Quantum Information and Convex Optimization
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A New Sletter from the Ins T Itute F OR Qu Antum C O Mput Ing , U N Ive R S
dition E pecial S | Bit Issue 20 Issue New A NEWSLETTER FROM THE INSTITUTE FOR QUANTUM COMPUTING, UNIVERSITY OF WATERLOO, WaTERLOO, ONTARIO, CANADA | C I S | NE CANADA NSTITUTE FOR QUANTUM QUANTUM FOR NSTITUTE PECIAL OMPUTING | NE NEWBIT ThisE is a state-of-the-art “ DITION | | research facility where I SSUE 20 SSUE scientistsW and students from many disciplinesBIT | will work Photos by Jonathan Bielaski W together toward the next BIT | I NSTITUTE FOR QUANTUM QUANTUM FOR NSTITUTE big breakthroughsI in science SSUE 20 | SSUE | and technology. SPECIAL EDITION I ” uantum Valley SSUE 20 | SSUE FERIDUN HAMDULLAHPUR, C OMPUTING, President, University of Waterloo Takes the Stage S S The science of the incredibly small has taken a giant leap at the Just asPECIAL the discoveries and PECIAL “ University of Waterloo. On Friday, Sept. 21 the MIKE & OPHELIA innovations at the Bell Labs LAZARIDIS QUANTUM-NANO CENTRE officially opened with a U led to the companies that ceremony attended by more than 1,200 guests and dignitaries, | NIVERSITY OF WATERLOO, ONTARIO, CANADA | NE CANADA ONTARIO, NIVERSITY OF WATERLOO, created SiliconI Valley, so will, including Prof. STEPHEN HAWKING. NSTITUTE F NSTITUTE E E I predict,DITION | the discoveries and DITION | innovationsC of the Quantum- OMPUTING, Nano Centre lead to the creation of companies that will lead to Distinguished I NSTITUTE FOR QUANTUM QUANTUM FOR NSTITUTE guests at the I Waterloo Region becoming NSTITUTE FOR QUANTUM QUANTUM FOR NSTITUTE O ribbon cutting of known as R QUANTUM the Quantum Valley. ” the Quantum-Nano Centre included Prof. U MIKE LAZARIDIS, STEPHEN HAWKING, NIVERSITY OF WATERLOO, ONTARIO, CANADA | NE CANADA ONTARIO, NIVERSITY OF WATERLOO, MPP JOHN MILLOY, Entrepreneur and philanthropist and MP PETER BRAID (behind). -
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Arxiv:1001.0543V2 [Quant-Ph] 10 Oct 2010
Optimal reconstruction of the states in qutrits system Fei Yan,1 Ming Yang∗,1 and Zhuo-Liang Cao2 1Key Laboratory of Opto-electronic Information Acquisition and Manipulation, Ministry of Education, School of Physics and Material Science, Anhui University, Hefei 230039, People's Republic of China 2Department of Physics & Electronic Engineering, Hefei Normal University, Hefei 230061, People's Republic of China Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is pre- sented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mutually unbiased basis with the standard computational basis. It is found that all those transformations can be decomposed into several basic implementable single- and two-qutrit unitary operations. For the three-qutrit system, there exist five different mutually unbiased-bases structures with different entanglement properties, so we introduce the concept of physical complexity to min- imize the number of nonlocal operations needed over the five different structures. This scheme is helpful for experimental scientists to realize the most economical reconstruction of quantum states in qutrit systems. PACS numbers: 03.65.Wj, 03.65.Ta, 03.65.Ud, 03.67.Mn I. INTRODUCTION is redundancy in these results in the previously used quantum tomography processes [12], which causes a resources waste. -
Mutually Unbiased Bases and Symmetric Informationally Complete Measurements in Bell Experiments
Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments Armin Tavakoli,1 Máté Farkas,2, 3 Denis Rosset,4 Jean-Daniel Bancal,1 and J˛edrzejKaniewski5 1Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland 2Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland 3International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdansk, Poland 4Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada 5Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland (Dated: October 21, 2020) Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are crucial to many conceptual and practical aspects of quantum theory. Here, we develop their role in quantum nonlocality by: i) introducing families of Bell inequalities that are maximally violated by d-dimensional MUBs and SICs respectively, ii) proving device-independent certification of natural operational notions of MUBs and SICs, and iii) using MUBs and SICs to develop optimal-rate and nearly optimal-rate protocols for device independent quantum key distribution and device-independent quantum random number generation respectively. Moreover, we also present the first example of an extremal point of the quantum set of correlations which admits physically inequivalent quantum realisations. Our results elaborately demonstrate the foundational and practical relevance of the two most important discrete Hilbert space structures to the field of quantum nonlocality. One-sentence summary — Quantum nonlocality is de- a single measurement. Since MUBs and SICs are both con- veloped based on the two most celebrated discrete struc- ceptually natural, elegant and (as it turns out) practically im- tures in quantum theory. -
