Transformations In Quantum Mechanics Research Articles

Contexts in Quantum Measurement Theory

State transformations in quantum mechanics are described by completely positive maps which are constructed from quantum channels. We call a finest sharp quantum channel a context. The result of a measurement depends on the context under which it is performed. Each context provides a viewpoint of the quantum system being measured. This gives only a partial picture of the system which may be distorted and in order to obtain a total accurate picture, various contexts need to be employed. We first discuss some basic definitions and results concerning quantum channels. We briefly describe the relationship between this work and ontological models that form the basis for contextuality studies. We then consider properties of channels and contexts. For example, we show that the set of sharp channels can be given a natural partial order in which contexts are the smallest elements. We also study properties of channel maps. The last section considers mutually unbiased contexts. These are related to mutually unbiased bases which have a large current literature. Finally, we connect them to completely random channel maps.

Modified correlation theorem for the linear canonical transform with representation transformation in quantum mechanics

As generalization of the fractional Fourier transform, the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties of the LCT are currently known including correlation theorem, but however, these do not generalize very nicely to the classical result for the FT. This paper presents a modified correlation theorem in the LCT domain with representation transformation in quantum mechanics. The derivation is direct and concise since the authors make full use of the Dirac’s representation theory. The proposed theorem is compared with the existing correlation theorem in the literature for the LCT and found to be better and befitting proposition. As an application, the proposed correlation theorem is used to determine the power spectral density of frequency-modulated (FM) signal, which shows that the same FM signal can be transmitted with less bandwidth requirement.

A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics

Abstract The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.

Quantum mechanics in non-inertial reference frames: Time-dependent rotations and loop prolongations

This is the fourth in a series of papers on developing a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group to include transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. In previous work, we have shown that there exist representations of the Galilean line group that uphold the non-relativistic equivalence principle as well as representations that violate the equivalence principle. In these previous studies, the focus was on linear accelerations. In this paper, we undertake an extension of the formulation to include rotational accelerations. We show that the incorporation of rotational accelerations requires a class of loop prolongations of the Galilean line group and their unitary cocycle representations. We recover the centrifugal and Coriolis force effects from these loop representations. Loops are more general than groups in that their multiplication law need not be associative. Hence, our broad theoretical claim is that a Galilean quantum theory that holds in arbitrary non-inertial reference frames requires going beyond groups and group representations, the well-established framework for implementing symmetry transformations in quantum mechanics.

Complex joint probabilities as expressions of reversible transformations in quantum mechanics

The density operator of a quantum state can be represented as a complex joint probability of any two observables whose eigenstates have non-zero mutual overlap. Transformations to a new basis set are then expressed in terms of complex conditional probabilities that describe the fundamental relation between precise statements about the three different observables. Since such transformations merely change the representation of the quantum state, these conditional probabilities provide a state-independent definition of the reversible and therefore effectively deterministic relations between the outcomes of different quantum measurements, including measurements of the same property performed at different times. In this paper, it is shown how classical reality emerges as an approximation to the fundamental laws of quantum determinism expressed by complex conditional probabilities. The quantum mechanical origin of phase spaces and trajectories is identified and implications for the interpretation of quantum measurements are considered. It is argued that the transformation laws of quantum determinism provide a fundamental description of the measurement dependence of empirical reality.

Quantum operations in probabilistic representation

Representing a single qubit by using a set of totally decohered mixed-state qubits is considered. A representation is constructed where physically realizable quantum mechanical transformations correspond approximately to unitary operations on the qubit. It is shown that the representation is valid for multiple qubit systems as well, and the precision of the approximation is sufficient for the realization of quantum algorithms.

Canonical transformations and molecular structure calculations

A canonical transformation method for use in molecular structure calculations is given whereby the original coordinates and momenta of the N-body system are expressed in closed form in terms of the transformed coordinates and momenta. The method is illustrated for a general N-dimensional (i) linear transformation and (ii) point transformation, in phase space. The difficulties in the generalization of this method to arbitrary canonical quantum mechanical transformations are discussed. This method is compared with that of using unitary operators, and its advantages over that method are described.

