Generally covariant quantum information

The Symmetries of Quantum and Classical Information. The Ressurected “Ether" of Quantum Information

The ability to transfer information securely is essential in the modern world we live in. While the field of cryptography has long researched ways to do this, events over the last couple of years have brought the issues of secure information transfer squarely into the public eye. As a consequence, security and cryptography have quickly become issues of social and political importance, sparking debates about the values of privacy and the role of government. An essential ingredient of modern cryptography is the generation of randomness: cryptographic techniques are built on the premise that one has access to random bits. It is well known, however, that computers cannot produce algorithmically random sequences – that is, sequences with maximal algorithmic information content – but are doomed to produce 'pseudorandomness'. This lack of randomness can be, and has been, exploited to compromise security, see [4]. The active field of quantum information theory has proposed approaches to provide supposedly 'unbreakable' security by exploiting various quantum phenomena. This security unfortunately relies on assumptions about the nature of quantum measurements and their ability to generate random bits. Anton Zeilinger summarises this by postulating that the simplest quantum systems, qubit, can hold only one bit of classical information [6]. This foundational principle is in line with a wider paradigm shift to view quantum information as an extension of classical information, but it is nonetheless unsatisfying to simply postulate this principle, given its importance in determining the practical advantages of quantum information theory. In order to understand better how quantum mechanics can help generate meaningful information, we instead look to relate the outcomes of quantum measurements to formal properties of the system based on more fundamental assumptions. Indeed, we have shown that a) some of the postulated properties of quantum information follow from the formal structure of the theory and b) a purely formal notion of information within a quantum world can generate, via measurement, meaningful and useful information in the macroscopic world (for example, in cryptography). The indeterminism of quantum measurements can be formalised via the notion of value indefiniteness. To explain this concept, let us consider an arbitrary quantum system and ask whether the outcome of a measurement of any observable quantity A (such as the energy of the system, its angular momentum (spin), etc.) is determined prior to the measurement. If this is the case, then we say the observable is value definite with value v(A); otherwise, the observable is value indefinite and v(A) is undefined. Mathematically, one can reduce all observable quantities to so-called projection observables, which can only take the values 0 or 1. Thus, the question of whether the outcome of several measurements can be simultaneously determined in advance can be rephrased in terms of the information 'carried' by a particular system in a definite quantum state. Classically, one expects that all quantities are determined in advance, and hence all observables are value definite. Quantum mechanically, however, the belief is that this is not the case, and the information content of quantum systems is limited. Formulating carefully the notion of value indefiniteness allows us to formalise the notion of (quantum) indeterminism; however, this doesn't help clarify whether quantum systems are indeed value indefinite or not. Staying in this formal framework, the Kochen-Specker theorem [5] provides a first positive result, showing that at least some observables must be value indefinite if one makes the assumption known as non-contextuality, which states that any definite values that exist must be independent of other compatible measurements that may or may not be performed on the system. Under the same assumption, this theorem can be strengthened to show that only one single observable can have the definite value 1 (see [2]). Furthermore, only observables that can be measured simultaneously with this single one can have the definite value 0; the rest must all be value indefinite. Since the preparation of a system involves precisely ensuring that, usually via measurement, the system is in a definite state with respect to some desired observable, this result shows that no other incompatible observable can be value definite. That is, preparing a system in a definite state by making the 'preparation' observable value definite specifies completely the information content of the quantum system. This is an example of syntactical quantum information acquiring meaning at the level of the quantum itself. Can the syntactical information at the level of the quantum generate meaning outside the quantum, that is, at the macroscopic level? The results cited above hold in the Hilbert-space framework of quantum mechanics and are formulated only in terms of a syntactical notion of information. Their real importance becomes evident when one interprets them in the context of quantum measurements. Specifically, if we prepare a quantum system in a known state, they allow us to 'locate' observables which we can measure, but which are value indefinite; that is, observables whose measurement results are not specified by any pre-existing property of the quantum system. Furthermore, with respect to a mathematical model of unpredictability which we developed in [3], the results of these measurements can be shown to be absolutely unpredictable. In this way we get a mathematical explanation and justification of the largely accepted intuition that quantum mechanics is inherently unpredictable, and that this unpredictability arises from the phenomenon of indefiniteness within the quantum world. Furthermore, if one considers a hypothetical infinite sequence generated by the repeated measurement of a quantum value indefinite observable, one can prove that these sequences must be strongly incomputable, technically 'bi-immune' [1]. Such sequences cannot be generated by any Turing machine or classical computer, showing that value indefiniteness leads to a clear classical/quantum split in a purely algorithmic context. Importantly, bi-immunity is a property of observed, macroscopic quantities, quite separate from the quantum framework in which the value indefiniteness is formalised. This form of macroscopic meaning created from the lack of syntactic information is precisely the scenario that quantum random number generators try to create, and which is essential for the development and certification of quantum cryptographic systems. The fact that the macroscopic information created goes beyond anything classically obtainable serves as a valuable practical resource, outside of and removed from the quantum formalism. To conclude, quantum information creates meaning within the quantum and via measure- ment, the lack of information within the quantum, creates meaning and valuable information at the macroscopic level. Acknowledgments The second author thanks Prof. S. Marcus for useful conversations on quantum information theory. This work was supported in part by Marie Curie FP7-PEOPLE-2010-IRSES Grant RANPHYS. References and Notes Abbott, A.A.; Calude, C.S.; Svozil, K. Strong Kochen-Specker theorem and incomputability of quantum randomness. Physical Review A 2012, 86, 062109. Abbott, A.A.; Calude, C.S.; Svozil, K. Value-indefinite observables are almost everywhere. Physical Review A 2013, 89, 032109. Abbott, A.A.; Calude, C.S.; Svozil, K. On the unpredictability of individual quantum measurement outcomes. CDMTCS Research Report 2014, 458. Calude, C.S. Quantum randomness & cryptology. CyberTalk 2013, 3, 42–43. Kochen, S.; Specker, E. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 1967, 17, 59–87. Zeilinger, A. A foundational principle for quantum mechanics. Foundations of Physics 1999, 29(4),631–643.

