The Minimal Modal Interpretation of Quantum Theory
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1 Postulate (QM4): Quantum Measurements
Part IIC Lent term 2019-2020 QUANTUM INFORMATION & COMPUTATION Nilanjana Datta, DAMTP Cambridge 1 Postulate (QM4): Quantum measurements An isolated (closed) quantum system has a unitary evolution. However, when an exper- iment is done to find out the properties of the system, there is an interaction between the system and the experimentalists and their equipment (i.e., the external physical world). So the system is no longer closed and its evolution is not necessarily unitary. The following postulate provides a means of describing the effects of a measurement on a quantum-mechanical system. In classical physics the state of any given physical system can always in principle be fully determined by suitable measurements on a single copy of the system, while leaving the original state intact. In quantum theory the corresponding situation is bizarrely different { quantum measurements generally have only probabilistic outcomes, they are \invasive", generally unavoidably corrupting the input state, and they reveal only a rather small amount of information about the (now irrevocably corrupted) input state identity. Furthermore the (probabilistic) change of state in a quantum measurement is (unlike normal time evolution) not a unitary process. Here we outline the associated mathematical formalism, which is at least, easy to apply. (QM4) Quantum measurements and the Born rule In Quantum Me- chanics one measures an observable, i.e. a self-adjoint operator. Let A be an observable acting on the state space V of a quantum system Since A is self-adjoint, its eigenvalues P are real. Let its spectral projection be given by A = n anPn, where fang denote the set of eigenvalues of A and Pn denotes the orthogonal projection onto the subspace of V spanned by eigenvectors of A corresponding to the eigenvalue Pn. -
Quantum Circuits with Mixed States Is Polynomially Equivalent in Computational Power to the Standard Unitary Model
Quantum Circuits with Mixed States Dorit Aharonov∗ Alexei Kitaev† Noam Nisan‡ February 1, 2008 Abstract Current formal models for quantum computation deal only with unitary gates operating on “pure quantum states”. In these models it is difficult or impossible to deal formally with several central issues: measurements in the middle of the computation; decoherence and noise, using probabilistic subroutines, and more. It turns out, that the restriction to unitary gates and pure states is unnecessary. In this paper we generalize the formal model of quantum circuits to a model in which the state can be a general quantum state, namely a mixed state, or a “density matrix”, and the gates can be general quantum operations, not necessarily unitary. The new model is shown to be equivalent in computational power to the standard one, and the problems mentioned above essentially disappear. The main result in this paper is a solution for the subroutine problem. The general function that a quantum circuit outputs is a probabilistic function. However, the question of using probabilistic functions as subroutines was not previously dealt with, the reason being that in the language of pure states, this simply can not be done. We define a natural notion of using general subroutines, and show that using general subroutines does not strengthen the model. As an example of the advantages of analyzing quantum complexity using density ma- trices, we prove a simple lower bound on depth of circuits that compute probabilistic func- arXiv:quant-ph/9806029v1 8 Jun 1998 tions. Finally, we deal with the question of inaccurate quantum computation with mixed states. -
Arxiv:1911.07386V2 [Quant-Ph] 25 Nov 2019
Ideas Abandoned en Route to QBism Blake C. Stacey1 1Physics Department, University of Massachusetts Boston (Dated: November 26, 2019) The interpretation of quantum mechanics known as QBism developed out of efforts to understand the probabilities arising in quantum physics as Bayesian in character. But this development was neither easy nor without casualties. Many ideas voiced, and even committed to print, during earlier stages of Quantum Bayesianism turn out to be quite fallacious when seen from the vantage point of QBism. If the profession of science history needed a motto, a good candidate would be, “I think you’ll find it’s a bit more complicated than that.” This essay explores a particular application of that catechism, in the generally inconclusive and dubiously reputable area known as quantum foundations. QBism is a research program that can briefly be defined as an interpretation of quantum mechanics in which the ideas of agent and ex- perience are fundamental. A “quantum measurement” is an act that an agent performs on the external world. A “quantum state” is an agent’s encoding of her own personal expectations for what she might experience as a consequence of her actions. Moreover, each measurement outcome is a personal event, an expe- rience specific to the agent who incites it. Subjective judgments thus comprise much of the quantum machinery, but the formalism of the theory establishes the standard to which agents should strive to hold their expectations, and that standard for the relations among beliefs is as objective as any other physical theory [1]. The first use of the term QBism itself in the literature was by Fuchs and Schack in June 2009 [2]. -