New Potentials for Old: The Darboux Transformation in Quantum Mechanics

The Darboux transformation in quantum mechanics is reviewed at a basic level. Examples of how this transformation leads to exactly solvable potentials related to the "particle in a box" and the harmonic oscillator are shown in detail. The connection between the Darboux transformation and some modern operator based approaches to quantum mechanics is also outlined.

Study Hankel Transforms and Properties of Bessel Function via Entangled StateRepresentation Transformation in Quantum Mechanics

In Phys. Lett. A 313 (2003) 343 we have found that theself-reciprocal Hankel transformation (HT) is embodied in quantummechanics by a transform between two entangled state representationsof continuum variables. In thiswork we study Hankel transforms and properties of Bessel function viaentangled state representations' transformation in quantum mechanics.

Quantized normal matrices: some exact results and collective field formulation

We formulate and study a class of U ( N ) -invariant quantum mechanical models of large normal matrices with arbitrary rotation-invariant matrix potentials. We concentrate on the U ( N ) singlet sector of these models. In the particular case of quadratic matrix potential, the singlet sector can be mapped by a similarity transformation onto the two-dimensional Calogero–Marchioro–Sutherland model at specific couplings. For this quadratic case we were able to solve the N-body Schrödinger equation and obtain infinite sets of singlet eigenstates of the matrix model with given total angular momentum. Our main object in this paper is to study the singlet sector in the collective field formalism, in the large- N limit. We obtain in this framework the ground state eigenvalue distribution and ground state energy for an arbitrary potential, and outline briefly the way to compute bona-fide quantum phase transitions in this class of models. As explicit examples, we analyze the models with quadratic and quartic potentials. In the quartic case, we also touch upon the disk-annulus quantum phase transition. In order to make our presentation self-contained, we also discuss, in a manner which is somewhat complementary to standard expositions, the theory of point canonical transformations in quantum mechanics for systems whose configuration space is endowed with non-Euclidean metric, which is the basis for constructing the collective field theory.

Conditionally exactly solvable potential and dual transformation in quantum mechanics

We comment that the conditionally exactly solvable potential of Dutt et al. and the exactly solvable potential from which it is derived form a dual system.

From limits of quantum operations to multicopy entanglement witnesses and state-spectrum estimation

Limits of state transformations in quantum mechanics are studied. Impossibility of physical implementation of the transformation ${\ensuremath{\varrho}}^{\ensuremath{\bigotimes}n}\ensuremath{\rightarrow}{\ensuremath{\varrho}}^{n}$ in quantum mechanics is proved. The most natural notions of structural completely positive approximation and structural physical approximations of nonphysical map are introduced. It is shown that these always exist for linear Hermitian maps and can be optimized under natural conditions. It is pointed out that it is physically possible to measure in a simple way the traces of $n\mathrm{th}$ power of quantum state $\mathrm{Tr}({\ensuremath{\varrho}}^{n})$ if only joint measurements on n copies of the system are allowed. This gives the interpretation of Tsallis entropies as values of ``multicopy'' observables. Following this observations, the notion of multicopy entanglement witnesses is defined and examples are provided. Finally, using notion of multicopy observable, a simple method of spectrum state estimation is discussed.

The Inconsistency of the Usual Galilean Transformation in Quantum Mechanics and How To Fix It

Abstract It is shown that the generally accepted statement that one cannot superpose states of different mass in non-relativistic quantum mechanics is inconsistent. It is pointed out that the extra phase induced in a moving system, which was previously thought to be unphysical, is merely the non-relativistic residue of the "twin-paradox" effect. In general, there are phase effects due to proper time differences between moving frames that do not vanish non-relativistically. There are also effects due to the equivalence of mass and energy in this limit. The remedy is to include both proper time and rest energy non-relativis-tically. This means generalizing the meaning of proper time beyond its classical meaning, and introduc­ ing the mass as its conjugate momentum. The result is an uncertainty principle between proper time and mass that is very general, and an integral role for both concepts as operators in non-relativistic physics.

Non-unitary transformations in quantum mechanics: an optical realization

Abstract We show how to construct devices which implement non-unitary transformations on quantum systems with a certain probability of success. The first transforms two specified non-orthogonal states to orthogonal ones while the second can be used to enhance the entanglement of quantum states. The states themselves are composed of qubits which are realized as photons in the dual-rail representation, and these photons serve as the inputs to a six-port. One of the outputs is measured, and if a null result is obtained, the desired non-unitary transformation has been carried out. The transformation which takes non-orthogonal to orthogonal states is an optical realization of an optimal quantum measurement for distinguishing the two non-orthogonal states.

