Comparison of the Mathematical Formalism of Associative ANN and Quantum Theory

7.1 Main Neuro-Quantum Analogies And Their Informational Significance

It is best to introduce quantum associative nets by first presenting parallels between quantum processes and neural-net processes. Many mathematical analogies of the theory of Hopfield- like associative ANN and the quantum theory can be found. Because we know that our

ANN simulations perform well (e.g., Perus, 2002, and refs. within), we can infer, using the correspondence list below, about similar effective information-processing capabilities on the quantum level also. Here is an overview of neuro-quantum analogies:

• 1.

  Neuronal State-Vector ↔ Quantum Wave-Function:

In artificial neural network (ANN) theory, the state of the system of neurons¹ is described by which denotes the activity of an individual neuron (located at ) at time t. Neuronal patterns are special neuronal configurations which represent some meaningful information. In quantum theory, the state of the quantum system at location r and time t is described by the wave-function Ψ () (Bohm, 1954; Messiah, 1965). They both represent a state vector describing a parallel-distributed configuration of a complex system.

• 2.

  Neuronal State Is a Superposition of Neuronal Patterns↔ Quantum Wave-Function Is a Superposition of Quantum Eigen-Wave-Functions:

A neuronal configuration q may be described as a linear combination of neuronal patterns (k = 1,..., P). P is the number of patterns represented simultaneously in the combination. Similarly, a wave-function Ψ can be described as a linear combination of eigen-states Ψ_k (”quantum patterns”):

Neuronal patterns and eigen-wave-functions can represent some object of conscious experience on different levels. So, as opposed to other configurations, they represent informational states which have a meaning, because they are correlated with some environmental objects.

• 3.

  Both sets of pattern-vectors, and y_k, usually have the properties of ORTHOGONALITY (mutual scalar product is zero) and NORMALITY (their ”length” is 1). The first property means that patterns do not have”anything in common”.

If this is satisfied, it ensures error-free recall of patterns, because it excludes overlap and mixing between patterns or corruption of a pattern because of this. So, the interference between patterns must be destroyed during recall in order to get one of them out properly, and this is realizable if patterns are orthogonal. However, interference must be eliminated during pattern-recall only; in associative memory, interference between patterns is essential. If eigenstates and y_k are not orthogonal completely, recall is not so pure, but associations between patterns may become more effective (cf., ch. 8.5).

(Quasi) orthogonality is a comparative property of biologically-processed images as far as their representations have high dimensionality (a lot of encoding units), or if their coding is sparse (Baird, 1990). This is often the case, unless very similar images are observed, which get anyway ”melted together into” the same attractor.