Pilot Quantum Error Correction for Global
Pilot Quantum Error Correction for Global- Scale Quantum Communications Laszlo Gyongyosi*1,2, Member, IEEE, Sandor Imre1, Member, IEEE 1Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences Hungarian Academy of Sciences H-1518, Budapest, Hungary *[email protected] Real global-scale quantum communications and quantum key distribution systems cannot be implemented by the current fiber and free-space links. These links have high attenuation, low polarization-preserving capability or extreme sensitivity to the environment. A potential solution to the problem is the space-earth quantum channels. These channels have no absorption since the signal states are propagated in empty space, however a small fraction of these channels is in the atmosphere, which causes slight depolarizing effect. Furthermore, the relative motion of the ground station and the satellite causes a rotation in the polarization of the quantum states. In the current approaches to compensate for these types of polarization errors, high computational costs and extra physical apparatuses are required. Here we introduce a novel approach which breaks with the traditional views of currently developed quantum-error correction schemes. The proposed solution can be applied to fix the polarization errors which are critical in space-earth quantum communication systems. The channel coding scheme provides capacity-achieving communication over slightly depolarizing space-earth channels. I. Introduction Quantum error-correction schemes use different techniques to correct the various possible errors which occur in a quantum channel. In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered [12-16], [19-22] however the efficient error- correction in quantum systems is still a challenge. -
Annual Report to Industry Canada Covering The
Annual Report to Industry Canada Covering the Objectives, Activities and Finances for the period August 1, 2008 to July 31, 2009 and Statement of Objectives for Next Year and the Future Perimeter Institute for Theoretical Physics 31 Caroline Street North Waterloo, Ontario N2L 2Y5 Table of Contents Pages Period A. August 1, 2008 to July 31, 2009 Objectives, Activities and Finances 2-52 Statement of Objectives, Introduction Objectives 1-12 with Related Activities and Achievements Financial Statements, Expenditures, Criteria and Investment Strategy Period B. August 1, 2009 and Beyond Statement of Objectives for Next Year and Future 53-54 1 Statement of Objectives Introduction In 2008-9, the Institute achieved many important objectives of its mandate, which is to advance pure research in specific areas of theoretical physics, and to provide high quality outreach programs that educate and inspire the Canadian public, particularly young people, about the importance of basic research, discovery and innovation. Full details are provided in the body of the report below, but it is worth highlighting several major milestones. These include: In October 2008, Prof. Neil Turok officially became Director of Perimeter Institute. Dr. Turok brings outstanding credentials both as a scientist and as a visionary leader, with the ability and ambition to position PI among the best theoretical physics research institutes in the world. Throughout the last year, Perimeter Institute‘s growing reputation and targeted recruitment activities led to an increased number of scientific visitors, and rapid growth of its research community. Chart 1. Growth of PI scientific staff and associated researchers since inception, 2001-2009. -
Morphophoric Povms, Generalised Qplexes, Ewline and 2-Designs
Morphophoric POVMs, generalised qplexes, and 2-designs Wojciech S lomczynski´ and Anna Szymusiak Institute of Mathematics, Jagiellonian University,Lojasiewicza 6, 30-348 Krak´ow, Poland We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into prob- ability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are re- placed by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case. 1 Introduction and preliminaries Over the last ten years in a series of papers [2,3, 30{32, 49, 68] Fuchs, Schack, Appleby, Stacey, Zhu and others have introduced first the idea of QBism (formerly quantum Bayesianism) and then its probabilistic embodiment - the (Hilbert) qplex, both based on the notion of symmetric informationally complete positive operator-valued measure -
