Measurement Theory

2019 ◽  
pp. 231-258
Author(s):  
P.J.E. Peebles
Keyword(s):  
Quantum Mechanics ◽  
State Vector ◽  
Measurement Theory ◽  
Albert Einstein ◽  
Experimental Tests ◽  
The State ◽  
Statistical Approximation ◽  
Deterministic Theory ◽  
Eugene Wigner

This chapter reviews measurement theory in quantum mechanics. The measurement prescription in quantum mechanics can be stated in a few lines and has found an enormous range of applications, in all of which it has proved to be consistent with logic and experimental tests. However, the implications seem so bizarre that people such as Albert Einstein and Eugene Wigner have argued that the theory cannot be physically complete as its stands. The chapter then extends the prescription to the case where the state vector is not known. It also discusses some of the “paradoxes” of quantum mechanics. Finally, the chapter presents Bell's theorem, which shows that there cannot be a local underlying deterministic theory for which quantum mechanics plays the role of a statistical approximation.

Quantenmechanik und Objektivierbarkeit/ Quantum Mechanics and Objective Reality

1970 ◽  
Vol 25 (12) ◽  
pp. 1954-1957 ◽  
Author(s):  
K. Baumann
Keyword(s):  
Quantum Mechanics ◽  
Density Matrix ◽  
State Vector ◽  
The State ◽  
Measuring Apparatus ◽  
Measuring Process ◽  
Objective Reality ◽  
Objective Description ◽  
The Universe

Abstract Quantum Mechanics and Objective Reality A Schrödinger function (or a density matrix) can he ascribed only to an object whose isolation time is larger than its time of revolution. This condition can never be satisfied for macroscopic bodies. Consequently, the "cut" between object and observer must not separate a macroscopic body (measuring apparatus) from the rest of the universe. Hence in an analysis of the measuring process, the state vector of the universe must be introduced. An interpretation of this state vector is given which provides an objective description of nature.

scholarly journals Measurement theory in classical mechanics

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
So Katagiri
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Uncertainty Relation ◽  
Measurement Theory ◽  
Classical Mechanics ◽  
The State ◽  
Measurement Device ◽  
Von Neumann ◽  
Relative Interpretation ◽  
Classical And Quantum Mechanics

Abstract We investigate measurement theory in classical mechanics in the formulation of classical mechanics by Koopman and von Neumann (KvN), which uses Hilbert space. We show a difference between classical and quantum mechanics in the “relative interpretation” of the state of the target of measurement and the state of the measurement device. We also derive the uncertainty relation in classical mechanics.

State vector and quantization in an over-all space-time view

1955 ◽  
Vol 229 (1177) ◽  
pp. 221-234 ◽  
Keyword(s):  
State Vector ◽  
Three Dimensional ◽  
External Source ◽  
Space Time ◽  
The State ◽  
Field Equations ◽  
Creation And Annihilation Operators ◽  
Alternative Description ◽  
Formal Solutions

The state vector of a quantized system of fields is defined as a functional of external source functions. Physical quantities are related to operators acting on such functionals. This corresponds to an over-all space-time view in which the state describes a space-time evolution and the state vector is not related to any special space-like surface. The field equations are derived by means of a simple formal quantization and are expressed as supplementary conditions restricting the state functionals. The equations are satisfied by generating functionals defined in the three-dimensional operator theory and were given previously. An investigation of the concept of functionals of anticommuting source functions leads immediately to the consideration of an antisymmetric tensor space with its creation and annihilation operators. The creation and annihilation operators introduced by Coester correspond to another representation and the connexion can be easily established. The relationship between different formal solutions of the field equations given by Coester, Nambu and Anderson, and Skyrme is made apparent. An interchange of the role of creation and annihilation operators, which amounts to a functional Fourier transformation, leads to an alternative description in which the state vector of the quantized system is given by a functional of fields which are functions of the four space-time co-ordinates.

scholarly journals A local deterministic model of quantum spin measurement

1995 ◽  
Vol 451 (1943) ◽  
pp. 585-608 ◽  
Keyword(s):  
State Vector ◽  
Quantum Physics ◽  
Deterministic Model ◽  
Quantum Spin ◽  
The State ◽  
Spin States ◽  
Truth Values ◽  
Conventional View ◽  
State Vector Reduction

