DOCSLIB.ORG

Entanglement Transformation Between Two-Qubit Mixed States by LOCC

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Entanglement Transformation Between Two-Qubit Mixed States by LOCC Physics Letters A 373 (2009) 3610–3613 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Entanglement transformation between two-qubit mixed states by LOCC ∗ De-Chao Li a,b, a School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316000, China b College of Automation, Northwestern Polytechnical University, Xi’an 710072, China article info abstract Article history: Based on a new set of entanglement monotones of two-qubit pure states, we give sufficient and Received 6 September 2008 necessary conditions that one two-qubit mixed state is transformed into another one by local operations Received in revised form 30 July 2009 and classical communication (LOCC). This result can be viewed as a generalization of Nielsen’s Accepted 30 July 2009 theorem Nielsen (1999) [1]. However, we find that it is more difficult to manipulate the entanglement Availableonline6August2009 transformation between single copy of two-qubit mixed states than to do between single copy of two- Communicated by P.R. Holland qubit pure ones. PACS: © 2009 Elsevier B.V. All rights reserved. 03.67.-a 03.67.Hk 03.65.Ud Keywords: Transformation Mixed state Entanglement 1. Introduction ory [4]. Obviously, for a pure bipartite state the entanglement is completely determined by its Schmidt coefficients according to the Entanglement has important applications in quantum informa- Schmidt decomposition [5]. Therefore, the Von Neumann entropy tion theory [2]. In particular, shared bipartite entanglement is an S(ρ) =−tr(ρ log ρ) has been accepted as a canonical measure of essential resource for processing and transmitting quantum infor- pure bipartite states entanglement. mation. Since most applications of quantum information theory re- However, in practical application people would have to deal quire the maximally entangled state in order to faithfully transmit with mixed states rather than pure ones due to decoherence. quantum information, it is vital necessary to develop the technique Hence, it is extremely important to quantify mixed states entan- of entanglement manipulation by LOCC to produce states with the glement for quantum information processing. This begs the ques- amount of entanglement as great as possible from partially entan- tion: Is majorization a suitable tool for transformation from one gled states [3]. There has been much attention recently concerning mixed state into another one by LOCC yet? If it is not true, what LOCC entanglement manipulation of a single copy of pure states. Nielsen is the condition of ρ −→ σ ?Afunction f : Rd → R is said to provided the necessary and sufficient conditions for pure bipar- be Schur-convex if x ≺ y ⇒ f (x) f (y) [6]. According to this tite states entanglement transformation by LOCC [1]. According to definition the Von Neumann entropy is Schur-concave, that is, Nielsen’s theorem the transformation between pure bipartite states λρ ≺ λσ ⇒ S(ρ) S(σ ). However, the Schumucher noiseless cod- LOCC |φ −→ | ψ can be performed by LOCC if and only if λφ ≺ λψ , ing theorem for quantum information [7] implies that the maxi- where λφ denotes the vector of Schmidt coefficients of trA (|φφ|) mum rate of error-free quantum information transmission is S(ρ) arranged in decreasing order. This result has been extended to qubits per signal. If S(ρ) S(σ ) holds, then the coding theorem LOCC will be violated clearly [8].Sowehavetodevelopothermeasures the case where the probabilistic transformation |φ −→ { p , |ψ } i i to quantify the entanglement of bipartite mixed states. Gour con- can be accomplished iff λ ≺ p λ and therefore majoriza- φ i i ψi sidered an essentially different type of measure of entanglement tion has received renewed attention in quantum information the- for two-qubit√ quantum√ systems [9]. For any two-qubit pure state |ϕ= λ0|00+ λ1|11 the author defined the entanglement x μ for x μ * Correspondence address: School of Mathematics, Physics and Information Sci- as follows: Eμ(|ϕ) = fμ(x) = with ∀0 < μ 1, ence, Zhejiang Ocean University, Zhoushan 316000, China. Tel.: +86 05808180913. 1forx > μ E-mail address: [email protected]. where x = 2min{λ0,λ1}. Based on this definition, Gour gave nec- 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.07.089 D.