DOCSLIB.ORG

Multipartite Quantum States and Their Marginals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multipartite Quantum States and Their Marginals Diss. ETH No. 22051 Multipartite Quantum States and their Marginals A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by Michael Walter Diplom-Mathematiker arXiv:1410.6820v1 [quant-ph] 24 Oct 2014 Georg-August-Universität Göttingen born May 3, 1985 citizen of Germany accepted on the recommendation of Prof. Dr. Matthias Christandl, examiner Prof. Dr. Gian Michele Graf, co-examiner Prof. Dr. Aram Harrow, co-examiner 2014 Abstract Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions – such as Pauli’s exclusion principle or the monogamy of quantum entanglement – which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part of this thesis (Chapters 2–6) we focus on the one-body marginals of multipartite quantum states. Starting from a novel ge- ometric solution of the compatibility problem, we then turn towards the phenomenon of quantum entanglement. We find that the one-body marginals through their local eigenvalues can characterize the entan- glement of multipartite quantum states, and we propose the notion of an entanglement polytope for its systematic study. Next, we consider random quantum states, where we describe a method for computing the joint probability distribution of the marginals. As an illustration of its versatility, we show that a discrete variant gives an efficient algorithm for the branching problem of compact connected Lie groups. A recurring theme throughout the first part is the reduction of quantum-physical problems to classical problems of largely combinatorial nature. In the second part of this thesis (Chapters 7–9), we study general quantum marginals from the perspective of the von Neumann entropy. We contribute two novel techniques for establishing entropy inequalities. The first technique is based on phase-space methods; it allows us to establish an infinite number of entropy inequalities for the class of stabilizer states. The second technique is based on a novel characterization of compatible marginals in terms of representation theory. We show how entropy inequalities can be understood in terms of symmetries, and illustrate the technique by giving a new, concise proof of the strong subadditivity of the von Neumann entropy. Zusammenfassung Teilsysteme zusammengesetzter Quantensysteme werden durch reduzierte Dichtematrizen, auch bekannt als Quantenmarginale, beschrieben. We- sentliche physikalische Eigenschaften hängen oft nicht von der gesamten Wellenfunktion ab, sondern nur von den reduzierten Dichtematrizen weniger Teilchen. Nicht alle Sätze von Dichtematrizen können als Mar- ginale eines globalen Quantenzustands auftreten. Tiefgreifende Kom- patibilitätsbedingungen wie das Pauli’sche Ausschlussprinzip oder die Monogamie der Quantenverschränkung beeinflussen auf fundamentale Weise die physikalischen Eigenschaften von Vielteilchensystemen und die Struktur von Quanteninformation. Ziel dieser Dissertation ist eine systematische und rigorose Untersuchung der allgemeinen Beziehung zwischen multipartiten Quantenzuständen, d.h. Quantenzuständen von Vielteilchensystemen, und ihren Marginalen. Im ersten Teil dieser Dissertation (Kapitel 2–6) betrachten wir Ein- teilchenmarginale multipartiter Quantenzustände. Wir beginnen mit einer neuen, geometrischen Lösung des Kompatibilitätsproblems. Danach wenden wir uns dem Phänomen der Quantenverschränkung zu. Wir zeigen, dass sich mittels der Eigenwerte der Einteilchendichtematrizen Aussagen über die Verschränkungseigenschaften des Gesamtzustands tre- ffen lassen. Zur systematischen Untersuchung dieses Phänomens führen wir das Konzept eines “Verschränkungspolytops” ein. Des Weiteren betra- chten