International Centre for Theoretical Physics
XA<? 6 H 0 <! W IC/95/320 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS BOHM'S THEORY VERSUS DYNAMICAL REDUCTION G.C. Ghirardi and INTERNATIONAL ATOMIC ENERGY R. Grassi AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE VQL 'l 7 Ha ° 7 IC/95/320 ABSTRACT International Atomic Energy Agency and United Nations Educational scientific and Cultural Organization made that theories which exhibit parameter , INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS BOHM'S THEORY VERSUS DYNAMICAL REDUCTION1 1. Introduction. We share today's widespread opinion that Standard Quantum Mechanics (SQM), in spite of its enormous successes, has failed in giving a G.C. Ghirardi satisfactory picture of the world, as we perceive it. The difficulties about International Centre for Theoretical Physics, Trieste, Italy, the conceptual foundations of the theory arising, as is well known, from and the so-called objectification problem, have stimulated various attempts to Department of Theoretical Physics. University di Trieste, Trieste, Italy overcome them. Among these one should mention the search for a and deterministic completion of the theory, the many worlds and many minds intepretations, the so called environment induced superselection rules, the R. Grassi quantum histories approach and the dynamical reduction program. In this Department of Civil Engineering, University of Udine, Udine, Italy. paper we will focus our attention on the only available and precisely formulated examples of a deterministic completion and of a stochastic and nonlinear modification of SQM, i.e., Bohm's theory and the spontaneous reduction models, respectively. It is useful to stress that while the first theory is fully equivalent, from a predictive point of view, to SQM, the second one qualifies itself as a rival of SQM but with empirical divergence so small that it can claim all the same experimental support. -
Arxiv:1511.01069V1 [Quant-Ph]
To appear in Contemporary Physics Copenhagen Quantum Mechanics Timothy J. Hollowood Department of Physics, Swansea University, Swansea SA2 8PP, U.K. (September 2015) In our quantum mechanics courses, measurement is usually taught in passing, as an ad-hoc procedure involving the ugly collapse of the wave function. No wonder we search for more satisfying alternatives to the Copenhagen interpretation. But this overlooks the fact that the approach fits very well with modern measurement theory with its notions of the conditioned state and quantum trajectory. In addition, what we know of as the Copenhagen interpretation is a later 1950’s development and some of the earlier pioneers like Bohr did not talk of wave function collapse. In fact, if one takes these earlier ideas and mixes them with later insights of decoherence, a much more satisfying version of Copenhagen quantum mechanics emerges, one for which the collapse of the wave function is seen to be a harmless book keeping device. Along the way, we explain why chaotic systems lead to wave functions that spread out quickly on macroscopic scales implying that Schr¨odinger cat states are the norm rather than curiosities generated in physicists’ laboratories. We then describe how the conditioned state of a quantum system depends crucially on how the system is monitored illustrating this with the example of a decaying atom monitored with a time of arrival photon detector, leading to Bohr’s quantum jumps. On the other hand, other kinds of detection lead to much smoother behaviour, providing yet another example of complementarity. Finally we explain how classical behaviour emerges, including classical mechanics but also thermodynamics. -
Demnächst Auch in Ihrem Laptop?
The Measurement Problem Johannes Kofler Quantum Foundations Seminar Max Planck Institute of Quantum Optics Munich, December 12th 2011 The measurement problem Different aspects: • How does the wavefunction collapse occur? Does it even occur at all? • How does one know whether something is a system or a measurement apparatus? When does one apply the Schrödinger equation (continuous and deterministic evolution) and when the projection postulate (discontinuous and probabilistic)? • How real is the wavefunction? Does the measurement only reveal pre-existing properties or “create reality”? Is quantum randomness reducible or irreducible? • Are there macroscopic superposition states (Schrödinger cats)? Where is the border between quantum mechanics and classical physics? The Kopenhagen interpretation • Wavefunction: mathematical tool, “catalogue of knowledge” (Schrödinger) not a description of objective reality • Measurement: leads to collapse of the wavefunction (Born rule) • Properties: wavefunction: non-realistic randomness: irreducible • “Heisenberg cut” between system and apparatus: “[T]here arises the necessity to draw a clear dividing line in the description of atomic processes, between the measuring apparatus of the observer which is described in classical concepts, and the object under observation, whose behavior is represented by a wave function.” (Heisenberg) • Classical physics as limiting case and as requirement: “It is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the -