-C. Li / Physics Letters A 373 (2009) 3610–3613 3611 essary and sufficient conditions for entanglement transformation Definition 2.2. The entanglement monotones of the mixed state ρ between two probability distribution of two-qubit pure states by is defined as the average entanglement of the pure states of the LOCC. However, this result cannot be viewed as the entanglement decomposition, minimized over all decompositions of ρ: transformation between two-qubit mixed states by LOCC. For ex- + + − − + + = ∀ ∈ ] ample, let = 1 |Φ Φ |+ 1 |Φ Φ | and =|Φ Φ | with Ek(ρ) min pi Ek(ψi), k (0, 1 . (3) ρ 2 2 σ {p ,|ψ } ± | ±| i i |Φ = 00√ 11 . It is well known that ρ cannot be transformed 2 to σ by LOCC, although they still meet the condition mentioned Definition 2.3. (See [11].) The concurrence of two-qubit mixed in Ref. [9]. In this Letter we first define a new set of entangle- state ρ is defined as ment monotones of two-qubit pure states. The necessary and suf- = ficient conditions for entanglement transformation between two- C(ρ) min pi C(ψi). (4) {pi ,|ψi } qubit mixed states by LOCC is then represented. Following the Wootters’ results [11], we can obtain that C(ρ) = 2. The condition for entanglement transformation between max{0,λ1 − λ2 − λ3 − λ4}, where the λi s are the square roots of ∗ two-qubit mixed states the eigenvalues of ρ(σ2 ⊗ σ2)ρ (σ2 ⊗ σ2) in decreasing order. Fur- thermore, for any mixed state ρ there always exists a pure-state We first briefly summarize some basic concepts and results ensemble with at most four states such that the following equa- that are needed for further treatment. An ensemble of pure states, tion holds: { | } 4 which is usually represented bypi, ψi , is characterized by a fi- = = nite set of positive numbers pi ( i pi 1) and a corresponding set C(ρ) pi C(ψi). (5) of normalized vectors |ψi of the Hilbert space H [10]. The density i=1 operator ρ (a trace one, semi-definite positive operator) associated By Eq. (5),wehave to {pi , |ψi} is defined as: Proposition 2.4. ρ = pi|ψiψi|, pi = 1, pi 0. (1) i=1 i=1 1 − 1 − C 2(ρ) Ek(ρ) = ∧ 1. (6) 2k Following the ideas presented in Ref. [9],wefirstdiscusstheen- { ;| } −→LOCC tanglement transformation between two ensembles pi ψi Proof. Note that there exists an ensemble {pi ;|ψi} such that √ {q ;|φ }. Suppose Alice and Bob share a pure state |ψ,andthen − − 4 2 j j 4 1− 1−C 2(ρ) 1 1 ( = pi C(ψi )) C = p C .So ∧ 1 = i 1 ∧ perform a quantum operation ε which outputs the pure states (ρ) i=1 √i (ψi ) 2k 2k − − 2 |ψi (i = 1, 2,...,n) with probability pi . We then consider that 4 1 1 C (ψi ) 4 1 = pi ∧ 1 = pi E(ψi ), where we use the Alice and Bob obtain an ensemble {p , |ψ }. Alice and Bob per- i 1 2k √ i i i − − 2 convex of function f x = 1 1 x ∧ 1 in the second inequality. form again quantum operations ε which output the pure states √( ) 2k i |φ with conditional probability p for all the possible out- 1− 1−C 2(ρ) j ji This means that ∧ 1 is a lower bound of p E(ψ ). | | = | | 2k √ i i i come states, that is, ε ( ψi ψi ) j p ji φ j φ j . Thus, the trans- i 1− 1−C 2(ρ) = Hence, we get E(ρ) ∧ 1. formation ε (ε1, ε2,...,εn), defined between the ensembles 2k √ { | } { | } | = − ∧ pi, ψi and q j, φ j , outputs the statesφ j with probability On the other hand, i pi C(ψ√i ) 2k i pi (1 E(ψi))E(ψi ) 1. = | | = | | = − ∧ q j i pi p ji, that is,ε ( i pi ψi ψi ) i, j pi p ji φ j φ j with Define a function g(x) 2k (1 x)x 1. It is easy to see = = that g(x) is concave and monotonically increasing on [0, 1 ].We q j i pi p ji, where j p ji 1. 2 | | = − ∧ √ We write√ an arbitrary two-qubit pure state ψ as ψ therefore have pi C(ψi ) 2k (1 i pi E(ψi ))( ipi E(ψi )) 1. x|00+ 1 − x|11 with x 1 according to Schmidt decompo- 2 Since g is monotonically increasing, min{pi ,|ψi } pi C(ψi ) sition. Firstly, we define a distinct measure of entanglement of { | } = f (min√ pi , ψi pi E(ψi )) f (E(ρ)) holds. Hence E(ρ) two-qubit pure state as follows: 1− 1−C 2(ρ) ∧ 2 2k 1. Definition 2.1. Let x be the minimal Schmidt coefficients of Remark 2. An ensemble that achieves the minimum in Eq. (4) is | | | trA ( ψ ψ ). We define entanglement of two-qubit pure state ψ referred to an optimal one in this Letter. as: Next we consider the entanglement transformation between = x ∧ ∀ ∈ ] Ek(ψ) 1, k (0, 1 , (2) two-qubit mixed states by LOCC. Let {p , |ψ } and {q ;|φ } be k i i j j two optimal ensembles of two-qubit mixed states ρ and σ ,re- where ∧ is the min operator. Especially, E0(ψ) = limk→0 Ek(ψ) = 1. spectively. We say −→LOCC { ;| } −→LOCC { ;| } Remark 1. It is easy to see that Ek(ψ) mentioned above is indeed Definition 2.5.