wir zufällige Quantenzustände. Hierzu entwickeln wir ein Verfahren, mit dessen Hilfe sich die gemeinsame Verteilung der Einteilchenmarginale exakt berechnen lässt. Wir illustrieren die Vielseitigkeit unseres Ansatzes, indem wir aus einer diskreten Variante einen effizienten Algorithmus für das Verzweigungsproblem kompakter, zusammenhängender Lie-Gruppen konstruieren. Ein wiederkehrendes Motiv im ersten Teil ist die Reduktion quantenphysikalischer Probleme auf im Wesentlichen kombinatorische, klassische Probleme. Im zweiten Teil dieser Dissertation (Kapitel 7–9) untersuchen wir allge- meine Marginale aus dem Blickwinkel der von Neumann’schen Entropie. Hier tragen wir zwei neue Techniken zum Beweis von Entropieungleichun- gen bei. Die erste Technik basiert auf Phasenraummethoden und erlaubt es uns, eine unendliche Zahl an Entropieungleichungen für die Klasse der Stabilisatorzustände zu beweisen. Die zweite Technik beruht auf einer neuen Charakterisierung kompatibler Marginale mittels Darstellungsthe- orie. Wir zeigen, dass Entropieungleichungen im Sinne von Symmetrien verstanden werden können, und geben zur Illustration einen neuen, prägnanten Beweis der starken Subadditivität der von Neumann’schen Entropie. Acknowledgements I am greatly indebted to my advisor Matthias Christandl for his support and enthusiasm in research and otherwise. I sincerely thank Gian Michele Graf and Aram Harrow for accepting to be my co-examiners and for their careful reading of this thesis. I also thank Jürg Fröhlich for his support. Much of my research work has been done in collaborations, some of whose results are incorporated in this thesis. It is a pleasure to thank my collaborators Matthias Christandl, Brent Doran, David Gross, Stavros Kousidis, Marek Kuś, Joseph M. Renes, Burak Şahinoğlu, Adam Sawicki, Sebastian Seehars, Roman Schmied, Volkher Scholz, Michèle Vergne, and Konstantin Wernli for making research such an enjoyable enterprise. I would also like to extend my gratitude to Philippe Biane, Alonso Botero, Fernando Brandão, Peter Bürgisser, Mikhail Gromov, Patrick Hayden, Christian Ikenmeyer, Wojciech Kamiński, Eckhard Meinrenken, Graeme Mitchison, Renato Renner, Nicolas Ressayre, Mary Beth Ruskai, Philipp Treutlein, Stephanie Wehner, Reinhard Werner, and Andreas Winter for many inspiring discussions, helpful advice and well-appreciated support. I am grateful to the Institute for Theoretical Physics at ETH Zürich for providing an excellent environment for research. I also thank all the members of the Quantum Information Theory group for creating such a pleasant and productive atmosphere. In the course of my graduate studies, I have had the good fortune of a number of travel opportunities. In particular, I want to thank the Isaac Newton Institute and the Institut des Hautes Études Scientifiques for their hospitality during my research visits; I am grateful to Richard Josza, Noah Linden, Peter Shor, and Andreas Winter and to Mikhail Gromov for making these visits possible. Finally, I want to thank Christian Majenz, Burak Şahinoğlu, Volkher Scholz, and Jonathan Skowera for their careful reading of parts of a preliminary version of this thesis. Contents Page 1 Introduction 1 2 The One-Body Quantum Marginal Problem 5 2.1 Lie Groups and their Representations . .8 2.2 Geometric Invariant Theory . 15 2.3 Consequences for the Quantum Marginal Problem . 23 2.4 Kronecker coefficients, Schur–Weyl duality, and Plethysms . 26 3 Solving The One-Body Quantum Marginal Problem 31 3.1 Summary of Results . 34 3.2 The Torus Action . 35 3.3 Facets of the Moment Polytope . 36 3.4 Examples . 42 3.5 Discussion . 47 4 Multipartite Entanglement 49 4.1 Summary of Results . 49 4.2 Classification of Entanglement . 54 4.3 Entanglement Polytopes . 56 4.4 Examples . 