DOCSLIB.ORG

Quantum Key Distribution Without the Wavefunction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Key Distribution Without the Wavefunction Quantum key distribution without the wavefunction Gerd Niestegge Zillertalstrasse 39, 81373 M¨unchen, Germany [email protected] Abstract A well-known feature of quantum mechanics is the secure exchange of secret bit strings which can then be used as keys to encrypt messages transmitted over any classical communication channel. It is demonstrated that this quantum key distribution allows a much more general and ab- stract access than commonly thought. The results include some gener- alizations for the Hilbert space version of quantum key distribution, but base upon a general non-classical extension of conditional probability. A special state-independent conditional probability is identified as origin of the superior security of quantum key distribution and may have more profound implications for the foundations and interpretation of quantum mechanics, quantum information theory, and the philosophical question what actually constitutes physical reality. Key Words: Quantum key distribution, No-cloning theorem, Founda- tions of quantum theory, Quantum logic PACS: 03.67.Dd, 03.65.Ta 1. Introduction In the past thirty years, quantum cryptography has been a large field of exten- sive theoretical and experimental research. Several quantum key distribution protocols have been elaborated, their technical feasibility has been discussed, experimental set-ups and some practical implementations have been realized. Quantum key distribution may be the first commercial application of quantum information theory. A vast amount of papers on these topics are available by now (Each of the two pioneering works [7] and [18] has more than ten thousand and almost eight thousand quotations, respectively; for a review, see [20] or [37]). The present paper, however, focuses on the theoretical foundations and concentrates on the two best-known quantum key distribution protocols. Quantum mechanics renders possible the secure exchange of secret bit strings which can then be used as keys to encrypt messages transmitted over a classical communication channel. The quantum no-cloning theorem [16, 43, 45] ensures that an eavesdropper cannot steal the key without being detected. These results follow from the usual quantum mechanical Hilbert space formalism. After it could recently be shown in Ref.[32] that the quantum no-cloning theorem can be extended to a much more general and abstract framework - the quantum 1 logics which possess a conditional probability calculus -, a similar extension of quantum key distribution shall be presented in this paper. A quantum logic E is a purely algebraic structure. It is quite common to use an orthomodular partially ordered set or lattice [5, 9, 25, 34]. Its elements can be interpreted in different ways: as events, propositions or properties of the quantum system under consideration. This is why they are called propositions by some authors and events, preferably in probabilistic approaches, by others. States are then defined in the same way as the classical probability measures, and conditional probabilities are postulated to behave like the classical ones on compatible subsets of E. Note that a subset is called compatible if it is contained in another subset of E forming a Boolean algebra (i.e., it is contained in a classical subsystem of E) [11]. Some quantum logics entail unique conditional probabilities, many others don't. The classical Boolean algebras and the Hilbert space quantum logic (consisting of the closed subspaces or, equivalently, the selfadjoint projection operators) do and, in the latter case, conditionalization becomes identical with the state transition of the L¨uders- von Neumann (i.e., projective) quantum measurement process [30]. Therefore, the quantum logics which possess unique conditional probabilities can be regarded as a generalized model of projective quantum measurement. Examples that are neither classical Boolean algebras nor Hilbert space quantum logics (nor sublogics of them) can be constructed using the exceptional real Jordan algebras [30, 31]. In this framework, a very special type of conditional probability emerges in the non-classical case [30, 31]. It describes the probability for the transition from a past event e to a future event f, independently of any underlying state, and results solely from the algebraic structure of the quantum logic E. This probability exists only for certain pairs e and f in E. It exists for all f 2 E, if e is minimal in E (i.e., e is an atom). The states resulting in this way are called atomic. They represent a generalization of the pure states in the usual quantum mechanical Hilbert space setting. After the early pioneering work by Birkhoff and von Neumann in 1936 [10], quantum logics have been studied extensively between 1960 and 1995 [5, 9, 25, 26, 33, 34, 39, 