Load more

Recommended publications
  • Free-Electron Qubits
    Free-Electron Qubits Ori Reinhardt†, Chen Mechel†, Morgan Lynch, and Ido Kaminer Department of Electrical Engineering and Solid State Institute, Technion - Israel Institute of Technology, 32000 Haifa, Israel † equal contributors Free-electron interactions with laser-driven nanophotonic nearfields can quantize the electrons’ energy spectrum and provide control over this quantized degree of freedom. We propose to use such interactions to promote free electrons as carriers of quantum information and find how to create a qubit on a free electron. We find how to implement the qubit’s non-commutative spin-algebra, then control and measure the qubit state with a universal set of 1-qubit gates. These gates are within the current capabilities of femtosecond-pulsed laser-driven transmission electron microscopy. Pulsed laser driving promise configurability by the laser intensity, polarizability, pulse duration, and arrival times. Most platforms for quantum computation today rely on light-matter interactions of bound electrons such as in ion traps [1], superconducting circuits [2], and electron spin systems [3,4]. These form a natural choice for implementing 2-level systems with spin algebra. Instead of using bound electrons for quantum information processing, in this letter we propose using free electrons and manipulating them with femtosecond laser pulses in optical frequencies. Compared to bound electrons, free electron systems enable accessing high energy scales and short time scales. Moreover, they possess quantized degrees of freedom that can take unbounded values, such as orbital angular momentum (OAM), which has also been proposed for information encoding [5-10]. Analogously, photons also have the capability to encode information in their OAM [11,12].
    [Show full text]
  • Design of a Qubit and a Decoder in Quantum Computing Based on a Spin Field Effect
    Design of a Qubit and a Decoder in Quantum Computing Based on a Spin Field Effect A. A. Suratgar1, S. Rafiei*2, A. A. Taherpour3, A. Babaei4 1 Assistant Professor, Electrical Engineering Department, Faculty of Engineering, Arak University, Arak, Iran. 1 Assistant Professor, Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran. 2 Young Researchers Club, Aligudarz Branch, Islamic Azad University, Aligudarz, Iran. *[email protected] 3 Professor, Chemistry Department, Faculty of Science, Islamic Azad University, Arak Branch, Arak, Iran. 4 faculty member of Islamic Azad University, Khomein Branch, Khomein, ABSTRACT In this paper we present a new method for designing a qubit and decoder in quantum computing based on the field effect in nuclear spin. In this method, the position of hydrogen has been studied in different external fields. The more we have different external field effects and electromagnetic radiation, the more we have different distribution ratios. Consequently, the quality of different distribution ratios has been applied to the suggested qubit and decoder model. We use the nuclear property of hydrogen in order to find a logical truth value. Computational results demonstrate the accuracy and efficiency that can be obtained with the use of these models. Keywords: quantum computing, qubit, decoder, gyromagnetic ratio, spin. 1. Introduction different hydrogen atoms in compound applied for the qubit and decoder designing. Up to now many papers deal with the possibility to realize a reversible computer based on the laws of 2. An overview on quantum concepts quantum mechanics [1]. In this chapter a short introduction is presented Modern quantum chemical methods provide into the interesting field of quantum in physics, powerful tools for theoretical modeling and Moore´s law and a summary of the quantum analysis of molecular electronic structures.
