter-Universal Entanglement http://dx.doi.org/10.5772/52012 197 posed by Einstein, Podolski and Rosen in a famous gedanken experiment [12]. Schrödinger also realized that entanglement is precisely the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought [64]. Let us briefly show it by following the example given in Ref. [66] (see also Ref. [82]). Let us consider the photo-disintegration of a Hg 2 molecule formed by two atoms of Hg with spin 1 2 . Before the disintegration, the molecule is taken to be in a state of zero angular momentum so that the composite state is given by | Hg 2 � = 1 √ 2 ( | ↑ 1 ↓ 2 � − | ↓ 1 ↑ 2 � ) , (42) where 1 and 2 refer to the atoms of Hg and | ↑ ( ↓ ) � refers to the value + 1 2 ( − 1 2 ) of the projection of their spin along the z -axis. After the photo-disintegration, performed with no disturbance of the angular momentum, the two atoms separate each other in opposite directions so we can make independent measurements on them. Before doing any measurement we do not know the particular value of the spin of each atom. However, we do anticipatedly know that if a measurement of the spin projection is performed on the atom 1 yielding a value + 1 2 ( − 1 2 ) , then, the spin projection of the atom 2 is to be − 1 2 (+ 1 2 ) . Furthermore, if it is performed a different measurement of the projection of the spin of the particle 1 along, say, the x -axis, we are determining the value of the spin projection of the particle 2 along the same axis, too. This non-local feature of the quantum theory is known as entanglement and the state (42) is called an entangled state .
- Louis Kauffman, Samuel J. Lomonaco(Authors)
- 2007(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 12 Quantum Entanglement: Concepts and Criteria Fu-li Li and M. Suhail Zubairy Abstract Entanglement is one of the key properties of the quantum me-chanical systems that is fundamentally different from a classical system. Quan-tum entanglement plays an important role in the debate concerning the founda-tions of physics but also is an important resource in various quantum informa-tion processes. In this article, we introduce basic concepts on EPR correlations and Quantum Entanglement, and review established entanglement criteria, and show how to use these criteria in the generation of entanglement by consid-ering coherence-induced entanglement and correlated spontaneous emission laser. 12.1 Introduction The concept of locality lies at the foundation of classical physics. The lo-cality implies that measurement performed on one of two systems which are spatially separated and without interaction cannot disturb one another. Based on the locality concept, Einstein, Podolsky and Rosen [1] (EPR) proposed a gedanken experiment that showed that both position and momentum variables could be simultaneously assigned to a single localized particle with certainty if two particles are initially prepared in the ideal position and momentum cor-related state. This obviously violates the principle of quantum mechanics that 349 350 12. Quantum Entanglement: CONCEPTS AND CRITERIA two physical quantities represented by noncommutable operators can not have simultaneous reality. Therefore, assuming that the locality concept is one of the universal physical principles, EPR argued that quantum mechanics is in-complete. In order to verify the EPR argument, Bell [2] derived inequalities of measurable physical quantities for all theories based on reality and locality. Standard quantum mechanics violates these inequalities.
- Jordi Boronat, Ferran Mazzanti, Gregory Astrakharchick(Authors)
- 2008(Publication Date)
- World Scientific(Publisher)
On one hand, entanglement is intimately tied to the inherent complexity of QIP, by constituting, in particular, a necessary resource for computa-tional speed-up in pure-state quantum algorithms. 5 On the other hand, critically re-assessing traditional many-body settings in the light of entanglement theory has al-ready resulted in a number of conceptual, computational, and information-theoretic developments. Notable advances include efficient representations of quantum states based on so-called projected entangled pair states , 6 improved renormalization-group methods for both static 2D and time-dependent 1D lattice systems, 7 as well as rig-orous results on the computational complexity of such methods and the solvability properties of a class of generalized mean-field Hamiltonians. 8 In this work, we focus on the problem of characterizing quantum critical models from a Generalized Entanglement (GE) perspective, 9,10 by continuing our earlier exploration with a twofold objective in mind: first, to further test the usefulness of GE-based criticality indicators in characterizing static quantum phase diagrams with a higher degree of complexity than considered so far (in particular, multi-ple competing phases); second, to start analyzing time-dependent, non-equilibrium QPTs, for which a number of outstanding physics questions remain. In this context, special emphasis will be devoted to establish the emergence and validity of universal scaling laws for non-equilibrium observables .
