DOCSLIB.ORG

Quantum Entanglement: Detection, Classification, and Quantification

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Entanglement: Detection, Classification, and Quantification Quantum Entanglement: Detection, Classification, and Quantification Diplomarbeit zur Erlangung des akademischen Grades ,,Magister der Naturwissenschaften'' an der UniversitatÄ Wien eingereicht von Philipp Krammer betreut von Ao. Univ. Prof. Dr. Reinhold A. Bertlmann Wien, Oktober 2005 CONTENTS 1. Introduction ::::::::::::::::::::::::::::: 4 2. Basic Mathematical Description :::::::::::::::: 6 2.1 Spaces, Operators and States in a Finite Dimensional Hilbert Space . 6 2.2 Bipartite Systems . 7 2.2.1 Qubits . 11 2.2.2 Qutrits . 12 2.3 Positive and Completely Positive Maps . 14 3. Detection of Entanglement ::::::::::::::::::: 16 3.1 Introduction . 16 3.2 Pure States . 17 3.3 General States . 17 3.3.1 Nonoperational Separability Criteria . 17 3.3.2 Operational Separability Criteria . 20 4. Classification of Entanglement :::::::::::::::: 32 4.1 Introduction . 32 4.2 Free and Bound Entanglement . 32 4.2.1 Distillation of Entangled States . 32 4.2.2 Bound Entanglement . 36 4.3 Locality vs. Non-locality . 42 4.3.1 EPR and Bell Inequalities . 42 4.3.2 General Bell Inequality . 44 4.3.3 Bell Inequalities and the Entanglement Witness Theorem 49 5. Quantification of Entanglement :::::::::::::::: 52 5.1 Introduction . 52 5.2 Pure States . 52 5.3 General States . 54 5.3.1 Entanglement of Formation . 54 Contents 3 5.3.2 Concurrence and Calculating the Entanglement of For- mation for 2 Qubits . 56 5.3.3 Entanglement of Distillation . 58 5.3.4 Distance Measures . 59 5.3.5 Comparison of Di®erent Entanglement Measures for the 2-Qubit Werner State . 64 6. Hilbert-Schmidt Measure and Entanglement Witness :: 66 6.1 Introduction . 66 6.2 Geometrical Considerations about the Hilbert-Schmidt Distance 66 6.3 The Bertlmann-Narnhofer-Thirring Theorem . 68 6.4 How to Check a Guess of the Nearest Separable State . 70 6.5 Examples . 72 6.5.1 Isotropic Qubit States . 72 6.5.2 Isotropic Qutrit States . 74 6.5.3 Isotropic States in Higher Dimensions . 76 7. Tripartite Systems ::::::::::::::::::::::::: 79 7.1 Introduction . 79 7.2 Basics . 79 7.3 Pure States . 81 7.3.1 Detection of Entangled Pure States . 81 7.3.2 Equivalence Classes of Pure Tripartite States . 81 7.4 General States . 87 7.4.1 Equivalence Classes of General Tripartite States . 87 7.4.2 Tripartite Witnesses . 88 8. Conclusion :::::::::::::::::::::::::::::: 90 1. INTRODUCTION What is Quantum Entanglement? If we look up the word `entanglement' in a dictionary, we ¯nd something like `state of being involved in complicated circumstances', the term also denotes an a®air between two people. Thus in quantum mechanics we could describe quantum entanglement literally as a `complicated a®air' between two or more particles. The ¯rst one to introduce the term was Erwin SchrÄodingerin Ref. [68]. Since this article was published in German, `entanglement' is a later transla- tion of the word `VerschrÄankung'.SchrÄodingerdoes not refer to a mathemat- ical de¯nition of entanglement. He introduces entanglement as a correlation of possible measurement outcomes and states the following: Maximal knowledge of a whole system does not necessarily include knowledge of all of its parts, even if these are totally divided from each other and do not influence each other at the present time. Note that `system' is always a generalized expression for some physical real- ization; in this context a system of two or more particles is meant. Nowadays the de¯nition of entanglement is a mathematical one and rather simple (see Chapter 2) { however, the phenomenological description of en- tanglement is still di±cult. Since J.S. Bell introduced his `Bell Inequality' [3] it has become clear that the correlations related to quantum entanglement can be stronger than merely classical