    [Show full text]
  • A Theoretical Study of Quantum Memories in Ensemble-Based Media
    A theoretical study of quantum memories in ensemble-based media Karl Bruno Surmacz St. Hugh's College, Oxford A thesis submitted to the Mathematical and Physical Sciences Division for the degree of Doctor of Philosophy in the University of Oxford Michaelmas Term, 2007 Atomic and Laser Physics, University of Oxford i A theoretical study of quantum memories in ensemble-based media Karl Bruno Surmacz, St. Hugh's College, Oxford Michaelmas Term 2007 Abstract The transfer of information from flying qubits to stationary qubits is a fundamental component of many quantum information processing and quantum communication schemes. The use of photons, which provide a fast and robust platform for encoding qubits, in such schemes relies on a quantum memory in which to store the photons, and retrieve them on-demand. Such a memory can consist of either a single absorber, or an ensemble of absorbers, with a ¤-type level structure, as well as other control ¯elds that a®ect the transfer of the quantum signal ¯eld to a material storage state. Ensembles have the advantage that the coupling of the signal ¯eld to the medium scales with the square root of the number of absorbers. In this thesis we theoretically study the use of ensembles of absorbers for a quantum memory. We characterize a general quantum memory in terms of its interaction with the signal and control ¯elds, and propose a ¯gure of merit that measures how well such a memory preserves entanglement. We derive an analytical expression for the entanglement ¯delity in terms of fluctuations in the stochastic Hamiltonian parameters, and show how this ¯gure could be measured experimentally.
    [Show full text]
  • Physical Implementations of Quantum Computing
    Physical implementations of quantum computing Andrew Daley Department of Physics and Astronomy University of Pittsburgh Overview (Review) Introduction • DiVincenzo Criteria • Characterising coherence times Survey of possible qubits and implementations • Neutral atoms • Trapped ions • Colour centres (e.g., NV-centers in diamond) • Electron spins (e.g,. quantum dots) • Superconducting qubits (charge, phase, flux) • NMR • Optical qubits • Topological qubits Back to the DiVincenzo Criteria: Requirements for the implementation of quantum computation 1. A scalable physical system with well characterized qubits 1 | i 0 | i 2. The ability to initialize the state of the qubits to a simple fiducial state, such as |000...⟩ 1 | i 0 | i 3. Long relevant decoherence times, much longer than the gate operation time 4. A “universal” set of quantum gates control target (single qubit rotations + C-Not / C-Phase / .... ) U U 5. A qubit-specific measurement capability D. P. DiVincenzo “The Physical Implementation of Quantum Computation”, Fortschritte der Physik 48, p. 771 (2000) arXiv:quant-ph/0002077 Neutral atoms Advantages: • Production of large quantum registers • Massive parallelism in gate operations • Long coherence times (>20s) Difficulties: • Gates typically slower than other implementations (~ms for collisional gates) (Rydberg gates can be somewhat faster) • Individual addressing (but recently achieved) Quantum Register with neutral atoms in an optical lattice 0 1 | | Requirements: • Long lived storage of qubits • Addressing of individual qubits • Single and two-qubit gate operations • Array of singly occupied sites • Qubits encoded in long-lived internal states (alkali atoms - electronic states, e.g., hyperfine) • Single-qubit via laser/RF field coupling • Entanglement via Rydberg gates or via controlled collisions in a spin-dependent lattice Rb: Group II Atoms 87Sr (I=9/2): Extensively developed, 1 • P1 e.g., optical clocks 3 • Degenerate gases of Yb, Ca,..
