Kinematics and Dynamics in Noninertial Quantum Frames of Reference
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Mixing Indistinguishable Systems Leads to a Quantum Gibbs Paradox ✉ ✉ ✉ Benjamin Yadin 1,2 , Benjamin Morris 1 & Gerardo Adesso 1
ARTICLE https://doi.org/10.1038/s41467-021-21620-7 OPEN Mixing indistinguishable systems leads to a quantum Gibbs paradox ✉ ✉ ✉ Benjamin Yadin 1,2 , Benjamin Morris 1 & Gerardo Adesso 1 The classical Gibbs paradox concerns the entropy change upon mixing two gases. Whether an observer assigns an entropy increase to the process depends on their ability to distinguish the gases. A resolution is that an “ignorant” observer, who cannot distinguish the gases, has 1234567890():,; no way of extracting work by mixing them. Moving the thought experiment into the quantum realm, we reveal new and surprising behaviour: the ignorant observer can extract work from mixing different gases, even if the gases cannot be directly distinguished. Moreover, in the macroscopic limit, the quantum case diverges from the classical ideal gas: as much work can be extracted as if the gases were fully distinguishable. We show that the ignorant observer assigns more microstates to the system than found by naive counting in semiclassical sta- tistical mechanics. This demonstrates the importance of accounting for the level of knowl- edge of an observer, and its implications for genuinely quantum modifications to thermodynamics. 1 School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University Park, University ✉ of Nottingham, Nottingham, UK. 2 Wolfson College, University of Oxford, Oxford, UK. email: [email protected]; benjamin. [email protected]; [email protected] NATURE COMMUNICATIONS | (2021) 12:1471 | https://doi.org/10.1038/s41467-021-21620-7 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-21620-7 espite its phenomenological beginnings, thermodynamics work from apparently indistinguishable gases as the informed Dhas been inextricably linked throughout the past century observer. -
Two-Dimensional Rotational Kinematics Rigid Bodies
Rigid Bodies A rigid body is an extended object in which the Two-Dimensional Rotational distance between any two points in the object is Kinematics constant in time. Springs or human bodies are non-rigid bodies. 8.01 W10D1 Rotation and Translation Recall: Translational Motion of of Rigid Body the Center of Mass Demonstration: Motion of a thrown baton • Total momentum of system of particles sys total pV= m cm • External force and acceleration of center of mass Translational motion: external force of gravity acts on center of mass sys totaldp totaldVcm total FAext==mm = cm Rotational Motion: object rotates about center of dt dt mass 1 Main Idea: Rotation of Rigid Two-Dimensional Rotation Body Torque produces angular acceleration about center of • Fixed axis rotation: mass Disc is rotating about axis τ total = I α passing through the cm cm cm center of the disc and is perpendicular to the I plane of the disc. cm is the moment of inertial about the center of mass • Plane of motion is fixed: α is the angular acceleration about center of mass cm For straight line motion, bicycle wheel rotates about fixed direction and center of mass is translating Rotational Kinematics Fixed Axis Rotation: Angular for Fixed Axis Rotation Velocity Angle variable θ A point like particle undergoing circular motion at a non-constant speed has SI unit: [rad] dθ ω ≡≡ω kkˆˆ (1)An angular velocity vector Angular velocity dt SI unit: −1 ⎣⎡rad⋅ s ⎦⎤ (2) an angular acceleration vector dθ Vector: ω ≡ Component dt dθ ω ≡ magnitude dt ω >+0, direction kˆ direction ω < 0, direction − kˆ 2 Fixed Axis Rotation: Angular Concept Question: Angular Acceleration Speed 2 ˆˆd θ Object A sits at the outer edge (rim) of a merry-go-round, and Angular acceleration: α ≡≡α kk2 object B sits halfway between the rim and the axis of rotation. -
Arxiv:1812.08053V1 [Quant-Ph] 19 Dec 2018 This Motivates the Study of Quantum Communication Without a Shared Reference Frame [4]
Communicating without shared reference frames Alexander R. H. Smith1, ∗ 1Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA (Dated: December 20, 2018) We generalize a quantum communication protocol introduced by Bartlett et al. [New. J. Phys. 11, 063013 (2009)] in which two parties communicating do not share a classical reference frame, to the case where changes of their reference frames form a one-dimensional noncompact Lie group. Alice sends to Bob the state ρR ⊗ρS , where ρS is the state of the system Alice wishes to communicate and ρR is the state of an ancillary system serving as a token of her reference frame. Because Bob is ignorant of the relationship between his reference frame and Alice's, he will describe the state ρR ⊗ ρS as an average over all possible reference frames. Bob measures the reference token and applies a correction to the system Alice wished to communicate conditioned on the outcome of the 0 measurement. The recovered state ρS is decohered with respect to ρS , the amount of decoherence depending on the properties of the reference token ρR. We present an example of this protocol when Alice and Bob do not share a reference frame associated with the one-dimensional translation group 0 and use the fidelity between ρS and ρS to quantify the success of the recovery operation. I. INTRODUCTION quire highly entangled states of many qubits. Another possibility for Alice and Bob to communicate Most quantum communication protocols assume that without a shared reference frame is for Alice to send Bob the parties communicating share a classical background a quantum system ρR to serve as a token of her reference reference frame. -
