General Probabilistic Theories Research Articles

Atomistic orthomodular lattices and a generalized probability theory

Atomistic orthomodular lattices and a generalized probability theory

Post-quantum steering is a stronger-than-quantum resource for information processing

We present the first instance where post-quantum steering is a stronger-than-quantum resource for information processing – remote state preparation. In addition, we show that the phenomenon of post-quantum steering is not just a mere mathematical curiosity allowed by the no-signalling principle, but it may arise within compositional theories beyond quantum theory, hence making its study fundamentally relevant. We show these results by formulating a new compositional general probabilistic theory – which we call Witworld – with strong post-quantum features, which proves to be a intuitive and useful tool for exploring steering and its applications beyond the quantum realm.

Incompatibility in General Probabilistic Theories, Generalized Spectrahedra, and Tensor Norms

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements.

Shannon theory beyond quantum: Information content of a source

The information content of a source is defined in terms of the minimum number of bits needed to store the output of the source in a perfectly recoverable way. A similar definition can be given in the case of quantum sources, with qubits replacing bits. In the mentioned cases the information content can be quantified through Shannon's and von Neumann's entropy, respectively. Here we extend the definition of information content to operational probabilistic theories, and prove relevant properties such as the subadditivity and the relation between purity and information content of a state. We prove the consistency of the present notion of information content when applied to the classical and the quantum case. Finally, the relation with one of the notions of entropy that can be introduced in general probabilistic theories, the maximum accessible information, is given in terms of a lower bound.

The operational foundations of PT-symmetric and quasi-Hermitian quantum theory

PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.

Entanglement and Superposition Are Equivalent Concepts in Any Physical Theory.

We prove that given any two general probabilistic theories (GPTs) the following are equivalent: (i)each theory is nonclassical, meaning that neither of their state spaces is a simplex; (ii)each theory satisfies a strong notion of incompatibility equivalent to the existence of "superpositions"; and (iii)the two theories are entangleable, in the sense that their composite exhibits either entangled states or entangled measurements. Intuitively, in the post-quantum GPT setting, a superposition is a set of two binary ensembles of states that are unambiguously distinguishable if the ensemble is revealed before the measurement has occurred, but not if it is revealed after. This notion is important because we show that, just like in quantum theory, superposition in the form of strong incompatibility is sufficient to realize the Bennett-Brassard 1984 protocol for secret key distribution.

Random access test as an identifier of nonclassicality* *This paper is dedicated to Stan Gudder.

Random access codes (RACs) are an intriguing class of communication tasks that reveal an operational and quantitative difference between classical and quantum information processing. We formulate a natural generalization of RACs and call them random access tests (RATs), defined for any finite collection of measurements in an arbitrary finite dimensional general probabilistic theory. These tests can be used to examine collective properties of collections of measurements. We show that the violation of a classical bound in a RAT is a signature of either measurement incompatibility or super information storability. The polygon theories are exhaustively analysed and a critical difference between even and odd polygon theories is revealed.

Foundational Problems of Quantum Theory: Novel Approach to Temporal Probability Density

<p><span lang="EN-US">Present study focuses on some foundational problems of quantum theory specifically deals with the concept of probability density and relating introductory problems. In this sense, the work initially investigates the origins of the general probability theory and re-examines the concepts of spatial and temporal probability densities based on genuine epistemological and ontological arguments. In order to tackle the foundational problems, standard theory is primarily memorised and criticized scientifically and philosophically in terms of foundationally disappearing term of time dependent potential energy within the time and space dependent Schrödinger wave equation. Based on those arguments, the problematic inconsistency between the spatial and temporal probability density functions is underlined. Given the problem, an original approach previously suggested, is concisely described and extended to resolve the existing problem. The novel approach, based on a novel time dependent Schrödinger wave equation, resolves the discrepancy with the classical wave equation and also leads to time dependent temporal probability densities even for the time free potential energies. Novel temporal probability density function is also normalized and has a fluctuation period of around 10<sup>-16 </sup>s which is very short compared to the atomic time scales.</span></p>

Incompatibility in restricted operational theories: connecting contextuality and steering

We investigate the connection between steering and contextuality in general probabilistic theories. We show that for a class of bipartite states the steerability of the state by given set of measurements is equivalent to non-existence of preparation noncontextual hidden variable model for certain restricted theory constructed from the given state and measurements. The connection between steering and contextuality is provided by the concept of incompatibility in restricted theories, which we also investigate.

