This document provides an overview of a lecture on classical and quantum information theory. It discusses topics such as Maxwell's demon, the laws of thermodynamics, Shannon information theory, quantum measurement, the qubit model, and differences between classical and quantum information theory. The lecture aims to compare classical and quantum information concepts and highlight new properties that emerge from quantum mechanics.
Branislav K. Nikoli
ć
Department of Physics and Astronomy, University of Delaware, U.S.A.
PHYS 624: Introduction to Solid State Physics
http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
Dielectrics are materials that have permanent electric dipole moments. All dielectrics are electrical insulators and are mainly used to store electrical energy by utilizing bound electric charges and dipoles within their molecular structure. Important properties of dielectrics include their electric intensity or field strength, electric flux density, dielectric parameters such as dielectric constant and electric dipole moment, and polarization processes including electronic, ionic, and orientation polarization. Dielectrics are characterized by their complex permittivity, which relates to their ability to transmit electric fields and is dependent on factors like frequency, temperature, and humidity that can influence dielectric losses.
Quantum Computing - History and Prospects (ppt slides)VGG Consulting
PowerPoint presentation slides containing a general overview of Quantum Computing its origin, history, and prospects presented on Dec. 14th, 2018 at the weekly seminar of the Ronin Institute for Independent Scholars (http://ronininstitute.org/).
Recording of the seminar talk on YouTube: https://youtu.be/HpGfNzPTKHI
Quantum computers have the potential to vastly outperform classical computers for certain problems. They make use of quantum bits (qubits) that can exist in superpositions of states and become entangled with each other. This allows quantum computers to perform calculations on all possible combinations of inputs simultaneously. However, building large-scale quantum computers faces challenges such as maintaining quantum coherence long enough to perform useful computations. Researchers are working to develop quantum algorithms and overcome issues like decoherence. If successful, quantum computers could solve problems in domains like cryptography, simulation, and machine learning that are intractable for classical computers.
Quantum information theory deals with integrating information theory with quantum mechanics by studying how information can be stored and retrieved from quantum systems. Quantum computing uses quantum physics and quantum bits (qubits) that can exist in superpositions of states to perform computations in parallel and solve problems like factoring prime numbers faster than classical computers. Key challenges for quantum computing include preventing decoherence and protecting fragile quantum states.
The document discusses the particle-wave duality in physics. It covers several key topics:
1) Early debates on the nature of light as either particles or waves, including experiments by Newton, Huygens, and Young.
2) Planck's work introducing the constant h and quantizing energy, laying foundations for quantum physics.
3) Einstein's explanation of the photoelectric effect supporting light behaving as particles called "light quanta".
4) De Broglie's hypothesis that all fundamental objects have both particle and wave properties, represented by his famous equation relating momentum and wavelength.
Quantum mechanics describes quantum states that can exist in superposition, where an element can exist in multiple states simultaneously. When measured, the element collapses into a single definite state. Quantum computing uses this principle of superposition, where qubits can represent 0 and 1 simultaneously, allowing massive parallelism that exceeds classical computers.
The document discusses photonic crystals, which are periodic electromagnetic media that can exhibit photonic band gaps. It provides examples of photonic crystals found in nature and techniques for fabricating synthetic photonic crystals, including layer-by-layer lithography, the woodpile crystal structure, two-photon lithography, and holographic lithography. Intentional defects in photonic crystals can be used to trap light and guide it in waveguides or cavities, allowing control of light propagation.
Branislav K. Nikoli
ć
Department of Physics and Astronomy, University of Delaware, U.S.A.
PHYS 624: Introduction to Solid State Physics
http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
Dielectrics are materials that have permanent electric dipole moments. All dielectrics are electrical insulators and are mainly used to store electrical energy by utilizing bound electric charges and dipoles within their molecular structure. Important properties of dielectrics include their electric intensity or field strength, electric flux density, dielectric parameters such as dielectric constant and electric dipole moment, and polarization processes including electronic, ionic, and orientation polarization. Dielectrics are characterized by their complex permittivity, which relates to their ability to transmit electric fields and is dependent on factors like frequency, temperature, and humidity that can influence dielectric losses.
Quantum Computing - History and Prospects (ppt slides)VGG Consulting
PowerPoint presentation slides containing a general overview of Quantum Computing its origin, history, and prospects presented on Dec. 14th, 2018 at the weekly seminar of the Ronin Institute for Independent Scholars (http://ronininstitute.org/).
Recording of the seminar talk on YouTube: https://youtu.be/HpGfNzPTKHI
Quantum computers have the potential to vastly outperform classical computers for certain problems. They make use of quantum bits (qubits) that can exist in superpositions of states and become entangled with each other. This allows quantum computers to perform calculations on all possible combinations of inputs simultaneously. However, building large-scale quantum computers faces challenges such as maintaining quantum coherence long enough to perform useful computations. Researchers are working to develop quantum algorithms and overcome issues like decoherence. If successful, quantum computers could solve problems in domains like cryptography, simulation, and machine learning that are intractable for classical computers.
Quantum information theory deals with integrating information theory with quantum mechanics by studying how information can be stored and retrieved from quantum systems. Quantum computing uses quantum physics and quantum bits (qubits) that can exist in superpositions of states to perform computations in parallel and solve problems like factoring prime numbers faster than classical computers. Key challenges for quantum computing include preventing decoherence and protecting fragile quantum states.
The document discusses the particle-wave duality in physics. It covers several key topics:
1) Early debates on the nature of light as either particles or waves, including experiments by Newton, Huygens, and Young.
2) Planck's work introducing the constant h and quantizing energy, laying foundations for quantum physics.
3) Einstein's explanation of the photoelectric effect supporting light behaving as particles called "light quanta".
4) De Broglie's hypothesis that all fundamental objects have both particle and wave properties, represented by his famous equation relating momentum and wavelength.
Quantum mechanics describes quantum states that can exist in superposition, where an element can exist in multiple states simultaneously. When measured, the element collapses into a single definite state. Quantum computing uses this principle of superposition, where qubits can represent 0 and 1 simultaneously, allowing massive parallelism that exceeds classical computers.
The document discusses photonic crystals, which are periodic electromagnetic media that can exhibit photonic band gaps. It provides examples of photonic crystals found in nature and techniques for fabricating synthetic photonic crystals, including layer-by-layer lithography, the woodpile crystal structure, two-photon lithography, and holographic lithography. Intentional defects in photonic crystals can be used to trap light and guide it in waveguides or cavities, allowing control of light propagation.
Quantum computation: EPR Paradox and Bell's InequalityStefano Franco
1) The document discusses quantum computation, including basic concepts like qubits, superposition, entanglement, and EPR paradox.
2) It explains that quantum computers can perform operations on data using quantum phenomena like superposition and entanglement. This allows for computations that classical computers cannot perform under the Church-Turing thesis.
3) Examples are given showing how a quantum protocol using an entangled EPR pair can solve a certain information processing task more efficiently than a classical protocol.
Quantum computing uses quantum mechanics phenomena like superposition and entanglement to perform calculations exponentially faster than classical computers for certain problems. While quantum computers have shown promise in areas like optimization, simulation, and encryption cracking, significant challenges remain in scaling up quantum bits and reducing noise and errors. Current research aims to build larger quantum registers of 50+ qubits to demonstrate quantum advantage and explore practical applications, with the future potential to revolutionize fields like artificial intelligence, materials design, and drug discovery if full-scale quantum computers can be realized.
