High-fidelity quantum logic operations using linear optical elements

Knill, Laflamme, and Milburn [Nature (London) 409, 46 ((2001))]] have shown that quantum logic operations can be performed using linear optical elements and additional ancilla photons. Their approach is probabilistic in the sense that the logic devices fail to produce an output with a failure rate that scales as 1/n, where n is the number of ancilla. Here we present an alternative approach in which the logic devices always produce an output with an intrinsic error rate that scales as 1/n(2), which may have several advantages in quantum computing applications.     

Automaton partition logic versus quantum logic

The propositional system of a general class of discrete deterministic systems is formally characterized. We find that any finite prime orthomodular lattice allowing two-valued states can be represented by an automaton logic.     

Fuzziness in abacus logic

The concept of abacus logic has recently been developed by the author (Malhas, n.d.). In this paper the relation of abacus logic to the concept of fuzziness is explored. It is shown that if a certain regularity condition is met, concepts from fuzzy set theory arise naturally within abacus logics. In particular it is shown that every abacus logic then has a pre-Zadeh orthocomplementation. It is also shown that it is then possible to associate a fuzzy set with every proposition of abacus logic and that the collection of all such sets satisfies natural conditions expected in systems of fuzzy logic. Finally, the relevance to quantum mechanics is discussed.     

A method of optical implementation of frequency encoded all optical logic gates based on multiphoton processes in non linear material

A novel frequency encoded all optical logic gates are proposed exploiting multiphoton processes in non linear optical medium. In the frequency encoding of the information the ‘0’ is represented by a frequency ω and ‘1’ is represented by another frequency 2ω. The gates proposed are NOT, OR, AND, NAND and NOR among which NAND and NOR are universal. Using these gates one can generate other important gates and logical function generating all optical devices. Two main three-photon processes, second harmonic generation (SHG) and parametric light generation (PLG) are used to implement the gates and the corresponding appropriate non linear material is LiB₃O₅ (LBO) which has wide operating and transparency range in the wavelength 350–3,200 nm. The source of optical frequency encoded signal may be derived from an external cavity diode laser generating a wavelength 1,560 nm for ω (‘0’ state of information) and its second harmonic 780 nm for 2ω (‘1’ state of information).     

Automaton logic

The experimental logic of Moore and Mealy-type automata is investigated.     

Conservative logic

Conservative logic is a comprehensive model of computation which explicitly reflects a number of fundamental principles of physics, such as the reversibility of the dynamical laws and the conservation of certainadditive quantities (among which energy plays a distinguished role). Because it more closely mirrors physics than traditional models of computation, conservative logic is in a better position to provide indications concerning the realization of high-performance computing systems, i.e., of systems that make very efficient use of the computing resources actually offered by nature. In particular, conservative logic shows that it is ideally possible to build sequential circuits with zero internal power dissipation. After establishing a general framework, we discuss two specific models of computation. The first uses binary variables and is the conservative-logic counterpart of switching theory; this model proves that universal computing capabilities are compatible with the reversibility and conservation constraints. The second model, which is a refinement of the first, constitutes a substantial breakthrough in establishing a correspondence between computation and physics. In fact, this model is based on elastic collisions of identical balls, and thus is formally identical with the atomic model that underlies the (classical) kinetic theory of perfect gases. Quite literally, the functional behavior of a general-purpose digital computer can be reproduced by a perfect gas placed in a suitably shaped container and given appropriate initial conditions.This research was supported by the Defense Advanced Research Projects Agency and was monitored by the Office of Naval Research under Contract No. N000 14-75-C-0661.     

Logical independence in quantum logic

The projection latticesP(₁),P(₂) of two von Neumann subalgebras ₁, ₂ of the von Neumann algebra are defined to be logically independent if A B0 for any 0AP(₁), 0BP(₂). After motivating this notion in independence, it is shown thatP(₁),P(₂) are logically independent if ₁ is a subfactor in a finite factor andP(₁),P(₂ commute. Also, logical independence is related to the statistical independence conditions called C^*-independence W^*-independence, and strict locality. Logical independence ofP(₁,P(₂ turns out to be equivalent to the C^*-independence of (₁,₂) for mutually commuting ₁,₂ and it is shown that if (₁,₂) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP(₁,P(₂ are logically independent in the following cases: is a finite-dimensional full-matrix algebra and ₁,₂ are C^*-independent; (₁,₂) is a W^*-independent pair; ₁,₂ have the property of strict locality.     

Quantum logic revisited

An adequate conjunction-implication pair is given for complete orthomodular lattices. The resulting conjunction is noncommutative in nature. We use the well-known lattice of closed subspaces of a Hilbert space, to give physical meaning to the given lattice operation.To the memory of Thomas A. Brody.     

Unified quantum logic

Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.On leave of absence from Department of Mathematics, University of Zagreb, Pot. Pret. 165, YU-41001 Zagreb, Yugoslavia.     