eBook - PDF- Ranee K. Brylinski, Goong Chen, Ranee K. Brylinski, Goong Chen(Authors)
- 2002(Publication Date)
- Chapman and Hall/CRC(Publisher)
[2], giving rise to the new field of quantum teleportation. There onwards followed a lot of interesting theoretical proposals [3] and exper-imental realizations [4] covering a wide range of quantum systems. Here we review several efforts made in the direction of employing quantum mechanical properties to achieve successful teleportation. Just to note, the quantum mechanical principles utilized have also given rise to very interesting fields like quantum computing [5], quantum cryptography [6] and communication [7]. Quantum teleportation primarily relies on Quantum Entanglement in a form considered by Einstein, Podolsky, and Rosen [8] and various new insights added by J. S. Bell [9, 10]. Quantum Entanglement essentially implies an intriguing property that two quantum correlated systems can-not be considered independent even if they are far apart in the sense that 13.1. INTRODUCTION 325 measuring the state of one of them allows one to predict something about the state of the other one. Nevertheless, the measurement statistics of each one is no different from the case in which there exists no correlation between them. Such an entangled pair of quantum systems constitutes an important element in any quantum teleportation scheme, and facili-tates the interesting feat of faithfully transferring an unknown quantum state to a distant location. The concept should come clear when we discuss the teleportation proposal for a two-level system in Section 13.2. The essentials of the quantum teleportation scheme can be summa-rized in three major steps: (1) The sender and receiver acquire one particle each off a pair of entangled particles through a quantum chan-nel. (2) The sender then performs a measurement of a special observable on the combined system of the shared entangled particle and the to-be-teleported particle and sends the classical result of the measurement to the receiver through a classical channel.
eBook - PDF- Franck Laloë(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
Nowadays, many experimental methods have been invented to obtain entanglement. Actually, Quantum Entanglement has now become an essential part of quantum information, quantum cryptography, teleportation, etc. (Chapter 8). All experiments already mentioned in §4.1.5 involve pairs of entangled particles, often photons with entangled polarization variables. 7.3.1 Entanglement created by local interactions As Schr¨ odinger had initially suggested (see quotation in §7.1), one way to obtain entanglement between physical subsystems is to make use of local interactions between particles. An atom emitting two photons in succession may provide such a scheme, which in fact has been used in many experiments. Initially, the atom is in an excited state, then emits a first photon, and reaches an intermediate state that depends on the polarization of the emitted photon; at this stage, the atom plus photon system is described by an entangled state, with coherent components on several polarization states associated with intermediate states of the atom. Each of these components then gives rise to the emission of a second photon with different polarizations, while the atom itself reaches a ground state that is independent of the polarizations of the emitted photons. This corresponds to the case |θ 1 〉 = |θ 2 〉 in (7.34), where the state of one of the three particles (here, the final state of the atom) factorizes; the atom leaves the Quantum Entanglement party, which allows the two photons to enter maximal entanglement (according to the monogamy rule). An often cited example is that of the atomic cascade J = 0 → 1 → 0 of the calcium atom. It provides, by successive spontaneous emissions, two photons in a state: | Ψ >= 1 √ 2 |1 : H; 2 : H〉 + |1 : V ; 2 : V 〉 (7.37) where | H〉 and |V 〉 are two states of polarization (horizontal and vertical) – for pho- tons 16 these states are analogous to states |+〉 and |−〉 for a spin 1/2.