Superconductivity is the ability of certain materials to conduct electric current with practically zero resistance. This capacity produces interesting and potentially useful effects. For a material to behave as a superconductor, low temperatures are required.
Dr. K. Ramya gave a lecture on superconductivity. The BCS theory proposed by Bardeen, Cooper and Schrieffer explains electron-phonon interaction in superconductors. In normal conductors, electrons scatter off vibrating atoms, increasing resistance. In superconductors, electron-phonon interaction decreases scattering, lowering energy. Electrons form Cooper pairs with equal and opposite momenta. Below a critical temperature, interaction between Cooper pairs and the positive ion core vanishes, resulting in zero resistivity and superconductivity. Applications of superconductivity include more efficient electrical generators, transformers, transmission lines, magnetic levitation, fast electrical switching, computer logic and storage, SQUIDs for magnetometry, and
Electronic band structures in crystals can be understood using Bloch's theorem. Bloch's theorem states that the eigenstates of electrons moving in a periodic potential can be written as a plane wave multiplied by a periodic function. This leads to the formation of allowed energy bands separated by forbidden band gaps. The energy bands arise because the electron momentum is restricted to the first Brillouin zone of the crystal lattice. Bloch's theorem provides insights into the distinction between metals, semiconductors and insulators by explaining whether the Fermi energy lies in an allowed band or forbidden band gap.
Introduction to Quantum Computation. Part - 1Arunabha Saha
Introduction to quantum computation. Here the very basic maths described needed for quantum information theory as well as computation. Postulates of quantum mechanics and the Hisenberg`s Uncertainty principle. Basic operator theories are described here.
Versão do seminário apresentado por Celia Olivero (Horiba) na seção UCS do Instituto Nacional de Engenharia de Superfícies no dia 28 de junho para um público de 18 estudantes, professores e profissionais de empresas.
Lecture 5: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
This document provides an overview of quantum computing, including its history, basic concepts, applications, advantages, difficulties, and future directions. It discusses how quantum computing originated in the 1980s with the goal of building a computer that is millions of times faster than classical computers and theoretically uses no energy. The basic concepts covered include quantum mechanics, superpositioning, qubits, quantum gates, and how quantum computers could perform calculations that are intractable on classical computers, such as factoring large numbers. The document also outlines some of the challenges facing quantum computing as well as potential future advances in the field.
This document summarizes a lecture on modern physics and quantum mechanics. It discusses infinite potential barriers, finite potential barriers, and quantum tunneling. For an infinite barrier, particles reflect completely. For a finite barrier, particles can partially penetrate the barrier due to quantum tunneling, with probability of penetration decreasing as the barrier height or width increases.
This slide starts from a basic explanation between Bit and Qubit. It then follows with a brief history behind Quantum Computer, current trends, and update with concerns to make the quantum computer practically useful.
Quantum computing and quantum communications utilize principles of quantum mechanics such as superposition and entanglement to process and transmit information in novel ways. Current research is exploring how to build reliable quantum computers and networks using technologies like ion traps, quantum dots, and optical methods. While still in early stages, quantum information science shows promise for solving computationally difficult problems in fields such as artificial intelligence, cybersecurity, and drug discovery. Pioneering work by groups like D-Wave, IBM, and China are helping advance our understanding of how to harness quantum effects for powerful new computing and communication applications.
Hs nanotechnology and electronics presentation updated_september_2011ananthababuanu
This document discusses applications of nanotechnology in electronics, computing, memory and storage, displays, and circuitry. It explains how nanotechnology allows for smaller transistors, circuitry, and memory, enabling increased processing power and decreased costs according to Moore's Law. Examples are given of how nanotechnology improves hard drive capacity and size over time. The document also explores uses of nanotechnology in transparent electrodes for displays and microscopy techniques for viewing circuits.
Nanotechnology Presentation For Electronic Industrytabirsir
Nanoelectronics aims to process, transmit, and store information using properties of matter at the nanoscale that are different from macroscale properties. Relevant length scales are a few nanometers for molecules acting as transistors or memory, and up to 999 nm for quantum dots using electron spin. While microelectronics uses gate sizes as small as 50 nm, it does not qualify as nanoelectronics as it does not exploit new physical properties related to reduced size.
Quantum computers have the potential to solve certain problems much faster than classical computers by exploiting principles of quantum mechanics, such as superposition and entanglement. However, building large-scale, reliable quantum computers faces challenges related to decoherence and controlling quantum systems. Current research aims to develop quantum algorithms and overcome issues in scaling up quantum hardware to perform more complex computations than today's most powerful supercomputers.
The document discusses heterojunctions and p-n junctions. It defines a heterojunction as the interface between two dissimilar semiconductors with different band gaps. There are three types of heterojunctions based on band alignment: type I where bands straddle, type II where bands are staggered, and type III where there is a broken gap. A p-n heterojunction diode forms when a p-doped and n-doped semiconductor meet; electrons flow from the higher to lower Fermi level side and holes in the opposite direction.
Quantum computing is a type of computation that harnesses the collective properties of quantum states, such as superposition, interference, and entanglement, to perform calculations.
This presentation is designed to elucidate about the Quantum Computing - History - Principles - QUBITS - Quantum Computing Models - Applications - Advantages and Disadvantages.
This document discusses molecular electronics and nanoelectronics. It describes two approaches to developing nanoelectronic devices - continuing to scale down existing solid-state devices, or developing new devices using molecules (molecular electronics). Molecular electronics involves using organic or organometallic molecules as the basic electronic components, such as in molecular wires, diodes, transistors, and LEDs. Challenges in molecular electronics include making reliable electrical contacts to organic molecules and improving carrier mobility and material stability.
Fundamental principle of information to-energy conversion.Fausto Intilla
Abstract. - The equivalence of 1 bit of information to entropy was given by Landauer in 1961 as kln2, k the Boltzmann constant. Erasing information implies heat dissipation and the energy of 1 bit would then be (the
Landauer´s limit) kT ln 2, T being the ambient temperature. From a quantum-cosmological point of view the minimum quantum of energy in the universe corresponds today to a temperature of 10^-29 ºK, probably forming a cosmic background of a Bose condensate [1]. Then, the bit with minimum energy today in the Universe is a quantum of energy 10^-45 ergs, with an equivalent mass of 10^-66 g. Low temperature implies low energy per bit and, of course, this is the way for faster and less energy dissipating computing devices. Our conjecture is this: the possibility of a future access to the CBBC (a coupling/channeling?) would mean a huge
jump in the performance of these devices.
Quantum computation: EPR Paradox and Bell's InequalityStefano Franco
1) The document discusses quantum computation, including basic concepts like qubits, superposition, entanglement, and EPR paradox.
2) It explains that quantum computers can perform operations on data using quantum phenomena like superposition and entanglement. This allows for computations that classical computers cannot perform under the Church-Turing thesis.
3) Examples are given showing how a quantum protocol using an entangled EPR pair can solve a certain information processing task more efficiently than a classical protocol.
Quantum computing uses quantum mechanics phenomena like superposition and entanglement to perform calculations exponentially faster than classical computers for certain problems. While quantum computers have shown promise in areas like optimization, simulation, and encryption cracking, significant challenges remain in scaling up quantum bits and reducing noise and errors. Current research aims to build larger quantum registers of 50+ qubits to demonstrate quantum advantage and explore practical applications, with the future potential to revolutionize fields like artificial intelligence, materials design, and drug discovery if full-scale quantum computers can be realized.
