Quantum Cryptography,Quantum Communication,and Quantum Computer in a Noisy Environment

First, we study several information theories based on quantum computing in a desirable noiseless situation. (1) We present quantum key distribution based on Deutsch’s algorithm using an entangled state. (2) We discuss the fact that the Bernstein-Vazirani algorithm can be used for quantum communication including an error correction. Finally, we discuss the main result. We study the Bernstein-Vazirani algorithm in a noisy environment. The original algorithm determines a noiseless function. Here we consider the case that the function has an environmental noise. We introduce a noise term into the function f(x). So we have another noisy function g(x). The relation between them is g(x) = f(x) ± O(??). Here O(??) ? 1 is the noise term. The goal is to determine the noisy function g(x) with a success probability. The algorithm overcomes classical counterpart by a factor of N in a noisy environment.     

Quantum Measurements,Energy Conservation and Quantum Clocks

A spin chain extending from Alice to Bob with nearest neighbors interactions, initially in its ground state, is considered. Assuming that Bob measures the last spin of the chain, the energy of the spin chain has to increase, at least on average, due to the measurement disturbance. Presumably, the energy is provided by Bob's measurement apparatus. Assuming that, simultaneously to Bob's measurement, Alice measures the first spin, it is shown that either energy is not conserved, – implausible – or the projection postulate doesn't apply, and that there is signalling. An explicit measurement model shows that energy is conserved (as expected), but that the spin chain energy increase is not provided by the measurement apparatus(es), that the projection postulate is not always valid – illustrating the Wigner–Araki–Yanase (WAY) theorem – and that there is signalling, indeed. The signalling is due to the non‐local interaction Hamiltonian. This raises the question of whether a suitable quantum‐information‐inspired model of such non‐local Hamiltonians can be developed.     

Quantum Interference, Quantum Theory of Measurement, and (In)completeness of Quantum Mechanics

The new techniques and ideas in quantum interferometry with neutrons, photons, atoms, electrons, and Bose condensates that fluorished in the last two decades have influenced in a decisive way the thinking and the research in the foundations and interpretation of quantum mechanics. The controversies existing among different schools on the reality of matter waves of quantum theory, the postulates of quantum measurement theory, and the (in)completeness of quantum mechanics have to be approached now in a new way. Our argumentation follows the spirit of the Paris school.     

Quantum Subgroups,Quantum Co-Set Spaces and Corresponding Differential Calculi

In this paper, quantum subgroups and quantum co-set spaces including corresponding dififerential calculi are discussed.     

Some Remarks on Unsharp Quantum Measurements,Quantum Non-Demolition,and All That

The theory of unsharp quantum measurements is reviewed as a generalization of the von Neumann-Lüders measurement theory and applied to measurements of continuous quantities. A generalized notion of repeatability is proposed which is applicable even to intrinsically unsharp continuous observables. As an illustration a precise formulation of a quantum non-demolition (QND) measurement performed on a harmonic oscillator is given.     

Quantum Chaos,Random Matrices,and Irreversibility in Interacting Many-Body Quantum Systems

The Pauli master equation describes the statistical equilibration of a closed quantum system. Simplifying and generalizing an approach developed in two previous papers, we present a derivation of that equation using concepts developed in quantum chaos and random-matrix theory. We assume that the system consists of subsystems with strong internal mixing. We can then model the system as an ensemble of random matrices. Equilibration results from averaging over the ensemble. The direction of the arrow of time is determined by an (ever-so-small) coupling to the outside world. The master equation holds for sufficiently large times if the average level densities in all subsystems are sufficiently smooth. These conditions are quantified in the text, and leading-order correction terms are given.     