Reference Frame Induced Symmetry Breaking on Holographic Screens
S S symmetry Article Reference Frame Induced Symmetry Breaking on Holographic Screens Chris Fields 1,* , James F. Glazebrook 2,† and Antonino Marcianò 3,‡ 1 23 Rue des Lavandières, 11160 Caunes Minervois, France 2 Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, IL 61920, USA; [email protected] 3 Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China; [email protected] * Correspondence: fi[email protected]; Tel.: +33-6-44-20-68-69 † Adjunct Faculty: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL 61820, USA. ‡ Also: Laboratori Nazionali di Frascati INFN, Frascati, Rome 00044, Italy. Abstract: Any interaction between finite quantum systems in a separable joint state can be viewed as encoding classical information on an induced holographic screen. Here we show that when such an interaction is represented as a measurement, the quantum reference frames (QRFs) deployed to identify systems and pick out their pointer states induce decoherence, breaking the symmetry of the holographic encoding in an observer-relative way. Observable entanglement, contextuality, and classical memory are, in this representation, logical and temporal relations between QRFs. Sharing entanglement as a resource requires a priori shared QRFs. Keywords: black hole; contextuality; decoherence; quantum error-correcting code; quantum refer- ence frame; system identification; channel theory Citation: Fields, C.; Glazebrook, J.F.; Marcianò, A. Reference Frame Induced Symmetry Breaking on Holographic Screens. Symmetry 2021, 1. Introduction 13, 408. https://doi.org/10.3390/ The holographic principle (HP) states, in its covariant formulation, that for any finite sym13030408 spacelike boundary B, open or closed, the classical, thermodynamic entropy S(L(B)) of Academic Editor: Kazuharu Bamba any light-sheet L(B) of B satisfies: S(L(B)) ≤ A(B) Received: 9 February 2021 /4 , (1) Accepted: 27 February 2021 where A(B) is the area of B in Planck units [1]. -
Kinematics, Impulse, and Human Running
Kinematics, Impulse, and Human Running Kinematics, Impulse, and Human Running Purpose This lesson explores how kinematics and impulse can be used to analyze human running performance. Students will explore how scientists determined the physical factors that allow elite runners to travel at speeds far beyond the average jogger. Audience This lesson was designed to be used in an introductory high school physics class. Lesson Objectives Upon completion of this lesson, students will be able to: ஃ describe the relationship between impulse and momentum. ஃ apply impulse-momentum theorem to explain the relationship between the force a runner applies to the ground, the time a runner is in contact with the ground, and a runner’s change in momentum. Key Words aerial phase, contact phase, momentum, impulse, force Big Question This lesson plan addresses the Big Question “What does it mean to observe?” Standard Alignments ஃ Science and Engineering Practices ஃ SP 4. Analyzing and interpreting data ஃ SP 5. Using mathematics and computational thinking ஃ MA Science and Technology/Engineering Standards (2016) ஃ HS-PS2-10(MA). Use algebraic expressions and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one dimension in various situations. ஃ HS-PS2-3. Apply scientific principles of motion and momentum to design, evaluate, and refine a device that minimizes the force on a macroscopic object during a collision. ஃ NGSS Standards (2013) HS-PS2-2. Use mathematical representations to support the claim that the total momentum of a system of objects is conserved when there is no net force on the system. -
Kinematics Big Ideas and Equations Position Vs
Kinematics Big Ideas and Equations Position vs. Time graphs Big Ideas: A straight line on a p vs t graph indicates constant speed (and velocity) An object at rest is indicated by a horizontal line The slope of a line on a p vs. t graph = velocity of the object (steeper = faster) The main difference between speed and velocity is that velocity incorporates direction o Speed will always be positive but velocity can be positive or negative o Positive vs. Negative slopes indicates the direction; toward the positive end of an axis (number line) or the negative end. For motion detectors this indicates toward (-) and away (+), but only because there is no negative side to the axis. An object with continuously changing speed will have a curved P vs. t graph; (curving steeper – speeding up, curving flatter – slowing down) (Instantaneous) velocity = average velocity if P vs. t is a (straight) line. Velocity vs Time graphs Big Ideas: The velocity can be found by calculating the slope of a position-time graph Constant velocity on a velocity vs. time graph is indicated with a horizontal line. Changing velocity is indicated by a line that slopes up or down. Speed increases when Velocity graph slopes away from v=0 axis. Speed decreases when velocity graph slopes toward v=0 axis. The velocity is negative if the object is moving toward the negative end of the axis - if the final position is less than the initial position (or for motion detectors if the object moves toward the detector) The slope of a velocity vs. -
Chapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 4.1 Introduction ............................................................................................................. 1 4.2 Position, Time Interval, Displacement .................................................................. 2 4.2.1 Position .............................................................................................................. 2 4.2.2 Time Interval .................................................................................................... 2 4.2.3 Displacement .................................................................................................... 2 4.3 Velocity .................................................................................................................... 3 4.3.1 Average Velocity .............................................................................................. 3 4.3.3 Instantaneous Velocity ..................................................................................... 3 Example 4.1 Determining Velocity from Position .................................................. 4 4.4 Acceleration ............................................................................................................. 5 4.4.1 Average Acceleration ....................................................................................... 5 4.4.2 Instantaneous Acceleration ............................................................................. 6 Example 4.2 Determining Acceleration from Velocity ......................................... -
Rotational Motion and Angular Momentum 317
CHAPTER 10 | ROTATIONAL MOTION AND ANGULAR MOMENTUM 317 10 ROTATIONAL MOTION AND ANGULAR MOMENTUM Figure 10.1 The mention of a tornado conjures up images of raw destructive power. Tornadoes blow houses away as if they were made of paper and have been known to pierce tree trunks with pieces of straw. They descend from clouds in funnel-like shapes that spin violently, particularly at the bottom where they are most narrow, producing winds as high as 500 km/h. (credit: Daphne Zaras, U.S. National Oceanic and Atmospheric Administration) Learning Objectives 10.1. Angular Acceleration • Describe uniform circular motion. • Explain non-uniform circular motion. • Calculate angular acceleration of an object. • Observe the link between linear and angular acceleration. 10.2. Kinematics of Rotational Motion • Observe the kinematics of rotational motion. • Derive rotational kinematic equations. • Evaluate problem solving strategies for rotational kinematics. 10.3. Dynamics of Rotational Motion: Rotational Inertia • Understand the relationship between force, mass and acceleration. • Study the turning effect of force. • Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration. 10.4. Rotational Kinetic Energy: Work and Energy Revisited • Derive the equation for rotational work. • Calculate rotational kinetic energy. • Demonstrate the Law of Conservation of Energy. 10.5. Angular Momentum and Its Conservation • Understand the analogy between angular momentum and linear momentum. • Observe the relationship between torque and angular momentum. • Apply the law of conservation of angular momentum. 10.6. Collisions of Extended Bodies in Two Dimensions • Observe collisions of extended bodies in two dimensions. • Examine collision at the point of percussion. -
Equivalence of the Symbol Grounding and Quantum System Identification Problems
Information 2014, 5, 172-189; doi:10.3390/info5010172 OPEN ACCESS information ISSN 2078-2489 www.mdpi.com/journal/information Article Equivalence of the Symbol Grounding and Quantum System Identification Problems Chris Fields 528 Zinnia Court, Sonoma, CA 95476, USA; E-Mail: fi[email protected]; Tel.: +1-707-980-5091 Received: 13 November 2013; in revised form: 4 December 2013 / Accepted: 26 February 2014 / Published: 27 February 2014 Abstract: The symbol grounding problem is the problem of specifying a semantics for the representations employed by a physical symbol system in a way that is neither circular nor regressive. The quantum system identification problem is the problem of relating observational outcomes to specific collections of physical degrees of freedom, i.e., to specific Hilbert spaces. It is shown that with reasonable physical assumptions these problems are equivalent. As the quantum system identification problem is demonstrably unsolvable by finite means, the symbol grounding problem is similarly unsolvable. Keywords: decoherence; decompositional equivalence; physical symbol system; observable-dependent exchange symmetry; semantics; tensor-product structure 1. Introduction The symbol grounding problem (SGP) was introduced by Harnad [1] as a generalization and clarification of the semantic under-determination issues raised by Searle [2] in his famous “Chinese room” argument. With reference to a “physical symbol system” model of cognition that posits purely syntactic operations (e.g., [3–6]), Harnad stated the SGP as follows: How can the semantic interpretation of a formal symbol system be made intrinsic to the system, rather than just parasitic on the meanings in our heads? How can the meanings of the meaningless symbol tokens, manipulated solely on the basis of their (arbitrary) shapes, be grounded in anything but other meaningless symbols? [1] (p. -