60 4.5 Gradient Flow . 69 4.6 Experimental Noise . 75 4.7 Discussion . 78 5 Random Marginals 81 5.1 Summary of Results . 81 5.2 Duistermaat–Heckman Measures . 83 5.3 The Abelian Measure . 86 5.4 The Derivative Principle . 90 5.5 Examples . 93 5.6 Discussion . 99 6 Interlude: Multiplicities of Representations 101 6.1 Summary of Results . 102 6.2 Weight Multiplicities . 105 6.3 The Finite Difference Formula . 107 6.4 Multiplicities for the Subgroup Restriction Problem . 109 6.5 Kronecker Coefficients . 112 6.6 Asymptotics . 113 6.7 Discussion . 119 7 The Search for Further Entropy Inequalities 121 8 Entropy Inequalities from Phase Space 127 8.1 Summary of Results . 127 8.2 Discrete Phase Space . 129 8.3 Stabilizer States in Phase Space . 131 8.4 A Classical Model for Stabilizer Entropies . 134 8.5 Gaussian States . 137 8.6 Discussion . 139 9 Entropy Inequalities from Recoupling Coefficients 141 9.1 Summary of Results . 141 9.2 Recoupling Coefficients . 143 9.3 Overlapping Marginals and Recoupling Coefficients . 146 9.4 Symmetry Properties of the Recoupling Coefficients . 149 9.5 Application: Strong Subadditivity of the von Neumann Entropy . 155 9.6 Semiclassicality . 156 9.7 Sums of Matrices and Quantum Marginals . 156 9.8 Discussion . 162 Bibliography 165 List of Symbols 191 Index 195 List of Publications 199 Chapter 1 Introduction The pure state of a quantum system is described by a vector in a Hilbert space, or, more precisely, by a point in the corresponding projective space. Since the Hilbert space for multiple particles is given by the tensor product of the
Load more

Recommended publications
  • Quantum Information
    Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding .
    [Show full text]
  • Lecture Notes: Qubit Representations and Rotations
    Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1.
    [Show full text]
  • A Representations of SU(2)
    A Representations of SU(2) In this appendix we provide details of the parameterization of the group SU(2) and differential forms on the group space. An arbitrary representation of the group SU(2) is given by the set of three generators Tk, which satisfy the Lie algebra [Ti,Tj ]=iεijkTk, with ε123 =1. The element of the group is given by the matrix U =exp{iT · ω} , (A.1) 1 where in the fundamental representation Tk = 2 σk, k =1, 2, 3, with 01 0 −i 10 σ = ,σ= ,σ= , 1 10 2 i 0 3 0 −1 standard Pauli matrices, which satisfy the relation σiσj = δij + iεijkσk . The vector ω has components ωk in a given coordinate frame. Geometrically, the matrices U are generators of spinor rotations in three- 3 dimensional space R and the parameters ωk are the corresponding angles of rotation. The Euler parameterization of an arbitrary matrix of SU(2) transformation is defined in terms of three angles θ, ϕ and ψ,as ϕ θ ψ iσ3 iσ2 iσ3 U(ϕ, θ, ψ)=Uz(ϕ)Uy(θ)Uz(ψ)=e 2 e 2 e 2 ϕ ψ ei 2 0 cos θ sin θ ei 2 0 = 2 2 −i ϕ θ θ − ψ 0 e 2 − sin cos 0 e i 2 2 2 i i θ 2 (ψ+ϕ) θ − 2 (ψ−ϕ) cos 2 e sin 2 e = i − − i . (A.2) − θ 2 (ψ ϕ) θ 2 (ψ+ϕ) sin 2 e cos 2 e Thus, the SU(2) group manifold is isomorphic to three-sphere S3.
    [Show full text]
  • Theory of Angular Momentum and Spin
    Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO(3) of the associated rotation matrices and the 1 corresponding transformation matrices of spin{ 2 states forming the group SU(2) occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors ~r in 3-dimensional space, i.e., ~r R3, 2 we will then consider transformations of wavefunctions (~r) of single particles in R3, and finally N transformations of products of wavefunctions like j(~rj) which represent a system of N (spin- Qj=1 zero) particles in R3. We will also review below the well-known fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states.