Five Isoentangled Mutually Unbiased Bases and Mixed States Designs Praca Licencjacka Na Kierunku fizyka Do´Swiadczalna I Teoretyczna
UNIWERSYTET JAGIELLONSKIW´ KRAKOWIE WYDZIAŁ FIZYKI,ASTRONOMII I INFORMATYKI STOSOWANEJ Jakub Czartowski Nr albumu: 1125510 Five isoentangled mutually unbiased bases and mixed states designs Praca licencjacka Na kierunku fizyka do´swiadczalna i teoretyczna Praca wykonana pod kierunkiem Prof. Karola Zyczkowskiego˙ Instytut Fizyki Kraków, 24 maja 2018 JAGIELLONIAN UNIVERSITYIN CRACOW FACULTY OF PHYSICS,ASTRONOMY AND APPLIED COMPUTER SCIENCE Jakub Czartowski Student number: 1125510 Five isoentangled mutually unbiased bases and mixed states designs Bachelor thesis on a programme Fizyka do´swiadczalnai teoretyczna Prepared under supervision of Prof. Karol Zyczkowski˙ Institute of Physics Kraków, 24 May 2018 Contents Abstract 5 Acknowledgements5 1 Introduction6 2 Space of pure quantum states7 3 Space of mixed quantum states7 4 Complex projective t-designs9 4.1 Symmetric Informationally Complete (SIC) POVM . 10 4.2 Mutually Unbiased Bases (MUB) . 10 5 Mixed states designs 10 5.1 Mean traces on the space of mixed states . 12 6 Notable examples 12 6.1 t = 1 - Positive Operator Valued Measure . 12 6.2 N=2, t=2, M=8 - Isoentangled SIC POVM for d 4................. 13 Æ 6.3 Short interlude - Numerical search for isoentangled MUB . 13 6.4 N=2, t=2, M=20 - Isoentangled MUB . 15 6.5 N=2, t=2, M=10 - The standard set of 5 MUB in H4 ................ 16 6.6 N=2, t=3, M=10 - Hoggar example number 24 . 17 6.7 N=2, t=3, M=20(40) - Witting polytope . 18 6.8 Interlude - Summary for N 2 ............................ 18 Æ 6.9 N=3, t=2, M=90 - MUB . 19 6.9.1 Standard representation . -
Some Open Problems in Quantum Information Theory
Some Open Problems in Quantum Information Theory Mary Beth Ruskai∗ Department of Mathematics, Tufts University, Medford, MA 02155 [email protected] June 14, 2007 Abstract Some open questions in quantum information theory (QIT) are described. Most of them were presented in Banff during the BIRS workshop on Operator Structures in QIT 11-16 February 2007. 1 Extreme points of CPT maps In QIT, a channel is represented by a completely-positive trace-preserving (CPT) map Φ: Md1 7→ Md2 , which is often written in the Choi-Kraus form X † X † Φ(ρ) = AkρAk with AkAk = Id1 . (1) k k The state representative or Choi matrix of Φ is 1 X Φ(|βihβ|) = d |ejihek|Φ(|ejihek|) (2) jk where |βi is a maximally entangled Bell state. Choi [3] showed that the Ak can be obtained from the eigenvectors of Φ(|βihβ|) with non-zero eigenvalues. The operators Ak in (1) are known to be defined only up to a partial isometry and are often called Kraus operators. When a minimal set is obtained from Choi’s prescription using eigenvectors of (2), they are defined up to mixing of those from degenerate eigenvalues ∗Partially supported by the National Science Foundation under Grant DMS-0604900 1 and we will refer to them as Choi-Kraus operators. Choi showed that Φ is an extreme † point of the set of CPT maps Φ : Md1 7→ Md2 if and only if the set {AjAk} is linearly independent in Md1 . This implies that the Choi matrix of an extreme CPT map has rank at most d1. -
Lecture 6: Quantum Error Correction and Quantum Capacity
Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde∗ The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally \quantum" theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum com- munication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem provides the intuition behind quantum error correction. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information|otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B) − H(E)|a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. -
Decoherence and the Transition from Quantum to Classical—Revisited
