Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by being based on two main postulates: 1. the theory should be logically consistent; 2. inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation.
Products and tensor products are linked by a universal property. Imposing the invariance of the laws of Nature under tensor composition along with Leibniz identity determines quantum and classical mechanics algebraic structure through the interplay between products, coproducts, and the tensor product. Violations of Bell's inequalities distinguishes quantum from classical mechanics.
Quantum and classical mechanics share a common algebraic formalism which is expressed naturally in the language of category theory. A third realization of this formalism is the so-called hyperbolic quantum mechanics where split-complex numbers replace the usual complex numbers. We introduce and explore the corresponding generalization of C*-algebras and prove that hyperbolic quantum mechanics is not a viable candidate for describing Nature. Quantum and classical mechanics are the only acceptable theories of Nature which are invariant under tensorial composition.
Quantum and classical mechanics are derived using four natural physical principles: (1) the laws of nature are invariant under time evolution, (2) the laws of nature are invariant under tensor composition, (3) the laws of nature are relational, and (4) positivity (the ability to define a physical state). Quantum mechanics is singled out by a fifth experimentally justified postulate: nature violates Bell's inequalities.
We discuss the reason why quantum mechanics is chosen as the most basic law of nature. Probability amplitude, which becomes a probability density after square it, is considered as one of the most essential ingredient of quantum mechanics. Code transfer experiments based on the probability amplitude is proved to be i) error of code transfer is minimum, ii) that error is independent of coding parameters and iii) non-trivial and non-local correlation is possible.
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide betwee…
It is argued that the nature of probability is essentially informational rather than physical and that quantum mechanical predictions should be viewed as logical inferences made on the basis of the information content of a given experimental situation. By implementing such a viewpoint, it is possible to maintain a sharp distinction between the physical and statistical aspects of quantum mechanics. The idea is applied to the double-beam interference experiment, reproducing the results of the standard formulation of quantum mechanics in a manner that renders the notion of wave-particle duality s…
Recent developments in quantum computing suggest that it could be possible to make conditional changes to the state of a quantum mechanical system without resorting to classical observation. It is accomplished through collective response of atoms comprising a lattice for the system and involves relative phase adjustments only. We exploit this possibility and describe experimental designs that could help elucidate quantum mechanical properties of these systems and distinguish between interpretations.
With the use of a second derivative test based on constrained second derivatives, a proof is given that excited states in nonrelativistic quantum mechanics are saddle points of the energy expectation value, and is shown, further, how to determine their Morse index.
A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being …
The wave function transformation of the quantum particle considered as a continuous medium was described by the evolution operator with the kernel in the form of path integral. It is shown that this approach allows considering not only Schrodinger evolution, but and the collapse phenomenon. The collapse entity is the nonlocal transformation of the quantum particle internal structure and it is not connected with its mechanical motion. Probably this fact allows the instantaneous signalling using the collapse without violating relativistic requirements.
We discuss quantum non-locality and contextuality, emphasising logical and structural aspects. We also show how the same mathematical structures arise in various areas of classical computation.
Discussed in the study are gravitational noise and the nature of entanglement states. Their role in forming of entangled states. Nonlocal nature of entangled states can be brought about by Polarization Variables. Polarization Variables consists of sum a background of random classical gravitational fields and waves, random electromagnets fields and so on known or unknown, distributed average isotropically in the space. This background is capable of correlating phases the oscillations of microobjects. From this follow, that entanglement polarizations states is the functions of Polarization Varia…
Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller (Journal of Philosophy, 1979), have thought that the answer to this question is No -- that the status of individual continuous quantities is very different in quantum mechanics than in classical mechanics. On the contrary, I shall show that the same subtle issues…