Peirce, Clifford, and Quantum Theory

Beginning in 1870 Charles Sanders Peirce published a series of papers on a logic of relations, which corresponded to a linear associative algebra. This algebra is related by a linear transformation to quaternions and thus to the C(3, 0) algebra of William Kingdon Clifford. This Clifford algebra contains the Pauli matrices and so constitutes an operator basis for the nonrelativistic quantum theory of spin one-half particles. A further unification is achieved by taking the wave functions themselves to be 2 × 2 matrices which are Peirce logical operators and also elements of the Clifford algebra. Thus we have discovered a direct path from the Peirce logic to quantum theory. A diagrammatic method follows from the Peirce/Clifford algebraic approach and is suitable for describing particle interactions.     

Weyl Geometries, Fisher Information and Quantum Entropy in Quantum Mechanics

It is known that quantum mechanics can be interpreted as a non-Euclidean deformation of the space-time geometries by means Weyl geometries. We propose here a dynamical explanation of such approach by deriving Bohm potential from minimum condition of Fisher information connected to the entropy of a quantum system.     

Information,Quantum Mechanics,and Gravity

This is a basically expository article, with some new observations, tracing connections of the quantum potential to Fisher information, to Kähler geometry of the projective Hilbert space of a quantum system, and to the Weyl-Ricci scalar curvature of a Riemannian flat spacetime with quantum matter.Á Denise     

Quantum Theory of the Classical: Einselection,Envariance, Quantum Darwinism and Extantons

Core quantum postulates including the superposition principle and the unitarity of evolutions are natural and strikingly simple. I show that—when supplemented with a limited version of predictability (captured in the textbook accounts by the repeatability postulate)—these core postulates can account for all the symptoms of classicality. In particular, both objective classical reality and elusive information about reality arise, via quantum Darwinism, from the quantum substrate. This approach shares with the Relative State Interpretation of Everett the view that collapse of the wavepacket reflects perception of the state of the rest of the Universe relative to the state of observer’s records. However, our “let quantum be quantum” approach poses questions absent in Bohr’s Copenhagen Interpretation that relied on the preexisting classical domain. Thus, one is now forced to seek preferred, predictable, hence effectively classical but ultimately quantum states that allow observers keep reliable records. Without such (i) preferred basis relative states are simply “too relative”, and the ensuing basis ambiguity makes it difficult to identify events (e.g., measurement outcomes). Moreover, universal validity of quantum theory raises the issue of (ii) the origin of Born’s rule, pk=|ψk|2, relating probabilities and amplitudes (that is simply postulated in textbooks). Last not least, even preferred pointer states (defined by einselection—environment—induced superselection)—are still quantum. Therefore, unlike classical states that exist objectively, quantum states of an individual system cannot be found out by an initially ignorant observer through direct measurement without being disrupted. So, to complete the ‘quantum theory of the classical’ one must identify (iii) quantum origin of objective existence and explain how the information about objectively existing states can appear to be essentially inconsequential for them (as it does for states in Newtonian physics) and yet matter in other settings (e.g., thermodynamics). I show how the mathematical structure of quantum theory supplemented by the only uncontroversial measurement postulate (that demands immediate repeatability—hence, predictability) leads to preferred states. These (i) pointer states correspond to measurement outcomes. Their stability is a prerequisite for objective existence of effectively classical states and for events such as quantum jumps. Events at hand, one can now enquire about their probability—the probability of a pointer state (or of a measurement record). I show that the symmetry of entangled states—(ii) entanglement—assisted invariance or envariance—implies Born’s rule. Envariance also accounts for the loss of phase coherence between pointer states. Thus, decoherence can be traced to symmetries of entanglement and understood without its usual tool—reduced density matrices. A simple and manifestly noncircular derivation of pk=|ψk|2 follows. Monitoring of the system by its environment in course of decoherence typically leaves behind multiple copies of its pointer states in the environment. Only pointer states can survive decoherence and can spawn such plentiful information-theoretic progeny. This (iii) quantum Darwinism allows observers to use environment as a witness—to find out pointer states indirectly, leaving systems of interest untouched. Quantum Darwinism shows how epistemic and ontic (coexisting in epiontic quantum state) separate into robust objective existence of pointer states and detached information about them, giving rise to extantons—composite objects with system of interest in the core and multiple records of its pointer states in the halo comprising of environment subsystems (e.g., photons) which disseminates that information throughout the Universe.     