The conventional view, that Einstein was wrong to believe that quantum physics is local and deterministic, is challenged. A parametrized model, ' Q ' for the state vector evolution of spin-1/2 particles during measurement is developed. Q draws on recent work on ‘riddled basins’ in dynamical systems theory, and is local, determin­istic, nonlinear and time asymmetric. Moreover, the evolution of the state vector to one of two chaotic attractors (taken to represent observed spin states) is effectively uncomputable. Motivation for considering this model arises from speculations about the (time asymmetric and uncomputable) nature of quantum gravity, and the (nonlinear) role of gravity in quantum state vector reduction. Although the evolution of Q s state vector cannot be determined by a numerical algorithm, the probability that initial states in some given region of phase space will evolve to one of these attractors, is itself computable. These probabilities can be made to correspond to observed quantum spin probabilities. In an ensemble sense, the evolution of the state vector to an attractor can be described by a diffusive random walk process, suggesting that deterministic dynamics may underlie recent attempts to model state vector evolution by stochastic equations. Bell’s theorem and a version of the Bell-Kochen-Specker quantum entanglement paradox, as illustrated by Penrose’s ‘magic dodecahedra’, are discussed using Q as a model of quantum spin measurement. It is shown that in both cases, proving an inconsistency with locality demands the existence of definite truth values to certain counterfactual propositions. In Q these deterministic propositions are uncomputable, and no non-algorithmic mathematical solution is either known or suspected. Adapt­ing the mathematical formalist approach, the non-existence of definite truth values to such counterfactual propositions is posited. No inconsistency with experiment is found. As a result, it is claimed that Q is not constrained by Bell’s inequality, locality and determinism notwithstanding.

EPR and Bell’s theorem, and quantum algorithms

Quantum 20/20 ◽  
2019 ◽  
pp. 163-180
Author(s):  
Ian R. Kenyon
Keyword(s):  
Quantum Mechanics ◽  
Quantum Cryptography ◽  
Quantum Algorithms ◽  
Hidden Variables ◽  
Experimental Tests ◽  
Deterministic Theory ◽  
It Implementation ◽  
Gate Structure ◽  
Shor's Algorithm ◽  
Over Time

EPR showed that quantum mechanics is not a local deterministic theory and on this account they argued that it is incomplete. Quantum mechanics predicts correlations over time-like separations. The suggested resolution in terms of local hidden variables is presented. Bell’s analysis leading to experimental tests is described. The experiment of Aspect, Grangier and Roger vindicating quantum mechanics is described. More refined experiments, avoiding conceivable biases, confirm this result. Then computing based on quantum principles is discussed. Bits with two states in a register would be replaced by qubits with values represented by points on the Bloch sphere. Basic gates are presented. Shor’s algorithm to decompose products of primes is described and a gate structure presented to implement it. Implementation would undermine current encryption methods. Quantum cryptography is described using the BB84 protocol. The no-cloning theorem protects this absolutely against attempts to intercept the encryption data.

scholarly journals Some Notes on Counterfactuals in Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 266
Author(s):  
Avshalom Elitzur ◽  
Eliahu Cohen
Keyword(s):  
Quantum Mechanics ◽  
State Vector ◽  
Explanatory Power ◽  
Causal Effects ◽  
Weak Values ◽  
Value Analysis ◽  
Unique Role ◽  
Weak Value ◽  
The Universe

Counterfactuals, i.e., events that could have occurred but eventually did not, play a unique role in quantum mechanics in that they exert causal effects despite their non-occurrence. They are therefore vital for a better understanding of quantum mechanics (QM) and possibly the universe as a whole. In earlier works, we have studied counterfactuals both conceptually and experimentally. A fruitful framework termed quantum oblivion has emerged, referring to situations where one particle seems to "forget" its interaction with other particles despite the latter being visibly affected. This framework proved to have significant explanatory power, which we now extend to tackle additional riddles. The time-symmetric causality employed by the Two State-Vector Formalism (TSVF) reveals a subtle realm ruled by “weak values,” already demonstrated by numerous experiments. They offer a realistic, simple and intuitively appealing explanation to the unique role of quantum non-events, as well as to the foundations of QM. In this spirit, we performed a weak value analysis of quantum oblivion and suggest some new avenues for further research.

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