40, 41]. Various forms of conditional probability have also been considered [5, 13, 17, 19, 23, 24]. However, the quantum logics which possess unique conditional probabilities and particularly the special type of the state- independent conditional probability had not received any attention before the author's work [29, 30, 31]. This special type of conditional probability is used in the present paper to study two well-known quantum key distribution protocols (including a variant of one of them). The first one was invented by Charles H. Bennett and Gilles Brassard in 1984 [7], based on earlier work in the 1970ies by Stephen Wiesner which was not published before 1983 [42]. Its variant is due to the first co- inventor [6]. The second protocol was proposed by Artur K. Ekert in 1991 [18]. These quantum key distribution protocols are generalized in a two-fold way. First, they are transferred to the quantum logics possessing the conditional 2 probability calculus; this shows that quantum key distribution does not nec- essarily require the full-blown Hilbert space formalism of quantum mechanics and works as well in a much more general and abstract model of the projective quantum measurement process. Second, system properties are used instead of states. Pure states, wavefunctions and their collapse are not needed. Before turning to quantum key distribution, the no-cloning theorem of Ref. [32] is revisited and extended by treating the cloning of system properties, since only the cloning of atomic states was considered in Ref. [32]. Moreover, the special type of conditional probability mentioned above is identified as the quantum origin of the no-cloning theorem as well as of the superior security of the key distribution processes. Other generalizations of the no-cloning theorem [1, 2] and quantum key distribution [4], using the generalized probabilistic theories, do not cover the results presented here, since they do not capture this special type of conditional probability. The paper is organized as follows. The next two sections restate some ma- terial from Refs. [30, 31, 32] as far as needed in the present paper. In section 2, the algebraic structure of a quantum logic is outlined and, in section 3, the conditional probability calculus. In section 4, the extension of the no-cloning theorem is introduced. The generalized versions of the quantum key distribu- tion protocols are presented in section 5. 2. Compatibility in orthomodular partially ordered sets In quantum mechanics, the measurable quantities of a physical system are re- presented by observables. Most simple are those observables where only the two values `true' and `false' (or `1' and `0') are possible as measurement outcome. They are elements of a mathematical structure called quantum logic, are usually called propositions, and they are called events in probabilistic approaches. The elements of the quantum logic can also be understood as potential properties of the system under consideration. If, for instance, this system is an electron, its spin along each spatial axis ~x can take one of the two values +~=2 and −~=2 and, for each ~x, `spin along ~x = +~=2' and `spin along ~x = −~=2' represent two elements of the quantum logic. In this paper, a quantum logic shall be an orthomodular partially ordered set E with the partial ordering ≤, the orthocomplementation 0, the smallest element 0 and the largest element I [5, 9, 25, 34]. This means that the following conditions are satisfied by all e; f 2 E: (A) e ≤ f implies f 0 ≤ e0. (B) (e0)0 = e. (C) e ≤ f 0 implies e _ f, the supremum of e and f, exists. (D) e _ e0 = I. (E) f ≤ e implies e = f _ (e ^ f 0). (orthomodular law) 3 Here, e ^ f denotes the infimum of e and f, which exists if and only if e0 _ f 0 exists. Two elements e; f 2 E are called orthogonal if e ≤ f 0 or, equivalently, f ≤ e0. An element e 6= 0 in E is called an atom if there is no element f in E with f ≤ e and 0 6= f 6= e. The interpretation of this mathematical terminology is as follows: two or- thogonal elements represent mutually exclusive events, propositions or system properties; e0 is the negation of e, and e_f is the disjunction of the two exclusive elements e and f. It is not assumed that E is a lattice (in a lattice, there is a smallest upper bound e _ f and largest lower bound e ^ f for any two elements e and f). If E were a lattice and satisfied the distributivity law e ^ (f _ g) = (e ^ f) _ (e ^ g) for all e; f; g 2 E, E would become a Boolean lattice or Boolean algebra. The orthomodular law is a weakening of the distributivity law. Classical probability theory uses Boolean algebras as mathematical structure for the random events, and it can be expected that those subsets of E, which are Boolean algebras, behave classically. Therefore, a subset E0 of E is called compatible if there is a Boolean algebra B with E0 ⊆ B ⊆ E. Two elements e and f in E are called compatible, if fe; fg forms a compatible subset.