Non-unitary transformations in quantum mechanics: An optical realization

We show how to construct devices which implement non-unitary transformations on quantum systems with a certain probability of success. The first transforms two specified non-orthogonal states to orthogonal ones while the second can be used to enhance the entanglement of quantum states. The states themselves are composed of qubits which are realized as photons in the dual-rail representation, and these photons serve as the inputs to a six-port. One of the outputs is measured, and if a null result is obtained, the desired non-unitary transformation has been carried out. The transformation which takes non-orthogonal to orthogonal states is an optical realization of an optimal quantum measurement for distinguishing the two non-orthogonal states.

A Theorem of Ludwig Revisited

Using a recent result of Busch and Gudder, we reconsider a theorem of Ludwig which allows one to identify a class of effect automorphisms as the symmetry transformations in quantum mechanics.

Comment on “What is a Gauge Transformation in Quantum Mechanics?”

Comment on “What is a Gauge Transformation in Quantum Mechanics?”

Gauge transformations in quantum mechanics and the unification of nonlinear Schrödinger equations

Beginning with ordinary quantum mechanics for spinless particles, together with the hypothesis that all experimental measurements consist of positional measurements at different times, we characterize directly a class of nonlinear quantum theories physically equivalent to linear quantum mechanics through nonlinear gauge transformations. We show that under two physically motivated assumptions, these transformations are uniquely determined: they are exactly the group of time-dependent, nonlinear gauge transformations introduced previously for a family of nonlinear Schrödinger equations. The general equation in this family, including terms considered by Kostin, by Bialynicki-Birula and Mycielski, and by Doebner and Goldin, with time-dependent coefficients, can be obtained from the linear Schrödinger equation through gauge transformation and a subsequent process we call gauge generalization. We thus unify, on fundamental grounds, a rather diverse set of nonlinear time evolutions in quantum mechanics.

Mapping approach to the semiclassical description of nonadiabatic quantum dynamics

A theoretical formulation is outlined that allows us to extend the semiclassical Van Vleck--Gutzwiller formulation to the description of nonadiabatic quantum dynamics on coupled potential-energy surfaces. In this formulation the problem of a classical treatment of discrete quantum degrees of freedom (DoF) such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. The mapping approach thus consists of two steps: an exact quantum-mechanical transformation of discrete onto continuous DoF (the ``mapping'') and a standard semiclassical treatment of the resulting dynamical problem. Extending previous work [G. Stock and M. Thoss, Phys. Rev. Lett. 78, 578 (1997)], various possibilities for obtaining a mapping from discrete to continuous DoF are investigated, in particular the Holstein-Primakoff transformation, Schwinger's theory of angular momentum [in Quantum Theory of Angular Momentum, edited by L. C. Biedenharn and H. V. Dam (Academic, New York, 1965)], and the spin-coherent-state representation. Although all these representations are exact on the quantum-mechanical level, the accuracy of their semiclassical evaluation is shown to differ considerably. In particular, it is shown that a generalization of Schwinger's theory appears to be the only transformation that provides an exact description of a general N-level system within a standard semiclassical evaluation. Exploiting the connection between spin-coherent states and Schwinger's representation for a two-level system, furthermore, a semiclassical initial-value representation of the corresponding spin-coherent-state propagator is derived. Although this propagator represents an approximation, its appealing numerical features make it a promising candidate for the semiclassical description of large molecular systems with many DoF. Adopting various spin-boson-type models (i.e., a two-level system coupled to a single or many DoF), computational studies are presented for Schwinger's and the spin-coherent-state representation, respectively. The performance of the semiclassical approximation in the case of regular and chaotic classical dynamics as well as for multimode electronic relaxation dynamics is discussed in some detail.

Erratum: What is a Gauge Transformation in Quantum Mechanics? [Phys. Rev. Lett. 80, 4613 (1998)

Erratum: What is a Gauge Transformation in Quantum Mechanics? [Phys. Rev. Lett. 80, 4613 (1998)