General Probabilistic Theories Research Articles

An Extension of the Data-Adaptive Probability-Based Electrical Resistivity Tomography Inversion Method (E-PERTI)

About a decade ago, the PERTI algorithm was launched as a tool for a data-adaptive probability-based analysis of electrical resistivity tomography datasets. It proved to be an easy and versatile inversion method providing estimates of the resistivity values within a surveyed volume as weighted averages of the whole apparent resistivity dataset. In this paper, with the aim of improving the interpretative process, the PERTI method is extended by exploiting some peculiar aspects of the general theory of probability. Bernoulli’s conceptual scheme is assumed to comply with any resistivity dataset, which allows a multiplicity of mutually independent subsets to be extracted and analysed singularly. A standard least squares procedure is at last adopted for the statistical determination of the model resistivity at each point of the surveyed volume as the slope of a linear equation that relates the multiplicity of the resistivity estimates from the extracted data subsets. A 2D synthetic test and a field apparent resistivity dataset collected for archaeological purposes are discussed using the new extended PERTI (E-PERTI) approach. The comparison with the results from the original PERTI shows that by the E-PERTI approach a significantly greater robustness against noise can be achieved, besides a general optimisation of the estimates of the most probable resistivity values.

General quantum theory—unification of classical and modal quantum theories

Inspired by classical (‘actual’) quantum theory over and modal quantum theory (MQT), which is a model of quantum theory over certain finite fields, we introduce general quantum theory as a quantum theory—in the København interpretation—over general division rings with involution, in which the inner product ‘is’ a (σ, 1)-Hermitian form φ. This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical quantum theory over and in some MQTs, hold for any general quantum theory. Our approach is totally different (and more general) than Hardy’s axiomatic approach [], and generalized probability theories. On the other hand, in many general quantum theories, a geometrical object which we call ‘quantum kernel’ arises, which is invariant under the unitary group U(V, φ), and which carries the geometry of a so-called polar space. This object cannot be seen in classical quantum theory over , but it is present, for instance, in all known MQTs. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic 0, every general quantum theory over an algebraically closed field behaves like classical quantum theory over at many levels, and that all such theories share one model, which we pin down as the ‘minimal model,’ which is countable and defined over . Moreover, to make the analogy with classical quantum theory even more striking, we show that Born’s rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for general quantum theories in characteristic 0 which allows one to extend any such theory over algebraically closed fields (such as classical complex quantum theory) to larger theories in which a quantum kernel is present. In this sense, these singular objects are always virtually around in abundance.

Self-Testing of Physical Theories, or, Is Quantum Theory Optimal with Respect to Some Information-Processing Task?

Self-testing usually refers to the task of taking a given set of observed correlations that are assumed to arise via a process that is accurately described by quantum theory, and trying to infer the quantum state and measurements. In other words it is concerned with the question of whether we can tell what quantum black-box devices are doing by looking only at their input-output behavior and is known to be possible in several cases. Here we introduce a more general question: is it possible to self-test a theory, and, in particular, quantum theory? More precisely, we ask whether within a particular causal structure there are tasks that can only be performed in theories that have the same correlations as quantum mechanics in any scenario. We present a candidate task for such a correlation self-test and analyze it in a range of generalized probabilistic theories (GPTs), showing that none of these perform better than quantum theory. A generalization of our results showing that all nonquantum GPTs are strictly inferior to quantum mechanics for this task would point to a new way to axiomatize quantum theory, and enable an experimental test that simultaneously rules out such GPTs.

Preparation uncertainty implies measurement uncertainty in a class of generalized probabilistic theories

In quantum theory, it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values and that there is no joint measurement of them. They are called preparation uncertainty and measurement uncertainty, respectively, and research has unveiled that they are not independent from but related with each other in a quantitative way. This study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories (GPTs). In particular, a certain class of GPTs is considered, which can be characterized by transitivity and self-duality and regarded as extensions of quantum theory. It is proved that there are close connections expressed quantitatively between two types of uncertainty on a pair observables also in those theories: if preparation uncertainty exists, then measurement uncertainty also exists, and they are described by similar inequalities. Our results manifest that their correspondences are not specific to quantum theory but more universal ones.