Are Right Angles Special? the Optimal Primitive Reference Frame
Optimal primitive reference frames. David Jennings1, 2 1Institute for Mathematical Sciences, Imperial College London, London SW7 2BW, United Kingdom 2QOLS, The Blackett Laboratory, Imperial College London, Prince Consort Road, SW7 2BW, United Kingdom (Dated: November 2, 2018) We consider the smallest possible directional reference frames allowed and determine the best one can ever do in preserving quantum information in various scenarios. We find that for the preservation of a single spin state, two orthogonal spins are optimal primitive reference frames, and in a product state do approximately 22% as well as an infinite-sized classical frame. By adding a small amount of entanglement to the reference frame this can be raised to 2(2/3)5 = 26%. Under the different criterion of entanglement-preservation a very similar optimal reference frame is found, however this time for spins aligned at an optimal angle of 87 degrees. In this case 24% of the negativity is preserved. The classical limit is considered numerically, and indicates under the criterion of entanglement preservation, that 90 degrees is selected out non-monotonically, with a peak optimal angle of 96.5 degrees for L = 3 spins. PACS numbers: 03.67.-a, 03.65.Ud, 03.65.Ta I. INTRODUCTION not. In the absence of a shared classical frame, the di- rectionality must be encoded in quantum systems QRFA In information processing it is generally assumed that and QRFB that accompany the two halves of the singlet a large, classical background reference frame is defined state. The quantum information, here entanglement, is and readily available. For example, to measure the spin only partially preserved in the relational properties of of a particle along a certain axis, a Cartesian frame is the composite state. -
Kinematics Definition: (A) the Branch of Mechanics Concerned with The
Chapter 3 – Kinematics Definition: (a) The branch of mechanics concerned with the motion of objects without reference to the forces that cause the motion. (b) The features or properties of motion in an object, regarded in such a way. “Kinematics can be used to find the possible range of motion for a given mechanism, or, working in reverse, can be used to design a mechanism that has a desired range of motion. The movement of a crane and the oscillations of a piston in an engine are both simple kinematic systems. The crane is a type of open kinematic chain, while the piston is part of a closed four‐bar linkage.” Distinguished from dynamics, e.g. F = ma Several considerations: 3.2 – 3.2.2 How do wheels motions affect robot motion (HW 2) 3.2.3 Wheel constraints (rolling and sliding) for Fixed standard wheel Steered standard wheel Castor wheel Swedish wheel Spherical wheel 3.2.4 Combine effects of all the wheels to determine the constraints on the robot. “Given a mobile robot with M wheels…” 3.2.5 Two examples drawn from 3.2.4: Differential drive. Yields same result as in 3.2 (HW 2) Omnidrive (3 Swedish wheels) Conclusion (pp. 76‐77): “We can see from the preceding examples that robot motion can be predicted by combining the rolling constraints of individual wheels.” “The sliding constraints can be used to …evaluate the maneuverability and workspace of the robot rather than just its predicted motion.” 3.3 Mobility and (vs.) Maneuverability Mobility: ability to directly move in the environment. -
Engineering Mechanics
Course Material in Dynamics by Dr.M.Madhavi,Professor,MED Course Material Engineering Mechanics Dynamics of Rigid Bodies by Dr.M.Madhavi, Professor, Department of Mechanical Engineering, M.V.S.R.Engineering College, Hyderabad. Course Material in Dynamics by Dr.M.Madhavi,Professor,MED Contents I. Kinematics of Rigid Bodies 1. Introduction 2. Types of Motions 3. Rotation of a rigid Body about a fixed axis. 4. General Plane motion. 5. Absolute and Relative Velocity in plane motion. 6. Instantaneous centre of rotation in plane motion. 7. Absolute and Relative Acceleration in plane motion. 8. Analysis of Plane motion in terms of a Parameter. 9. Coriolis Acceleration. 10.Problems II.Kinetics of Rigid Bodies 11. Introduction 12.Analysis of Plane Motion. 13.Fixed axis rotation. 14.Rolling References I. Kinematics of Rigid Bodies I.1 Introduction Course Material in Dynamics by Dr.M.Madhavi,Professor,MED In this topic ,we study the characteristics of motion of a rigid body and its related kinematic equations to obtain displacement, velocity and acceleration. Rigid Body: A rigid body is a combination of a large number of particles occupying fixed positions with respect to each other. A rigid body being defined as one which does not deform. 2.0 Types of Motions 1. Translation : A motion is said to be a translation if any straight line inside the body keeps the same direction during the motion. It can also be observed that in a translation all the particles forming the body move along parallel paths. If these paths are straight lines. The motion is said to be a rectilinear translation (Fig 1); If the paths are curved lines, the motion is a curvilinear translation.