Traditionally, certain physical theories have been thought to have relativity principles associated with them. Associated with Newtonian mechanics is the principle of Galilean relativity, associated with special relativity is the special or restricted principle of relativity, associated with general relativity is the general principle of relativity, etc. Such relativity principles are often expressed in terms of groups of transformations. The principle of Galilean relativity is expressed by the ‘invariance’ of Newtonian mechanics under the Galilean group, that of special relativity by the ‘invariance’ of special relativity under the Lorentz group, and that of general relativity by the ‘invariance’ (or ‘covariance’) of general relativity under the group of all 1–1 transformations with non-vanishing Jacobian. Unfortunately, just what these various groups are groups of, and exactly how they are associated with given physical theories, has been far from clear. Similarly, the nature and role of the various relativity principles has been correspondingly unclear.

Quantum mechanical irrebersibility

A history of the discovery of `new' quantum mechanics and theparadoxes of its probabilistic interpretation are brieflyreviewed from the modern point of view of quantum probabilityand information. Modern quantum theory, which has beendeveloped during the last 20 years for the treatment of quantum opensystems including quantum noise, decoherence, quantum diffusionsand spontaneous jumps occurring under continuous in timeobservation, is not yet a part of the standard curriculum ofquantum physics. It is argued that the conventional formalism ofquantum mechanics is insufficient for the description of quantumevents, such as spontaneous decays say, and the new experimentalphenomena related to individual quantum measurements, but theyhave all received an adequate mathematical treatment in quantumstochastics of open systems.Moreover, the only reasonable probabilistic interpretation ofquantum mechanics put forward by Max Born was, in fact, inirreconcilable contradiction with traditional mechanical realityand causality. This led to numerous quantum paradoxes, some ofthem due to the great inventors of quantum theory such asEinstein and Schrödinger. They are reconsidered in thispaper from the modern point of view of quantum stochastics and information.The development of quantum measurement theory, initiated by vonNeumann, indicated a possibility for resolution of thisinterpretational crisis by divorcing the algebra of thedynamical generators and the algebra of the actual observables,or Bell's beables. It is shown that within this approachquantum causality can be rehabilitated in the form of asuperselection rule for compatibility of the actual historieswith the potential future. This rule, together with theself-compatibility of the measurements ensuring the consistencyof the histories, is called the nondemolition, or causalityprinciple in modern quantum theory. The application of this rulein the form of dynamical commutation relations leads to thederivation of the von Neumann projection postulate, and also tothe more general reductions, instantaneous, spontaneous, andeven continuous in time. This gives a dynamical solution, in theform of the quantum stochastic filtering equations, of thenotorious measurement problem which was tackled unsuccessfullyby many famous physicists starting with Schrödinger andBohr.It has been recently proved that the quantum stochastic modelfor the continuous in time measurements is equivalent to a Diractype boundary-value problem for the secondary quantized input`offer waves from future' in one extra dimension, and to areduction of the algebra of the consistent histories of pastevents to an Abelian subalgebra for the `trajectories of theoutput particles'. This supports the corpuscular-wave duality inthe form of the thesis that everything in the future arequantized waves, while everything in the past are trajectories of therecorded particles.

