Related papers: The Representation of Natural Numbers in Quantum M…
Is quantum mechanics about 'states'? Or is it basically another kind of probability theory? It is argued that the elementary formalism of quantum mechanics operates as a well-justified alternative to 'classical' instantiations of a…
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic…
A certain generalization of the mathematical formalism of quantum mechanics beyond operator algebras is considered. The approach is based on the concept of conditional probability and the interpretation of the Lueders - von Neumann quantum…
The Variational Quantum Eigensolver approach to the electronic structure problem on a quantum computer involves measurement of the Hamiltonian expectation value. Formally, quantum mechanics allows one to measure all mutually commuting or…
Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathematical structure of quantum measurements, the formula for…
The mathematical formulation of Quantum Mechanics is derived from purely operational axioms based on a general definition of "experiment" as a set of transformations. The main ingredient of the mathematical construction is the postulated…
Dirac talked about q-numbers versus c-numbers. Quantum observables are q-number variables that generally do not commute among themselves. He was proposing to have a generalized form of numbers as elements of a noncommutative algebra. That…
The mathematical formalism of Quantum Mechanics is derived or "reconstructed" from more basic considerations of probability theory and information geometry. The starting point is the recognition that probabilities are central to QM: the…
A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear…
Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a…
Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work…
The use of geometric methods has proved useful in the hamiltonian description of classical constrained systems. In this note we provide the first steps toward the description of the geometry of quantum constrained systems. We make use of…
In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of…
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcomes; dually, states can be modelled as functions from the algebra of observables to outcomes. The…
The mechanism of describing quantum states by standard probability (tomographic one) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of the identity…
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields…
Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems, their dynamics, and interaction. Since the inception of quantum theory, it has been…
In this paper, we extend the standard formalism of quantum mechanics to a quantum theory for a total system including one internal measuring apparatus. The internality of the measuring apparatus implies that different decomposition of a…
A proof is given for the Fourier transform for functions in a quantum mechanical Hilbert space on a non-compact manifold in general relativity. In the (configuration space) Newton-Wigner representation we discuss the spectral decomposition…
One of the main challenges of quantum many-body physics is that the dimensionality of the Hilbert space grows exponentially with the system size, which makes it extremely difficult to solve the Schr\"{o}dinger equations of the system. But…