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Citations Publications citing this paper. Extreme event statistics of daily rainfall: dynamical systems approach G. Fluctuating noise drives Brownian transport. Yoshihiko Hasegawa , Masanori Arita. Generalized statistical mechanics for superstatistical systems. Christian Beck. Hardcover; fading and shelf wear to exterior; bumps to the lower corners; otherwise in good condition with clean text and tight binding.

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## [PDF] Superpositions of probability distributions. - Semantic Scholar

No Jacket. Published by Interscience Publishers, London From: T. About this Item: Interscience Publishers, London, Condition: Near Fine. Original cloth. Xlib: ink stamp on top edge, front flyleaf, and rear flyleaf. Ink number on verso of title page. Lower corners of covers slightly bumped. In the former case, again by considering C'x' as. So, there cannot exist an X-predictor for Z, just as there cannot be a cloner for Z, because Z is not an information variable. Thus, unpredictability is predicted by the superinformation theory's deterministic17 laws.

Its physical explanation is given by the subsidiary theory. In Everettian quantum theory, it is that there are different ' observed outcomes' across the multiverse. But constructor theory has emancipated unpredictability from observers', relative states' and universes', stating it as a qualitative information-theoretic property, just as no-cloning is.

## Topics in a Probability and Statistics Course

Quantum systems exhibit the appearance of stochasticity, which is more than mere unpredictability. Successive measurements of X on N instances of S, each identically prepared in a. But what justifies the expectation in i? By contrast, in unitary quantum theory, it is the decision-theoretic approach that establishes the same conclusion— without probabilistic assumptions.

I shall now give sufficient conditions on superinformation theories for a generalization of those equivalence classes, which I shall call X-indistinguishability classes, to exist on the set of. As the 'trace' operator need not be available in superinformation theories, to construct such equivalence classes I shall deploy fictitious ensembles. This is a novel mathematical construction; properties of such ensembles will be used to define properties of single systems without, of course, any probabilistic or frequentist interpretation.

At this stage, the fx are only labels of equivalence classes. Additional conditions will therefore be needed for the fx to inform decisions in the way that probabilities are assumed to do in stochastic theories including traditional quantum theory via the Born Rule. No conclusion about decisions could possibly follow merely from what the results of measurements on an infinite ensemble would be, which is what my formal definition of the fx is about.

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Fix an N. For any attribute x in X, I define a constructor DN for the task of counting the number of replicas that hold a sharp value x of X:. Thus, whenever presented with the substrate S N on which X N is sharp with value s i. It could be realized, for instance, by measuring the observable 'X' on each of the N substrates in S N , and then adding one unit to the output substrate, initially at 0, for each 'x' detected figure 4.

In quantum theory, it effects a unitary operation:. N to define attributes of the substrate S N , whose limit for N -. ON sharp in output, with value fN. Therefore, crucially, for a given x, F x N can be sharp even if the observable XN is not. Let me now recall the formal expression of the convergence property in quantum theory [31].

## Statistics and applied probability

The convergence property is that for any positive, arbitrarily small e:. A sufficient condition on a superinformation theory for a generalization of these X-indistinguishability classes to exist under it, on the set of all generalized mixtures of attributes of a given observable X, is that it satisfies the following requirements:. The existence of the limit implies that those attributes do not intersect, i.

I shall call [f z x]xeX the X-partition of unity for the attribute z. If z has such a partition of unity, it must be unique because of E1. An X-indistinguishability equivalence class is defined as the set of all attributes with the same X-partition of unity: any two attributes within that class cannot be distinguished by measuring only the observable X on each individual substrate, even in the limit of an infinite ensemble. A superinformation theory admits X-partitions of unity on the set of generalized mixtures of attributes in X if conditions E1 and E2 are satisfied.

A key innovation of this paper is showing how the mathematical construction of an abstract infinite ensemble culminating in the property E1 can define structure on individual systems via property E2 —the attributes Xf and the X-partition of unity—without recourse to the frequency interpretation of probability or any other probabilistic assumption. In quantum theory, the reduced density matrices of the source and target substrate as delivered by an X-measurer acting on z still have the same partition of unity.

The same holds in constructor theory. Thus, the construction that would classify z as being in a certain X-partition of unity can be reinterpreted as providing a classification of [az]x, under the same labellings: the two classifications must coincide. If y has a given X-partition of unity, the X-intrinsic part [az]x of az must have the same one. Likewise for the intrinsic part [bz]x of bz obtained as the output attribute on the target of the X-measurement applied to y : the 'X'-partition of unity of [bz]x is numerically the same as the X-partition of unity of z.

Even though they are numbers between 0 and 1, summing to unity, they need not satisfy other axioms of the probability calculus: for instance, in quantum interference experiments they do not obey the axiom of additivity of probabilities of mutually exclusive events [33]. I shall now establish sufficient conditions on superinformation theories to support the decision-theory argument, thus characterizing decision-supporting superinformation theories.

I shall introduce one of the conditions via the special case of quantum theory.

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Consider the x- and y-components of a qubit spin, X and Y. There exist eigenstates of X, i. Moreover, there exist quantum states on the composite system of two qubits, which are likewise equally weighted' and have the special property that. I shall now require that the analogous property holds in superinformation theories.

While in quantum theory it is straightforward to express this via the powerful tools of linear superpositions, in constructor theory expressing the same conditions will require careful definition in terms of 'generalized mixtures'. The conditions for decision-supporting information theories is that there exist two complementary information observables X and Y such that:. A superinformation theory is decision-supporting if there exist complementary observablesX and Y such that:.

The same, mutatis mutandis, holds for xi, x2.