Classic Probability Revisited (I): Mathematical Models of an Extended Probability Theory

Part I of this paper presents a set of extended mathematical models of probability theory in order to explain the nature, properties, and rules of general probability. It is found that probability is a hyperstructure beyond those of the traditional monotonic and one-dimensional discrete structures. The sample space of probability is not invariant in general cases. Types of vents in the sample space may be refined as joint or disjoint and dependent, independent, or mutuallyexclusive. These newly identified properties lead to a three-dimensional dynamic model of probability structures constrained by the type of sample spaces, the relation of events, and the dependency of events. A set of algebraic operators on the mathematical structures of the general probability theory is derived based on the extended mathematical models of probability. It is revealed that the Bayes’ law needs to be extended in order to fit more general contexts on variant sample spaces and complex event properties in fundamental probability theories. The revisited probability theory enables a rigorous treatment of uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft computing, and brain informatics.

Classic Probability Revisited (II): Algebraic Operations of the Extended Probability Theory

Part II of this paper presents a set of comprehensive algebraic operators on the extended mathematical structures of the general probability theory. It is recognized that the classic probability theory is cyclically defined among a small set of highly coupled operations. In order to solve this fundamental problem, a reductive framework of the general probability theory is introduced. It is found that conditional probability operation on consecutive events is the key to independently manipulate other probability operations. This leads to a revisited framework of rigorous manipulations on general probabilities. It also provides a proof for a revisited Bayes’ law fitting in more general contexts of variant sample spaces and complex event relations in fundamental probability theories. The revisited probability theory enables a rigorous treatment of uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft computing, and brain informatics.

Experimentally bounding deviations from quantum theory for a photonic three-level system using theory-agnostic tomography

If one seeks to test quantum theory against many alternatives in a landscape of possible physical theories, then it is crucial to be able to analyze experimental data in a theory-agnostic way. This can be achieved using the framework of Generalized Probabilistic Theories (GPTs). Here, we implement GPT tomography on a three-level system corresponding to a single photon shared among three modes. This scheme achieves a GPT characterization of each of the preparations and measurements implemented in the experiment without requiring any prior characterization of either. Assuming that the sets of realized preparations and measurements are tomographically complete, our analysis identifies the most likely dimension of the GPT vector space describing the three-level system to be nine, in agreement with the value predicted by quantum theory. Relative to this dimension, we infer the scope of GPTs that are consistent with our experimental data by identifying polytopes that provide inner and outer bounds for the state and effect spaces of the true GPT. From these, we are able to determine quantitative bounds on possible deviations from quantum theory. In particular, we bound the degree to which the no-restriction hypothesis might be violated for our three-level system.

The Extended Data-Adaptive Probability-Based Electrical Resistivity Tomography Inversion Method (E-PERTI) for the Characterization of the Buried Ditch of the Ancient Egnazia (Puglia, Italy)

A geoelectrical survey was carried out outside the walls of the ancient Egnazia (Puglia, Italy) with the aim of enriching the knowledge about its defense system. Nine Electrical Resistivity Tomographies (ERTs) were realized using the dipole-dipole (DD) electrode array, approximately transversal to the walls and equally spaced. The new Extended data-adaptive Probability-based Electrical Resistivity Tomography Inversion Method (E-PERTI) was applied, for the first time, to model the resistivity distribution of a large dataset. Considering some peculiar aspects of the general theory of probability, an optimization of results was reached giving major emphasis to one dataset portion rather than another and inspecting selectively vertical or lateral resistivity variations. In this way, sets of aligned resistivity lows attributable to the trace of an ancient ditch were found.

Operational Theories in Phase Space: Toy Model for the Harmonic Oscillator.

We show how to construct general probabilistic theories that contain an energy observable dependent on position and momentum. The construction is in accordance with classical and quantum theory and allows for physical predictions, such as the probability distribution for position, momentum, and energy. We demonstrate the construction by formulating a toy model for the harmonic oscillator that is neither classical nor quantum. The model features a discrete energy spectrum, a ground state with sharp position and momentum, an eigenstate with a nonpositive Wigner function as well as a state that has tunneling properties. The toy model demonstrates that operational theories can be a viable alternative approach for formulating physical theories.

Analysis of Relationships between Common Distributions Based on Computing Science

Random variables are the most fundamental concept in probability theory and mathematical statistics. Each random variable can have a distribution, different random variables can have different distributions, and different random variables can have the same distribution. There are many random variables, and there are many distributions that can be formed, but there are not many commonly used distributions. In general probability theory and mathematical statistics textbooks, each distribution is explained separately, and the relationship between distributions is occasionally discussed, and there is no detailed description. Establishing the correspondence between distributions plays an important role in mastering the theory and relationship of common distributions. This paper studies the relationship between distributions, and establishes some common distribution diagrams on the basis of distribution relationships, so as to illustrate the connections between some distributions with distribution diagrams.