correlations. Classical correlations are those that are explainable by a local realistic theory, and it was propagated by Einstein, Podolsky, and Rosen [31] that quantum mechanics should also be a local realistic theory (see Sec. 4.3). The mysteriousness inherent to quantum entanglement mainly comes from the fact that often in cannot be explained with a classical deterministic model [3, 4] and so underlines the `new physics' that comes with quantum mechanics and distinguishes it from classical physics. Why do we need quantum entanglement? What at ¯rst seemed to be a more philosophical investigation became of practical use in recent years. With the development of quantum information theory a new `quantum' way of informa- tion processing and communication was initiated which makes direct use of 1. Introduction 5 quantum entanglement and takes advantage of it (see, e.g., Refs. [16, 13, 45]). There are various tasks involving quantum entanglement that are an improve- ment to classical information theory, for example quantum teleportation and cryptography (see, e.g., Refs. [5, 17, 15, 32]). In the course of years comput- ing has become and still becomes more and more e±cient, information has to be encoded into less physical material. To be able to keep pace with the technological demand, quantum information theory could serve as the future concept of information processing and communication devices. It is therefore not only of philosophical but also of practical use to deepen and extend the description of quantum entanglement. Aim and Structure of this Work. The aim of this work is to provide a basic mathematical overview of quantum entanglement which includes the fundamental aspects of detecting, classifying, and quantifying entanglement. Several examples should give insight of the explicit application of the given theory. There is no emphasis on detailed proofs. Nevertheless some proofs that are useful to be explicitly mentioned and do not take too long are given, otherwise the reader is refered to other literature. The main part of the work is concerned with bipartite systems, these are systems that consist of two parts (i.e. particles in experimental application). The work is organized as follows: In Chapter 2 we start with basic math- ematical concepts. Next, in Chapter 3, we address the problem of detecting entanglement; in Chapter 4 entanglement is classi¯ed according to certain properties, and in Chapter 5 we discuss several methods to quantify entan- glement. In Chapter 6 we combine the concept of detecting and quantifying entanglement. Finally, in Chapter 7, we briefly take a look on tripartite systems. 2. BASIC MATHEMATICAL DESCRIPTION 2.1 Spaces, Operators and States in a Finite Dimensional Hilbert Space Operators act on the Hilbert space H of a quantum mechanical system, they make up a Hilbert Space themselves, called Hilbert-Schmidt space A. We are only interested in ¯nite dimensional Hilbert spaces, so that in fact A can be regarded as a space of matrices, taking into account that in ¯nite dimensions operators can be written in matrix form. A scalar product de¯ned on A is (A; B 2 A) hA; Bi = TrAyB; (2.1) with the corresponding Hilbert-Schmidt norm p kAk = hA; Ai A 2 A : (2.2) Matrix Notation. Generally, any operator A 2 A can be expressed as a matrix with the elements Aij = heij A jeji ; (2.3) where ei and ej are vectors of an arbitrary basis feig of the Hilbert Space. Of course the same holds for states, since they are operators. De¯nition of a State. An operator ½ is called `state' (or density operator or density matrix) if 1 Tr½ = 1; ½ ¸ 0 ; (2.4) where ½ ¸ 0 means that ½ is a positive operator (more precise: positive semide¯nite), that is, if all its eigenvalues are larger than or equal to zero. Positivity of ½ can be equivalently expressed as Tr½P ¸ 0 8P; (2.5) where P is any projector, de¯ned by P 2 = P . 