Optical logic redux

Twenty years ago IBM physicist Robert Keyes published a paper entitled “Optical Logic—in the light of computer technology.” It caused an instant furor in the fledgling optical logic community. Now, 20 years after that devastating critique, the field of optical logic has grown enormously. There are literally thousands of papers. Many of them are collected in a bibliography given here. Was Keyes’ critique wrong? Have opticists simply ignored what Keyes pointed out? Have new developments made some of his remarks not quite so relevant? We argue here that

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Keyes was and still is mostly correct, but that may change in a few years

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Many researchers have indeed simply ignored what he said

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New developments in both optical logic and its applications open niches for optical logic that Keyes did not (and probably could not) anticipate

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New and anticipated developments in electronics may increase the role for optics

     

Coherent quantum logic

The von Neumann quantum logic lacks two basic symmetries of classical logic, that between sets and classes, and that between lower and higher order predicates. Similarly, the structural parallel between the set algebra and linear algebra of Grassmann and Peano was left incomplete by them in two respects. In this work a linear algebra is constructed that completes this correspondence and is interpreted as a new quantum logic that restores these invariances, and as a quantum set theory. It applies to experiments with coherent quantum phase relations between the quantum and the apparatus. The quantum set theory is applied to model a Lorentz-invariant quantum time-space complex.     

On quantum logic

The status and justification of quantum logic are reviewed. On the basis of several independent arguments it is concluded that it cannot be a logic in the philosophical sense of a general theory concerning the structure of valid inferences. Taken as a calculus for combining quantum mechanical propositions, it leaves a number of significant aspects of quantum physics unaccounted for. It is shown, moreover, that quantum logic, far from being more general than Boolean logic, forms a subset of a slight and natural extension of Boolean logic, a subset which corresponds to incomplete statements. The philosophical background of this unsatisfactory state of affairs is briefly explored.     

Infinitesimals without logic

We introduce a ring of the so-called Fermat reals, which is an extension of the real field containing nilpotent infinitesimals. The construction is inspired by Smooth Infinitesimal Analysis (SIA) and provides a powerful theory of actual infinitesimals without any background in mathematical logic. In particular, in contrast to SIA, which admits models in intuitionistic logic only, the theory of Fermat reals is consistent with the classical logic. We face the problem of deciding whether or not a product of powers of nilpotent infinitesimals vanishes, study the identity principle for polynomials, and discuss the definition and properties of the total order relation. The construction is highly constructive, and every Fermat real admits a clear and order-preserving geometrical representation. Using nilpotent infinitesimals, every smooth function becomes a polynomial because the remainder in Taylor’s formulas is now zero. Finally, we present several applications to informal classical calculations used in physics, and all these calculations now become rigorous, and at the same time, formally equal to the informal ones. In particular, an interesting rigorous deduction of the wave equation is given, which clarifies how to formalize the approximations tied with Hooke’s law using the language of nilpotent infinitesimals.     

Relativistic quantum logic

On the basis of the well-known quantum logic and quantum probability a formal language of relativistic quantum physics is developed. This language incorporates quantum logical as well as relativistic restrictions. It is shown that relativity imposes serious restrictions on the validity regions of propositions in space-time. By an additional postulate this relativistic quantum logic can be made consistent. The results of this paper are derived exclusively within the formal quantum language; they are, however, in accordance with well-known facts of relativistic quantum physics in Hilbert space.     

Three-valued Brouwer-zadeh logic

A semantic investigation of a particular form of Brouwer-Zadeh logic (three-valued Brouwer-Zadeh logic) is presented and it is shown that this logic can be characterized by means of Kripke-style semantics. Some connections of Brouwer-Zadeh logics with unsharp quantum mechanics are also investigated.     

Magnetic-bubble conservative logic

Among integrated-circuit devices, magnetic bubbles are a particularly interesting candidate to implement the Fredkin gate and conservative logic. The magnetostatic repulsion of magnetic bubbles simulates the bouncing-ball model of conservative logic.     

The Leibniz principle in quantum logic

The principle of the identity of indiscernibles (Leibniz Principle) is investigated within the framework of the formal language of quantum physics, which is given by an orthomodular lattice. We show that the validity of this principle is based on very strong preconditions (concerning the existence of convenient predicates) which are given in the language of classical physics but which cannot be fulfilled in orthomodular quantum logic.     

Propositions and orthocomplementation in quantum logic

We briefly analyze two partial order relations that are usually introduced in quantum logic by making use of the concepts of yes-no experiment and of preparation as fundamental. We show that two distinct posetsE andL can be defined, the latter being identifiable with the lattice of quantum logic. We consider the posetE and find that it contains a subsetE ₀ which can easily be orthocomplemented. These results are used, together with suitable assumptions, in order to show that an Orthocomplementation inL can be deduced by the Orthocomplementation defined inE ₀, and also to give a rule to find the orthocomplement of any element ofL.Research sponsored by C.N.R. (Italy).