The space P of pure states of any physical system, classical or quantum, is identified as a Poisson space with a transition probability. The latter is a function p: PxP -> [0,1]; in addition, a Poisson bracket is defined for functions on P. These two structures are connected through unitarity. Classical and quantum mechanics are each characterized by a simple axiom on the transition probability p. Unitarity then determines the Poisson bracket of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Superselection rules are naturally incorporated.
The classical Hilbert space formulation of the axioms of Quantum Mechanics appears to leave open the question whether the Hermitian operators which are associated with the observables of a finite non-relativistic quantum system are uniquely determined.
An axiomatic formalism for a minimal irreversible quantum mechanics is introduced. It is shown that a quantum equilibrium and the decoherence phenomenon are consequences of the axioms and that Lyapunov variables, exponential survival probabilities, and a classisal conditional never-decreasing entropy can be defined.
The axioms of nonrelativistic quantum mechanics lack clear physical meaning. In particular, they say nothing about nonlocality. Yet quantum mechanics is not only nonlocal, it is twice nonlocal: there are nonlocal quantum correlations, and there is the Aharonov-Bohm effect, which implies that an electric or magnetic field h e r e may act on an electron t h e r e. Can we invert the logical hierarchy? That is, can we adopt nonlocality as an axiom for quantum mechanics and derive quantum mechanics from this axiom and an additional axiom of causality? Three versions of these two axioms lead to thre…
Conventional quantum mechanics with a complex Hilbert space and the Born Rule is derived from five axioms describing properties of probability distributions for the outcome of measurements. Axioms I,II,III are common to quantum mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first noted by Turing and von Neumann, in which the increase in entropy resulting from a measurement is reduced by a suitable intermediate measurement. This is shown to be impossible for local hidden variable theories. Axiom IV, together with the first three, almost suffice to deduce the conventio…
We present an axiomatization of non-relativistic Quantum Mechanics for a system with an arbitrary number of components. The interpretation of our system of axioms is realistic and objective. The EPR paradox and its relation with realism is discussed in this framework. It is shown that there is no contradiction between realism and recent experimental results.
Recent developments in the ZX-Calculus have resulted in complete axiomatisations first for an approximately universal restriction of the language, and then for the whole language. The main drawbacks were that the axioms that were added to achieve completeness were numerous, tedious to manipulate and lacked a physical interpretation. We present in this paper two complete axiomatisations for the general ZX-Calculus, that we believe are optimal, in that all their equations are necessary and moreover have a nice physical interpretation.
The ZX-Calculus is a graphical language for quantum mechanics. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clifford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structur…
Having analyzed the formal aspects of Wallace's proof of the Born rule, we now discuss the concepts and axioms upon which it is built. Justification for most axioms is shown to be problematic, and at times contradictory. Some of the problems are caused by ambiguities in the concepts used. We conclude the axioms are not reasonable enough to be taken as mandates of rationality in Everettian Quantum Mechanics. This invalidates the interpretation of Wallace's result as meaning it would be rational for Everettian agents to decide using the Born rule.
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…
Quantum mechanics postulates the existence of states determined by a particle position at a single time. This very concept, in conjunction with superposition, induces much of the quantum-mechanical structure. In particular, it implies the time evolution to obey the Schroedinger equation, and it can be used to complete a truely basic derivation of the statistical axiom as recently proposed by Deutsch.
A C*-algebra formulation of Quantum Mechanics is derived from purely operational axioms in which the primary role is played by the "transformations" that the system undergoes in the course of an "experiment". The notion of the {\em adjoint} of a transformation is based on the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus.
In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in this derivation are the co-ordinatization of generalized geometries and a theorem of Solér which characterizes Hilbert spaces among the orthomodular spaces. A generalized Wigner theorem is applied to reduce some of the assumptions of the theorem of Solér to the theory of symmetry in quantum mechanics. Since this reduction is only partial we also point out the r…
We introduce the first complete and approximatively universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding four new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that t…
Quantum mechanics permits nonlocality - both nonlocal correlations and nonlocal equations of motion - while respecting relativistic causality. Is quantum mechanics the unique theory that reconciles nonlocality and causality? We consider two models, going beyond quantum mechanics, of nonlocality: "superquantum" correlations, and nonlocal "jamming" of correlations. These models are consistent with some definitions of nonlocality and causality.
