scholarly journals General Probabilistic Theories with a Gleason-type Theorem

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 588
Author(s):  
Victoria J Wright ◽  
Stefan Weigert
Keyword(s):  
Quantum Theory ◽  
State Space ◽  
Quantum State ◽  
Type Theorem ◽  
General Probabilistic Theories ◽  
Probabilistic Theories

Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment. We identify the class of general probabilistic theories which also admit Gleason-type theorems. It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes. Our result also implies that the standard no-restriction hypothesis applied to effects is not equivalent to the dual no-restriction hypothesis applied to states which is found to be less restrictive.

scholarly journals No-free-information principle in general probabilistic theories

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 157 ◽  
Author(s):  
Teiko Heinosaari ◽  
Leevi Leppäjärvi ◽  
Martin Plávala
Keyword(s):  
Quantum Theory ◽  
State Spaces ◽  
Coin Tossing ◽  
Free Information ◽  
General Probabilistic Theories ◽  
Probabilistic Theories

In quantum theory, the no-information-without-disturbance and no-free-information theorems express that those observables that do not disturb the measurement of another observable and those that can be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these statements do not hold in general and continue to completely specify these two classes of observables. In this way, we obtain characterizations of the probabilistic theories where these statements hold. As a particular class of state spaces we consider the polygon state spaces, in which we demonstrate our results and show that while the no-information-without-disturbance principle always holds, the validity of the no-free-information principle depends on the parity of the number of vertices of the polygons.

scholarly journals Almost Quantum Correlations are Inconsistent with Specker's Principle

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 87 ◽  
Author(s):  
Tomáš Gonda ◽  
Ravi Kunjwal ◽  
David Schmid ◽  
Elie Wolfe ◽  
Ana Belén Sainz
Keyword(s):  
Quantum Theory ◽  
Quantum Correlations ◽  
Popular Belief ◽  
Joint Measurability ◽  
Rule Out ◽  
General Probabilistic Theories ◽  
Probabilistic Theories

Ernst Specker considered a particular feature of quantum theory to be especially fundamental, namely that pairwise joint measurability of sharp measurements implies their global joint measurability (vimeo.com/52923835). To date, Specker's principle seemed incapable of singling out quantum theory from the space of all general probabilistic theories. In particular, its well-known consequence for experimental statistics, the principle of consistent exclusivity, does not rule out the set of correlations known as almost quantum, which is strictly larger than the set of quantum correlations. Here we show that, contrary to the popular belief, Specker's principle cannot be satisfied in any theory that yields almost quantum correlations.

Probability and Explanation

2017 ◽  
Author(s):  
Richard Healey
Keyword(s):  
Quantum Theory ◽  
Quantum State ◽  
Causal Explanation ◽  
Born Rule ◽  
Correct Application

We can use quantum theory to explain an enormous variety of phenomena by showing why they were to be expected and what they depend on. These explanations of probabilistic phenomena involve applications of the Born rule: to accept quantum theory is to let relevant Born probabilities guide one’s credences about presently inaccessible events. We use quantum theory to explain a probabilistic phenomenon by showing how its probabilities follow from a correct application of the Born rule, thereby exhibiting the phenomenon’s dependence on the quantum state to be assigned in circumstances of that type. This is not a causal explanation since a probabilistic phenomenon is not constituted by events that may manifest it: but each of those events does depend causally on events that actually occur in those circumstances. Born probabilities are objective and sui generis, but not all Born probabilities are chances.

Entanglement

2017 ◽  
Author(s):  
Richard Healey
Keyword(s):  
Quantum Theory ◽  
Quantum State ◽  
Physical Condition ◽  
Physical System ◽  
Polarization State ◽  
Quantum Systems ◽  
Ghz State ◽  
Mathematical Representation ◽  
Entangled State ◽  
Physical Relation

Often a pair of quantum systems may be represented mathematically (by a vector) in a way each system alone cannot: the mathematical representation of the pair is said to be non-separable: Schrödinger called this feature of quantum theory entanglement. It would reflect a physical relation between a pair of systems only if a system’s mathematical representation were to describe its physical condition. Einstein and colleagues used an entangled state to argue that its quantum state does not completely describe the physical condition of a system to which it is assigned. A single physical system may be assigned a non-separable quantum state, as may a large number of systems, including electrons, photons, and ions. The GHZ state is an example of an entangled polarization state that may be assigned to three photons.

scholarly journals The Invariant Set Postulate: a new geometric framework for the foundations of quantum theory and the role played by gravity

2009 ◽  
Vol 465 (2110) ◽  
pp. 3165-3185 ◽  
Author(s):  
T. N. Palmer
Keyword(s):  
Quantum Theory ◽  
Science Fiction ◽  
State Space ◽  
Fractal Geometry ◽  
Quantum Physics ◽  
Nonlinear Dynamical Systems ◽  
Physical Reality ◽  
Invariant Set ◽  
Geometric Framework

A new law of physics is proposed, defined on the cosmological scale but with significant implications for the microscale. Motivated by nonlinear dynamical systems theory and black-hole thermodynamics, the Invariant Set Postulate proposes that cosmological states of physical reality belong to a non-computable fractal state-space geometry I , invariant under the action of some subordinate deterministic causal dynamics D I . An exploratory analysis is made of a possible causal realistic framework for quantum physics based on key properties of I . For example, sparseness is used to relate generic counterfactual states to points p ∉ I of unreality, thus providing a geometric basis for the essential contextuality of quantum physics and the role of the abstract Hilbert Space in quantum theory. Also, self-similarity, described in a symbolic setting, provides a possible realistic perspective on the essential role of complex numbers and quaternions in quantum theory. A new interpretation is given to the standard ‘mysteries’ of quantum theory: superposition, measurement, non-locality, emergence of classicality and so on. It is proposed that heterogeneities in the fractal geometry of I are manifestations of the phenomenon of gravity. Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that a successful quantum theory of gravity should unify the causal non-Euclidean geometry of space–time with the atemporal fractal geometry of state space. The task is not to make sense of the quantum axioms by heaping more structure, more definitions, more science fiction imagery on top of them, but to throw them away wholesale and start afresh. We should be relentless in asking ourselves: From what deep physical principles might we derive this exquisite structure? These principles should be crisp, they should be compelling. They should stir the soul. Chris Fuchs ( Gilder 2008 , p. 335)

scholarly journals Spooky action-at-a-distance in general probabilistic theories

2012 ◽  
Vol 376 (45) ◽  
pp. 2926-2930 ◽  
Author(s):  
Giacomo Mauro DʼAriano ◽  
Franco Manessi ◽  
Paolo Perinotti
Keyword(s):  
Action At A Distance ◽  
General Probabilistic Theories ◽  
Probabilistic Theories
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