    [Show full text]
  • Two-State Systems
    1 TWO-STATE SYSTEMS Introduction. Relative to some/any discretely indexed orthonormal basis |n) | ∂ | the abstract Schr¨odinger equation H ψ)=i ∂t ψ) can be represented | | | ∂ | (m H n)(n ψ)=i ∂t(m ψ) n ∂ which can be notated Hmnψn = i ∂tψm n H | ∂ | or again ψ = i ∂t ψ We found it to be the fundamental commutation relation [x, p]=i I which forced the matrices/vectors thus encountered to be ∞-dimensional. If we are willing • to live without continuous spectra (therefore without x) • to live without analogs/implications of the fundamental commutator then it becomes possible to contemplate “toy quantum theories” in which all matrices/vectors are finite-dimensional. One loses some physics, it need hardly be said, but surprisingly much of genuine physical interest does survive. And one gains the advantage of sharpened analytical power: “finite-dimensional quantum mechanics” provides a methodological laboratory in which, not infrequently, the essentials of complicated computational procedures can be exposed with closed-form transparency. Finally, the toy theory serves to identify some unanticipated formal links—permitting ideas to flow back and forth— between quantum mechanics and other branches of physics. Here we will carry the technique to the limit: we will look to “2-dimensional quantum mechanics.” The theory preserves the linearity that dominates the full-blown theory, and is of the least-possible size in which it is possible for the effects of non-commutivity to become manifest. 2 Quantum theory of 2-state systems We have seen that quantum mechanics can be portrayed as a theory in which • states are represented by self-adjoint linear operators ρ ; • motion is generated by self-adjoint linear operators H; • measurement devices are represented by self-adjoint linear operators A.
    [Show full text]
  • 10 Group Theory
    10 Group theory 10.1 What is a group? A group G is a set of elements f, g, h, ... and an operation called multipli- cation such that for all elements f,g, and h in the group G: 1. The product fg is in the group G (closure); 2. f(gh)=(fg)h (associativity); 3. there is an identity element e in the group G such that ge = eg = g; 1 1 1 4. every g in G has an inverse g− in G such that gg− = g− g = e. Physical transformations naturally form groups. The elements of a group might be all physical transformations on a given set of objects that leave invariant a chosen property of the set of objects. For instance, the objects might be the points (x, y) in a plane. The chosen property could be their distances x2 + y2 from the origin. The physical transformations that leave unchanged these distances are the rotations about the origin p x cos ✓ sin ✓ x 0 = . (10.1) y sin ✓ cos ✓ y ✓ 0◆ ✓− ◆✓ ◆ These rotations form the special orthogonal group in 2 dimensions, SO(2). More generally, suppose the transformations T,T0,T00,... change a set of objects in ways that leave invariant a chosen property property of the objects. Suppose the product T 0 T of the transformations T and T 0 represents the action of T followed by the action of T 0 on the objects. Since both T and T 0 leave the chosen property unchanged, so will their product T 0 T . Thus the closure condition is satisfied.
    [Show full text]
  • Spectral Quantum Tomography
    www.nature.com/npjqi ARTICLE OPEN Spectral quantum tomography Jonas Helsen 1, Francesco Battistel1 and Barbara M. Terhal1,2 We introduce spectral quantum tomography, a simple method to extract the eigenvalues of a noisy few-qubit gate, represented by a trace-preserving superoperator, in a SPAM-resistant fashion, using low resources in terms of gate sequence length. The eigenvalues provide detailed gate information, supplementary to known gate-quality measures such as the gate fidelity, and can be used as a gate diagnostic tool. We apply our method to one- and two-qubit gates on two different superconducting systems available in the cloud, namely the QuTech Quantum Infinity and the IBM Quantum Experience. We discuss how cross-talk, leakage and non-Markovian errors affect the eigenvalue data. npj Quantum Information (2019) 5:74 ; https://doi.org/10.1038/s41534-019-0189-0 INTRODUCTION of the form λ = exp(−γ)exp(iϕ), contain information about the A central challenge on the path towards large-scale quantum quality of the implemented gate. Intuitively, the parameter γ computing is the engineering of high-quality quantum gates. To captures how much the noisy gate deviates from unitarity due to achieve this goal, many methods that accurately and reliably entanglement with an environment, while the angle ϕ can be characterize quantum gates have been developed. Some of these compared to the rotation angles of the targeted gate U. Hence ϕ methods are scalable, meaning that they require an effort which gives information about how much one over- or under-rotates. scales polynomially in the number of qubits on which the gates The spectrum of S can also be related to familiar gate-quality 1–8 act.