Load more

Recommended publications
  • Arxiv:1707.06910V2 [Physics.Hist-Ph] 22 Feb 2018 Xlntoscnenn H Eei Ftebte-Nw Pa Better-Known the of Genesis D the Park’S Concerning Analyze Explanations I States
    Twelve years before the quantum no-cloning theorem Juan Ortigoso∗ Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain (Dated: January 22, 2018) Abstract The celebrated quantum no-cloning theorem establishes the impossibility of making a perfect copy of an unknown quantum state. The discovery of this important theorem for the field of quantum information is currently dated 1982. I show here that an article published in 1970 [J. L. Park, Foundations of Physics, 1, 23-33 (1970)] contained an explicit mathematical proof of the impossibility of cloning quantum states. I analyze Park’s demonstration in the light of published explanations concerning the genesis of the better-known papers on no-cloning. arXiv:1707.06910v2 [physics.hist-ph] 22 Feb 2018 1 I. INTRODUCTION The no-cloning theorem of quantum mechanics establishes that an arbitrary unknown quantum state cannot be copied.1 A modern proof,2 based on the linearity of quantum mechanics, takes two lines. Suppose that a device can implement a transformation T for copying two orthogonal states ψ and φ of a qubit: T ψ 0 = ψ ψ and T φ 0 = φ φ , | i | i | i| i | i| i | i| i | i| i where 0 is the ready state of the target system. It follows, from linearity, that | i T (a ψ + b φ ) 0 = aT ψ 0 + bT φ 0 = a ψ ψ + b φ φ . (1) | i | i | i | i| i | i| i | i| i | i| i But if the transformation T can clone arbitrary states, it should give, for any a, b values T (a ψ +b φ ) 0 =(a ψ +b φ )(a ψ +b φ )= a2 ψ ψ +b2 φ φ +ab ψ φ +ab φ ψ , (2) | i | i | i | i | i | i | i | i| i | i| i | i| i | i| i which is different from Eq.
    [Show full text]
  • Photonic Quantum Simulator for Unbiased Phase Covariant Cloning
    Applied Physics B manuscript No. (will be inserted by the editor) Photonic quantum simulator for unbiased phase covariant cloning Laura T. Knoll12, Ignacio H. L´opez Grande12, Miguel A. Larotonda123 1 DEILAP, CITEDEF-CONICET, J.B. de La Salle 4397, 1603 Villa Martelli, Buenos Aires, Argentina 2 Departamento de F´ısica, FCEyN, UBA. Ciudad Universitaria, 1428 Buenos Aires, Argentina 3 CONICET, Argentina e-mail: [email protected] The date of receipt and acceptance will be inserted by the editor Abstract We present the results of a linear optics pho- applying a trace-preserving, completely-positive map to tonic implementation of a quantum circuit that simu- the composite system [6,7,8,9,10]. lates a phase covariant cloner, by using two different de- Among the family of Quantum Cloning Machines grees of freedom of a single photon. We experimentally (QCM), one can make a distinction between universal simulate the action of two mirrored 1 ! 2 cloners, each cloning machines (UQCM), which copy all the states of them biasing the cloned states into opposite regions with the same fidelity F , regardless of the state j i of the Bloch sphere. We show that by applying a ran- to be cloned, and state-dependent QCM. For qubits, dom sequence of these two cloners, an eavesdropper can the optimum UQCM can achieve a cloning fidelity F of mitigate the amount of noise added to the original in- 5=6 = 0:833 [6,7,8,11]. UQCMs have been experimen- put state and therefore prepare clones with no bias but tally realized in photon stimulated emission setups [12, with the same individual fidelity, masking its presence in 13,14], linear optics [15] and NMR systems [16].