    [Show full text]
  • Electron Spin Qubits in Quantum Dots
    Electron spin qubits in quantum dots R. Hanson, J.M. Elzerman, L.H. Willems van Beveren, L.M.K. Vandersypen, and L.P. Kouwenhoven' 'Kavli Institute of NanoScience Delft, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands (Dated: September 24, 2004) We review our experimental progress on the spintronics proposal for quantum computing where the quantum bits (quhits) are implemented with electron spins confined in semiconductor quantum dots. Out of the five criteria for a scalable quantum computer, three have already been satisfied. We have fabricated and characterized a double quantum dot circuit with an integrated electrometer. The dots can be tuned to contain a single electron each. We have resolved the two basis states of the qubit by electron transport measurements. Furthermore, initialization and singleshot read-out of the spin statc have been achieved. The single-spin relaxation time was found to be very long, but the decoherence time is still unknown. We present concrete ideas on how to proceed towards coherent spin operations and two-qubit operations. PACS numbers: INTRODUCTION depicted in Fig. la. In Fig. 1b we map out the charging diagram of the The interest in quantum computing [l]derives from double dot using the electrometer. Horizontal (vertical) the hope to outperform classical computers using new lines in this diagram correspond to changes in the number quantum algorithms. A natural candidate for the qubit of electrons in the left (right) dot. In the top right of the is the electron spin because the only two possible spin figure lines are absent, indicating that the double dot orientations 17) and 11) correspond to the basis states of is completely depleted of electrons.
    [Show full text]
  • Quantum Computing : a Gentle Introduction / Eleanor Rieffel and Wolfgang Polak
    QUANTUM COMPUTING A Gentle Introduction Eleanor Rieffel and Wolfgang Polak The MIT Press Cambridge, Massachusetts London, England ©2011 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email [email protected] This book was set in Syntax and Times Roman by Westchester Book Group. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Rieffel, Eleanor, 1965– Quantum computing : a gentle introduction / Eleanor Rieffel and Wolfgang Polak. p. cm.—(Scientific and engineering computation) Includes bibliographical references and index. ISBN 978-0-262-01506-6 (hardcover : alk. paper) 1. Quantum computers. 2. Quantum theory. I. Polak, Wolfgang, 1950– II. Title. QA76.889.R54 2011 004.1—dc22 2010022682 10987654321 Contents Preface xi 1 Introduction 1 I QUANTUM BUILDING BLOCKS 7 2 Single-Qubit Quantum Systems 9 2.1 The Quantum Mechanics of Photon Polarization 9 2.1.1 A Simple Experiment 10 2.1.2 A Quantum Explanation 11 2.2 Single Quantum Bits 13 2.3 Single-Qubit Measurement 16 2.4 A Quantum Key Distribution Protocol 18 2.5 The State Space of a Single-Qubit System 21 2.5.1 Relative Phases versus Global Phases 21 2.5.2 Geometric Views of the State Space of a Single Qubit 23 2.5.3 Comments on General Quantum State Spaces
    [Show full text]
  • Computational Difficulty of Computing the Density of States
    Computational Difficulty of Computing the Density of States Brielin Brown,1, 2 Steven T. Flammia,2 and Norbert Schuch3 1University of Virginia, Departments of Physics and Computer Science, Charlottesville, Virginia 22904, USA 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 3California Institute of Technology, Institute for Quantum Information, MC 305-16, Pasadena, California 91125, USA We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP, which is the counting version of the quantum complexity class QMA. We show that #BQP is not harder than its classical counting counterpart #P, which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians. Understanding the physical properties of correlated quantum computer. We show that both problems, com- quantum many-body systems is a problem of central im- puting the density of states and counting the ground state portance in condensed matter physics. The density of degeneracy, are complete problems for the class #BQP, states, defined as the number of energy eigenstates per i.e., they are among the hardest problems in this class. energy interval, plays a particularly crucial role in this Having quantified the difficulty of computing the den- endeavor. It is a key ingredient when deriving many sity of states and counting the number of ground states, thermodynamic properties from microscopic models, in- we proceed to relate #BQP to known classical counting cluding specific heat capacity, thermal conductivity, band complexity classes, and show that #BQP equals #P (un- structure, and (near the Fermi energy) most electronic der weakly parsimonious reductions).