Quantum mechanics is an extremely successful theory of nature and yet it lacks an intuitive axiomatization. In contrast, the special theory of relativity is well understood and is rooted into natural or experimentally justified postulates. Here we introduce an axiomatization approach to quantum mechanics which is very similar to special theory of relativity derivation. The core idea is that a composed system obeys the same laws of nature as its components. This leads to a Jordan-Lie algebraic formulation of quantum mechanics. The starting assumptions are minimal: the laws of nature are invaria…
While we have intuitive notions of structure and complexity, the formalization of this intuition is non-trivial. The statistical complexity is a popular candidate. It is based on the idea that the complexity of a process can be quantified by the complexity of its simplest mathematical model - the model that requires the least past information for optimal future prediction. Here we review how such models, known as $ε$-machines can be further simplified through quantum logic, and explore the resulting consequences for understanding complexity. In particular, we propose a new measure of complexit…
We make comments on some shortcomings of the non-unitary-invariant and non-bi-invariant complexity in quantum mechanics/field theory and argue that the unitary-invariant and bi-invariant complexity is still a competitive candidate in quantum mechanics/field theory, contrary to quantum circuits in quantum computation. Based on the unitary-invariance of the complexity and intuitions from the holographic complexity, we propose a novel complexity formula between two states. Our proposal shows that i) the complexity between certain states in two dimensional CFTs is given by the Liouville action, wh…
Ontological theories of quantum mechanics describe a single system by means of well-defined classical variables and attribute the quantum uncertainties to our ignorance about the underlying reality represented by these variables. We consider the general class of ontological theories describing a quantum system by a set of variables with Markovian (either deterministic or stochastic) evolution. We provide the first proof that the number of continuous variables can not be smaller than 2N-2, N being the Hilbert space dimension. Thus, any ontological Markovian theory of quantum mechanics requires …
This paper presents a new modified quantum mechanics, Critical Complexity Quantum Mechanics, which includes a new account of wavefunction collapse. This modified quantum mechanics is shown to arise naturally from a fully discrete physics, where all physical quantities are discrete rather than continuous. I compare this theory with the spontaneous collapse theories of Ghirardi, Rimini, Weber and Pearle and discuss some implications of the theory for a realist view of the quantum realm.
The notion of "closed systems" in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum-mechanical system $P$ spatially far separated from the "rest of the universe" $Q$. Under reasonable assumptions on the interaction between $P$ and $Q$, we show that the system $P$ behaves as a closed system if the initial state of $P \vee Q$ belongs to a large class of states, including ones exhibiting entanglement between $P$ and $Q$. We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initia…
We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory (QFT) and quantum mechanics (QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to one-dimensional quadratic Hamiltonians is equivalent to AdS$_3$ spacetime. Here, the requirement that the complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and the speed of a particle is not superluminal. Our proposal prove…
The Schrödinger's wave function can naturally be realized as an 'instantaneous resonant spatial mode' in which quantum particle moves and hence the Born's rule is derived after identifying its origin. This realization facilitates the visualization of `what's really going on?' in the Young's double-slit experiment which is known to be the central mystery of quantum mechanics. Also, an actual mechanism underlying the `spooky-action-at-a-distance', another mystery regarding the entangled quantum particles, is revealed. Wheeler's delayed choice experiments, delayed choice quantum eraser experiment…
The physical consequences of the analysis performed in Parts I-IV are outlined within a scheme of the complete quantum (wave) mechanics called quantum field mechanics and completing the original ideas of Louis de Broglie by the dynamic complexity concept. The total picture includes the formally complete description at the level of the "average" wave function of Schroedinger type that shows dynamically chaotic behaviour in the form of either quantum chaos (Parts I-III), or quantum measurement (Part IV) with causal indeterminacy and wave reduction. This level is only an approximation, though oft…
Quantum computation teaches us that quantum mechanics exhibits exponential complexity. We argue that the standard scientific paradigm of "predict and verify" cannot be applied to testing quantum mechanics in this limit of high complexity. We describe how QM can be tested in this regime by extending the usual scientific paradigm to include {\it interactive experiments}.
We propose a new diagnostic for quantum chaos. We show that time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behaviour.