2 1 Certainly the presented conditions refer to the matrix form of a state ½. 2 Eq. (2.5) follows from the fact that the eigenvalues are nonnegative, since ½ can be written in appropriate matrix notation in which it is diagonal, where the eigenvalues are 2. Basic Mathematical Description 7 Remark. In early quantum mechanics (pure) states are represented as vec- tors jÃi in Hilbert Space. This concept is widened with the introduction of mixed states, so that in general states are viewed as operators. If one is inter- ested in pure states only, either the vector representation jÃi or the operator representation ½pure = jÃi hÃj can be used. Note that Eqs. (2.4) and (2.5) imply Tr½2 · 1.3 In particular we have Tr½2 = 1 ) ½ is a pure state ; Tr½2 < 1 ) ½ is a mixed state : (2.6) 2.2 Bipartite Systems In all chapters but the last we will consider bipartite systems. Following the convention of quantum communication, the two parties are usually referred to as `Alice' and `Bob'. For bipartite systems the Hilbert space is denoted as d1 d2 HA ­ HB , where d1 is the dimension of Alice's subspace and d2 is Bob's, or just HA ­ HB when there need not be a special indication to the dimensions. We may also drop for convenience the indices `A' and `B', e.g. we will often consider the Hilbert Space H2 ­ H2, states on this space are called 2-qubit states. Matrix Notation. In general, we can write a state ½ as a matrix according to Eq. (2.3). However, often we have to use a product basis, to guarantee that certain calculations etc. make sense. In this case for the matrix notation of d1 d2 a state ½ on HA ­ HB we have ½m¹;nº = hem ­ f¹j ½ jen ­ fºi : (2.7) Here feig and ffig are bases of Alice's and Bob's subspaces. Reduced Density Matrices. The notation in a product basis is for example needed to calculate the reduced density matrices of a state ½. These are obtained if Alice neglects Bob's system, or vice versa, which mathematically means she takes a partial trace of the density matrix, she \traces out" Bob's the diagonal elements.
Load more

Recommended publications
  • Modern Quantum Technologies of Information Security
    MODERN QUANTUM TECHNOLOGIES OF INFORMATION SECURITY Oleksandr Korchenko 1, Yevhen Vasiliu 2, Sergiy Gnatyuk 3 1,3 Dept of Information Security Technologies, National Aviation University, Kosmonavta Komarova Ave 1, 03680 Kyiv, Ukraine 2Dept of Information Technologies and Control Systems, Odesa National Academy of Telecommunications n.a. O.S. Popov, Koval`ska Str 1, 65029 Odesa, Ukraine E-mails: [email protected], [email protected], [email protected] In this paper, the systematisation and classification of modern quantum technologies of information security against cyber-terrorist attack are carried out. The characteristic of the basic directions of quantum cryptography from the viewpoint of the quantum technologies used is given. A qualitative analysis of the advantages and disadvantages of concrete quantum protocols is made. The current status of the problem of practical quantum cryptography use in telecommunication networks is considered. In particular, a short review of existing commercial systems of quantum key distribution is given. 1. Introduction Today there is virtually no area where information technology ( ІТ ) is not used in some way. Computers support banking systems, control the work of nuclear reactors, and control aircraft, satellites and spacecraft. The high level of automation therefore depends on the security level of IT. The latest achievements in communication systems are now applied in aviation. These achievements are public switched telephone network (PSTN), circuit switched public data network (CSPDN), packet switched public data network (PSPDN), local area network (LAN), and integrated services digital network (ISDN) [73]. These technologies provide data transmission systems of various types: surface-to-surface, surface-to-air, air-to-air, and space telecommunication.
