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- Aerts, D. and Czachor, M. (2007). Cartoon computation: Quantum-like algorithms without quantum mechanics,
Journal of Physics A: Mathematical and Theoretical,40, F259-F266, Fast Track Communication. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0611279. doi: 10.1088/1751-8113/40/13/F01. download pdf.

Abstract: We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.- Aerts, D. and Czachor, M. (2007). Two-state dynamics for replicating two-strand systems.
Open Systems & Information Dynamics,14, 397-410. Archive reference and link: http://uk.arxiv.org/abs/q-bio/0512048. doi: 10.1007/s11080-007-9064-0. download pdf.

Abstract: We propose a formalism for describing two-strand systems of a DNA type by means of soliton von Neumann equations, and illustrate how it works on a simple example exactly solvably by a Darboux transformation. The main idea behind the construction is the link between solutions of von Neumann equations and entangled states of systems consisting of two subsystems evolving in time in opposite directions. Such a time evolution has analogies in realistic DNA where the polymerazes move on leading and lagging strands in opposite directions.- Aerts, D., Czachor, M., Dehaene, J., De Moor, B. and D'Hooghe, B. (2007). Macroscopic models for quantum systems and computers.
Journal of Physics: Conference Series,70, 012001. doi: 10.1088/1742-6596/70/1/012001. download pdf.

Abstract: We present examples of macroscopic systems entailing a quantum mechanical structure. One of our examples has a structure which is isomorphic to the spin structure for a spin 1/2 and another system entails a structure isomorphic to the structure of two spin 1/2 in the entangled singlet state. We elaborate this system by showing that an arbitrary tensor product state representing two entangled qubits can be described in a complete way by a specific internal constraint between the ray or density states of the two qubits, which describes the behavior of the state of one of the spins if measurements are executed on the other spin. Since any n-qubit unitary operation can be decomposed into 2-qubit gates and unary operations, we argue that our representation of 2-qubit entanglement contributes to a better understanding of the role of n-qubit entanglement in quantum computation. We illustrate our approach on two 2-qubit algorithms proposed by Deutsch, respectively Arvind et al. One of the advantages of the 2-qubit case besides its relative simplicity is that it allows for a nice geometrical representation of entanglement, which contributes to a more intuitive grasp of what is going on in a 2-qubit quantum computation.- Aerts, D., Czachor, M. and Pawlowski, M. (2007). Security in quantum cryptography vs. nonlocal hidden variables.
AIP Conference Proceedings,889, pp. 71-78. doi: 10.1063/1.2713448. download pdf.

Abstract: In order to prove equivalence of quantum mechanics with nonlocal hidden-variable theories of a Bohm type one assumes that all the possible measurements belong to a restricted class: (a) we measure only positions of particles and (b) have no access to exact values of initial conditions for Bohm's trajectories. However, in any computer simulation based on Bohm's equations one relaxes the assumption (b) and yet obtains agreement with quantum predictions concerning the results of positional measurements. Therefore a theory where (b) is relaxed, although in principle allowing for measurements of a more general type, cannot be experimentally falsified within the current experimental paradigm. Such generalized measurements have not been invented, or have been invented but the information is qualified, but we cannot exclude their possibility on the basis of known experimental data. Since the measurements would simultaneously allow for eavesdropping in standard quantum cryptosystems, the arguments for security of quantum cryptography become logically circular: Bohm-type theories do not allow for eavesdropping because they are fully equivalent to quantum mechanics, but the equivalence follows from the assumption that we cannot measure hidden variables, which would be equivalent to the possibility of eavesdropping... Here we break the vicious circle by a simple modification of entangled-state protocols that makes them secure even if our enemies have more imagination and know how to measure hidden-variable initial conditions with arbitrary precision.- Gabora, L. and Aerts, D. (2007). A cross-disciplinary framework for the description of contextually mediated change.
Electronic Journal of Theoretical Physics,4, 1-22.

Abstract: We present a mathematical framework (referred to as Context-driven Actualization of Potential, or CAP) for describing how entities change over time under the influence of a context. The approach facilitates comparison of change of state of entities studied in different disciplines. Processes are seen to differ according to the degree of nondeterminism, and the degree to which they are sensitive to, internalize, and depend upon a particular context. Our analysis suggests that the dynamical evolution of a quantum entity described by the Schrodinger equation is not fundamentally different from change provoked by a measurement often referred to as collapse but a limiting case, with only one way to collapse. The biological transition to coded replication is seen as a means of preserving structure in the face of context, and sexual replication as a means of increasing potentiality thus enhancing diversity through interaction with context. The framework sheds light on concepts like selection and fitness, reveals how exceptional Darwinian evolution is as a means of 'change of state', and clarifies in what sense culture (and the creative process underlying it) are Darwinian.

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