    [Show full text]
  • Multi-Level Control in Large-Anharmonicity High-Coherence Capacitively Shunted Flux Quantum Circuits
    Multi-level Control in Large-anharmonicity High-coherence Capacitively Shunted Flux Quantum Circuits by Muhammet Ali Yurtalan A thesis presented to the University of Waterloo in fulfillment of the thesis requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering (Quantum Information) Waterloo, Ontario, Canada, 2019 c Muhammet Ali Yurtalan 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner Max Hofheinz Associate Professor Supervisor Adrian Lupascu Associate Professor Supervisor Zbigniew Wasilewski Professor Internal Member Bo Cui Associate Professor Internal Member Guoxing Miao Associate Professor Internal-External Member Jonathan Baugh Associate Professor ii This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions Most of the material in Chapter 5 consists of co-authored content. Muhammet Ali Yur- talan and Adrian Lupascu worked on the design and modeling of the device. Muhammet Ali Yurtalan conducted the experiments and performed device simulations. Muhammet Ali Yurtalan and Jiahao Shi fabricated the sample and performed data analysis. Muhammet Ali Yurtalan and Graydon Flatt performed numerical simulations of randomized bench- marking protocol. Adrian Lupascu supervised the experimental work. Most of the material in Chapter 6 consists of co-authored content. Muhammet Ali Yurtalan conducted the experiments and performed numerical simulations.
    [Show full text]
  • Quantum Cloning and Deletion in Quantum Information Theory
    Quantum Cloning and Deletion in Quantum Information Theory THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (SCIENCE) OF BENGAL ENGINEERING AND SCIENCE UNIVERSITY BY SATYABRATA ADHIKARI arXiv:0902.1622v1 [quant-ph] 10 Feb 2009 DEPARTMENT OF MATHEMATICS BENGAL ENGINEERING AND SCIENCE UNIVERSITY UNIVERSITY HOWRAH- 711103, West Bengal INDIA 2006 i DECLARATION I declare that the thesis entitled “Quantum Cloning and Deletion in Quantum Information Theory” is composed by me and that no part of this thesis has formed the basis for the award of any Degree, Diploma, Associateship, Fellowship or any other similar title to me. (Satyabrata Adhikari) Date: Department of Mathematics, Bengal Engineering and Science University, Shibpur Howrah- 711103, West Bengal India ii CERTIFICATE This is to certify that the thesis entitled “Quantum Cloning and Deletion in Quantum Information Theory” submitted by Satyabrata Adhikari who got his name registered on 21.12.2002 for the award of Ph.D.(Science) degree of Bengal Engi- neering and Science University, is absolutely based upon his own work under my super- vision in the Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, and that neither this thesis nor any part of it has been sub- mitted for any degree / diploma or any other academic award anywhere before. (Binayak Samaddar Choudhury) Date: Professor Department of Mathematics, Bengal Engineering and Science University, Shibpur Howrah- 711103, West Bengal India iii Acknowledgements A journey is easier when you travel together. Interdependence is certainly more valuable than independence. This thesis is the result of five years of work whereby I have been accompanied and supported by many people.
    [Show full text]
  • Experimental Investigation of High-Dimensional Quantum Key Distribution Protocols with Twisted Photons
    Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons Frédéric Bouchard1, Khabat Heshami2, Duncan England2, Robert Fickler1, Robert W. Boyd1 3 4, Berthold-Georg Englert5 6 7, Luis L. Sánchez-Soto3 8, and Ebrahim Karimi1 3 9 1Department of physics, University of Ottawa, Advanced Research Complex, 25 Templeton, Ottawa ON Canada, K1N 6N5 2National Research Council of Canada, 100 Sussex Drive, Ottawa ON Canada, K1A 0R6 3Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany 4Institute of Optics, University of Rochester, Rochester, NY 14627, USA 5Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore 6Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore. 7MajuLab, CNRS-UNS-NUS-NTU International Joint Unit, UMI 3654, Singapore. 8Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain 9Department of Physics, Institute for Advanced Studies in Basic Sciences, 45137-66731 Zanjan, Iran. November 30, 2018 Quantum key distribution is on the verge 1 Introduction of real world applications, where perfectly secure information can be distributed Quantum cryptography allows for the broadcast- among multiple parties. Several quantum ing of information between multiple parties in cryptographic protocols have been theo- a perfectly secure manner under the sole as- retically proposed and independently real- sumption that the laws of quantum physics are ized in different experimental conditions. valid [1]. Quantum Key Distribution (QKD) [2, Here, we develop an experimental plat- 3] is arguably the most well-known and studied form based on high-dimensional orbital an- quantum cryptographic protocol to date.
