Related papers: Measures and their random reals
In a recent work, arXiv:2503.05884, we proposed a unified notion of nonclassicality that applies to arbitrary processes in quantum theory, including individual quantum states, measurements, channels, set of these, etc. This notion is…
A sequence $x_1,\dots,x_n,\dots$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $\mu$. After observing each outcome, one is required to give conditional probabilities of the next…
An a priori semimeasure (also known as "algorithmic probability" or "the Solomonoff prior" in the context of inductive inference) is defined as the transformation, by a given universal monotone Turing machine, of the uniform measure on the…
A formulation towards quantifying resource count used in a measurement, that is independent of the model of the measurement dynamics(Quantum/Classical), is considered. For any general measurement with $(M+1)$ discrete outcomes, it is found…
The theory of measurement is employed to elucidate the physical basis of general relativity. For measurements involving phenomena with intrinsic length or time scales, such scales must in general be negligible compared to the (translational…
We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if…
In this paper, the defining properties of a valid measure of the dependence between two random variables are reviewed and complemented with two original ones, shown to be more fundamental than other usual postulates. While other popular…
In this mainly pedagogical article, we discuss under what circumstances measurements play a special role in quantum processes. In particular we discuss the following facts which appear to be a common area of confusion: i) from a fundamental…
Identifying when observed statistics cannot be explained by any reasonable classical model is a central problem in quantum foundations. A principled and universally applicable approach to defining and identifying nonclassicality is given by…
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise…
A non-Boolean extension of the classical probability model is proposed. The non-Boolean probabilities reproduce typical quantum phenomena. The proposed model is more general and more abstract, but easier to interpret, than the quantum…
We show that the spectral measure of any non-commutative polynomial of a non-commutative $n$-tuple cannot have atoms if the free entropy dimension of that $n$-tuple is $n$ (see also work of Mai, Speicher, and Weber). Under stronger…
Unlike Martin-L\"of randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and…
A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This result generalises the ones given up to date.
It is shown that any theory that has certain properties has a measurement problem, in the sense that it makes predictions that are incompatible with measurement outcomes being absolute (that is, unique and non-relational). These properties…
Quantum-enhanced measurements exploit quantum mechanical effects to provide ultra-precise estimates of physical variables for use in advanced technologies, such as frequency calibration of atomic clocks, gravitational waves detection, and…
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its…
In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…
A new, realist interpretation of the quantum measurement processes is given. In this scenario a quantum measurement is a non-equilibrium phase transition in a ``resonant cavity'' formed by the entire physical universe including all its…
We show that one may interpret physical reality as random fields in space-time. These have a probability given by the expectation of a coherent state projection operator, called the Q-function. The resulting dynamical evolution includes…