Operational Restrictions in General Probabilistic Theories

The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further demonstrate that there are physically meaningful restrictions that fall into the third class. The presented study of operational restrictions provides a better understanding on how accessible measurements modify general probabilistic theories and quantum theory in particular.

Asymptotic properties for Markovian dynamics in quantum theory and general probabilistic theories

We address asymptotic decoupling in the context of Markovian quantum dynamics. Asymptotic decoupling is an asymptotic property on a bipartite quantum system, and asserts that any correlation between two quantum systems is broken after a sufficiently long time passes. In this paper, we show that asymptotic decoupling is equivalent to local mixing which asserts the convergence to a unique stationary state on at least one quantum system. If dynamics is asymptotically decoupling, any correlation between two quantum systems is broken exponentially. Also, we give a criterion of mixing that is a system of linear equations. All results in this paper are proved in the framework of general probabilistic theories, but we also summarize them in quantum theory.

Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories

In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.

Universal Gaps for XOR Games from Estimates on Tensor Norm Ratios

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus connected to a universal function r(n, m) called ‘projective–injective ratio’. This is defined as the minimal constant rho such that Vert cdot Vert _{Xotimes _pi Y}leqslant rho ,Vert cdot Vert _{Xotimes _varepsilon Y} holds for all Banach spaces of dimensions dim X=n and dim Y=m, where Xotimes _pi Y and X otimes _varepsilon Y are the projective and injective tensor products. By requiring that X=Y, one obtains a symmetrised version of the above ratio, denoted by r_s(n). We prove that r(n,m)geqslant 19/18 for all n,mgeqslant 2, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of r(n, m) and r_s(n), showing that, up to log factors: r_s(n) is of the order sqrt{n} (which is sharp); r(n, n) is at least of the order n^{1/6}; and r(n, m) grows at least as min {n,m}^{1/8}. These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an ‘ell _1/ell _2/ell _{infty } trichotomy theorem’ based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements.

Multi-agent paradoxes beyond quantum theory

Which theories lead to a contradiction between simple reasoning principles and modelling observers’ memories as physical systems? Frauchiger and Renner have shown that this is the case for quantum theory (Frauchiger and Renner 2018 Nat. Commun. 9 3711). Here we generalize the conditions of the Frauchiger–Renner result so that they can be applied to arbitrary physical theories, and in particular to those expressed as generalized probabilistic theories (GPTs) (Hardy 2001 arXiv:quant-ph/0101012; Barrett 2007 Phys. Rev. A 75 032304). We then apply them to a particular GPT, box world, and find a deterministic contradiction in the case where agents may share a PR box (Popescu and Rohrlich 1994 Found. Phys. 24 379–85), which is stronger than the quantum paradox, in that it does not rely on post-selection. Obtaining an inconsistency for the framework of GPTs broadens the landscape of theories which are affected by the application of classical rules of reasoning to physical agents. In addition, we model how observers’ memories may evolve in box world, in a way consistent with Barrett’s criteria for allowed operations (Barrett 2007 Phys. Rev. A 75 032304; Gross et al 2010 Phys. Rev. Lett. 104 080402). Ordinary readers, forgive my paradoxes: one must make them when one reflects; and whatever you may say, I prefer being a man with paradoxes than a man with prejudices.Jean-Jacques Rousseau, Emile or On Education

Perfect discrimination of non-orthogonal separable pure states on bipartite system in general probabilistic theory

We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states perfectly, and find that some non-orthogonal states are perfectly distinguishable. However, the above framework does not improve the capacity, namely, the maximum number of states that are simultaneously and perfectly distinguishable.