    [Show full text]
  • Intercept-Resend Attack on SARG04 Protocol: an Extended Work
    Polytechnic Journal. 2020. 10(1): 88-92 ISSN: 2313-5727 http://journals.epu.edu.iq/index.php/polytechnic RESEARCH ARTICLE Intercept-Resend Attack on SARG04 Protocol: An Extended Work Ali H. Yousif*, Omar S. Mustafa, Dana F. Abdulqadir, Farah S. Khoshaba Department of Information System Engineering, Erbil Technical Engineering College, Erbil Polytechnic University, Erbil, Kurdistan Region, Iraq *Correspondence author: ABSTRACT Ali H. Yousif, Department of Information In this paper, intercept/resend eavesdropper attack over SARG04 quantum key distribution protocol is System Engineering, investigated by bounding the information of an eavesdropper; then, the attack has been analyzed. In Erbil Technical Engineering 2019, simulation and enhancement of the performance of SARG04 protocol have been done by the College, Erbil Polytechnic same research group in terms of error correction stage using multiparity rather than single parity (Omar, University, Erbil, Kurdistan 2019). The probability of detecting the case in the random secret key by eavesdropper is estimated. Region, Iraq, The results of intercept/resend eavesdropper attack proved that the attack has a significant impact E-mail: [email protected] on the operation of the SARG04 protocol in terms of the final key length. Received: 28 October 2019 Keywords: Eavesdropper; Intercept-resend; IR; Quantum; SARG04 Accepted: 05 February 2020 Published: 30 June 2020 DOI 10.25156/ptj.v10n1y2020.pp88-92 INTRODUCTION promises to revolutionize secure communication using the key distribution. Nowadays, the information security is the most important concern of people and entity. One of the main issues Quantum key distribution (QKD) enables two parties to of practical quantum communication is the information establish a secure random secret key depending on the security under the impact of disturbances or quantum principles of quantum mechanics.
    [Show full text]
  • Distinguishing Different Classes of Entanglement for Three Qubit Pure States
    Distinguishing different classes of entanglement for three qubit pure states Chandan Datta Institute of Physics, Bhubaneswar [email protected] YouQu-2017, HRI Chandan Datta (IOP) Tripartite Entanglement YouQu-2017, HRI 1 / 23 Overview 1 Introduction 2 Entanglement 3 LOCC 4 SLOCC 5 Classification of three qubit pure state 6 A Teleportation Scheme 7 Conclusion Chandan Datta (IOP) Tripartite Entanglement YouQu-2017, HRI 2 / 23 Introduction History In 1935, Einstein, Podolsky and Rosen (EPR)1 encountered a spooky feature of quantum mechanics. Apparently, this feature is the most nonclassical manifestation of quantum mechanics. Later, Schr¨odingercoined the term entanglement to describe the feature2. The outcome of the EPR paper is that quantum mechanical description of physical reality is not complete. Later, in 1964 Bell formalized the idea of EPR in terms of local hidden variable model3. He showed that any local realistic hidden variable theory is inconsistent with quantum mechanics. 1Phys. Rev. 47, 777 (1935). 2Naturwiss. 23, 807 (1935). 3Physics (Long Island City, N.Y.) 1, 195 (1964). Chandan Datta (IOP) Tripartite Entanglement YouQu-2017, HRI 3 / 23 Introduction Motivation It is an essential resource for many information processing tasks, such as quantum cryptography4, teleportation5, super-dense coding6 etc. With the recent advancement in this field, it is clear that entanglement can perform those task which is impossible using a classical resource. Also from the foundational perspective of quantum mechanics, entanglement is unparalleled for its supreme importance. Hence, its characterization and quantification is very important from both theoretical as well as experimental point of view. 4Rev. Mod. Phys. 74, 145 (2002).
    [Show full text]
  • Topological Quantum Computation Zhenghan Wang
    Topological Quantum Computation Zhenghan Wang Microsoft Research Station Q, CNSI Bldg Rm 2237, University of California, Santa Barbara, CA 93106-6105, U.S.A. E-mail address: [email protected] 2010 Mathematics Subject Classification. Primary 57-02, 18-02; Secondary 68-02, 81-02 Key words and phrases. Temperley-Lieb category, Jones polynomial, quantum circuit model, modular tensor category, topological quantum field theory, fractional quantum Hall effect, anyonic system, topological phases of matter To my parents, who gave me life. To my teachers, who changed my life. To my family and Station Q, where I belong. Contents Preface xi Acknowledgments xv Chapter 1. Temperley-Lieb-Jones Theories 1 1.1. Generic Temperley-Lieb-Jones algebroids 1 1.2. Jones algebroids 13 1.3. Yang-Lee theory 16 1.4. Unitarity 17 1.5. Ising and Fibonacci theory 19 1.6. Yamada and chromatic polynomials 22 1.7. Yang-Baxter equation 22 Chapter 2. Quantum Circuit Model 25 2.1. Quantum framework 26 2.2. Qubits 27 2.3. n-qubits and computing problems 29 2.4. Universal gate set 29 2.5. Quantum circuit model 31 2.6. Simulating quantum physics 32 Chapter 3. Approximation of the Jones Polynomial 35 3.1. Jones evaluation as a computing problem 35 3.2. FP#P-completeness of Jones evaluation 36 3.3. Quantum approximation 37 3.4. Distribution of Jones evaluations 39 Chapter 4. Ribbon Fusion Categories 41 4.1. Fusion rules and fusion categories 41 4.2. Graphical calculus of RFCs 44 4.3. Unitary fusion categories 48 4.4. Link and 3-manifold invariants 49 4.5.