We discuss the axiomatic basis of quantum mechanics and show that it is neither general nor consistent, since its axioms are incompatible with each other and moreover it does not incorporate the magnetic quantization as in the cyclotron motion. A general and consistent system of axioms is conjectured which incorporates also the magnetic quantization.
A realistic axiomatic formulation of nonrelativistic quantum mechanics for a single microsystem with spin is presented, from which the most important theorems of the theory can be deduced. In comparison with previous formulations, the formal aspect has been improved by the use of certain mathematical theories, such as the theory of equipped spaces, and group theory. The standard formalism is naturally obtained from the latter, starting from a central primitive concept: the Galilei group.
The list of basic axioms of quantum mechanics as it was formulated by von Neumann includes only the mathematical formalism of the Hilbert space and its statistical interpretation. We point out that such an approach is too general to be considered as the foundation of quantum mechanics. In particular in this approach any finite-dimensional Hilbert space describes a quantum system. Though such a treatment might be a convenient approximation it can not be considered as a fundamental description of a quantum system and moreover it leads to some paradoxes like Bell's theorem. I present a list from …
We present an analysis of quantum mechanics and its problems and paradoxes taking into account the results that have been obtained during the last two decades by investigations in the field of `quantum structures research'. We concentrate mostly on the results of our group FUND at Brussels Free University. By means of a spin 1/2 model where the quantum probability is generated by the presence of fluctuations on the interactions between measuring apparatus and physical system, we show that the quantum structure can find its origin in the presence of these fluctuations. This appraoch, that we ha…
The standard axiomatization of quantum mechanics (QM) is not fully explicit about the role of the time-parameter. Especially, the time reference within the probability algorithm (the Born Rule, BR) is unclear. Using a plausible principle P1, about the role of probability in a physical theory, and a second principle P2 affording a most natural way to make BR precise, a logical conflict with the standard expression for the completeness of QM can be derived. Rejecting P1 is implausible. Rejecting P2 leads to unphysical results and to a conflict with a generalization of P2, a principle P3. It is t…
We consider the situation of a physical entity that is the compound entity consisting of two 'separated' quantum entities. In earlier work it has been proven by one of the authors that such a physical entity cannot be described by standard quantum mechanics. More precisely, it was shown that two of the axioms of traditional quantum axiomatics are at the origin of the impossibility for standard quantum mechanics to describe this type of compound entity. One of these axioms is equivalent with the superposition principle, which means that separated quantum entities put the linearity of quantum me…
I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is a…
The debate on the nature of quantum probabilities in relation to Quantum Non Locality has elevated Quantum Mechanics to the level of an "Operational Epistemic Theory". In such context the quantum superposition principle has an extraneous non epistemic nature. This leads us to seek purely operational foundations for Quantum Mechanics, from which to derive the current mathematical axiomatization based on Hilbert spaces. In the present work I present a set of axioms of purely operational nature, based on a general definition of "the experiment", the operational/epistemic archetype of information …
The role of operational quantum mechanics, quantum axiomatics and quantum structures in general is presented as a contribution to a compendium on quantum physics, its history and philosophy.
We expand on a recent development by Hardy, in which quantum mechanics is derived from classical probability theory supplemented by a single new axiom, Hardy's Axiom 5. Our scenario involves a `pretend world' with a `pretend' Heisenberg who seeks to construct a dynamical theory of probabilities and is lead -- seemingly inevitably -- to the Principles of Quantum Mechanics.