Superconductivity is the ability of certain materials to conduct electric current with practically zero resistance. This capacity produces interesting and potentially useful effects. For a material to behave as a superconductor, low temperatures are required.
Dr. K. Ramya gave a lecture on superconductivity. The BCS theory proposed by Bardeen, Cooper and Schrieffer explains electron-phonon interaction in superconductors. In normal conductors, electrons scatter off vibrating atoms, increasing resistance. In superconductors, electron-phonon interaction decreases scattering, lowering energy. Electrons form Cooper pairs with equal and opposite momenta. Below a critical temperature, interaction between Cooper pairs and the positive ion core vanishes, resulting in zero resistivity and superconductivity. Applications of superconductivity include more efficient electrical generators, transformers, transmission lines, magnetic levitation, fast electrical switching, computer logic and storage, SQUIDs for magnetometry, and
Electronic band structures in crystals can be understood using Bloch's theorem. Bloch's theorem states that the eigenstates of electrons moving in a periodic potential can be written as a plane wave multiplied by a periodic function. This leads to the formation of allowed energy bands separated by forbidden band gaps. The energy bands arise because the electron momentum is restricted to the first Brillouin zone of the crystal lattice. Bloch's theorem provides insights into the distinction between metals, semiconductors and insulators by explaining whether the Fermi energy lies in an allowed band or forbidden band gap.
Introduction to Quantum Computation. Part - 1Arunabha Saha
Introduction to quantum computation. Here the very basic maths described needed for quantum information theory as well as computation. Postulates of quantum mechanics and the Hisenberg`s Uncertainty principle. Basic operator theories are described here.
Versão do seminário apresentado por Celia Olivero (Horiba) na seção UCS do Instituto Nacional de Engenharia de Superfícies no dia 28 de junho para um público de 18 estudantes, professores e profissionais de empresas.
Lecture 5: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
This document provides an overview of quantum computing, including its history, basic concepts, applications, advantages, difficulties, and future directions. It discusses how quantum computing originated in the 1980s with the goal of building a computer that is millions of times faster than classical computers and theoretically uses no energy. The basic concepts covered include quantum mechanics, superpositioning, qubits, quantum gates, and how quantum computers could perform calculations that are intractable on classical computers, such as factoring large numbers. The document also outlines some of the challenges facing quantum computing as well as potential future advances in the field.
This document summarizes a lecture on modern physics and quantum mechanics. It discusses infinite potential barriers, finite potential barriers, and quantum tunneling. For an infinite barrier, particles reflect completely. For a finite barrier, particles can partially penetrate the barrier due to quantum tunneling, with probability of penetration decreasing as the barrier height or width increases.
This slide starts from a basic explanation between Bit and Qubit. It then follows with a brief history behind Quantum Computer, current trends, and update with concerns to make the quantum computer practically useful.
Quantum computing and quantum communications utilize principles of quantum mechanics such as superposition and entanglement to process and transmit information in novel ways. Current research is exploring how to build reliable quantum computers and networks using technologies like ion traps, quantum dots, and optical methods. While still in early stages, quantum information science shows promise for solving computationally difficult problems in fields such as artificial intelligence, cybersecurity, and drug discovery. Pioneering work by groups like D-Wave, IBM, and China are helping advance our understanding of how to harness quantum effects for powerful new computing and communication applications.
Hs nanotechnology and electronics presentation updated_september_2011ananthababuanu
This document discusses applications of nanotechnology in electronics, computing, memory and storage, displays, and circuitry. It explains how nanotechnology allows for smaller transistors, circuitry, and memory, enabling increased processing power and decreased costs according to Moore's Law. Examples are given of how nanotechnology improves hard drive capacity and size over time. The document also explores uses of nanotechnology in transparent electrodes for displays and microscopy techniques for viewing circuits.
Nanotechnology Presentation For Electronic Industrytabirsir
Nanoelectronics aims to process, transmit, and store information using properties of matter at the nanoscale that are different from macroscale properties. Relevant length scales are a few nanometers for molecules acting as transistors or memory, and up to 999 nm for quantum dots using electron spin. While microelectronics uses gate sizes as small as 50 nm, it does not qualify as nanoelectronics as it does not exploit new physical properties related to reduced size.
Quantum computers have the potential to solve certain problems much faster than classical computers by exploiting principles of quantum mechanics, such as superposition and entanglement. However, building large-scale, reliable quantum computers faces challenges related to decoherence and controlling quantum systems. Current research aims to develop quantum algorithms and overcome issues in scaling up quantum hardware to perform more complex computations than today's most powerful supercomputers.
The document discusses heterojunctions and p-n junctions. It defines a heterojunction as the interface between two dissimilar semiconductors with different band gaps. There are three types of heterojunctions based on band alignment: type I where bands straddle, type II where bands are staggered, and type III where there is a broken gap. A p-n heterojunction diode forms when a p-doped and n-doped semiconductor meet; electrons flow from the higher to lower Fermi level side and holes in the opposite direction.
Quantum computing is a type of computation that harnesses the collective properties of quantum states, such as superposition, interference, and entanglement, to perform calculations.
This presentation is designed to elucidate about the Quantum Computing - History - Principles - QUBITS - Quantum Computing Models - Applications - Advantages and Disadvantages.
This document discusses molecular electronics and nanoelectronics. It describes two approaches to developing nanoelectronic devices - continuing to scale down existing solid-state devices, or developing new devices using molecules (molecular electronics). Molecular electronics involves using organic or organometallic molecules as the basic electronic components, such as in molecular wires, diodes, transistors, and LEDs. Challenges in molecular electronics include making reliable electrical contacts to organic molecules and improving carrier mobility and material stability.
Fundamental principle of information to-energy conversion.Fausto Intilla
Abstract. - The equivalence of 1 bit of information to entropy was given by Landauer in 1961 as kln2, k the Boltzmann constant. Erasing information implies heat dissipation and the energy of 1 bit would then be (the
Landauer´s limit) kT ln 2, T being the ambient temperature. From a quantum-cosmological point of view the minimum quantum of energy in the universe corresponds today to a temperature of 10^-29 ºK, probably forming a cosmic background of a Bose condensate [1]. Then, the bit with minimum energy today in the Universe is a quantum of energy 10^-45 ergs, with an equivalent mass of 10^-66 g. Low temperature implies low energy per bit and, of course, this is the way for faster and less energy dissipating computing devices. Our conjecture is this: the possibility of a future access to the CBBC (a coupling/channeling?) would mean a huge
jump in the performance of these devices.
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This document provides an overview of classical and quantum mechanics concepts relevant to statistical thermodynamics. It begins with an introduction to classical mechanics using Lagrangian and Hamiltonian formulations. It then discusses limitations of classical mechanics and introduces key concepts in quantum mechanics, including the Schrodinger equation. Examples are provided of a particle in a box and harmonic oscillator to illustrate differences between classical and quantum descriptions of particle behavior.
Max Planck's and his Major contributions in Quantum MechanicsRUSHABHSHAH310
Max Planck was a German physicist born in 1858 who made pioneering contributions to quantum theory and is considered the founder of quantum mechanics. Some of his major accomplishments include deducing the relationship between energy and frequency of radiation, which led to his famous Planck's constant and Planck's radiation law. He also helped establish quantum theory and the need for quantum mechanics to describe phenomena at atomic and subatomic scales. Planck received numerous honors for his revolutionary work, including the Nobel Prize in Physics in 1918.
