Cartoon Computation: Quantum-like computing without quantum mechanics

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.Comment: version accepted in J. Phys.A (Letter to the Editor

Geometric Algebra Model of Distributed Representations

Formalism based on GA is an alternative to distributed representation models developed so far --- Smolensky's tensor product, Holographic Reduced Representations (HRR) and Binary Spatter Code (BSC). Convolutions are replaced by geometric products, interpretable in terms of geometry which seems to be the most natural language for visualization of higher concepts. This paper recalls the main ideas behind the GA model and investigates recognition test results using both inner product and a clipped version of matrix representation. The influence of accidental blade equality on recognition is also studied. Finally, the efficiency of the GA model is compared to that of previously developed models.Comment: 30 pages, 19 figure

On classical models of spin

The reason for recalling this old paper is the ongoing discussion on the attempts of circumventing certain assumptions leading to the Bell theorem (Hess-Philipp, Accardi). If I correctly understand the intentions of these Authors, the idea is to make use of the following logical loophole inherent in the proof of the Bell theorem: Probabilities of counterfactual events A and A' do not have to coincide with actually measured probabilities if measurements of A and A' disturb each other, or for any other fundamental reason cannot be performed simulaneously. It is generally believed that in the context of classical probability theory (i.e. realistic hidden variables) probabilities of counterfactual events can be identified with those of actually measured events. In the paper I give an explicit counterexample to this belief. The "first variation" on the Aerts model shows that counterfactual and actual problems formulated for the same classical system may be unrelated. In the model the first probability does not violate any classical inequality whereas the second does. Pecularity of the Bell inequality is that on the basis of an in principle unobservable probability one derives probabilities of jointly measurable random variables, the fact additionally obscuring the logical meaning of the construction. The existence of the loophole does not change the fact that I was not able to construct a local model violating the inequality with all the other loopholes eliminated.Comment: published as Found. Phys. Lett. 3 (1992) 24

Quantum Aspects of Semantic Analysis and Symbolic Artificial Intelligence

Modern approaches to semanic analysis if reformulated as Hilbert-space problems reveal formal structures known from quantum mechanics. Similar situation is found in distributed representations of cognitive structures developed for the purposes of neural networks. We take a closer look at similarites and differences between the above two fields and quantum information theory.Comment: version accepted in J. Phys. A (Letter to the Editor

Hidden measurements, hidden variables and the volume representation of transition probabilities

We construct, for any finite dimension $n$, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For $n=2$ our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions $n \geq 3$. We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our hidden variables scheme and their relationship with the results of Fine.Comment: 23 pages, 1 figur

Teleportation of geometric structures in 3D

Simplest quantum teleportation algorithms can be represented in geometric terms in spaces of dimensions 3 (for real state-vectors) and 4 (for complex state-vectors). The geometric representation is based on geometric-algebra coding, a geometric alternative to the tensor-product coding typical of quantum mechanics. We discuss all the elementary ingredients of the geometric version of the algorithm: Geometric analogs of states and controlled Pauli gates. Fully geometric presentation is possible if one employs a nonstandard representation of directed magnitudes, formulated in terms of colors defined via stereographic projection of a color wheel, and not by means of directed volumes.Comment: typos corrected, one plot remove