Probabilistic Process Algebra to Unifying Quantum and Classical Computing in Closed Systems

We have unified quantum and classical computing in open quantum systems called qACP which is a quantum generalization of process algebra ACP. But, an axiomatization of quantum and classical processes with an assumption of closed quantum systems is still missing. For closed quantum systems, unitary operator, quantum measurement and quantum entanglement are three basic components of quantum computing. This leads to probability unavoidable. Along the solution of qACP to unify quantum and classical computing in open quantum systems, we unify quantum and classical computing with an assumption of closed systems under the framework of ACP-like probabilistic process algebra. This unification make it can be used widely in verification of quantum and classical computing mixed systems, such as most quantum communication protocols.

     

Strong and Weak Optimizations in Classical and Quantum Models of Stochastic Processes

Journal of Statistical Physics - Among the predictive hidden Markov models that describe a given stochastic process, the $$\epsilon \text{-machine }$$ is strongly minimal in that it minimizes every...     

Foundation for Quantum Computing

In this paper we introduce a minimal formal intuitionistic propositional Gentzen sequent calculus for handling quantum types, quantum storage being introduced syntactically along the lines of Girard's of course operator !. The intuitionistic fragment of orthologic is found to be translatable into this calculus by means of a quantum version of the Heyting paradigm. When realized in the category of finite dimensional Hilbert spaces, the familiar qubit arises spontaneously as the irreducible storage capable quantum computational unit, and the necessary involvement of quantum entanglement in the quantum duplication process is plainly and explicitly visible. Quantum computation is modelled by a single extra axiom, and reproduces the standard notion when interpreted in a larger category.     

Non-traditional Approach to Quantum Computing

The N-qubit system characterized by an effective spin \(S = 2^{N - 1} - {1/2}\) is carried out in the representation of two coupled harmonic oscillators. It is shown that quantum computing results obtained with spinor algebra can be obtained also using the algebra of two coupled harmonic oscillators which is a convenient formalism, especially in the case of large number of qubits. In this formalism the non-abelian and abelian groups of the order of 16 related to one- and two-qubit systems were found. The structure of Cayley tables of those groups is different due to different commutation (anticommutation) relations for operators forming each group.     

Foundation for Quantum Computing II

In this continuation of an earlier paper we develop further the theme of quantum logical specification and derive from it some apparently physically viable instantiations of potential quantum computing devices. Specifically, in the case of a one-parameter set of terms (or labels)—read as instants of time—we find, emerging quite naturally from the algebraic setup, the paradigm for a single qubit epitomized by the case of a two-state fermion interacting with an external single mode boson. This covers the cases: cavity QED, trapped ions, and, when the qubits are multiplexed appropriately, NMR based systems. (This case degenerates to one in which only bosons are relevant as in the case of pure bosonic harmonic oscillator models in the “dual rail” representation. Such models fly in the face of the logic itself, thus clearly revealing even at this level their well-known shortcomings as practical quantum computing devices. Here as elsewhere logical constraints apparently dominate physical ones.) In a final section we indicate briefly how this process exactly generalizes, in the case of a manifold of terms more general than the one-parameter case, to yield the notion of holonomic quantum computation. In the course of this investigation we find an interpretation of path integrals as limits of sequences of logical CUTS, thus establishing a link—though still tenuous—between ensembles of acts of quantum computation and Lagrangians.     

Quantum and Classical Brackets

We describe p-mechanical (Kisil, V. V. (1996). Journal of Natural Geometry 9(1), 1–14; Kisil, V. V. (1999). Advances in Mathematics 147(1), 35–73; Prezhdo, O. V. and Kisil, V. V. (1997). Physical Review A 56(1), 162–175) brackets that generate quantum (commutator) and classical (Poisson) brackets in corresponding representations of the Heisenberg group. We do not use any kind of semiclassical approximation or limiting procedure for 0     

Localized-Particles Approach for Classical and Quantum Crystals

This paper is the first review devoted to the localized-particles approach for strongly anharmonic crystals. We present mathematical basises of such an approach for classical and quantum cases and we discuss its different applications. In the framework of this method various technical tricks and detailes could, of course, be changed, but the main idea should be conserved: that of the localization of atoms, forming a solid, near their lattice sites. The localized-particles approach describes not only collective excitations as the phonons but the state of particles themselves too; that is why it is much more convenient for describing crystals with defects, crystal-gas intersurfaces, disordered solids and some phase transitions.     

Identity Rule for Classical and Quantum Theories

It is shown that the identity rule — arule of inference which has the form of modus ponens butwith the operation of identity substituted for theoperation of implication — turns any ortholatticeinto either an orthomodular lattice (a model of a quantumtheory) or a distributive lattice (a model of aclassical theory). It is also shown that — asopposed to the implication algebras — one cannotconstruct an identity algebra although the identity rule contains theoperation of identity as the only operation.     