International Centre for Theoretical Physics
... „. •::^i'— IC/94/93 S r- INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS QUANTUM MECHANICS WITH SPONTANEOUS LOCALIZATION AND EXPERIMENTS F. Benatti INTERNATIONAL ATOMIC ENERGY G.C. Ghirardi AGENCY and R. Grassi UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE 1 f I \ •» !r 10/94/93 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization ABSTRACT INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS We examine from an experimental point of v'iew the recently proposed models of spontaneous reduction. We compare their implications about decoherence with those of environmental effects. We discuss the treatment, within the considered models, of the so called quantum telegraph phenomenon and we show that, contrary to what has QUANTUM MECHANICS heen recently stated, no problems are met, Finally, we review recent interesting work WITH SPONTANEOUS LOCALIZATION investigating the implications of dynamical reduction for the proton decay. AND EXPERIMENTS ' F. Heiiatti Dip.irtimento di I-'iska Tcwicii, Univorsita di Trieste. Trieste, July, G.C. (ihirardi 1. INTRODUCTION International Centre for Theoretical Physics, Trieste, Italy One of the most intriguing features of Quantum Mechanics (QM) is that it provides and too good a description of microphenomena to be considered a mere recipe to predict Dipartiniento di Fisica Tc-orica, Univorsita di Trieste, Trieste, Italy probabilities of prospective measurement outcomes from given initial conditions and, and yet, as a fundamental theory of reality, it gives rise to serious conceptual problems, among which the impossibility, in its orthodox interpretation, to think of all physical II. CSrasfli observables of the system under consideration as possessing in all instances definite Dipartiinento di Kisira, Universita di Udinc, Udrne, Italy. -
Quantum Zeno Dynamics from General Quantum Operations
Quantum Zeno Dynamics from General Quantum Operations Daniel Burgarth1, Paolo Facchi2,3, Hiromichi Nakazato4, Saverio Pascazio2,3, and Kazuya Yuasa4 1Center for Engineered Quantum Systems, Dept. of Physics & Astronomy, Macquarie University, 2109 NSW, Australia 2Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy 3INFN, Sezione di Bari, I-70126 Bari, Italy 4Department of Physics, Waseda University, Tokyo 169-8555, Japan June 30, 2020 We consider the evolution of an arbitrary quantum dynamical semigroup of a finite-dimensional quantum system under frequent kicks, where each kick is a generic quantum operation. We develop a generalization of the Baker- Campbell-Hausdorff formula allowing to reformulate such pulsed dynamics as a continuous one. This reveals an adiabatic evolution. We obtain a general type of quantum Zeno dynamics, which unifies all known manifestations in the literature as well as describing new types. 1 Introduction Physics is a science that is often based on approximations. From high-energy physics to the quantum world, from relativity to thermodynamics, approximations not only help us to solve equations of motion, but also to reduce the model complexity and focus on im- portant effects. Among the largest success stories of such approximations are the effective generators of dynamics (Hamiltonians, Lindbladians), which can be derived in quantum mechanics and condensed-matter physics. The key element in the techniques employed for their derivation is the separation of different time scales or energy scales. Recently, in quantum technology, a more active approach to condensed-matter physics and quantum mechanics has been taken. Generators of dynamics are reversely engineered by tuning system parameters and device design. -
Advanced Quantum Theory AMATH473/673, PHYS454
Advanced Quantum Theory AMATH473/673, PHYS454 Achim Kempf Department of Applied Mathematics University of Waterloo Canada c Achim Kempf, September 2017 (Please do not copy: textbook in progress) 2 Contents 1 A brief history of quantum theory 9 1.1 The classical period . 9 1.2 Planck and the \Ultraviolet Catastrophe" . 9 1.3 Discovery of h ................................ 10 1.4 Mounting evidence for the fundamental importance of h . 11 1.5 The discovery of quantum theory . 11 1.6 Relativistic quantum mechanics . 13 1.7 Quantum field theory . 14 1.8 Beyond quantum field theory? . 16 1.9 Experiment and theory . 18 2 Classical mechanics in Hamiltonian form 21 2.1 Newton's laws for classical mechanics cannot be upgraded . 21 2.2 Levels of abstraction . 22 2.3 Classical mechanics in Hamiltonian formulation . 23 2.3.1 The energy function H contains all information . 23 2.3.2 The Poisson bracket . 25 2.3.3 The Hamilton equations . 27 2.3.4 Symmetries and Conservation laws . 29 2.3.5 A representation of the Poisson bracket . 31 2.4 Summary: The laws of classical mechanics . 32 2.5 Classical field theory . 33 3 Quantum mechanics in Hamiltonian form 35 3.1 Reconsidering the nature of observables . 36 3.2 The canonical commutation relations . 37 3.3 From the Hamiltonian to the equations of motion . 40 3.4 From the Hamiltonian to predictions of numbers . 44 3.4.1 Linear maps . 44 3.4.2 Choices of representation . 45 3.4.3 A matrix representation . 46 3 4 CONTENTS 3.4.4 Example: Solving the equations of motion for a free particle with matrix-valued functions . -
Perceived Reality, Quantum Mechanics, and Consciousness