Starting from a simple classical framework and employing some stochastic concepts, the basic ingredients of the quantum formalism are recovered. It has been shown that the traditional axiomatic structure of quantum mechanics can be rebuilt, so that the quantum mechanical framework resembles to a large extent that of the classical statistical mechanics. The main assumption used here is the existence of a random irrotational component in the classical momentum. Various basic elements of the quantum formalism (calculation of expectation values, the Heisenberg uncertainty principle, the correspond…
The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (``NP-complete'') and those which are likel…
We show that for positive integer values $l$ of the parameter in the conformal mechanics model the system possesses a hidden nonlinear superconformal symmetry, in which reflection plays a role of the grading operator. In addition to the even $so(1,2)\oplus u(1)$-generators, the superalgebra includes $2l+1$ odd integrals, which form the pair of spin-$(l+{1/2})$ representations of the bosonic subalgebra and anticommute for order $2l+1$ polynomials of the even generators. This hidden symmetry, however, is broken at the level of the states in such a way that the action of the odd generators violat…
Quantum mechanics and the theory of gravity are presently not compatible. A particular question is whether gravity causes decoherence - an unavoidable source of noise. Several models for gravitational decoherence have been proposed, not all of which can be described quantum mechanically. In parallel, several experiments have been proposed to test some of these models, where the data obtained by such experiments is analyzed assuming quantum mechanics. Since we may need to modify quantum mechanics to account for gravity, however, one may question the validity of using quantum mechanics as a calc…
Classical mechanics is based upon a mechanical picture of nature that is fundamentally incorrect. It has been replaced at the basic level by a radically different theory: quantum mechanics. This change entails an enormous shift in our basic conception of nature, one that can profoundly alter the scientific image of man himself. Self-image is the foundation of values, and the replacement of the mechanistic self-image derived from classical mechanics by one concordant with quantum mechanics may provide the foundation of a moral order better suited to our times, a self-image that endows human lif…
The quest for finding the right interpretation of Quantum Mechanics (QM) is as old als QM and still has not ended, and may never end. The question what an interpretation of QM is has hardly ever been raised explicitly, let alone answered. We raise it and answer it. Then the quest for the right interpretation can continue self-consciously, for we then know exactly what we are after. We present a list of minimal requirements that something has to meet in order to qualify as an interpretation of QM. We also raise, as a side issue, the question how the discourse on the interpretation of QM relates…
We propose a new theory on a relation between diffusive and coherent nature in one dimensional wave mechanics based on a quantum walk. It is known that the quantum walk in homogeneous matrices provides the coherent property of wave mechanics. Using the recent result of a localization phenomenon in a one-dimensional quantum walk (Konno, Quantum Inf. Proc. (2010) 9, 405-418), we numerically show that the randomized localized matrices suppress the coherence and give diffusive nature.
I explore whether it is possible to make sense of the quantum mechanical description of physical reality by taking the proper subject of physics to be correlation and only correlation, and by separating the problem of understanding the nature of quantum mechanics from the hard problem of understanding the nature of objective probability in individual systems, and the even harder problem of understanding the nature of conscious awareness. The resulting perspective on quantum mechanics is supported by some elementary but insufficiently emphasized theorems. Whether or not it is adequate as a new …
At the 1927 Como conference Bohr spoke the now famous words "It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature." However, if the Copenhagen interpretation really holds on to this motto, why then is there this feeling of conflict when comparing it with realist interpretations? Surely what one can say about nature should in a certain sense be interpretation independent. In this paper I take Bohr's motto seriously and develop a quantum logic that avoids assuming any form of realism as much as possible. To illustrate the non-t…
I feel compelled to respond to the frequent references to spooky action at a distance that often accompany reports of experiments investigating entangled quantum mechanical states. Most, but not all, of these articles have appeared in the popular press. As an experimentalist I have great admiration for such experiments and the concomitant advances in quantum information and quantum computing, but accompanying claims of action at a distance are quite simply nonsense. Some physicists and philosophers of science have bought into the story by promoting the nonlocal nature of quantum mechanics. In …
The pigeonhole principle: "If you put three pigeons in two pigeonholes at least two of the pigeons end up in the same hole" is an obvious yet fundamental principle of Nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above "quantum pigeonhole principle" is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoti…
Beginning with the Everett-DeWitt many-worlds interpretation of quantum mechanics, there have been a series of proposals for how the state vector of a quantum system might be split at any instant into orthogonal branches, each of which exhibits approximately classical behavior. Here we propose a decomposition of a state vector into branches by finding the minimum of a measure of the net quantum complexity of the branch decomposition. We then propose a method for finding an ensemble of possible initial state vectors from which a randomly selected member, if evolved by ordinary Hamiltonian time …
It is argued on the basis of certain mathematical characteristics that classical mechanics is not constitutionally suited to accomodate consciousness, whereas quantum mechanics is. These mathematical characteristics pertain to the nature of the information represented in the state of the brain, and the way this information enters into the dynamics.
