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It is well-known in quantum information theory that a positive operator valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalised POVM, namely a function on certain subsets…
Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has…
Von Neumann projections are the main operations by which information can be extracted from the quantum to the classical realm. They are however static processes that do not adapt to the states they measure. Advances in the field of adaptive…
Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in…
We present a mathematical framework based on quantum interval-valued probability measures to study the effect of experimental imperfections and finite precision measurements on defining aspects of quantum mechanics such as contextuality and…
The problem of using observed correlations to infer causal relations is relevant to a wide variety of scientific disciplines. Yet given correlations between just two classical variables, it is impossible to determine whether they arose from…
We investigate the bounding problem of causal effects in experimental studies in which the outcome is truncated by death, meaning that the subject dies before the outcome can be measured. Causal effects cannot be point identified without…
The quantum mechanics postulate called the Born Rule attributes a probabilistic meaning to a wave function. This paper derives the Born Rule from other quantum principles along with a model of the measurement process. The nondeterministic…
Probabilities enter quantum mechanics via Born's rule, the uniqueness of which was proven by Gleason. Busch subsequently relaxed the assumptions of this proof, expanding its domain of applicability in the process. Extending this work to…
It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a von Neumann algebra has a $C^*$-algebra structure. This extends the notion of non-commutative Poisson boundary by including…
This paper investigates system identification problems with Gaussian inputs and quantized observations under fixed thresholds. By reinterpreting the nonlinear effects induced by quantization as the product of the unknown parameter and an…
Starting with the first-order singular Lagrangian, the canonical structures of the noncommutative quantum system on a submanifold embedded in the higher-dimensional Euclidean space are investigated with the projection operator method (POM)…
I argue that the rules of unitary quantum mechanics imply that observers who will themselves be subject to measurements in a linear combination of macroscopic states (``cat" measurements) cannot make reliable predictions on the results of…
Sequential experimental design to discover interventions that achieve a desired outcome is a key problem in various domains including science, engineering and public policy. When the space of possible interventions is large, making an…
In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVM's into POVM's, generally irreversibly, thus loosing some of the…
Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives…
We characterize a value of an observable by a `sum rule' for generally non-commuting observables and a `product rule' when restricted to a maximal commuting subalgebra of observables together with the requirement that the value is unity for…
In a quantum-Bayesian take on quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue…
Computationalism provides a framework for understanding how a mathematically describable physical world could give rise to conscious observations without the need for dualism. A criterion is proposed for the implementation of computations…
Bayesian Optimization (BO) methods are useful for optimizing functions that are expen- sive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the…