The Higgs boson (or Higgs particle) produced by the quantum excitation of the Higgs field, that was confirmed on 2012 in the ATLAS detector at CERN is supposed to be the explanation for the mass of elementary particles. In this paper I will explain why this Higgs field is a new dimension which I refer to as the Grid dimensions (or Grid extra dimensions). This paper will explain what are the expected measurements regarding the Higgs particles based on this assumption. In this paper I will show what will be the future measured evidence that he Higgs particle measured at the particle accelerators is a quantum excitation of the Grid dimensions themselves.
σT 4
where σ is the Stefan-Boltzmann constant.
1) The document discusses a computer simulation called Starsmasher that astrophysicists use to model binary star mergers like that of V1309 Scorpii.
2) Starsmasher uses smoothed particle hydrodynamics (SPH) which treats fluids as interacting parcels to efficiently simulate gas dynamics in stellar events.
3) The document provides details on how Starsmasher simulations work and the goals of modeling the light curve and visual appearance of V1309 Scorpii's merger event.
Planck was able to account for the measured spectral distribution of radiation from a thermal source by postulating that the energies of harmonic oscillators are quantized. Einstein then used this idea to explain the photoelectric effect. The Planck radiation law provides the frequency distribution of stored energy in a resonator in thermal equilibrium. It avoids the ultraviolet catastrophe seen in the Rayleigh-Jeans law. Einstein introduced phenomenological coefficients (A and B) to describe absorption, stimulated emission, and spontaneous emission in a two-level system, which relate to the Planck radiation law.
MASSIVE PHOTON HYPOTHESIS OPENS DOORS TO NEW FIELDS OF RESEARCHijrap
1) A massive photon hypothesis is proposed, where the photon mass is directly calculated from kinetic gas theory to be 1.25605 x 10-39 kg.
2) This photon mass explains various experiments like light deflection near the Sun and the gravitational redshift.
3) The photon gas is found to behave as a perfect blackbody and ideal gas, with photons having 6 degrees of freedom.
4) The thermal de Broglie wavelength of this photon gas is calculated to be 1.75967 x 10-3 m, matching the wavelength of the cosmic microwave background radiation.
5) This links the CMB radiation to being continuously generated by the photon gas permeating space, rather than being a relic of
Informational Nature of Dark Matter and Dark Energy and the Cosmological Cons...OlivierDenis15
In this article, realistic quantitative estimation of dark matter and dark energy considered as informational phenomena
have been computed, thereby explaining certain anomalies and effects within the universe. Moreover, by the same conceptual
approach, the cosmological constant problem has been reduced by almost 120 orders of magnitude in the prediction of the
vacuum energy from a quantum point of view. We argue that dark matter is an informational field with finite and quantifiable
negative mass, distinct from the conventional fields of matter of quantum field theory and associated with the number of bits of
information in the observable universe, while dark energy is negative energy, calculated as the energy associated with dark
matter. Since dark energy is vacuum energy, it emerges from dark matter as a collective potential of all particles with their
individual zero-point energy via Landauer's principle
This document provides an overview of quantum mechanics. It begins by explaining that quantum mechanics describes the motion of subatomic particles and is needed to understand the properties of atoms and molecules. It then discusses some key developments in quantum mechanics, including Planck's quantum theory of radiation, Einstein's explanation of the photoelectric effect, de Broglie's hypothesis of matter waves, Heisenberg's uncertainty principle, and Schrodinger's wave equation. The document also compares classical and quantum mechanics and provides examples of quantum mechanical applications like atomic orbitals and black body radiation.
This document provides an overview of dimuon analyses at the LHC and discusses big data challenges. It outlines the Standard Model and motivations for new physics searches. The CMS detector is described, focusing on muon reconstruction challenges. Data selection and efficiency measurements are discussed. The analysis philosophy of searching for a narrow resonance over the Drell-Yan continuum is presented.
This document provides a summary of quantum mechanical and solid state physics concepts:
1. It reviews the Schrodinger equation and wavefunction solutions for quantum systems like electrons and atoms. The wavefunction Ψ relates to real physical measurements through the Born interpretation of probability density.
2. When atoms come together to form solids, the discrete energy levels of individual atoms overlap and form continuous energy bands. This allows electrons in solids to take on only certain allowed energy values.
3. Models like the Kronig-Penny model illustrate how the periodic potential of a solid crystal results in an electronic band structure with permitted energy bands separated by forbidden gaps.
This document summarizes an upcoming presentation on using computational modeling and experimental testing to better understand atmospheric entry of spacecraft. It discusses how different facilities can simulate some but not all entry conditions, and how multidisciplinary modeling is needed due to the complex coupled physics involved. Experimental testing in plasma wind tunnels can characterize the high-temperature reacting flow environment, while computational modeling requires approaches that span continuum to rarefied regimes to fully capture the multi-scale physics. Improving predictive capabilities will help design future planetary missions.
This document provides an overview of classical physics in the late 19th century, including mechanics, electromagnetism, thermodynamics, and the atomic theory of matter. It discusses outstanding questions like the structure of atoms and blackbody radiation. New discoveries like X-rays and radioactivity added complications. This set the stage for the development of modern physics theories like relativity and quantum mechanics to resolve issues classical physics could not.
This document provides a summary of quantum mechanical concepts and solid state physics. It begins with a review of quantum mechanics and the Schrodinger equation. It then discusses the wave nature of electrons and how the Schrodinger equation describes the wavefunction and probability of finding an electron. It also covers energy band diagrams and how the periodic potential in solids leads to the formation of allowed energy bands. It discusses these concepts for isolated atoms, silicon crystals, and the one-dimensional Kronig-Penny model.
This document discusses thermodynamic principles and concepts. It defines key thermodynamic terms like system, environment, isolated system, open system, closed system, state parameters, and equations of state. The first law of thermodynamics states that the change in internal energy of a system equals the heat transferred plus work done on the system. The second law states that the entropy of any isolated system always increases and approaches a maximum value. Entropy is a measure of disorder in a system and is related to the number of microscopic arrangements that can produce a given macrostate.
This document discusses blackbody radiation and the key laws that govern it. It introduces blackbody radiation as electromagnetic radiation emitted by a perfect absorber and emitter of energy based solely on its temperature. It describes Stefan's Law, which relates the total power radiated to temperature, and Wien's Law, which connects the peak wavelength to temperature. It also explains Planck's Law, which provided an accurate description of the spectral distribution of blackbody radiation and was crucial for the development of quantum theory. The document concludes that understanding the principles of blackbody radiation is important for applications in physics, astrophysics, and engineering.
Final parsec problem of black hole mergers and ultralight dark matterSérgio Sacani
When two galaxies merge, they often produce a supermassive black hole binary (SMBHB) at
their center. Numerical simulations with cold dark matter show that SMBHBs typically stall out
at a distance of a few parsecs apart, and take billions of years to coalesce. This is known as the
final parsec problem. We suggest that ultralight dark matter (ULDM) halos around SMBHBs can
generate dark matter waves due to gravitational cooling. These waves can effectively carry away
orbital energy from the black holes, rapidly driving them together. To test this hypothesis, we
performed numerical simulations of black hole binaries inside ULDM halos. Our results imply that
ULDM waves can lead to the rapid orbital decay of black hole binaries.