Quantum mechanics and relativistic causality together imply nonlocality: nonlocal correlations (that violate the CHSH inequality) and nonlocal equations of motion (the Aharonov-Bohm effect). Can we invert the logical order? We consider a conjecture that nonlocality and relativistic causality together imply quantum mechanics. We show that correlations preserving relativistic causality can violate the CHSH inequality more strongly than quantum correlations. Also, we describe nonlocal equations of motion, preserving relativistic causality, that do not arise in quantum mechanics. In these nonlocal…
From the standpoint of Hilbert's Sixth Problem, which is the axiomatisation of Physics, the famous paper of Lucien Hardy's, Quantum Theory from Five Reasonable Axioms, is not relevant. The present paper argues that Hardy does not give a physical definition of `limit', and if we assume the usual mathematical definition of limit of a sequence, he fails to define a sequence in physical terms to which the usual definition is applicable. We argue that one should not, in fact, try to define probability in terms of the usual notion of limit of a sequence of results of a measurement because of seeming…
Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive nonrelativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical st…
A non-relativistic quantum mechanical theory is proposed that combines elements of Bohmian mechanics and of Everett's "many-worlds" interpretation. The resulting theory has the advantage of resolving known issues of both theories, as well as those of standard quantum mechanics. It has a clear ontology and a set of precisely defined postulates from where the predictions of standard quantum mechanics can be derived. Most importantly, the Born rule can be derived by straightforward application of the Laplacian rule, without reliance on a "quantum equilibrium hypothesis" that is crucial for Bohmia…
A necessary and natural change in our application of quantum mechanics to separated systems is shown to reconcile quantum mechanics and local realism. An analysis of separation and localization justifies the proposed change in application of quantum mechanics. An important EPRB experiment is reconsidered and it is seen that when it is correctly interpreted it supports local realism. This reconciliation of quantum mechanics with local realism allows the axiom sets of quantum mechanics, probability, and special relativity to be joined in a consistent global axiom set for physics.
The statistical model of quantum mechanics is based on the mapping between operators on the Hilbert space and functions on the phase space. This map can be implemented by an operator that satisfies physically motivated Stratonovich-Weyl axioms. Arguments are given in favour of a certain extension of the axioms, provided that there is a priori knowledge about the composite nature of the quantum system.
In this article, the axioms presented in the first one are reformulated according to the special theory of relativity. Using these axioms, quantum mechanic's relativistic equations are obtained in the presence of electromagnetic fields for both the density function and the probability amplitude. It is shown that, within the present theory's scope, Dirac's second order equation should be considered the fundamental one in spite of the first order equation. A relativistic expression is obtained for the statistical potential. Axioms are again altered and made compatible with the general theory of …
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatiza…
In 1964, John Bell proved that quantum mechanics is "unreasonable" (to use Einstein's term): there are nonlocal bipartite quantum correlations. But they are not the most nonlocal bipartite correlations consistent with relativistic causality ("no superluminal signalling"): also maximally nonlocal "superquantum" (or "PR-box") correlations are consistent with relativistic causality. I show that---unlike quantum correlations---these correlations do not have a classical limit consistent with relativistic causality. The generalization of this result to all stronger-than-quantum nonlocal correlations…
Hardy (quant-ph/0101012) conjectures in his Axiom 2 that K=K(N), and that in classical probability K=N, while in quantum mechanics K=N^2. We offer an example in classical probability for which K=NV, V the number of independent complete variables; with N=V this classical example satisfies the purported quantal relation K=N^2.
A Newtonian mechanics model is essentially the model of a point body in an inertial reference frame. How to describe extended bodies in non-inertial (vibrational) reference frames with the random initial conditions? One of the most general description (known as the higher derivatives formalism) consists in taking into account the infinite number of the higher order temporal derivatives of the coordinates in the Lagrange function. Such formalism describes physical objects in the infinite dimensional space does not contradict quantum mechanics and infinite dimensional Hilbert space.
We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except the one that expresses the uncertainty principle, are shared with the classical event space. The only models for the set of axioms are lattices of subspaces of inner product spaces over a field K. (b) A…
We present a heuristic derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. This approach naturally leads to the usual quantum formalism, within a new realistic conceptual framework that is discussed in details. Physically, the structure of Quantum Mechanics appears as a result of the interplay between the quantized number of "modalities" accessible to a quantum system, and the continuum of "contexts" that are required to define these modalities. Mathematically, the Hilbert space structure appear…
It is supposed the alternative to Quantum Mechanics Axiomatic. Fluctuational Theory save the Mathematics of Quantum Mechanic without change, naming this Mathematics as Method of Indirect Computation. Fluctuational Theory is delete the axiomatic of Quantum Mechanics and replaces it by the assumption of Gravitational Noise. This assumption is connects the Method of Indirect Computation to the Classical Physics. Physical fluctuations of classical gravitational fields are mathematically expressed through geometric fluctuations of metrics of Riemann Space. Metrics Fluctuational Theory and Quantum M…
In a recent paper, Nagata [1] claims to derive inconsistencies from quantum mechanics. In this paper, we show that the inconsistencies do not come from quantum mechanics, but from extra assumptions about the reality of observables.