    [Show full text]
  • Quantum Communication Jubilee of Teleportation
    Quantum Communication Quantum • Rotem Liss and Tal Mor Liss • Rotem and Tal Quantum Communication Celebrating the Silver Jubilee of Teleportation Edited by Rotem Liss and Tal Mor Printed Edition of the Special Issue Published in Entropy www.mdpi.com/journal/entropy Quantum Communication—Celebrating the Silver Jubilee of Teleportation Quantum Communication—Celebrating the Silver Jubilee of Teleportation Editors Rotem Liss Tal Mor MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editors Rotem Liss Tal Mor Technion–Israel Institute of Technology Technion–Israel Institute of Technology Israel Israel Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Entropy (ISSN 1099-4300) (available at: https://www.mdpi.com/journal/entropy/special issues/Quantum Communication). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03943-026-0 (Hbk) ISBN 978-3-03943-027-7 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editors .............................................
    [Show full text]
  • Quantum Cryptography with Several Cloning Attacks
    Journal of Computer Science 6 (7): 682-686, 2010 ISSN 1549-3636 © 2010 Science Publications Quantum Cryptography with Several Cloning Attacks Mustapha Dehmani, Hamid Ez-Zahraouy and Abdelilah Benyoussef Laboratory of Magnetism and High Energy Physics, Faculty of Science, University Mohammed V-Agdal, Rabat, Morocco Abstract: Problem statement: In a previous research, we investigated the quantum key distribution of the well known BB84 protocol with several intercept and resend attacks. In the present research, we studied the effect of many eavesdroppers cloning attacks of the Bennett-Brassard cryptographic protocol on the quantum error and mutual information between honest parties and information with sender for each eavesdropper. Approach: The quantum error and the mutual information were calculated analytically and computed for arbitrary number of cloning attacks. Our objective in this study was to know if the number of the eavesdroppers and their angle of cloning act on the safety of information. Results: It was found that the quantum error and the secured/no secured transition depend strongly on the number of eavesdropper and their angle of attacks. The particular cases where all eavesdroppers collaborate were also investigated. Conclusion: Furthermore, the cloning attack’s quantum error is lower than the intercept and resends attacks one, which means that the cloning attacks is the optimal one for arbitrary number of eavesdropper. Key words: Quantum, cryptography, cloning, eavesdroppers INTRODUCTION bases of measurement. In impossibility of doing it, the eavesdropper can decide to copy the qubits in an The quantum mechanics is used by Wiesner (1983) optimal approximate way. It is thus essential for the to serve safety of information and he introduces the transmitter and the receiver to know what eavesdropper concept of quantum conjugate coding.