This document discusses whether quantum mechanics is involved in the early evolution of the universe and if a Machian relationship between gravitons and gravitinos can help answer this question. It proposes that:
1) Gravitons and gravitinos carry information and their relationship, described as a Mach's principle, conserves this information from the electroweak era to today. This suggests quantum mechanics may not be essential in early universe formation.
2) A minimum amount of initial information, such as a small value for Planck's constant, is needed to set fundamental cosmological parameters and could be transferred from a prior universe.
3) Early spacetime may have had a pre-quantum state with low entropy and degrees of freedom
Similar to Lecture on classical and quantum information (20)
The document introduces the Keldysh technique, which is used to calculate non-equilibrium Green's functions. It discusses equilibrium Green's functions, the Keldysh contour used to calculate non-equilibrium Green's functions, and the application of the Keldysh technique. Key aspects covered include Green's functions on the Keldysh contour, the Larkin representation, identities among Green's functions, equations of motion for Green's functions on the Keldysh contour, and Keldysh Green's functions. The goal is to provide an overview of the Keldysh technique.
Preparation and properties of polycrystalline YBa2Cu3o7-x and Fe mixturesKrzysztof Pomorski
The polycrystalline samples of YBa2Cu3O7-x High-Temperature Superconductor (HTS), also called „YBCO-123”, were prepared by mixing (II) oxide (CuO), carbonate (BaCO3) and yttrium (III) oxide (Y2O3) powders and followed by a heat treatment high temperature (900 °C - 950 °C) flowing oxygen. The polycrystalline samples of YBCO-Fe composites were prepared by grinding the mixture of single phased YBCO-123 and small of iron (1% and 3% wt.), followed over by a heat treatment . The results of structural (SEM, EDS, Raman spectroscopy), magnetic (AC susceptibility and magnetization measurements) and magneto-transport on produced composites will be presented. Scanning electron microscopy (SEM) for YBCO and Fe mixtures showed iron particles homogeneously placed on YBCO grains boundaries. As the concentration of iron particles increased the critical temperature decreased. The magnetization measurements LN temperature revealed transition from diamagnetic to paramagnetic behaviour of YBCO-Fe samples originated from the iron grains.
From embodied Artificial Intelligence to Artificial LifeKrzysztof Pomorski
The methodological stages presented in embodied Artificial Intelligence are given. Systematically we broaden the concept AI so finally we can approach systems related to Artificial Life.
Justification of canonical quantization of Josephson effect in various physic...Krzysztof Pomorski
This document discusses the justification of canonical quantization of the Josephson effect and modifications due to large capacitance energy. It presents the commonly used canonical quantization approach and derives the Josephson junction Hamiltonian in second quantization. It also discusses corrections to the Cooper pair box model that arise when the capacitance energy is comparable to the superconducting gap.
Brief introduction is given to Rapid Single Flux Quantum (RSFQ) electronics. It can be useful both for physicist and electrical engineer. Idea of classical superconducting computer is explained and such computer has also potential to be integrated with superconducting quantum computer.
Field Induced Josephson Junction (FIJJ) is defined as the physical system made by placement of ferromagnetic strip directly or indirectly [insulator layer in-between] on the top of superconducting strip [3, 4, 7]. The analysis conducted in extended Ginzburg-Landau, Bogoliubov-de Gennes and RCSJ [11] models essentially points that the system is in most case a weak-link Josephson junction [2] and sometimes has features of tunneling Josephson junction [1]. Generalization of Field Induced Josephson junctions leads to the case of network of robust coupled field induced Josephson junctions [4] that interact in inductive way. Also the scheme of superconducting Random Access Memory (RAM) for Rapid Single Flux [8, 9] quantum (RSFQ) computer is drawn [6, 10] using the concept of tunneling Josephson junction [1] and Field Induced Josephson junction [3, 4].
The given presentation is also available by YouTube (https://www.youtube.com/watch?v=uIqXqiwDsSM).
Literature
[1]. B.D.Josephson, Possible new effects in superconductive tunnelling, PL, Vol.1, No. 251, 1962
[2]. K.Likharev, Josephson junctions Superconducting weak links, RMP, Vol. 51, No. 101, 1979
[3]. K.Pomorski and P.Prokopow, Possible existence of field induced Josephson junctions, PSS B, Vol.249, No.9, 2012
[4]. K.Pomorski, PhD thesis: Physical description of unconventional Josephson junction, Jagiellonian University, 2015
[4]. K.Pomorski, H.Akaike, A.Fujimaki, Towards robust coupled field induced Josephson junctions, arxiv:1607.05013, 2016
[6]. K.Pomorski, H.Akaike, A.Fujimaki, Relaxation method in description of RAM memory cell in RSFQ computer, Procedings of Applied Conference 2016 (in progress)
[7]. J.Gelhausen and M.Eschrig, Theory of a weak-link superconductor-ferromagnet Josephson structure, PRB, Vol.94, 2016
[8]. K.K. Likharev, Rapid Single Flux Quantum Logic (http://pavel.physics.sunysb.edu/RSFQ/Research/WhatIs/rsfqre2m.html)
[9]. Proceedings of Applied Superconductivity Confence 2016, plenary talk by N.Yoshikawa, Low-energy high-performance computing based on superconducting technology (http://ieeecsc.org/pages/plenary-series-applied-superconductivity-conference-2016-asc-2016#Plenary7)
[10]. A.Y.Herr and Q.P.Herr, Josephson magnetic random access memory system and method, International patent nr:8 270 209 B2, 2012
[11]. J.A.Blackburn, M.Cirillo, N.Gronbech-Jensen, A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures, arXiv:1602.05316v1, 2016
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Lecture on classical and quantum information
1. Lecture on classical and quantum information theory
Krzysztof Pomorski
University of Warsaw
kdvpomorski@gmail.com
30 marca 2017
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 1 / 66
2. Zakres wykładu:
Omówione zastaną wybrane zagadnienia z klasycznej i kwantowej teorii
informacji. Potwierdzona zostanie teza Landauera. Wskazane zostaną
procesy utraty informacji w trakcie przekazywania danych na odległość.
Nakreślone zostanie odwołanie do termodynamiki. Wskazana zostanie
między innymi teza Żurka o nieklonowalności kubitu.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 2 / 66
3. 1 Motivation
2 Interlink between information and physics
Maxwell’s demon
The four laws of thermodynamics
Thermodynamics of big systems
3 Classical information theory
Shannon theory
Classical measurement
4 Quantum information theory
Analogy between QM and Statistical Physics
Quantum measurement
Qubit as Bloch sphere
New properties from quantum mechanics
Interaction of 2 quantum systems
5 Classical vs quantum information theory
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 3 / 66
4. Motivation
1. China has just launched the world’s first quantum communications
satellite [600kg]. The satellite is both an extreme test of the weird
properties of quantum mechanics, and a technology tested for what could
be the start of a global, unhackable communications network.
2. Quantum Cryptography is like computing all over again. We cannot
possibly tell what the implications may be. Andrew Hilton, director of CSFI
3. Quantum computer going beyond classical computer.
4. Construction of supersensitive devices.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 4 / 66
5. Information and physics [4]
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 5 / 66
6. Chaos vs order in classical systems [12]
Detection of chaos in simple classical systems.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 6 / 66
7. Maxwell’s demon
Maxwell’s demon is the name given to a thought experiment designed to
question the possibility of violating the second law of thermodynamics. It
was formulated and named after the Scottish physicist James Clerk
Maxwell in 1867.