Decoherence and the Transition from Quantum to Classical—Revisited Wojciech H. Zurek This paper has a somewhat unusual origin and, as a consequence, an unusual structure. It is built on the principle embraced by families who outgrow their dwellings and decide to add a few rooms to their existing structures instead of start- ing from scratch. These additions usually “show,” but the whole can still be quite pleasing to the eye, combining the old and the new in a functional way. What follows is such a “remodeling” of the paper I wrote a dozen years ago for Physics Today (1991). The old text (with some modifications) is interwoven with the new text, but the additions are set off in boxes throughout this article and serve as a commentary on new developments as they relate to the original. The references appear together at the end. In 1991, the study of decoherence was still a rather new subject, but already at that time, I had developed a feeling that most implications about the system’s “immersion” in the environment had been discovered in the preceding 10 years, so a review was in order. While writing it, I had, however, come to suspect that the small gaps in the landscape of the border territory between the quantum and the classical were actually not that small after all and that they presented excellent opportunities for further advances. Indeed, I am surprised and gratified by how much the field has evolved over the last decade. The role of decoherence was recognized by a wide spectrum of practic- 86 Los Alamos Science Number 27 2002 ing physicists as well as, beyond physics proper, by material scientists and philosophers. -
Capacity Estimates Via Comparison with TRO Channels
Commun. Math. Phys. 364, 83–121 (2018) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-018-3249-y Mathematical Physics Capacity Estimates via Comparison with TRO Channels Li Gao1 , Marius Junge1, Nicholas LaRacuente2 1 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. E-mail: [email protected]; [email protected] 2 Department of Physics, University of Illinois, Urbana, IL 61801, USA. E-mail: [email protected] Received: 1 November 2017 / Accepted: 27 July 2018 Published online: 8 September 2018 – © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract: A ternary ring of operators (TRO) in finite dimensions is an operator space as an orthogonal sum of rectangular matrices. TROs correspond to quantum channels that are diagonal sums of partial traces, we call TRO channels. TRO channels have simple, single-letter entropy expressions for quantum, private, and classical capacity. Using operator space and interpolation techniques, we perturbatively estimate capacities, capacity regions, and strong converse rates for a wider class of quantum channels by comparison to TRO channels. 1. Introduction Channel capacity, introduced by Shannon in his foundational paper [45], is the ultimate rate at which information can be reliably transmitted over a communication channel. During the last decades, Shannon’s theory on noisy channels has been adapted to the framework of quantum physics. A quantum channel has various capacities depending on different communication tasks, such as quantum capacity for transmitting qubits, and private capacity for transmitting classical bits with physically ensured security. The coding theorems, which characterize these capacities by entropic expressions, were major successes in quantum information theory (see e.g. -
QIP 2010 Tutorial and Scientific Programmes
QIP 2010 15th – 22nd January, Zürich, Switzerland Tutorial and Scientific Programmes asymptotically large number of channel uses. Such “regularized” formulas tell us Friday, 15th January very little. The purpose of this talk is to give an overview of what we know about 10:00 – 17:10 Jiannis Pachos (Univ. Leeds) this need for regularization, when and why it happens, and what it means. I will Why should anyone care about computing with anyons? focus on the quantum capacity of a quantum channel, which is the case we understand best. This is a short course in topological quantum computation. The topics to be covered include: 1. Introduction to anyons and topological models. 15:00 – 16:55 Daniel Nagaj (Slovak Academy of Sciences) 2. Quantum Double Models. These are stabilizer codes, that can be described Local Hamiltonians in quantum computation very much like quantum error correcting codes. They include the toric code This talk is about two Hamiltonian Complexity questions. First, how hard is it to and various extensions. compute the ground state properties of quantum systems with local Hamiltonians? 3. The Jones polynomials, their relation to anyons and their approximation by Second, which spin systems with time-independent (and perhaps, translationally- quantum algorithms. invariant) local interactions could be used for universal computation? 4. Overview of current state of topological quantum computation and open Aiming at a participant without previous understanding of complexity theory, we will discuss two locally-constrained quantum problems: k-local Hamiltonian and questions. quantum k-SAT. Learning the techniques of Kitaev and others along the way, our first goal is the understanding of QMA-completeness of these problems.