Риск как гражданско-правовая категория: доктринальные подходы и законодательные позиции

The scientific research is devoted to the law characterization of risk. Risk as a law category has a rather contradictory assessment in matters of subjects activities of civil law, including in situations of bringing them to civil law liability. We present doctrinal opinions regarding the risk concept, as well as the provisions of Russian civil law containing regulatory prescriptions regarding risky activities. We point out that the risk theory correlates with the general scientific theory of probability, and illustrative examples are given that prove such a connection between them. The purpose of the study: to conduct a doctrinal and legal analysis of risk as a civil category, as well as to formulate the author’s definition of the concept under study. The methodological basis of the work is based on such methods as: comparison method, complex analysis method, as well as a number of special scientific cognition methods: historical, comparative legal, formal legal. The author’s formulation of risk concept, including a number of identifying features, is proposed. We indicate that risk is a category of subjective-objective property. We conclude that the fact of the existence of the so-called increased responsibility is presumed by the nature of such activity, which is expressed in an increased risk of its implementation, and which this person is aware of, accepted, and because of its objectivity in the activities of a person cannot be completely excluded.

No Preferred Reference Frame at the Foundation of Quantum Mechanics.

Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information Invariance and Continuity, at the foundation of those axiomatic reconstructions, maps to “no preferred reference frame” (NPRF, aka “the relativity principle”) as it pertains to the invariant measurement of Planck’s constant h for Stern-Gerlach (SG) spin measurements. This is in exact analogy to the relativity principle as it pertains to the invariant measurement of the speed of light c at the foundation of special relativity (SR). Essentially, quantum information theorists have extended Einstein’s use of NPRF from the boost invariance of measurements of c to include the SO(3) invariance of measurements of h between different reference frames of mutually complementary spin measurements via the principle of Information Invariance and Continuity. Consequently, the “mystery” of the Bell states is understood to result from conservation per Information Invariance and Continuity between different reference frames of mutually complementary qubit measurements, and this maps to conservation per NPRF in spacetime. If one falsely conflates the relativity principle with the classical theory of SR, then it may seem impossible that the relativity principle resides at the foundation of non-relativisitic QM. In fact, there is nothing inherently classical or quantum about NPRF. Thus, the axiomatic reconstructions of QM have succeeded in producing a principle account of QM that reveals as much about Nature as the postulates of SR.

General Probabilistic Theories with a Gleason-type Theorem

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.

Any physical theory of nature must be boundlessly multipartite nonlocal

We introduce the class of Genuinely Local Operation and Shared Randomness (LOSR) Multipartite Nonlocal correlations, that is, correlations between N parties that cannot be obtained from unlimited shared randomness supplemented by any composition of (N-1)-shared causal Generalized-Probabilistic-Theory (GPT) resources. We then show that noisy N-partite GHZ quantum states as well as the 3-partite W quantum state can produce such correlations. This proves, if the operational predictions of quantum theory are correct, that Nature's nonlocality must be boundlessly multipartite in any causal GPT. We develop a computational method which certifies that a noisy N=3 GHZ quantum state with fidelity 85 percent satisfies this property, making an experimental demonstration of our results within reach. We motivate our definition and contrast it with preexisting notions of genuine multipartite nonlocality. This work extends a more compact parallel letter [Phys. Rev. Lett. 127, 200401 (2021)] on the same subject and provides all the required technical proofs.

Belief eXtropy: Measure uncertainty from negation

As the duality and complement of the Shannon entropy, the eXtropy is proposed in this years. The eXtropy also can be explained from the perspective of negation, which can be seen as an entropy with one-step negation operation of elements. The Dempster–Shafer theory (DST), as the generalization of probability theory (PT), it can express the more uncertain information than the Bayesian distribution. It contains the uncertainty of the event probability expressed in the Bayesian distribution and the uncertainty of the Bayesian distribution. Based on above, this article proposes a new belief entropy in DST called SU entropy, which is the first belief entropy to use the elements rather than mass functions. And then, a new negation of non-specificity part in DST is proposed, which is the first time to research the negation of non-specificity alone. Finally, a new belief eXtropy was proposed. It is different from the traditional eXtropy thinking, but from the perspective of elements’ negation. After verification, it has a satisfying performance in measurement uncertainty in DST. At the end of the paper, we discuss how to understand eXtropy from the perspective of negation, and the future research direction for the new belief eXtropy.