Maxwell’s demon demonstration turns information into energy [if we
accomodate all gas particle on one side thanks to information on position
of particles].
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 7 / 66
8. Small quantum systems
The Ohm law or Heat Flow law does not work properly in small systems
since electron or phonon flow is not fully deterministic [it works on
average].
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 8 / 66
9. The four laws of thermodynamics
Zeroth law of thermodynamics: If two systems are in thermal
equilibrium with a third system, they are in thermal equilibrium with each
other. This law helps define the notion of temperature.
First law of thermodynamics: When energy passes, as work, as heat, or
with matter, into or out from a system, the system’s internal energy
changes in accord with the law of conservation of energy.
Second law of thermodynamics: In a natural thermodynamic process,
the sum of the entropies of the interacting thermodynamic systems
increases.
Third law of thermodynamics: The entropy of a system approaches a
constant value as the temperature approaches absolute zero.With the
exception of non-crystalline solids (glasses) the entropy of a system at
absolute zero is typically close to zero, and is equal to the logarithm of the
product of the quantum ground states.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 9 / 66
10. Thermodynamics of big systems [4]
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 10 / 66
11. Gibbs entropy [11]
In statistical thermodynamics, entropy S is a measure of the number
of microscopic configurations Ω that a thermodynamic system can
have when in a state as specified by some macroscopic variables.
Specifically, assuming that each of the microscopic configurations is
equally probable, the entropy of the system is the natural logarithm of that
number of configurations, multiplied by the Boltzmann constant kb. Let us
consider the system with W degeneracies for given energy. For fixed
internal energy U, volume V, number of particles N,
S = −kb
i
pi log(pi ) = −kb
i
(1/wi )log(1/wi ) = kblogW (U, V , N)
(1)
We can view W = Ω as a measure of our lack of knowledge about
a system [microcanonical ensemble]. 3 type of ensembles are given as:
(1) The Microcanonical ensemble is an isolated system.
(2) The Canonical ensemble is a system in contact with a heat bath.
(3) The Grand Canonical ensemble is a system in contact with a heat and
particle bath.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 11 / 66
12. Schematic representation of communication channel
Rysunek: From Shannon ’s A Mathematical Theory of Communication, page 3.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 12 / 66
13. The Shannon information content h(x) of an outcome x is defined to be
h(x) = log2(
1
P(x)
). (2)
It is measured in bits. [The word ’bit’ is also used to denote a variable
whose value is 0 or 1].
The entropy of an ensemble X is defined to be the average Shannon
information content of an outcome:
H(x) =
1
P(x)
log2(
1
P(x)
), (3)
with the convention for P(x)=0 that 0 × log(1/0) = 0, since
limΘ→0+ Θlog(1/Theta) = 0.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 13 / 66
14. Definition of channel capacity C
The capacity C of a discrete channel is given by
C = LimT→∞
logN(T)
T
, (4)
where N(T) is the number of allowed signals of duration T.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 14 / 66
15. The first theorem deals with communication over a
noiseless channel.
Let a source have entropy H(bits per symbol) and a channel have a
capacity C (bits per transmit at the average rate C/H - symbols per
second over the channel where is arbitrarily small. It is not possible to
transmit at an average rate greater than C/H.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 15 / 66
16. Shannon-Hartley theorem
It states that the channel capacity C, meaning the theoretical
tightest upper bound on the information rate of data that can be
communicated at an arbitrarily low error rate using an average
received signal power S through an analog communication channel
subject to additive white Gaussian noise of power N:
C = B log2 1 +
S
N
(5)
where: C - the channel capacity in bits per second, a theoretical upper
bound on the net bit rate (information rate, sometimes denoted I)
excluding error-correction codes; B - the bandwidth of the channel in hertz
(passband bandwidth in case of a bandpass signal); S - the average
received signal power over the bandwidth (in case of a carrier-modulated
passband transmission, often denoted C), measured in watts (or volts
squared); N - the average power of the noise and interference over the
bandwidth, measured in watts (or volts squared).
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 16 / 66
17. Classical channel with white noise
The capacity of a channel of band W perturbed by white thermal noise
power N when the average transmitter power is limited to P is given by
C = W × log(P + S/N) (6)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 17 / 66
18. Landauer principle: .Each time a single bit of information is erased it the
amount of energy dissipated to environment is kBTln2. where T is the
temperature of enviroment and kB is Boltzmann constant.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 18 / 66
19. Derivation of Landauer principle [4]
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 19 / 66
20. Classical measurement-example
Imagine two wooden balls of Radius R1 and R2 in space [as far from
Pluton ”planet”] that are charged with Q1 and Q2 where Q1 and Q2 have
opposite signs and that they have bounded state. They are analogical to
Newton gravitational problem of 2 bodies. We can photograph two moving
balls by shining radiation on this system. In such case we perturbed the
already existing [because of photon pressure] state by changing balls
trajectory, moving some of charge to vacuum and so on. It is impossible to
copy dynamical behavior of the system without changing it by small
perturbation!!!! In very real sense it is example of classical
non-cloning theorem since we cannot copy the exact classical state
[only with certain approximation]. It is desirable to refer to [17].
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 20 / 66
21. Analogy between Quantum Mechanics and Statistical
Physics
Superfluid liquid helium flowing out of container against gravitational field.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 21 / 66
22. Hamiltonian and Lagrangian formalism
Hamiltonian equations of motion:
d
dt
pk = −
dH
dqk
,
d
dt
qk = −
dH
dpk
.L =
k
d
dt
(q)p − H (7)
Lagrangian equations of motions:
dL
dqk
=
d
dt
dL
d(dqk
dt )
. (8)
H = Ek + Ep =
1
2
m(
d
dt
y)2
+ mgy, L = Ek − Ep =
1
2
m(
d
dt
y)2
− mgy. (9)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 22 / 66
23. Moving from classical to quantum mechanics
We have given equations of motion in Hamiltonian formalism. Suppose
that f (p, q, t) is a function on the manifold. Then
d
dt
f (p, q, t) =
∂f
∂q
dq
dt
+
∂f
∂p
dp
dt
+
∂f
∂t
. Further, one may take p = p(t) and
q = q(t) to be solutions to Hamilton’s equations; that is,
˙q = ∂H
∂p = {q, H}
˙q = ∂H
∂p = {q, H}
˙p = −∂H
∂q = {p, H}
Then
d
dt
f (p, q, t) =
∂f
∂q
∂H
∂p
−
∂f
∂p
∂H
∂q
+
∂f
∂t
= {f , H} +
∂f
∂t
.
Quantization is by replacement Poisson bracket {A, H} with
commutator[A, H] = AH − HA. All observables in Quantum Mechanics
as x, p or any other are operators and they do not need to commute and
they cannot be fully determined at the same time [as it is in classical
mechanics] in general!!! [For example ˆx = x and ˆpx = i
d
dx ] Since
[ˆx, ˆp] = i I they do not commute and cannot be measured at the same
time. For non-commuting variables x and p we have∆x∆p >= /2.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 23 / 66
24. Quantum harmonic oscillator
(H0(x) + V (x))ψ(x) = (
1
2m
ˆp2
+ ˆV (x))ψ(x) = Eψ(x) (10)
(−
2
2m
d2
dx2
+
1
2
kx2
)ψ(x) = Eψ(x) (11)
Here ψ(x) is eigenfunction and E is eigenvalue.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 24 / 66
25. Quantum measurement-example
Particle is at certain measurement with output A1 with certain probability.