    [Show full text]
  • Quantum Teleportation Nicolas Gisin H
    Quantum teleportation Nicolas Gisin H. Deriedmatten, I. Marcikic, R. Thew, W. Tittel, H. Zbinden Groupe de Physique Appliquée-unité d’Optique Université de Genène • The science and the science-fiction of quantum teleportation • Intuitive and mathematical introduction • What is teleported ? • Q fax? The no-cloning theorem • The “teleportation channel”: Entanglement • The Geneva experiment Geneva University • Telecom wavelengths • Time-bin qubits • partial Bell measurement and tests of Bell inequality • Applications: Quantum Key Distribution GAP Optique • Simplifications, limitations • Q relays and Q repeaters 1 The Geneva Teleportation experiment over 3x2 km Photon = particle (atom) of light Polarized photon Unpolarized photon ≈ ( structured photon) (≈ unstructured ≈ dust) Geneva University GAP Optique 2 55 metres 2 km of 2 km of optical fibre optical fibre Geneva University Two entangled photons GAP Optique 3 55 metres 2 km of Geneva University optical fibre GAP Optique 4 55 metres Bell measurement (partial) the 2 photons Geneva University interact 4 possible results: 0, 90, 180, 270 degrees GAP Optique 5 55 metres Bell measurement (partial) the 2 photons ion Geneva University at interact rel Cor ect erf 4 possible results: P The correlation is independent of the quantum 0, 90, 180, 270 degrees GAP Optique state which may be unknown or even entangled with a fourth photon 6 Quantum teleportation ψ 2 bits U Bell ψ ⊗ EPR = + ⊗ + (c 0 0 c1 1 ) ( 0,0 1,1 ) / 2 11 = ((00,0,0 ++ 11,1,1 )) ⊗ ( c 0 + c 1 ) 2 0 1 Ψ 2 2 123 1 Geneva University + ((00,0,0 −− 11,1,1 )) ⊗ ( c 0 − c 1 ) 2 0 1 σ Ψ 2 2 123 z 1 + ((00,1,1 ++ 11,0,0 )) ⊗ ( c 0 + c 1 ) 2 22 1 0 σ Ψ x GAP Optique 123 1 + ((00,1,1 −− 11,0,0 )) ⊗ ( c 0 − c 1 ) 2 22 1 0 σ Ψ 123 y 7 What is teleported ? According to Aristotle, objects are constituted by matter and form, ie by elementary particles and quantum states.
    [Show full text]
  • Fundamentals of Quantum Information Theory Michael Keyl
    Physics Reports 369 (2002) 431–548 www.elsevier.com/locate/physrep Fundamentals of quantum information theory Michael Keyl TU-Braunschweig, Institute of Mathematical Physics, Mendelssohnstrae 3, D-38106 Braunschweig, Germany Received 3 June 2002 editor: J. Eichler Abstract In this paper we give a self-contained introduction to the conceptional and mathematical foundations of quantum information theory. In the ÿrst part we introduce the basic notions like entanglement, channels, teleportation, etc. and their mathematical description. The second part is focused on a presentation of the quantitative aspects of the theory. Topics discussed in this context include: entanglement measures, chan- nel capacities, relations between both, additivity and continuity properties and asymptotic rates of quantum operations. Finally, we give an overview on some recent developments and open questions. c 2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.−a; 03.65.−w Contents 1. Introduction ........................................................................................ 433 1.1. What is quantum information? ................................................................... 434 1.2. Tasks of quantum information .................................................................... 436 1.3. Experimental realizations ........................................................................ 438 2. Basic concepts ...................................................................................... 439 2.1. Systems, states and e=ects ......................................................................
    [Show full text]
  • Twenty Years of Quantum State Teleportation at the Sapienza University in Rome
    entropy Review Twenty Years of Quantum State Teleportation at the Sapienza University in Rome Francesco De Martini and Fabio Sciarrino * Dipartimento di Fisica—Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy * Correspondence: [email protected] Received: 31 July 2018; Accepted: 14 July 2019; Published: 6 August 2019 Abstract: Quantum teleportation is one of the most striking consequence of quantum mechanics and is defined as the transmission and reconstruction of an unknown quantum state over arbitrary distances. This concept was introduced for the first time in 1993 by Charles Bennett and coworkers, it has then been experimentally demonstrated by several groups under different conditions of distance, amount of particles and even with feed forward. After 20 years from its first realization, this contribution reviews the experimental implementations realized at the Quantum Optics Group of the University of Rome La Sapienza. Keywords: quantum teleportation; entanglement; photonics; quantum information 1. Introduction Entanglement is today the core of many key discoveries ranging from quantum teleportation [1], to quantum dense coding [2], quantum computation [3–5] and quantum cryptography [6,7]. Quantum communication protocols such as device-independent quantum key distribution [8] are heavily based on entanglement to reach nonlocality-based communication security [9]. The prototype for quantum information transfer using entanglement as a communication channel is the quantum state teleportation (QST) protocol, introduced for the first time in 1993 by Charles Bennett et al. [1], where a sender and a receiver share a maximally entangled state which they can use to perfectly transfer an unknown quantum state. Undoubtedly, quantum teleportation is one of the most counterintituitive consequences of quantum mechanics and it is defined as the transmission and reconstruction over arbitrary distances of an unknown quantum state.