    [Show full text]
  • Quantum Information in the Posner Model of Quantum Cognition
    Quantum information in quantum cognition Nicole Yunger Halpern1, 2 and Elizabeth Crosson1 1Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: November 15, 2017) Matthew Fisher recently postulated a mechanism by which quantum phenomena could influence cognition: Phosphorus nuclear spins may resist decoherence for long times. The spins would serve as biological qubits. The qubits may resist decoherence longer when in Posner molecules. We imagine that Fisher postulates correctly. How adroitly could biological systems process quantum information (QI)? We establish a framework for answering. Additionally, we apply biological qubits in quantum error correction, quantum communication, and quantum computation. First, we posit how the QI encoded by the spins transforms as Posner molecules form. The transformation points to a natural computational basis for qubits in Posner molecules. From the basis, we construct a quantum code that detects arbitrary single-qubit errors. Each molecule encodes one qutrit. Shifting from information storage to computation, we define the model of Posner quantum computation. To illustrate the model's quantum-communication ability, we show how it can teleport information in- coherently: A state's weights are teleported; the coherences are not. The dephasing results from the entangling operation's simulation of a coarse-grained Bell measurement. Whether Posner quantum computation is universal remains an open question. However, the model's operations can efficiently prepare a Posner state usable as a resource in universal measurement-based quantum computation. The state results from deforming the Affleck-Lieb-Kennedy-Tasaki (AKLT) state and is a projected entangled-pair state (PEPS).
    [Show full text]
  • Motion of the Reduced Density Operator
    Motion of the Reduced Density Operator Nicholas Wheeler, Reed College Physics Department Spring 2009 Introduction. Quantum mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The observable properties of the system are latent then in the reduced density operator ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ. Concerning the specific meaning of (1). Let n) be an orthonormal basis | in the state space H of the (open) system, and N) be an orthonormal basis s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗ tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e The dynamics of the composite system is generated by Hamiltonian of the form H = H s + H e + H i 2 Motion of the reduced density operator where H = h I s s ⊗ e = m h n m N n N $ | s| % | % ⊗ | % · $ | ⊗ $ | m,n N # # $% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · $ | ⊗ $ | $ | e| % n M,N # # $% & % &' H = m M m M H n N n N i | % ⊗ | % $ | ⊗ $ | i | % ⊗ | % $ | ⊗ $ | m,n M,N # # % &$% & % &'% & —all components of which we will assume to be time-independent.
    [Show full text]
  • Quantum Cryptography: from Theory to Practice
    Quantum cryptography: from theory to practice by Xiongfeng Ma A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Thesis Graduate Department of Department of Physics University of Toronto Copyright °c 2008 by Xiongfeng Ma Abstract Quantum cryptography: from theory to practice Xiongfeng Ma Doctor of Philosophy Thesis Graduate Department of Department of Physics University of Toronto 2008 Quantum cryptography or quantum key distribution (QKD) applies fundamental laws of quantum physics to guarantee secure communication. The security of quantum cryptog- raphy was proven in the last decade. Many security analyses are based on the assumption that QKD system components are idealized. In practice, inevitable device imperfections may compromise security unless these imperfections are well investigated. A highly attenuated laser pulse which gives a weak coherent state is widely used in QKD experiments. A weak coherent state has multi-photon components, which opens up a security loophole to the sophisticated eavesdropper. With a small adjustment of the hardware, we will prove that the decoy state method can close this loophole and substantially improve the QKD performance. We also propose a few practical decoy state protocols, study statistical fluctuations and perform experimental demonstrations. Moreover, we will apply the methods from entanglement distillation protocols based on two-way classical communication to improve the decoy state QKD performance. Fur- thermore, we study the decoy state methods for other single photon sources, such as triggering parametric down-conversion (PDC) source. Note that our work, decoy state protocol, has attracted a lot of scienti¯c and media interest. The decoy state QKD becomes a standard technique for prepare-and-measure QKD schemes.