Once the measurement says A1 value the whole wavefunctions collapses
and particle in only this state with probability 1.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 25 / 66
26. Qubit and quantum-mechanics
Rysunek: Qubit in hydrogen atom.
ˆH = ( ˆH0 + V (x))|ψ >= E|ψ > (12)
|ψ(x) >=
ψ1(x)
ψ2(x)
...
ψm(x)
(13)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 26 / 66
27. |ψ >= cos(Θ/2)|0 > +sin(Θ/2)eiϕ
|1 >, (14)
|ψ >= cos(Θ/2)
0
1
+ sin(Θ/2)eiϕ 1
0
, (15)
Bloch sphere representation of qubit, where Θ is the ’latitude’ and ϕ the
’longitude’ angles of ϕ on the Bloch sphere.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 27 / 66
28. Physical implementation of qubit
Rysunek: Josephson junction as system of two coupling superconductors.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 28 / 66
29. T1 is the average time that the system takes for its excited state |1 > to
decay to the ground state |0 >. T2 represents the average time over which
the qubit energy-level difference does not vary [10].
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 29 / 66
31. Quantum parallelism
Rysunek: Photon distribution in system.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 31 / 66
32. Quantum entanglement
|ψ >=
1
2
(|0 > |1 > −|1 > |0 >) (16)
Entanglement is the strange phenomenon in which two quantum particles
become so deeply linked that they share the same existence. When this
happens, a measurement on one particle immediately influences the other,
regardless of the distance between them. Measurement with 1 in first qubit
gives state
(|1 >< 1|×I)|ψ >= (|1 >< 1|×I)
1
2
(|0 > |1 > −|1 > |0 >) = −
1
2
|1 > |0 >
(17)
while the measurement of 0 in first qubit gives the state
(|0 >< 0|×I)|ψ >= (|0 >< 0|×I)
1
2
(|0 > |1 > −|1 > |0 >) = +
1
2
|0 > |1 > .
(18)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 32 / 66
34. Interaction of two quantum systems A and B [as qubits]
ˆH = ˆHa + ˆHb + ˆHab = Ha(2times2) × I2by2 + I2by2 × Hb(2times2) + Hab(4times4)
(19)
Here × denotes tensor product and I is identity matrix. The eigenstate of
two isolated (non-interacting systems ˆHab = 0) A and B is given as
|ψ >= |ψA > |ψB >=
ψ1A
ψ2A
ψ1B
ψ2B
. (20)
Consequently we have
(Ha + Hb)|ψ >=
HA(1, 1) HA(1, 2) 0 0
HA(2, 1) HA(2, 2) 0 0
0 0 HB(1, 1) HB(1, 2)
0 0 HB(1, 1) HB(1, 2)
= E|ψ > .
(21)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 34 / 66
35. Idea of quantum computer
The DiVincenzo criteria is a list of conditions that are necessary for
constructing a quantum computer proposed by the theoretical physicist
David P. DiVincenzo in his 2000 paper ”The Physical Implementation of
Quantum Computation”. Quantum computation was first proposed by
Richard Feynman (1982) as a means to efficiently simulate quantum
systems. There have been many proposals of how to construct a quantum
computer, all of which have varying degrees of success against the
different challenges of constructing quantum devices. Some of these
proposals involve using superconducting qubits, trapped ions, liquid and
solid state nuclear magnetic resonance or optical cluster states all of which
have remarkable prospects, however, they all have issues that prevent
practical implementation. The DiVincenzo criteria are a list of conditions
that are necessary for constructing the quantum computer as proposed by
Feynman.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 35 / 66
36. Idea of quantum computer and DiVincenzo criteria
In order to construct a quantum computer the following conditions must
be met by the experimental setup. The first five are necessary for quantum
computation and the remaining two are necessary for quantum
communication.
1. A scalable physical system with well characterised qubits.
2. The ability to initialise the state of the qubits to a simple fiducial state.
3. Long relevant decoherence times.
4. A universal set of quantum gates.
5. A qubit-specific measurement capability.
6. The ability to interconvert stationary and flying qubits.
7. The ability to faithfully transmit flying qubits between specified
locations.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 36 / 66
42. Physical system implementing quantum annealing
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 42 / 66
43. Entanglement in Josephson junction system
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 43 / 66
44. Unitary quantum gates
The diagram representing the action of a unitary matrix U corresponding
to a quantum gate on a qubit in a state UU† = 1.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 44 / 66
50. Quantum teleportation
Quantum reexportation is a process by which quantum information (e.g.
the exact state of an atom or photon) can be transmitted (exactly, in
principle) from one location to another, with the help of classical
communication and previously shared quantum entanglement between the
sending and receiving location.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 50 / 66
51. Example of weak measurement
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 51 / 66
52. Concept of density matrix
The density matrix is generalization of description given by Schrodinger
equation and formally is defined as the outer product of the wavefunction
and its conjugate so it is matrix of the form as ρ(t) = |ψ(t) >< ψ(t)|. We
have H|ψ(t) >= E|ψ(t) > and < ψ(t)|H† =< ψ(t)|E. Liouville-Von
Neumann equation describes the dynamics of density matrix with time:
d
dt
ρ =
i
[ρ, H]. (22)
Average of observable A is given as
< A >= tr(Aρ). (23)
In case of thermal ensemble in equilibrium we have density matrix given as
Z = tr(e−H/(kT))-partition function, ρequilibrium = e−En/kT /Z
[thermodynamical ensemble].
ρ2
=
1 for pure state
< 1 for mixed state
(24)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 52 / 66
54. Quantum Entropy (Von Neumann entropy)
Definition of Entanglement in density matrix picture
Entangled state is when its density matrix cannot be written as tensor
product of two density matrices that is ρ = ρA × ρB.
Quantum entropy is given by formula
H(ρ) = Tr[ρlog(1/ρ)] = −Tr[ρlog(ρ)]. (25)
If ρ is the joint state of two quantum systems A and B then the quantum
mutual information is
I(A, B) = H(ρA) + H(ρB) − H(ρ). (26)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 54 / 66
55. QM equation of motion with presence of dissipation
Decoherence and equations of motion.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 55 / 66
56. QM equation of motion with use of Wigner functions
W (x, p) =
1
π
+∞
−∞
ψ†
(x + y)ψ(x − y)e
2iyp
dy (27)
Equations of motion for Wigner function.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 56 / 66
57. Quantum information theory
Quantum information theory deals with four main topics:
(1) Transmission of classical information over quantum channels.
(2) The tradeoff between acquisition of information about a quantum state
and disturbance of the state (briefly included in quantum cryptography).
(3) Quantifying quantum entanglement.
(4) Transmission of quantum information over quantum channels.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 57 / 66
58. Properties of the Isolated Quantum System with Finite Volume and a
Finite Number of Particles [7] 1) These quantum systems evolve for reversible
equations of motion (Schrodinger’s equation)
2) Poincare’s theorem is also accurate for these systems.
3) For quantum systems, it is also possible to define the entropy of ensemble.
Entropy is a measure of uncertainty about the state of a system. A pure state
provides a maximally complete description of a quantum system. Therefore, any
pure state entropy is zero by definition. For the mixed-state case, the system
corresponds to a set of pure states. Therefore, entropy already exceeds zero.