If the statement by Einstein, Podolsky and Rosen on incompleteness of Quantum-Mechanical description of nature is correct, then we can regard Quantum Mechanics as a Method of Indirect Computation. The problem is, whether the theory is incomplete or the nature itself does not allow complete description? And if the first option is correct, how is it possible to complete the Quantum-Mechanical description? Here we try to complement de-Broglie's idea on wave-pilot the stochastic gravitation gives origin to. We assume that de-Broglie's wave-pilots are gravitational stochastic ones, and we shall reg…
We study the dynamical behavior of nonlinear coupling in a quantum wave equation of a logarithmic type. Using statistical mechanical arguments for a large class of many-body systems, this coupling is shown to be related to temperature which is a thermodynamic conjugate to the Everett-Hirschman's quantum information entropy. A combined quantum-mechanical and field-theoretical model is proposed, which leads to a logarithmic equation with variable nonlinear coupling. We study its properties and present arguments regarding its nature and interpretation, including the connection to Landauer's princ…
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 true dynamical randomness is obtained as a natural fundamental property of deterministic quantum systems. It provides quantum chaos passing to the classical dynamical chaos under the ordinary semiclassical transition, which extends the correspondence principle to chaotic systems. In return one should accept the modified form of quantum formalism (presented by the Schroedinger equation) which, however, does not contradict the ordinary form and the main postulates of quantum mechanics. It introduces the principle of the fundamental dynamic multivaluedness (redundance) extending the quantum p…
Relational mechanics is a reformulation of mechanics (classical or quantum) for which space is relational. This means that the configuration of an $N$-particle system is a shape, which is what remains when the effects of rotations, translations, and dilations are quotiented out. This reformulation of mechanics naturally leads to a relational notion of time as well, in which a history of the universe is just a curve in shape space without any reference to a special parametrization of the curve given by an absolute Newtonian time. When relational mechanics (classical or quantum) is regarded as f…
Recently, Brassard et. al. conjectured that the fact that the maximal possible correlations between two non-local parties are the quantum-mechanical ones is linked to a reasonable restriction on communication complexity. We provide further support for the conjecture in the multipartite case. We show that any multipartite communication complexity problem could be reduced to triviality, had Nature been more non-local than quantum-mechanics by a quite small gap for any number of parties. Intriguingly, the multipartite nonlocal-box that we use to show the result corresponds to the generalized Bell…
The qualitatively new concept of dynamic complexity in quantum mechanics is based on a new paradigm appearing within a nonperturbational analysis of the Schroedinger equation for a generic Hamiltonian system. The unreduced analysis explicitly provides the complete, consistent solution as a set of many incompatible components ('realisations') which should permanently and probabilistically replace one another, since each of them is 'complete' in the ordinary sense. This discovery leads to the universally applicable concept of dynamic complexity and self-consistent, realistic resolution of the st…
Mathematical models are an essential component of quantitative science. They generate predictions about the future, based on information available in the present. In the spirit of Occam's razor, simpler is better; should two models make identical predictions, the one that requires less input is preferred. Yet, for almost all stochastic processes, even the provably optimal classical models waste information. The amount of input information they demand exceeds the amount of predictive information they output. We systematically construct quantum models that break this classical bound, and show th…
We propose a model CCS (complex-conjugate-space) to understand the inner and outer product nature of wave functions in non-hermitian PT-symmetry model in quantum mechanics considering (NxN) matrix model. Further we reflect the correct nature of C-symmetry ,P-parity and original Hamiltonian matrix for any arbitrary values of N. Interestingly the present result on N=2 , remains the same reported earlier by Bender,Brody and Jones model PT-symmetry operator. In non-conventional way one can notice that wave functions in a PT-symmetry model satisfies similar relations as in hermitian operator .
Generally a central role has been assigned to an unavoidable physical interaction between the measuring instrument and the physical entity measured in the change in the wave function that often occurs in measurement in quantum mechanics. A survey of textbooks on quantum mechanics by authors such as Dicke and Witke (1960), Eisberg and Resnick (1985), Gasiorowicz (1974), Goswami (1992), and Liboff (1993) supports this point. Furthermore, in line with the view of Bohr and Feynman, generally the unavoidable interaction between a measuring instrument and the physical entity measured is considered r…
The quantum nature of bulk ensemble NMR quantum computing --the center of recent heated debate, is addressed. Concepts of the mixed state and entanglement are examined, and the data in a 2 qubit liquid NMR quantum computation are analyzed. It is pointed out that the key problem in the current debate is the understanding of entanglement in a mixed state system. The following points are concluded in this Letter: 1)Density matrix describes the "state" of an average particle in an ensemble. It can not describe the state of an individual particle in an ensemble in detail; 2) Entanglement is a prope…
We raise fundamental questions about the very meaning of conservation laws in quantum mechanics and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.
We consider non-local properties of quanternionic quantum mechanics, in which the complex numbers are replaced by the quaternions as the underlying algebra. Specifically, we show that it is possible to construct a non-local box. This allows one to rule out quaternionic quantum mechanics using assumptions about communication complexity or information causality.
We address the question: Why are dynamical laws governing in quantum mechanics and in neuroscience of probabilistic nature instead of being deterministic? We discuss some ideas showing that the probabilistic option offers advantages over the deterministic one.