    [Show full text]
  • Quantum Computing a New Paradigm in Science and Technology
    Quantum computing a new paradigm in science and technology Part Ib: Quantum computing. General documentary. A stroll in an incompletely explored and known world.1 Dumitru Dragoş Cioclov 3. Quantum Computer and its Architecture It is fair to assert that the exact mechanism of quantum entanglement is, nowadays explained on the base of elusive A quantum computer is a machine conceived to use quantum conjectures, already evoked in the previous sections, but mechanics effects to perform computation and simulation this state-of- art it has not impeded to illuminate ideas and of behavior of matter, in the context of natural or man-made imaginative experiments in quantum information theory. On this interactions. The drive of the quantum computers are the line, is worth to mention the teleportation concept/effect, deeply implemented quantum algorithms. Although large scale general- purpose quantum computers do not exist in a sense of classical involved in modern cryptography, prone to transmit quantum digital electronic computers, the theory of quantum computers information, accurately, in principle, over very large distances. and associated algorithms has been studied intensely in the last Summarizing, quantum effects, like interference and three decades. entanglement, obviously involve three states, assessable by The basic logic unit in contemporary computers is a bit. It is zero, one and both indices, similarly like a numerical base the fundamental unit of information, quantified, digitally, by the two (see, e.g. West Jacob (2003). These features, at quantum, numbers 0 or 1. In this format bits are implemented in computers level prompted the basic idea underlying the hole quantum (hardware), by a physic effect generated by a macroscopic computation paradigm.
    [Show full text]
  • Parafermions in a Kagome Lattice of Qubits for Topological Quantum Computation
    PHYSICAL REVIEW X 5, 041040 (2015) Parafermions in a Kagome Lattice of Qubits for Topological Quantum Computation Adrian Hutter, James R. Wootton, and Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Received 3 June 2015; revised manuscript received 16 September 2015; published 14 December 2015) Engineering complex non-Abelian anyon models with simple physical systems is crucial for topological quantum computation. Unfortunately, the simplest systems are typically restricted to Majorana zero modes (Ising anyons). Here, we go beyond this barrier, showing that the Z4 parafermion model of non-Abelian anyons can be realized on a qubit lattice. Our system additionally contains the Abelian DðZ4Þ anyons as low-energetic excitations. We show that braiding of these parafermions with each other and with the DðZ4Þ anyons allows the entire d ¼ 4 Clifford group to be generated. The error-correction problem for our model is also studied in detail, guaranteeing fault tolerance of the topological operations. Crucially, since the non- Abelian anyons are engineered through defect lines rather than as excitations, non-Abelian error correction is not required. Instead, the error-correction problem is performed on the underlying Abelian model, allowing high noise thresholds to be realized. DOI: 10.1103/PhysRevX.5.041040 Subject Areas: Condensed Matter Physics, Quantum Physics, Quantum Information I. INTRODUCTION Non-Abelian anyons supported by a qubit system Non-Abelian anyons exhibit exotic physics that would typically are Majorana zero modes, also known as Ising – make them an ideal basis for topological quantum compu- anyons [12 15]. A variety of proposals for experimental tation [1–3].