Entropy of mixing exists only for classical and quantum-like theories among the regular polygon theories

The thermodynamical entropy of a system which consists of different kinds of ideal gases is known to be defined successfully in the case when the differences are described by classical or quantum theory. Since these theories are special examples in the framework of generalized probabilistic theories (GPTs), it is natural to generalize the notion of thermodynamical entropy to systems where the internal degrees of particles are described by other possible theories. In this paper, we consider thermodynamical entropy of mixing in a specific series of theories of GPTs called the regular polygon theories, which can be regarded from a geometrical perspective as intermediate theories between a classical trit and a quantum bit with real coefficients. We prove that the operationally natural thermodynamical entropy of mixing does not exist in those inbetween theories, that is, the existence of the natural entropy results in classical and quantum-like theories among the regular polygon theories.

General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks

We establish an operational characterization of general convex resource theories -- describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -- in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as well as establishing that such discrimination problems can fully characterize the allowed transformations within the given resource theory. Specifically, we introduce quantifiers of resourcefulness of states, measurements, and channels in any GPT based on the generalized robustness, and show that they exactly characterize the maximum advantage that a given resource provides over all free states, measurements, or channels in a class of state or channel discrimination tasks. In quantum mechanics, we show that the robustness of measurement can be alternatively understood as the maximal increase in one-shot accessible information when compared to free measurements. We furthermore endow the standard robustness of a state with an operational meaning as the quantifier of the maximum advantage in binary channel discrimination tasks. Finally, we show that several classes of channel and state discrimination tasks can form complete families of monotones fully characterizing the transformations of states and measurements under any chosen class of free operations. Our results establish a fundamental connection between operational tasks of discrimination and core concepts of resource theories, valid for all physical theories with no additional assumptions about the structure of the GPT required.

Memory cost of temporal correlations

A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general.

Hierarchical axioms for quantum mechanics

The origin of nonclassicality in quantum mechanics (QM) has been investigated recently by a number of authors with a view to identifying axioms that would single out quantum mechanics as a special theory within a broader framework such as convex operational theories. In these studies, the axioms tend to be logically independent in the sense that no specific ordering of the axioms is implied. Here, we identify a hierarchy of five nonclassical features that separate QM from a classical theory: (Q1) Incompatibility and indeterminism; (Q2) Contextuality; (Q3) Entanglement; (Q4) Nonlocality and (Q5) Indistinguishability of identical particles. Such a hierarchy isn't obvious when viewed from within the quantum mechanical framework, but, from the perspective of generalized probability theories (GPTs), the later axioms can be regarded as further structure introduced on top of earlier axioms. Relevant toy GPTs are introduced at each layer when useful to illustrate the action of the nonclassical features associated with the particular layer.

Information Transfer in Generalized Probabilistic Theories Based on Weak Repeatability

Information transfer in generalized probabilistic theories (GPT) is an important problem. We have dealt with the problem based on repeatability postulate, which generalizes Zurek’s result to the GPT framework (Wu et al., Phys. Lett. A 379, 2694, 2015). A natural question arises: can we deduce the information transfer result under weaker assumptions? In this paper, we generalize Zurek’s result to the framework of GPT using weak repeatability postulate. We show that if distinguishable information can be transferred from a physical system to a series of apparatuses under the weak repeatability postulate in GPT, then the initial states of the physical system must be completely distinguishable. Moreover, after each step of invertible transformation, the composite states of the composite system composed of the physical systems and the apparatuses must also be completely distinguishable.

Ruling out the class of statistical processes involving two noninteracting identical particles in two modes

In the framework of Generalized probabilistic theories (GPT), we illustrate a class of statistical processes in case of two noninteracting identical particles in two modes that satisfies a well motivated notion of physicality conditions namely the double stochasticity and the no-interaction condition proposed by Karczewski et. al. (Phys. Rev. Lett. 120, 080401 (2018)), which can not be realized through a quantum mechanical process. This class of statistical process is ruled out by an additional requirement called the evolution condition imposed on two particle evolution. We also show that any statistical process of two noninteracting identical particles in two modes that satisfies all of the three physicality conditions can be realized within quantum mechanics using the beam splitter operation.