    [Show full text]
  • Quantum Error Correction Codes-From Qubit to Qudit
    Quantum Error Correction Codes-From Qubit to Qudit Xiaoyi Tang, Paul McGuirk Outline • Introduction to quantum error correction codes (QECC) • Qudits and Qudit Gates • Generalizing QECC to Qudit computing Need for QEC in Quantum Computation • Sources of Error – Environment noise • Cannot have complete isolation from environment entanglement with environment random changes in environment cause undesirable changes in quantum system – Control Error • e.g. timing error for X gate in spin resonance • Cannot have reliable quantum computer without QEC Error Models • Bit flip |0> |1>, |1> |0> Pauli X • Phase flip |0> |0>, |1> -|1> Pauli Z • Bit and phase flip Y = iXZ • General unitary error operator I, X, Y, Z form a basis for single qubit unitary operator. Correctable if I, X, Y, Z are. QECC • Achieved by adding redundancy. – Transmit or store n qubits for every k qubits. • 3 qubit bit flip code Simple repetition code |0> |000>, |1> |111> that can correct up to 1 bit flip error. • Phase flip code – Phase flip in |0>, |1> basis is bit flip in |+>, |-> basis. a|0> + b|1> a|0>-b|1> (a+b)|+> + (a-b)|-> (a- b)|+> + (a+b) |-> – 3 qubit bit flip code can be used to correct 1 phase flip error after changing basis by H gate. QECC • Shor code: combine bit flip and phase flip codes to correct arbitrary error on a single qubit |0> (|000>+|111>) (|000>+|111>) (|000> +|111>)/2sqrt(2) |1> (|000>-|111>) (|000>-|111>) (|000>-| 111>)/2sqrt(2) Stabilizer Codes • Group theoretical framework for QEC analysis • Pauli Group – I, X, Y, Z form a basis for operator on single qubit – G1= {aE | a is 1, -1, i, -i and E is I, X, Y, Z} is a group – Gn is n-fold tensor of G1 • S: an Abelian (commutative) subgroup of Pauli Group Gn • Stabilized: g|φ> = |φ> (i.e.
    [Show full text]
  • A Quantum-Logic Gate Between Distant Quantum-Network Modules
    A Quantum-Logic Gate between Distant Quantum-Network Modules Severin Daiss,1, ∗, y Stefan Langenfeld,1, y Stephan Welte,1 Emanuele Distante,1, 2 Philip Thomas,1 Lukas Hartung,1 Olivier Morin,1 and Gerhard Rempe1 1Max-Planck-Institut fur¨ Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany 2ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain The big challenge in quantum computing is to realize scalable multi-qubit systems with cross-talk free addressability and efficient coupling of arbitrarily selected qubits. Quantum networks promise a solution by integrating smaller qubit modules to a larger computing cluster. Such a distributed architecture, however, requires the capability to execute quantum-logic gates between distant qubits. Here we experimentally realize such a gate over a distance of 60m. We employ an ancillary photon that we successively reflect from two remote qubit modules, followed by a heralding photon detection which triggers a final qubit rotation. We use the gate for remote entanglement creation of all four Bell states. Our non-local quantum-logic gate could be extended both to multiple qubits and many modules for a tailor-made multi-qubit computing register. Quantum computers promise the capability to solve problems classical communication [15]. The general working principle that cannot be dealt with by classical machines [1]. Despite re- has first been demonstrated with photonic systems based on cent progress with local qubit arrays [2–4], foreseeable imple- linear optical quantum computing [16, 17] and, more recently, mentations of quantum computers are limited in the number of in isolated setups with material qubits like superconductors in individually controllable and mutually coupleable qubits held a single cryostat [18] and ions in a single Paul trap [19].
    [Show full text]