    [Show full text]
  • Density Matrix Description of NMR
    Density Matrix Description of NMR BCMB/CHEM 8190 Operators in Matrix Notation • It will be important, and convenient, to express the commonly used operators in matrix form • Consider the operator Iz and the single spin functions α and β - recall ˆ ˆ ˆ ˆ ˆ ˆ Ix α = 1 2 β Ix β = 1 2α Iy α = 1 2 iβ Iy β = −1 2 iα Iz α = +1 2α Iz β = −1 2 β α α = β β =1 α β = β α = 0 - recall the expectation value for an observable Q = ψ Qˆ ψ = ∫ ψ∗Qˆψ dτ Qˆ - some operator ψ - some wavefunction - the matrix representation is the possible expectation values for the basis functions α β α ⎡ α Iˆ α α Iˆ β ⎤ ⎢ z z ⎥ ⎢ ˆ ˆ ⎥ β ⎣ β Iz α β Iz β ⎦ ⎡ α Iˆ α α Iˆ β ⎤ ⎡1 2 α α −1 2 α β ⎤ ⎡1 2 0 ⎤ ⎡1 0 ⎤ ˆ ⎢ z z ⎥ 1 Iz = = ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ ⎢ ˆ ˆ ⎥ 1 2 β α −1 2 β β 0 −1 2 2 0 −1 ⎣ β Iz α β Iz β ⎦ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎣ ⎦ • This is convenient, as the operator is just expressed as a matrix of numbers – no need to derive it again, just store it in computer Operators in Matrix Notation • The matrices for Ix, Iy,and Iz are called the Pauli spin matrices ˆ ⎡ 0 1 2⎤ 1 ⎡0 1 ⎤ ˆ ⎡0 −1 2 i⎤ 1 ⎡0 −i ⎤ ˆ ⎡1 2 0 ⎤ 1 ⎡1 0 ⎤ Ix = ⎢ ⎥ = ⎢ ⎥ Iy = ⎢ ⎥ = ⎢ ⎥ Iz = ⎢ ⎥ = ⎢ ⎥ ⎣1 2 0 ⎦ 2 ⎣1 0⎦ ⎣1 2 i 0 ⎦ 2 ⎣i 0 ⎦ ⎣ 0 −1 2⎦ 2 ⎣0 −1 ⎦ • Express α , β , α and β as 1×2 column and 2×1 row vectors ⎡1⎤ ⎡0⎤ α = ⎢ ⎥ β = ⎢ ⎥ α = [1 0] β = [0 1] ⎣0⎦ ⎣1⎦ • Using matrices, the operations of Ix, Iy, and Iz on α and β , and the orthonormality relationships, are shown below ˆ 1 ⎡0 1⎤⎡1⎤ 1 ⎡0⎤ 1 ˆ 1 ⎡0 1⎤⎡0⎤ 1 ⎡1⎤ 1 Ix α = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = β Ix β = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = α 2⎣1 0⎦⎣0⎦ 2⎣1⎦ 2 2⎣1 0⎦⎣1⎦ 2⎣0⎦ 2 ⎡1⎤ ⎡0⎤ α α = [1 0]⎢ ⎥
    [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]
  • {Dоwnlоаd/Rеаd PDF Bооk} Entangled Kindle
    ENTANGLED PDF, EPUB, EBOOK Cat Clarke | 256 pages | 06 Jan 2011 | Hachette Children's Group | 9781849163941 | English | London, United Kingdom Entangled PDF Book A Trick of the Tail is the seventh studio album by English progressive rock band Genesis. Why the Name Entangled? Shor, John A. Annals of Physics. Bibcode : arXiv The Living Years. Suhail Bennett, David P. Part of a series on. Examples of entanglement in a Sentence his life is greatly complicated by his romantic entanglements. Take the quiz Forms of Government Quiz Name that government! These four pure states are all maximally entangled according to the entropy of entanglement and form an orthonormal basis linear algebra of the Hilbert space of the two qubits. The researchers used a single source of photon pairs that had been entangle d, which means their quantum states are intrinsically linked and any change or measurement of one is mirrored in the other. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit which is now a LOCC operation, since both qubits are in B's lab and then teleporting the qubit back to A. Looking for some great streaming picks? In earlier tests, it couldn't be absolutely ruled out that the test result at one point could have been subtly transmitted to the remote point, affecting the outcome at the second location. Thesis University of California at Berkeley, The Hilbert space of the composite system is the tensor product.