eBook - PDFAn Introduction to Quantum Optics
Photon and Biphoton Physics
- Yanhua Shih, Yanhua Shih(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
10 Quantum Entanglement In quantum theory, a particle is allowed to be in a state of coherent super-position between a set of orthogonal states. A vivid picture of this concept is the Schrödinger cat: a cat is alive and dead simultaneously. This would be no surprise if one means a large number of an ensemble of cats with 50% alive and 50% dead. But, Schrödinger cat is a cat . We are talking about a cat being alive and dead simultaneously. In mathematics, the best word to characterize the states of “alive” and “dead” is perhaps “orthogo-nal.” In quantum mechanics, the superposition of orthogonal states is used to describe the state of a quantum object, or a particle. The superposi-tion principle is indeed a mystery compared to our everyday classical life experience. In this chapter, we turn to another surprising consequence of quantum mechanics, namely, the Quantum Entanglement. Quantum Entanglement involves a multiparticle system and coherent superposition of orthogonal multiparticle states. The Schrödinger cat is perhaps still the best example to picturize Quantum Entanglement. Now, we are talking about a pair of Schrödinger cats “propagating” to distant locations. The two cats are non-classical by means of the following two features: (1) each of the cats is in the state of alive and dead simultaneously; (2) the two have to be observed both alive or both dead whenever we look at them, despite the distance between the two. There would be probably no surprise if our observation is based on a large number of twin cats that are prepared to be half alive– alive and another half dead–dead. In this case, obviously, we have 50% chance to observe an alive–alive pair and 50% chance to observe a dead– dead pair for each joint-observation. However, we are talking about a pair of cats, i.e., each pair of cats, to be in the state of alive–alive and dead–dead simultaneously, and, in addition, each of the cats in the pair must be alive and dead simultaneously.
eBook - PDF- Eduardo Fradkin(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
A problem in which the ideas of Quantum Entanglement will very likely play an important role is the behavior of disordered quantum systems. By disorder here we do not mean uniform systems without long-range order but rather systems that are physically disordered and are best regarded as random systems. This is a notori- ously difficult problem, which we have not discussed in this book. It has remained 796 Quantum Entanglement an open problem even in classical statistical mechanics, which in spite of decades of effort still has a host of so far poorly understood problems such as spin glasses and random field systems. The quantum version of these problems is certainly no less difficult and the study of the behavior (and role) of Quantum Entanglement in these systems is in its begin- nings. Chakravarty has shown that the scaling of the entanglement entropy can be used to study the Anderson localization–delocalization transition in disordered sys- tems (Jia et al., 2008; Chakravarty, 2010). In a pioneering series of papers Refael and Moore showed that, in the case of random spin chains at their infinite-disorder fixed point (Fisher, 1994, 1995), the (ensemble-average) entanglement entropy of 1D random critical spin chains has a logarithmic dependence on the linear size of the entangling region, S A ∼ A ln (Refael and Moore, 2004, 2009). The con- stant A turns out to be universal and to be different only for different universality classes of random fixed points. Although this result has the same form as that in the case of 1D CFT, the random fixed points are not conformal and there is no natural definition of a central charge. So the connection between the universal prefactor A and the critical behavior of local operators at random fixed points is not so far understood. One question that we have not discussed is that of the so-called entanglement spectrum. The entanglement spectrum is the spectrum of the reduced density matrix of the entangling region.
eBook - PDF- Arjun Berera, Luigi Del Debbio(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
15.4.2 Quantum Teleportation of a Qubit Suppose the observer in location A, who for this problem is usually called Alice, has a quantum system, such as a spin- 1 2 particle, photon or some such identical particle, in a quantum state | χ. Although she is in possession of this system, the precise state of this system need not be known to her. Suppose at location B another observer, usually called Bob, is in possession of one of these identical particles. Alice wants to send the complete information about the quantum state | χ over to Bob and input it into the identical particle he has. The end result being that Bob’s particle is now in exactly the same state as 238 15 Quantum Entanglement EPR shared entanglement Classical channel B |Â〉 |Â〉 A t Fig. 15.9 Teleportation of state | χ from A to B requiring both a classical and a quantum channel. the particle Alice was in possession of. Thus even though no matter is transferred over, only information, the end result is as if Bob has received the exact system Alice had in her possession. This process is called quantum teleportation 6 (secretly we know the participants are Kirk and Scotty of course). Thus, in the quantum teleportation process only information is transferred from one location to another, but no matter is being transferred. To explain the quantum teleportation process, we will focus on the simplest case where the state | χ is from a two-state system, such as a spin- 1 2 identical particle (or equally the example could apply to the polarisation of a photon). The way quantum teleportation works is first two spin- 1 2 particles are prepared in the Bell state: | Ψ − AB = 1 √ 2 [ | ↑ A ↓ B − | ↓ A ↑ B ] . (15.65) The subscripts A and B correspond to who received the respective spin- 1 2 particle, Alice or Bob. This one ebit of entanglement is thus shared between Alice and Bob. Both Alice and Bob had agreed beforehand on the above entangled state.
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