    [Show full text]
  • Singles out a Specific Basis
    Quantum Information and Quantum Noise Gabriel T. Landi University of Sao˜ Paulo July 3, 2018 Contents 1 Review of quantum mechanics1 1.1 Hilbert spaces and states........................2 1.2 Qubits and Bloch’s sphere.......................3 1.3 Outer product and completeness....................5 1.4 Operators................................7 1.5 Eigenvalues and eigenvectors......................8 1.6 Unitary matrices.............................9 1.7 Projective measurements and expectation values............ 10 1.8 Pauli matrices.............................. 11 1.9 General two-level systems....................... 13 1.10 Functions of operators......................... 14 1.11 The Trace................................ 17 1.12 Schrodinger’s¨ equation......................... 18 1.13 The Schrodinger¨ Lagrangian...................... 20 2 Density matrices and composite systems 24 2.1 The density matrix........................... 24 2.2 Bloch’s sphere and coherence...................... 29 2.3 Composite systems and the almighty kron............... 32 2.4 Entanglement.............................. 35 2.5 Mixed states and entanglement..................... 37 2.6 The partial trace............................. 39 2.7 Reduced density matrices........................ 42 2.8 Singular value and Schmidt decompositions.............. 44 2.9 Entropy and mutual information.................... 50 2.10 Generalized measurements and POVMs................ 62 3 Continuous variables 68 3.1 Creation and annihilation operators................... 68 3.2 Some important
    [Show full text]
  • Operators and Matrices
    Operators and Matrices Let º be an inner-product vector space with an ONB fjejig, so that 8jxi 2 º there exists a unique representation X jxi = xjjeji ; xj = hejjxi : (1) j [We remember that if the dimensionality of the vector space is ¯nite, then the sum is ¯nite. In an in¯nite-dimensional vector space, like, e.g., L2, the sum is in¯nite and is called Fourier series.] The isomorphism of the vector spaces allows one to map our space º onto the vector spaceº ~ formed by the coordinates of the vector with respect to our ¯xed ONB (see section Vector Spaces): jxi Ã! (x1; x2; x3;:::) 2 º~ : (2) In the context of this isomorphism|which is very important for practical applications, since it allows one to work with just the numbers instead of abstract vectors|the following question naturally arises. Suppose A is some linear operator that acts on some jxi 2 º producing corresponding vector jyi 2 º: jyi = Ajxi : (3) By the isomorphism, jyi Ã! (y1; y2; y3;:::) 2 º~ ; (4) and we realize that it is crucially important to understand how (y1; y2; y3;:::) is obtained from (x1; x2; x3;:::) directly in the spaceº ~. To this end, we note that X yj = hejjyi = hejjAjxi = hejjA xsjesi ; (5) s and, taking into account linearity of the operator A, ¯nd that X yj = Ajs xs ; (6) s where Ajs = hejjAjesi : (7) We see that the set of numbers fAjsg de¯ned in accordance with Eq. (7) completely de¯nes the action of the operator A on any vector jxi (in terms 1 of the coordinates with respect to a given ONB).