    [Show full text]
  • (Ab Initio) Pathintegral Molecular Dynamics
    (Ab initio) path-integral Molecular Dynamics The double slit experiment Sum over paths: Suppose only two paths: interference The double slit experiment ● Introduce of a large number of intermediate gratings, each containing many slits. ● Electrons may pass through any sequence of slits before reaching the detector ● Take the limit in which infinitely many gratings → empty space Do nuclei really behave classically ? Do nuclei really behave classically ? Do nuclei really behave classically ? Do nuclei really behave classically ? Example: proton transfer in malonaldehyde only one quantum proton Tuckerman and Marx, Phys. Rev. Lett. (2001) Example: proton transfer in water + + Classical H5O2 Quantum H5O2 Marx et al. Nature (1997) Proton inWater-Hydroxyl (Ice) Overlayers on Metal Surfaces T = 160 K Li et al., Phys. Rev. Lett. (2010) Proton inWater-Hydroxyl (Ice) Overlayers on Metal Surfaces Li et al., Phys. Rev. Lett. (2010) The density matrix: definition Ensemble of states: Ensemble average: Let©s define: ρ is hermitian: real eigenvalues The density matrix: time evolution and equilibrium At equilibrium: ρ can be expressed as pure function of H and diagonalized simultaneously with H The density matrix: canonical ensemble Canonical ensemble: Canonical density matrix: Path integral formulation One particle one dimension: K and Φ do not commute, thus, Trotter decomposition: Canonical density matrix: Path integral formulation Evaluation of: Canonical density matrix: Path integral formulation Matrix elements of in space coordinates: acts on the eigenstates from the left: Canonical density matrix: Path integral formulation Where it was used: It can be Monte Carlo sampled, but for MD we need momenta! Path integral isomorphism Introducing a chain frequency and an effective potential: Path integral molecular dynamics Fictitious momenta: introducing a set of P Gaussian integrals : convenient parameter Gaussian integrals known: (adjust prefactor ) Does it work? i.
    [Show full text]
  • Chapter 4 Information Theory
    Chapter Information Theory Intro duction This lecture covers entropy joint entropy mutual information and minimum descrip tion length See the texts by Cover and Mackay for a more comprehensive treatment Measures of Information Information on a computer is represented by binary bit strings Decimal numb ers can b e represented using the following enco ding The p osition of the binary digit 3 2 1 0 Bit Bit Bit Bit Decimal Table Binary encoding indicates its decimal equivalent such that if there are N bits the ith bit represents N i the decimal numb er Bit is referred to as the most signicant bit and bit N as the least signicant bit To enco de M dierent messages requires log M bits 2 Signal Pro cessing Course WD Penny April Entropy The table b elow shows the probability of o ccurrence px to two decimal places of i selected letters x in the English alphab et These statistics were taken from Mackays i b o ok on Information Theory The table also shows the information content of a x px hx i i i a e j q t z Table Probability and Information content of letters letter hx log i px i which is a measure of surprise if we had to guess what a randomly chosen letter of the English alphab et was going to b e wed say it was an A E T or other frequently o ccuring letter If it turned out to b e a Z wed b e surprised The letter E is so common that it is unusual to nd a sentence without one An exception is the page novel Gadsby by Ernest Vincent Wright in which
    [Show full text]
  • Guide for the Use of the International System of Units (SI)
    Guide for the Use of the International System of Units (SI) m kg s cd SI mol K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. Taylor NIST Special Publication 811 2008 Edition Guide for the Use of the International System of Units (SI) Ambler Thompson Technology Services and Barry N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899 (Supersedes NIST Special Publication 811, 1995 Edition, April 1995) March 2008 U.S. Department of Commerce Carlos M. Gutierrez, Secretary National Institute of Standards and Technology James M. Turner, Acting Director National Institute of Standards and Technology Special Publication 811, 2008 Edition (Supersedes NIST Special Publication 811, April 1995 Edition) Natl. Inst. Stand. Technol. Spec. Publ. 811, 2008 Ed., 85 pages (March 2008; 2nd printing November 2008) CODEN: NSPUE3 Note on 2nd printing: This 2nd printing dated November 2008 of NIST SP811 corrects a number of minor typographical errors present in the 1st printing dated March 2008. Guide for the Use of the International System of Units (SI) Preface The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce. The Omnibus Trade and Competitiveness Act of August 1988 [Public Law (PL) 100-418] changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S.
    [Show full text]