Assume that the probability of a pure state is near 1. This mixed state is almost
pure and its entropy is almost zero. On the other hand, when all pure states of
the mixed state have equivalent probabilities, entropy reaches a maximum.
4) During the evolution of a quantum system, the pure state can evolve to a pure
state only. In the mixed state, the probabilities of pure states also remain
unchanged. Thus, the entropy of ensemble does not change during the evolution
of a quantum system.
5) We can represent a large quantum system by a small number of parameters
named macroscopic parameters. A large set of pure states defined by microscopic
parameters corresponds to this mixed macroscopic state. The entropy of a
macroscopic state can be calculated based on this pure set. We define this
entropy as macroscopic entropy. In contrast with the entropy of ensemble, the
macroscopic entropy should not be conserved during the evolution of a qsystem.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 58 / 66
59. 6) A quantum system will not be considered to be an isolated system due
to its interaction with the measuring device. Its initial pure state evolves to
a mixed state and its microscopic entropy increases. This evolution cannot
be reversed by inversion of the measured system as inversion of the
measuring device is also necessary.
Poincare recurrence theorem [ from Wikipedia ]
Any dynamical system defined by an ordinary differential equation
determines a flow map f t mapping phase space on itself. The system is
said to be volume-preserving if the volume of a set in phase space is
invariant under the flow. For instance, all Hamiltonian systems are
volume-preserving because of Liouville’s theorem. The theorem is then: If a
flow preserves volume and has only bounded orbits, then for each open set
there exist orbits that intersect the set infinitely often.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 59 / 66
60. Quantum vs classical information
Quantum information differs strongly from classical information,
epitomized by the bit, in many striking and unfamiliar ways. Among these
are the following:
A unit of quantum information is the qubit. Unlike classical digital states
(which are discrete), a qubit is continuous-valued, describable by a
direction on the Bloch sphere. Despite being continuously valued in this
way, a qubit is the smallest possible unit of quantum information. The
reason for this indivisibility is due to the Heisenberg uncertainty principle:
despite the qubit state being continuously-valued, it is impossible to
measure the value precisely. A qubit cannot be (wholly) converted into
classical bits; that is, it cannot be ’read’. This is the no-teleportation
theorem. Despite the awkwardly-named no-teleportation theorem, qubits
can be moved from one physical particle to another, by means of quantum
teleportation. That is, qubits can be transported, independently of the
underlying physical particle. An arbitrary qubit can neither be copied, nor
destroyed. This is the content of the no cloning theorem and the
no-deleting theorem.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 60 / 66
61. Although a single qubit can be transported from place to place (e.g.
via quantum teleportation), it cannot be delivered to multiple
recipients; this is the no-broadcast theorem, and is essentially
implied by the no-cloning theorem. Qubits can be changed, by applying
linear transformations or quantum gates to them, to alter their state.
Classical bits may be combined with and extracted from configurations of
multiple qubits, through the use of quantum gates. That is, two or more
qubits can be arranged in such a way as to convey classical bits. The
simplest such configuration is the Bell state, which consists of two qubits
and four classical bits (i.e. requires two qubits and four classical bits to
fully describe). Quantum information can be moved about, in a quantum
channel, analogous to the concept of a classical communications channel.
Quantum messages have a finite size, measured in qubits; quantum
channels have a finite channel capacity, measured in qubits per second.
Multiple qubits can be used to carry classical bits.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 61 / 66
62. Although n qubits can carry more than n classical bits of
information, the greatest amount of classical information that can
be retrieved is n. This is Holevo’s theorem. Quantum information,
and changes in quantum information, can be quantitatively
measured by using an analogue of Shannon entropy, called the von
Neumann entropy.
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 62 / 66
63. Many of the same entropy measures in classical information theory can
also be generalized to the quantum case, such as Holevo entropy and the
conditional quantum entropy. Quantum algorithms have a different
computational complexity than classical algorithms. The most famous
example of this is Shor’s factoring algorithm, which is not known to have a
polynomial time classical algorithm, but does have a polynomial time
quantum algorithm. Other examples include Grover’s search algorithm,
where the quantum algorithm gives a quadratic speed-up over the best
possible classical algorithm. Quantum key distribution allows
unconditionally secure transmission of classical information, unlike classical
encryption, which can always be broken in principle, if not in practice.
(Note that certain subtle points are hotly debated).
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 63 / 66
64. Holevo bound
Initially the sender, Alice, holds a long classical message. She encodes
letter i (which appears with probability pi ) of this message into a pure
state which, during the transmission, is turned into a possibly mixed
quantum state q i due to the incomplete knowledge of the environment or
of Eve’s actions. These quantum states are then passed on to the receiver,
Bob, who then has the task to infer Alice’s classical message from these
quantum states. The upper bound for the capacity for such a transmission,
i.e. the information I that Bob can obtain about Alice’ s message per sent
quantum state, is known as the Holevo bound
I <= S(ρ) −
i
pi S(ρi ) (28)
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 64 / 66
65. References
[1]. http://www.ieee.ca/millennium/radio/radio_differences.html
[2]. Jakob Foerster, Lecture 1 of the Course on Information Theory, Pattern
Recognition, and Neural Networks. Produced by: David Mac Kay (University of
Cambridge) https://www.youtube.com/watch?v=BCiZc0n6COY
[2]. Elementary gates for quantum computation, Adriano Barenco et al. ,
https://arxiv.org/abs/quant-ph/9503016
[3]. Entanglement in a Quantum Annealing Processor, PRX 4, 2014
[4]. Presentation: The physics of information: from MaxwellŹs demon to
Landauer by Eric Lutz from University of Erlangen-Nrnberg
[5]. Physics of Information F. Alexander Bais J. Doyne Farmer
http://samoa.santafe.edu/media/workingpapers/07-08-029.pdf.
[6]. https://ru.coursera.org/learn/quantum-optics-single-photon/
lecture/eo9Ym/7-3-one-photon-polarization-as-a-qubit
[7]. Basic Paradoxes of Statistical Classical Physics and Quantum Mechanics by
Oleg Kupervasser.
[8]. http://web.physics.ucsb.edu/~quniverse/dhqm-exprob.html
[9]. Quantum memories in atomic ensembles-G.Braunbeck
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 65 / 66
66. [10]. Superconducting circuits and quantum information-Physics today, J.You and
F.Nori, 2005
[11]. Wikipedia
[12]. Deterministic chaos-Shuster
[13]. Landauer and noise
[14]. Anne Hillebrand, PhD thesis: Quantum Protocols involving Multiparticle
Entanglement and their Representations in the zx-calculus, 2011
[15]. Lecture on statistical physics,
http://www.physics.udel.edu/~glyde/PHYS813/Lectures/chapter_6.pdf
[16]. A single quantum cannot be cloned, W.K.Wootters, W.H.Zurek, Nature,
Vol.299, 1982
[17]. Physical Review Letters, Vol. 88., No.21.,2002, Classical No-Cloning
Theorem, A. Daffertshofer et al.
[18]. Teleportation breakthrough made, Paul Rincon, 2004
http://news.bbc.co.uk/2/hi/science/nature/3811785.stm
[19]. Quantum information theory and Holevo bound
https://www.cs.cmu.edu/~odonnell/quantum15/lecture18.pdf
Krzysztof Pomorski (UW) Classical and quantum information theory 30 marca 2017 66 / 66