    [Show full text]
  • Arxiv:1905.07138V2 [Quant-Ph] 1 Jan 2020 Asnumbers: W PACS Chain 4-Spin a in Evolution Such Using Instance, for Data Input Chain
    Solving systems of linear algebraic equations via unitary transformations on quantum processor of IBM Quantum Experience. S.I.Doronin, E.B.Fel’dman and A.I.Zenchuk Corresponding author: A.I.Zenchuk, [email protected] Institute of Problems of Chemical Physics RAS, Chernogolovka, Moscow reg., 142432, Russia Abstract We propose a protocol for solving systems of linear algebraic equations via quantum mechanical methods using the minimal number of qubits. We show that (M + 1)-qubit system is enough to solve a system of M equations for one of the variables leaving other variables unknown provided that the matrix of a linear system satisfies certain conditions. In this case, the vector of input data (the rhs of a linear system) is encoded into the initial state of the quantum system. This protocol is realized on the 5-qubit superconducting quantum processor of IBM Quantum Experience for particular linear systems of three equations. We also show that the solution of a linear algebraic system can be obtained as the result of a natural evolution of an inhomogeneous spin-1/2 chain in an inhomogeneous external magnetic field with the input data encoded into the initial state of this chain. For instance, using such evolution in a 4-spin chain we solve a system of three equations. PACS numbers: arXiv:1905.07138v2 [quant-ph] 1 Jan 2020 1 I. INTRODUCTION The creation of quantum counterparts of classical algorithms solving various algebraic problems, and their programming on IBM quantum computers is an important directions in development of quantum information processing. In our paper, we refer to a problem of solving a linear system of algebraic equations Ax = b via the quntum-mechanical approach.
    [Show full text]
  • Calculating the Pauli Matrix Equivalent for Spin-1 Particles and Further
    Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System Rajdeep Tah To cite this version: Rajdeep Tah. Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System. 2020. hal-02909703 HAL Id: hal-02909703 https://hal.archives-ouvertes.fr/hal-02909703 Preprint submitted on 31 Jul 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System Rajdeep Tah1, ∗ 1School of Physical Sciences, National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni, P.O.-752050, Odisha, India (Dated: July, 2020) Here, we derive the Pauli Matrix Equivalent for Spin-1 particles (mainly Z-Boson and W-Boson). 1 Pauli Matrices are generally associated with Spin- 2 particles and it is used for determining the 1 properties of many Spin- 2 particles.
    [Show full text]
  • Alternating Sign Matrices, Completely Packed Loops and Plane Partitions Tiago Fonseca
    Alternating Sign Matrices, completely Packed Loops and Plane Partitions Tiago Fonseca To cite this version: Tiago Fonseca. Alternating Sign Matrices, completely Packed Loops and Plane Partitions. Mathemat- ical Physics [math-ph]. Université Pierre et Marie Curie - Paris VI, 2010. English. tel-00521884v1 HAL Id: tel-00521884 https://tel.archives-ouvertes.fr/tel-00521884v1 Submitted on 28 Sep 2010 (v1), last revised 17 Nov 2010 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Th`esepresent´eepar Tiago DINIS DA FONSECA Pour obtenir le grade de Docteur de l’Universit´ePierre et Marie Curie Sp´ecialit´e: Physique Math´ematique Sujet: Matrices `asignes alternants, boucles denses et partitions planes Soutenue le 24 septembre 2010 devant le jury compos´ede M. Alain LASCOUX, examinateur, M. Bernard NIENHUIS, examinateur, M. Christian KRATTENTHALER, rapporteur, M. Jean-Bernard ZUBER, examinateur, M. Nikolai KITANINE, rapporteur, M. Paul ZINN-JUSTIN, directeur de th`ese. Remerciements Avant tout je remercie Paul qui m’a guid´edans ma th`eseet qui m’a propos´eplein de probl`emesint´eressants, quelques uns que j’ai su r´esoudre et d’autres non. Pendant trois ans il a ´et´etoujours pr´esent et disponible pour discuter et j’esp`ere,sinc`erement, qu’on pourra continuer `atravailler ensemble.
    [Show full text]