Within the lore of nearly every scientific theory, among the facts, laws, theorems, and such that constitute the “truths” it expresses, are one or more principles. Physics textbooks, I know, usually mention them at the beginning, or during some interlude given over to general remarks or speculation. As examples, the uncertainty principle and correspondance principle of quantum mechanics come to mind (see, e.g., Messiah 1960). And associated with the general theory of relativity, which is particularly rich in principles, are a principle of general covariance, a principle of equivalence (or two), Mach's principle, and a general principle of relativity (see, e.g., Misner, Thome, and Wheeler 1973).Where do principles come from? Generally, as I said, they are supposed to be “truths” about the working of the world: observations and experiments are appealed to as their sources. But this doesn't distinguish them from facts and laws.

The extension of the principle of relativity to general coordinate systems is based on the hypothesis that an accelerated observer is locally equivalent to a hypothetical inertial observer with the same velocity as the noninertial observer. This hypothesis of locality is expected to be valid for classical particle phenomena as well as for classical wave phenomena but only in the short-wavelength approximation. The generally covariant theory is therefore expected to be in conflict with the quantum theory which is based on wave-particle duality. This is explicitly demonstrated for the frequency of electromagnetic radiation measured by a uniformly rotating observer. The standard Doppler formula is shown to be valid only in the geometric optics approximation. A new definition for the frequency is proposed, and the resulting formula for the frequency measured by the rotating observer is shown to be consistent with expectations based on the classical theory of electrons. A tentative quantum theory is developed on the basis of the generalization of the Bohr frequency condition to include accelerated observers. The description of the causal sequence of events is assumed to be independent of the motion of the observer. Furthermore, the quantum hypothesis is supposed to be valid for all observers. The implications of this theory are critically examined. The new formula for frequency, which is still based on the hypothesis of locality, leads to the observation of negative energy quanta by the rotating observer and is therefore in conflict with the quantum theory.

This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be be addressed as well in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac's work, which, in particular Dirac's derivation of his relativistic equation for the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall, however, also consider Heisenberg's derivation of quantum mechanics, which inspired Dirac. I argue that Heisenberg's and Dirac's work alike was guided by their adherence to and confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by G. M. D' Ariano and his coworkers on the principles of quantum information theory, which extends quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac's equation from these principles alone, without using the principles of relativity.

Various approaches to quantum measurement problem within the framework of usual unitary quantum dynamics are considered. It is argued that neither decoherence theory nor many-worlds interpretation of quantum mechanics do provide ultimate solution of the measurement problem: they cannot solve the problem of an alternative selection within the framework of unitary quantum dynamics. It is argued that the selection postulate in quantum theory is a very fundamental entity tightly connected with the nature of mathematics and with the nature of the mind, while the probability rules are more technical things admitting various approaches based on various sets of axioms.

Quantum mechanics is the most ground-breaking and fascinating theoretical concept developed in physics during the past century. Much of our present understanding of the microscopic world and its extension into the macroscopic world, including modern technical applications, is based upon quantum mechanics. We have experienced a remarkable development of information and communication technology during the past two decades, to a large extent depending upon successful fabrication of smaller and smaller components and circuits. However, we are finally approaching the physical limits of component miniaturization as we enter a microscopic world ruled by quantum mechanics.