    [Show full text]
  • Entanglement and Quantum Cryptography
    Entanglement and Quantum Cryptography Joonwoo Bae Barcelona, January, 2007 Universitat de Barcelona Departament d’Estructura i Constituents de la Mat`eria Tesis presentada para optar al grado de doctor, January 2007. Programa de doctorado: F´ısicaAvanzada. Bienio: 2003-2005. Thesis advisor: Dr. Antonio Ac´ınDal Maschio( Assistant Professor in ICFO and ICREA Junior) Dear friend, I pray that you may enjoy good health And that all may go well with you, Even as your soul is getting along well. Acknowledgement I am grateful to all the people who helped me throughout my education. My advisor, Antonio Ac´ın,has patiently guided me with his creative insights. In particular, many thanks for his kind care and ceaseless encouragement. He allowed me to have the journey in quantum information when I was just a beginner, and I am grateful for his clear, ideaful, and well-motivated expla- nations. I thank Ignacio Cirac who allowed me to join Barcelona quantum information. Also, thanks for the opportunity to visit MPQ. I thank Jos´e Ignacio Latorre for his kind support as my university tutor. I would also like to thank collaborators, Maciej Lewenstein, Anna San- pera, Emili Bagan, Ramon Mu˜noz-Tapia, Maria Baig, Lluis Masanes, Ujjwal Sen, Aditi Sen De, and Sofyan Iblisdir. I have enjoyed attending discussions; these helped me improve immensely. Many thanks go to Nicolas Gisin, So- fyan Iblisdir, Renato Renner, Andreas Buchleitner, Anna Sanpera, Maciej Lewenstein, and Ramon Mu˜noz-Tapia, for being on the thesis committee. I am very grateful for the people who assisted me while I was traveling as well.
    [Show full text]
  • Cloning a Qutrit
    Cloning a Qutrit ∗ Nicolas J. Cerf,1,2 Thomas Durt,3 and Nicolas Gisin4 1 Ecole Polytechnique, CP 165, Universit´eLibre de Bruxelles, 1050 Brussels, Belgium 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA 3 Toegepaste Natuurkunde en Fotonica, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium 4 Group of Applied Physics - Optique, Universit´ede Gen`eve, 20 rue de l’Ecole de M´edecine, Gen`eve 4, Switzerland August 2001 Abstract We investigate several classes of state-dependent quantum cloners for three-level systems. These cloners optimally duplicate some of the four maximally-conjugate bases with an equal fidelity, thereby extending the phase-covariant qubit cloner to qutrits. Three distinct classes of qutrit cloners can be distinguished, depending on two, three, or four maximally-conjugate bases are cloned as well (the latter case simply corresponds to the universal qutrit cloner). These results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can also be analyzed. 1 Introduction arXiv:quant-ph/0110092v1 15 Oct 2001 Since its inception, quantum information theory has traditionally been concerned with informational processes involving two-level quantum systems, known as qubits. For example, quantum teleportation, quantum cryp- tography, quantum computation, or quantum cloning were all developed using qubits as fundamental units of quantum information[1]. Over the last few years, however, there has been a growing interest in quantum informational processes based on multi-level or even continuous-spectrum systems. There are several reasons for this. Firstly, higher-dimensional quantum informational processes seem to be more efficient in certain sit- uations.
    [Show full text]