Abstract The collapse of a quantum state can be understood as a mathematical way to construct a joint probability density even for operators that do not commute . We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. The familiar collapse picture , in which a quantum state collapses after each measurement as a way to construct a joint probability density for consecutive measurements, is equivalent to a no-collapse picture in which Lüders transformers applied to subsequent measurements construct a quantum-mechanics—free subsystem of quantum non-demolition operators, not as a dynamical process but as an alternative mathematical model for the same consecutive measurements. The no-collapse picture is particularly simpler when we apply signal analysis to millions or billions of consecutive measurements.
Abstract The collapse of a quantum state can be understood as a mathematical way to construct a joint probability density even for operators that do not commute . We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. The familiar collapse picture , in which a quantum state collapses after each measurement as a way to construct a joint probability density for consecutive measurements, is equivalent to a no-collapse picture in which Lüders transformers applied to subsequent measurements construct a quantum-mechanics—free subsystem of quantum non-demolition operators, not as a dynamical process but as an alternative mathematical model for the same consecutive measurements. The no-collapse picture is particularly simpler when we apply signal analysis to millions or billions of consecutive measurements.
The collapse of a quantum state is a mathematical sleight of hand that allows the construction of a joint probability density even for operators that are noncommutative. We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. A collapse picture, in which a quantum state collapses after each measurement, can therefore be equivalent to a no-collapse picture that uses a different state and different, mutually commutative, Quantum Non-Demolition operators to construct the same joint probability density for consecutive measurements. The collapse of a state is a mathematical construction that can also be a useful tool when using the Koopman formalism for classical mechanics and signal analysis.
Consecutive quantum measurements performed on the same system can reveal fundamental insights into quantum theory's causal structure, and probe different aspects of the quantum measurement problem. According to the Copenhagen interpretation, measurements affect the quantum system in such a way that the quantum superposition collapses after the measurement, erasing any knowledge of the prior state. We show that a sequence of measurements in a collapse picture is equivalent to a quantum Markov chain, and that considering the unitary evolution of quantum wavefunctions interacting consecutively with more than two detectors reveals an experimentally measurable difference between a collapse and unitary picture. The non-Markovian nature of sequential measurements that we report is consistent with earlier discoveries in optimal quantum state discrimination.
This paper explores the limitations of Kolmogorov's measure-theoretic foundation of probability in the context of quantum mechanics and proposes an extension towards a non-commutative framework. We delve into the mathematical structures underlying quantum probability spaces, focusing on operator algebras and their associated states. We analyze the challenges in defining joint probabilities for non-commuting observables and present an overview of alternative approaches based on operator algebras, quantum logic, and non-commutative integration. Furthermore, we investigate the applications of non-commutative measure theory in quantum information theory, quantum field theory, and the foundations of quantum mechanics, highlighting the potential for a more comprehensive understanding of probabilistic phenomena in quantum systems.
We use a two states quantum spin system S, and thus considering the particular case of three anticommuting elements and the measurement of . We evidence that, during the wave collapse, we have a transition of standard commutation relation of the spin to new commutation relations and this occurs during the interaction of the S system with the macroscopic measurement system M. The reason to accept such view point is that it causes the destruction of the interferential factors and of the fermion creation and annihilation operators of the S system without recourse to further elaborations based on the use of Hamiltonians or other methods. By this formulation we propose a new method in attempting to solve the problem of wave function collapse. The concept of Observable , in use in standard quantum mechanics, is resolved in an abstract entity to which is connected a linear hermitean operator that signs mathematically the operation that we must perform on the wave function in order to obtain the potential and possible values of the observable. It does not commute with a number of other operators characterizing the system and the non commuting rules have a fundamental role in quantum mechanics .They have a logic that must be analyzed in each phase of the non measuring and the measuring processes. When we consider the dynamics of wave function collapse we must account that the observed observable becomes a number ,with proper unity of measurements ,during the measurement, thus the linear hermitean operator to which is connected before the measurement, disappears and in its place it appears a new operator that maintain the non commutativity with the other operators to which the old and disappeared operator was connected.
Quantum non-locality is normally defined via violations of Bell's inequalities that exclude certain classical hidden variable theories from explaining quantum correlations. Another definition of non-locality refers to the wave function collapse thereby one can prepare a quantum state from arbitrary far away. In both cases one can debate on whether non-locality is a real physical phenomenon, e.g. , one can employ formulations of quantum mechanics that do not use collapse, or one can simply refrain from explaining quantum correlations via classical hidden variables. Here we point out that there is a non-local effect within quantum mechanics, i.e. , without involving hidden variables or collapse. This effect is seen via imprecise ( i.e. , interval-valued) joint probability of two observables, which replaces the ill-defined notion of the precise joint probability for non-commuting observables. It is consistent with all requirements for the joint probability, e.g. , those for commuting observales. The non-locality amounts to a fact that (in a two-particle system) the joint imprecise probability of non-commuting two-particle observables ( i.e. , tensor product of single-particle observables) does not factorize into single-particle contributions, even for uncorrelated states of the two-particle system. The factorization is recovered for a less precise ( i.e. , the one involving a wider interval) joint probability. This approach to non-locality reconciles it with locality, since the latter emerges as a less precise description.
A persistent focus on the concept of emergence as a core element of the scientific method allows a clean separation, insofar as this is possible, of the physical and philosophical aspects of the problem of outcomes in quantum mechanics. The philosophical part of the problem is to explain why a closed system has definite experimental outcomes. The physical part is to show mathematically that there exists a limit in which the contradiction between unitary Schroedinger dynamics and a reduction process leading to distinct outcomes becomes negligible according to an explicitly stated criterion, and to make this criterion as objective as possible. The physical problem is solved here by redefining the notion of a quantum state and finding a suitable measure for the change of state upon reduction. The appropriate definition of the quantum state is not as a ray or density operator in Hilbert space, but rather as an equivalence class consisting of all density operators in a given subspace, the members of which all describe the same experimental outcome. For systems containing only subsystems that are integrated with their environments, these equivalence classes can be represented mathematically by projection operators, and the resulting formalism is closely related to that used by von Neumann to study the increase of entropy predicted by the second law of thermodynamics. However, nearly isolated subsystems are reduced only indirectly, as a consequence of their interaction with integrated subsystems. The reduced states of isolated subsystems are the same conditional states used in the definition of quantum discord. The key concepts of decoherence theory can all be adapted to fit this definition of a quantum state, resulting in a unified theory capable of resolving, in principle, all aspects of the quantum measurement problem. The theory thus obtained is weakly objective but not strongly objective.