No-free-information principle in general 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.

Given Keynes's Definitions of Logical Probability and Evidential Weight, It Is Impossible for Keynes’s Approach to Measurement to Be an Ordinal Theory; His 'Non Numerical' Probabilities Must Be Based on Inexact and Imprecise Measurement Using Approximation Involving Boolean Upper and Lower Probability Bounds and Bounded Outcomes

Keynes spent all of his introductory Chapter One of the A Treatise on Probability emphasizing that his logical relation of probability (a relation of similarity – dissimilarity based on analogy) P(a/h)=α, where α came in degrees and 0≤α≤1. In chapter 6, Keynes introduced his logical relation of the evidential weight of the argument, V(a/h), but did not specify what it equaled, although he clearly stated that he would integrate both probability and weight together later. In chapter 26, Keynes specified that V(a/h)= w, where we came in degrees and 0≤w≤1. Keynes showed how to integrate both probability and weight together in his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. Probability and weight were also integrated together by Keynes using the theory of inexact measurement or approximation based on intervals with upper and lower bounds or limits. Ordinal probability can do none of these things. Advocates of the claim that Keynes’s theory was an ordinal theory of probability concentrate exclusively on a misinterpreted diagram that appears near the end of chapter 3 of the A Treatise on Probability. Thus, while Keynes’s theory can easily deal with ordinal probability with the aid of the Principle of Indifference, it is not part of Keynes’s main or major analysis in the A Treatise on Probability, which was interval probability. The same conclusion holds for numerical probability and rational expectations. Rational expectations are a very special case that occurs if w=1. The heterodox, institutionalist, or Post Keynesian approaches, involving irreducible, fundamental, or radical uncertainty, are a very, very special case that occurs only if w=0 in the very long run. Their argument, that measurement is not possible at all, be it exact or inexact, is rejected by Keynes in chapter 4 of the General Theory. Thus, while Keynes’s theory can easily deal with ordinal probability, it is impossible for the ordinal probability to deal with inexact and imprecise interval-valued probability measurement. Keynes’s theory of measurement involves interval-valued probability in the A Treatise on Probability with the numerical and ordinal probability being very special cases of his general theory of interval-valued probability. Keynes’s approach to measurement in the General Theory also involved inexact and imprecise measurement as deployed by Keynes in chapter 4 of the General Theory. No ordinal probability approach to measurement is used by Keynes in the General Theory. The great value of the 1936-1938 Keynes-Townshend exchanges is that they show that Keynes presented an approach to measurement that was imprecise and inexact in both the A Treatise on Probability and the General Theory.

Research on Some Problems of Inland River Channel Width Determination

In general, Generally speaking, most of the inland waterway transportation channels are naturally formed, while a few are artificially excavated. On both sides of the channel, the terrain is complex, and the channel itself has a large length, so it is difficult to find the rule whether the water level changes or the hydrological conditions change. The width of inland waterway affects the shipping safety of ships, so it is very necessary to study the method of determining the width of inland waterway. In this paper, the determination method of distance between ships and distance between ship and bank are studied by using the general layout design theory and probability theory. The method of determining the drift and the distance between ship and bridge caused by the flow in the bend is studied by using hydrodynamic theory. The establishment of inland waterway width should comprehensively use various theories.

Choquet calculus using operational matrices

In generalized probability theory, Choquet calculus on a real line, including Choquet integral equations and differential equations with respect to non-additive measures, has been recently discussed by Sugeno, Torra and Narukawa. These studies were based on Laplace transformation as a basic tool. However, in many problems, it is not possible to provide Laplace transformation. On the other hand, in many problems, differential equations with respect to non-additive measures often do not have exact solutions. These reasons permit us to propose an alternative approach for numerical Choquet calculus based on operational matrices, which also readily gives the approximate numerical solution of Choquet integral equations and differential equations with respect to non-additive measures. Finally, we present some illustrative examples for numerical Choquet calculus based on operational matrices which the Laplace transformation does not seem to be applicable in these cases.