We propose four information-theoretic axioms for the foundations of statistical mechanics in general physical theories. The axioms---Causality, Purity Preservation, Pure Sharpness, and Purification---identify a class of theories where every mixed state can be modelled as the marginal of a pure entangled state and where every unsharp measurement can be modelled as a sharp measurement on a composite system. This class of theories, called sharp theories with purification, includes quantum theory both with complex and real amplitudes, as well as a suitable extension of classical probability theory…
In Ref. [1] one of the authors proposed postulates for axiomatizing Quantum Mechanics as a "fair operational framework", namely regarding the theory as a set of rules that allow the experimenter to predict future events on the basis of suitable tests, having local control and low experimental complexity. In addition to causality, the following postulates have been considered: PFAITH (existence of a pure preparationally faithful state), and FAITHE (existence of a faithful effect). These postulates have exhibited an unexpected theoretical power, excluding all known nonquantum probabilistic theor…
One of Yakir Aharonov's endlessly captivating physics ideas is the conjecture that two axioms, namely relativistic causality ("no superluminal signalling") and nonlocality, so nearly contradict each other that a unique theory - quantum mechanics - reconciles them. But superquantum (or "PR-box") correlations imply that quantum mechanics is not the most nonlocal theory (in the sense of nonlocal correlations) consistent with relativistic causality. Let us consider supplementing these two axioms with a minimal third axiom: there exists a classical limit in which macroscopic observables commute. Th…
We make a first attempt to axiomatically formulate the Montevideo interpretation of quantum mechanics. In this interpretation environmental decoherence is supplemented with loss of coherence due to the use of realistic clocks to measure time to solve the measurement problem. The resulting formulation is framed entirely in terms of quantum objects without having to invoke the existence of measurable classical quantities like the time in ordinary quantum mechanics. The formulation eliminates any privileged role to the measurement process giving an objective definition of when an event occurs in …
In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the ques…
Authoritative appraisals qualified this book as an axiomatic theory. However, being its essential content no more than an analogy, its theoretical organization cannot be an axiomatic one. In fact, in the first edition Dirac declares to avoid an axiomatic presentation. Moreover, I show that the text is aimed at solving a basic problem: How quantum mechanics is similar to classical mechanics. A previous paper analyzed all past theories of physics, chemistry and mathematics, presented by the respective authors in a non-axiomatic way. Four characteristic features of a new model of organizing a the…
This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This work is limited to k-ary representations of length L and to the axioms for arithmetic modulo k^{L}. A model of the axioms is described based on states in and operators on an abstract L fold tensor product Hilbert space H^{arith}. Unitary maps of t…
Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability f…
Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a "quantum theory without observers," alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics fo…
We formulate physically-motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to Quantum Mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory respectively. We show that even if the second group of axioms is dropped, there are no deformations of Quantum Mechanics which preserve the kinematic axioms. Thus any theory going beyond Quantum Mec…
The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -- i.e. the ability to derive any true equation -- is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -- called cyclotomic supplementarity -- which consists in merging n subdiagrams at o…
Recently, we gave a complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. Based on this result, here we also obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by restricting the ring of complex numbers to its subring corresponding to the Clifford+T fragment resting on the completeness theorem of the ZW-calculus for arbitrary commutative ring. In contrast to the first complete axiomatisation of the ZX-calculus for the Clifford+T fragment, we have two new generators as features rather than novelties: the triangle can be em…
Quantum information and computation may serve as a source of useful axioms and ideas for the quantum logic/quantum structures project of characterizing and classifying types of physical theories, including quantum mechanics and classical mechanics. The axiomatic approach of quantum structures may help isolate what aspects of quantum mechanics are responsible for what aspects of its greater-than-classical information processing power, and whether more general physical theories may escape some common limitations of classical and quantum theories. Also, by by helping us understand how existing qu…
Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs $\pm iε$ of the Lippmann…