7/28/2015 Cosmology About Contents Abstracting Processing Editorial Manuscript Submit Your Book/Journal the All & Indexing Charges Guidelines & Preparation Manuscript Sales Contact Journal Volumes Review Order from Amazon Order from Amazon Order from Amazon Order from Amazon Order from Amazon Cosmology, 2014, Vol. 18. 231-245 Cosmology.com, 2014 Perceived Reality, Quantum Mechanics, and Consciousness Subhash Kak1, Deepak Chopra2, and Menas Kafatos3 1Oklahoma State University, Stillwater, OK 74078 2Chopra Foundation, 2013 Costa Del Mar Road, Carlsbad, CA 92009 3Chapman University, Orange, CA 92866 Abstract: Our sense of reality is different from its mathematical basis as given by physical theories. Although nature at its deepest level is quantum mechanical and nonlocal, it appears to our minds in everyday experience as local and classical. Since the same laws should govern all phenomena, we propose this difference in the nature of perceived reality is due to the principle of veiled nonlocality that is associated with consciousness. Veiled nonlocality allows consciousness to operate and present what we experience as objective reality. In other words, this principle allows us to consider consciousness indirectly, in terms of how consciousness operates. We consider different theoretical models commonly used in physics and neuroscience to describe veiled nonlocality. Furthermore, if consciousness as an entity leaves a physical trace, then laboratory searches for such a trace should be sought for in nonlocality, where probabilities do not conform to local expectations. Keywords: quantum physics, neuroscience, nonlocality, mental time travel, time Introduction Our perceived reality is classical, that is it consists of material objects and their fields. On the other hand, reality at the quantum level is different in as much as it is nonlocal, which implies that objects are superpositions of other entities and, therefore, their underlying structure is wave-like, that is it is smeared out. -
Pilot-Wave Theory, Bohmian Metaphysics, and the Foundations of Quantum Mechanics Lecture 6 Calculating Things with Quantum Trajectories
Pilot-wave theory, Bohmian metaphysics, and the foundations of quantum mechanics Lecture 6 Calculating things with quantum trajectories Mike Towler TCM Group, Cavendish Laboratory, University of Cambridge www.tcm.phy.cam.ac.uk/∼mdt26 and www.vallico.net/tti/tti.html [email protected] – Typeset by FoilTEX – 1 Acknowledgements The material in this lecture is to a large extent a summary of publications by Peter Holland, R.E. Wyatt, D.A. Deckert, R. Tumulka, D.J. Tannor, D. Bohm, B.J. Hiley, I.P. Christov and J.D. Watson. Other sources used and many other interesting papers are listed on the course web page: www.tcm.phy.cam.ac.uk/∼mdt26/pilot waves.html MDT – Typeset by FoilTEX – 2 On anticlimaxes.. Up to now we have enjoyed ourselves freewheeling through the highs and lows of fundamental quantum and relativistic physics whilst slagging off Bohr, Heisenberg, Pauli, Wheeler, Oppenheimer, Born, Feynman and other physics heroes (last week we even disagreed with Einstein - an attitude that since the dawn of the 20th century has been the ultimate sign of gibbering insanity). All tremendous fun. This week - we shall learn about finite differencing and least squares fitting..! Cough. “Dr. Towler, please. You’re not allowed to use the sprinkler system to keep the audience awake.” – Typeset by FoilTEX – 3 QM computations with trajectories Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation P. Holland (2004) “The notion that the concept of a continuous material orbit is incompatible with a full wave theory of microphysical systems was central to the genesis of wave mechanics. -
Singles out a Specific Basis
Quantum Information and Quantum Noise Gabriel T. Landi University of Sao˜ Paulo July 3, 2018 Contents 1 Review of quantum mechanics1 1.1 Hilbert spaces and states........................2 1.2 Qubits and Bloch’s sphere.......................3 1.3 Outer product and completeness....................5 1.4 Operators................................7 1.5 Eigenvalues and eigenvectors......................8 1.6 Unitary matrices.............................9 1.7 Projective measurements and expectation values............ 10 1.8 Pauli matrices.............................. 11 1.9 General two-level systems....................... 13 1.10 Functions of operators......................... 14 1.11 The Trace................................ 17 1.12 Schrodinger’s¨ equation......................... 18 1.13 The Schrodinger¨ Lagrangian...................... 20 2 Density matrices and composite systems 24 2.1 The density matrix........................... 24 2.2 Bloch’s sphere and coherence...................... 29 2.3 Composite systems and the almighty kron............... 32 2.4 Entanglement.............................. 35 2.5 Mixed states and entanglement..................... 37 2.6 The partial trace............................. 39 2.7 Reduced density matrices........................ 42 2.8 Singular value and Schmidt decompositions.............. 44 2.9 Entropy and mutual information.................... 50 2.10 Generalized measurements and POVMs................ 62 3 Continuous variables 68 3.1 Creation and annihilation operators................... 68 3.2 Some important