No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics

Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis $\alpha _{i}$ chosen by Alice, irrespective of Bob’s axis $\beta _{j}$ (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s and Bob’s spins are identified as $A_{ij}$ and $B_{ij}$ , even though their distributions are determined by, respectively, $\alpha _{i}$ alone and $\beta _{j}$ alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins $A_{ij},B_{ij}$ across all values of $\alpha _{i},\beta _{j}$ are constrained by fixing distributions of some subsets thereof. Of special interest among these subsets is the set of probabilistic connections, defined as the pairs $\left( A_{ij},A_{ij'}\right) $ and $\left( B_{ij},B_{i'j}\right) $ with $\alpha _{i}\not =\alpha _{i'}$ and $\beta _{j}\not =\beta _{j'}$ (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of $A_{ij}\not =A_{ij'}$ and $B_{ij}\not =B_{i'j}$ ). Thus, one can achieve a complete KPT characterization of the Bell-type inequalities, or Tsirelson’s inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs $\left( A_{ij},B_{ij}\right) $ that are subject to these inequalities. We show, however, that quantum-mechanical (QM) constraints are special. No-forcing theorem says that if a set of probabilistic connections is not compatible with correlations violating QM, then it is compatible only with the classical–mechanical correlations. No-matching theorem says that there are no subsets of the spin variables $A_{ij},B_{ij}$ whose distributions can be fixed to be compatible with and only with QM-compliant correlations.     

On Classical and Quantum Objectivity

We propose a conceptual framework for understanding the relationship between observables and operators in mechanics. To do so, we introduce a postulate that establishes a correspondence between the objective properties permitting to identify physical states and the symmetry transformations that modify their gauge dependant properties. We show that the uncertainty principle results from a faithful—or equivariant—realization of this correspondence. It is a consequence of the proposed postulate that the quantum notion of objective physical states is not incomplete, but rather that the classical notion is overdetermined.     

Classical and Quantum Coherent States

p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously (V. V. Kisil, p-Mechanics as a physical theory. An Introduction, E-print:arXiv:quant-ph/0212101, 2002; International Journal of Theoretical Physics 41(1), 63–77, 2002). We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allows us to evaluate classical observables at any point of phase space and simultaneously to evaluate quantum probability amplitudes. The example of the forced harmonic oscillator is used to demonstrate these concepts.     

Classical Limit and Quantum Logic

The analysis of the classical limit of quantum mechanics usually focuses on the state of the system. The general idea is to explain the disappearance of the interference terms of quantum states appealing to the decoherence process induced by the environment. However, in these approaches it is not explained how the structure of quantum properties becomes classical. In this paper, we consider the classical limit from a different perspective. We consider the set of properties of a quantum system and we study the quantum-to-classical transition of its logical structure. The aim is to open the door to a new study based on dynamical logics, that is, logics that change over time. In particular, we appeal to the notion of hybrid logics to describe semiclassical systems. Moreover, we consider systems with many characteristic decoherence times, whose sublattices of properties become distributive at different times.     

Separating Classical and Quantum Correlations

We examine to what extent the correlation between two quantum observables at a mixed state can be separated into a classical and a quantum term. The nonunique decomposition of quantum mixed states into pure states makes such a separation ambiguous. We outline this fact by a simple example, which also shows that classical and quantum correlations may cancel each other out.     

Classical and Quantum Radiation Reaction for Linear Acceleration

We investigate the effect of radiation reaction on the motion of a wave packet of a charged scalar particle linearly accelerated in quantum electrodynamics (QED). We give the details of the calculations for the case where the particle is accelerated by a static potential that were outlined in Higuchi and Martin Phys. Rev. D 70 (2004) 081701(R) and present similar results in the case of a time-dependent but space-independent potential. In particular, we calculate the expectation value of the position of the charged particle after the acceleration, to first-order in the fine structure constant in the ℏ→ 0 limit, and find that the change in the expectation value of the position (the position shift) due to radiation reaction agrees exactly with the result obtained using the Lorentz-Dirac force in classical electrodynamics for both potentials.     

Creating Very True Quantum Algorithms for Quantum Energy Based Computing

An interpretation of quantum mechanics is discussed. It is assumed that quantum is energy. An algorithm by means of the energy interpretation is discussed. An algorithm, based on the energy interpretation, for fast determining a homogeneous linear function f(x) := s.x = s ₁ x ₁ + s ₂ x ₂ + ? + s _N x _N is proposed. Here x = (x ₁, … , x _N ), x _j ∈ R and the coefficients s = (s ₁, … , s _N ), s _j ∈ N. Given the interpolation values \((f(1), f(2),...,f(N))=\vec {y}\), the unknown coefficients \(s = (s_{1}(\vec {y}),\dots , s_{N}(\vec {y}))\) of the linear function shall be determined, simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of N. Our method is based on the generalized Bernstein-Vazirani algorithm to qudit systems. Next, by using M parallel quantum systems, M homogeneous linear functions are determined, simultaneously. The speed of obtaining the set of M homogeneous linear functions is shown to outperform the classical case by a factor of N × M.     

Quantum and Classical Disparity and Accord

Discrepancies and accords between quantum (QM) and classical mechanics (CM) related to expectation values and periods are generally found for both the harmonic oscillator (SHO) and a free particle in a box (FPB), which may apply generally. These indicate non-locality is expected throughout QM. The FPB energy states violate the Correspondence Principle. Previously unexpected accords are found and proven that 〈x ²〉 _CM =〈x ²〉 _QM and τ _CM =τ _QMb (beat period i.e. beats between the phases for adjoining energy states) for the SHO for all quantum numbers, n. However, for the FPB the beat periods differ at small n. It is shown that a particle’s velocity in an infinite square well varies, no matter how wide the box, nor how far the particle is from the walls. The quantum free particle variances share an indirect commonality with the Aharonov-Bohm and Aharonov-Casher effects in that there is a quantum action in the absence of a force. The concept of an “Expectation Value over a Partial Well Width” is introduced. This paper raises the question as to whether these inconsistencies are undetectable, or can be empirically ascertained. These inherent variances may need to be fixed, or nature is manifestly more non-classical than expected.