Classical Systems, Standard Quantum Systems, and Mixed Quantum Systems in Hilbert Space

Traditionally, there has been a clear distinction between classical systems and quantum systems, particularly in the mathematical theories used to describe them. In our recent work on macroscopic quantum systems, this distinction has become blurred, making a unified mathematical formulation desirable, so as to show up both the similarities and the fundamental differences between quantum and classical systems. This paper serves this purpose, with explicit formulations and a number of examples in the form of superconducting circuit systems. We introduce three classes of physical systems with finite degrees of freedom: classical, standard quantum, and mixed quantum, and present a unified Hilbert space treatment of all three types of system. We consider the classical/quantum divide and the relationship between standard quantum and mixed quantum systems, illustrating the latter with a derivation of a superselection rule in superconducting systems.     

Quantum dynamics, measurement and entropy

The aim of this paper is to present a line of ideas, centred around entropy production andquantum dynamics, emerging from von Neumann's work on foundations of quantum mechanics and leading to current research. The concepts of measurement, dynamical evolution and entropy were central in J. von Neumann's work. Further developments led to the introduction of generalized measurements in terms of positive operator-valued measures, closely connected to the theory of open systems. Fundamental properties of quantum entropy were derived and Kolmogorov and Sinai related the chaotic properties of classical dynamical systems with asymptotic entropy production. Finally, entropy production in quantum dynamical systems was linked with repeated measurement processes and a whole research area on nonequilibrium phenomena in quantum dynamical systems seems to emerge.     

Quantum mechanics,relativity and time

A discussion on quantum mechanics, general relativity and their relations is introduced. The assumption of the absolute validity of conservation laws and the extension to a 5D-space lead to reconsider several shortcomings and paradoxes of modern physics under a new light without the necessity to take into account symmetry breakings. In this picture, starting from first principles, and after a reduction procedure from 5D to 4D, dynamics leads to the natural emergence of two time arrows and ofa scalar-tensor theory of gravity. In this framework, phenomena like entanglement of systems and topology changes can be naturally accounted and, furthermore, several experimental evidences as gamma ray bursts, sizes of astrophysical structures and the observed values of cosmological parameters can be explained. The identification, thanks to conservation laws, of a covariant symplectic structure as a general feature also for gravity can be seen as a deep link common to all the interactions.     

Quantum measurements,sequential and latent

The results of a hypothetical experiment requiring a sequence of quantum measurements are obtained retrospectively, after the experiment has been completed, from a single reading of an apparatus register. The experiment is carried out reversibly and Schrödinger's equation is satisfied until the terminal reading of the register. The technique is illustrated using a feasible method of measuring photon spin as the quantum object observable and using the photon energy as the apparatus register. The technique is used to discuss the watchdog effect, the effect of repeated measurements inhibiting quantum jumps.     

Quantum Geometry on Quantum Spacetime: Distance,Area and Volume Operators

We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes.     

Planck, the Quantum, and the Historians

In late 1900, the German theoretical physicist Max Planck derived an expression for the spectrum of black-body radiation. That derivation was the first step in the introduction of quantum concepts into physics. But how did Planck think about his result in the early years of the twentieth century? Did he assume that his derivation was consistent with the continuous energies inherent in Maxwellian electrodynamics and Newtonian mechanics? Or did he see the beginnings, however tentative and uncertain, of the quantum revolution to come? Historians of physics have debated this question for over twenty years. In this article, I review that debate and, at the same time, present Planck's achievement in its historical context.     

Quantum Correlations,Nuclearity, and Spacetime Curvature

It is proved for a Haag–Araki–Kastler quantum field theory, that gravitation reduces the correlations in the vacuum state. Secondly, we prove Bell's inequalities by nuclearity assumptions. The so-called -content of certain compact mappings restricts the size of the set of measurements which violate Bell's inequalities.