Relational Quantum Mechanics (RQM) claims to be an interpretation of quantum theory \cite{Rovelli.21}. However, there are significant departures from quantum theory: (i) in RQM measurement outcomes arise from interactions which entangle a system S and an observer A without decoherence, and (ii) such an outcome is a "fact" relative to the observer A, but it is not a fact relative to another observer B who has not interacted with S or A during the foregoing measurement process. For B the system S⊗A remains entangled. We derive a GHZ-like contradiction showing that relative facts described by these statements are incompatible with quantum theory. Hence Relational Quantum Mechanics should not be considered an interpretation of quantum theory, according to a criterion for interpretations that we have introduced. The criterion states that whenever an interpretation introduces a notion of outcomes, these outcomes, whatever they are, must follow the probability distribution specified by the Born rule.

The bulk of sociological theory is based on a notion of “state”: mechanical-organic solidarity, feudalism, capitalism, socialism, folk-urban, and so forth. We propose a theory that social systems consist of actors and groups in constant motion. Further, we propose that the duality of actor and society generate a system that can be understood by using the concepts of complementarity, indeterminacy and the principle of relativity. Such a system is defined and the issue of social change is addressed. Basically, change is defined as an alteration in the motion of actors relative to the motion of groups. As the number of groups increases relative to the number of actors, the velocity of actors increases and the velocity of groups decreases. The increasing disparity of these velocities indicates greater social change.

Entanglement is one of the most useful resources in quantum information processing. It is effectively the quantum correlation between different subsystems of a composite system. Mathematically, one of the most hard tasks in quantum mechanics is to quantify entanglement. However, progress in this field is remarkable but not complete yet. There are many things to do with quantification of entanglement. In this review, we will discuss some of the important measures of bipartite entanglement.

Quantum information technology mainly includes quantum communication, quantum radar, quantum computer these three aspects, in recent years, rapid development, many countries invested heavily in development. The awarding of the Nobel Prize in Physics in 2022 to three scientists for their work in quantum informatics has further stimulated people's interest in quantum theory, which has been studied and discussed. This paper not only reviews the historical situation, but also thinks and innovates in theory. The main contents of this paper are as follows: It is pointed out that there are fundamental contradictions between relativity and quantum mechanics; The wave function, quantum statistics and uncertainty principle are discussed in detail. The hidden variable theory is reviewed and the Bell inequality is discussed. The Aspect two-photon experiment was analyzed. The rationality of Bohm experiment scheme is discussed. The development of Bell type experiment is discussed. It is pointed out that the entangled state is not acting at a distance but propagating faster than the speed of light. Reviews the Copenhagen interpretation of quantum mechanics; Quantum communication and Wootters theorem are discussed. etc. This paper holds that quantum mechanics has been finalized from 1926 to 1928, and its basic content has not changed much. But for some of the accusations, it is necessary to answer them theoretically. For example, we believe that the Copenhagen interpretation fundamentally changes our understanding of nature and marks a profound revolution in physics. And no other theory has since emerged that has such a profound understanding and wide application of microscopic phenomena as this interpretation. Another example is Bohm's two-particle correlation spin scheme in quantum entanglement experiments, which has been proved to be effective in a series of experiments, which is an important contribution in Bohm's life. This paper holds that the so-called quantum field theory is a failure, and the original quantum mechanics should still be advocated today. Current theoretical research should pay great attention to quantum entangled states, because its nature is still unclear. Understanding this "first mystery of the physical world" not only has scientific significance, but also has great philosophical significance for understanding the universe.

This paper discusses a particular type of quantum-like literary models, which are conceptual, rather than mathematical, in character. These models share with quantum mechanics the difficulties of applying the concepts of reality and causality at the ultimately ontological levels they consider, analogous to the level of quantum objects and processes in quantum mechanics. They respond to this difficulty by suspending and even precluding the application of both concepts, as do certain interpretations of quantum mechanics. I call such models and such interpretations "nonclassical ", in juxtaposition to "classical " models, which retain realism and causality at the ultimate level of description, even when considering random events. While I offer a sketch of Western thinking concerning the subject, I focus on certain philosophical and literary quantum-like thinking of the late-eighteenth and early-nineteenth centuries, associated with Romantic literature, which shows particular affinities with quantum-theoretical thinking later on. I also consider, in closing, the literary model, found in Beckett's plays, that was developed after quantum mechanics and that shares with it features that earlier literary quantum-like models do not possess.