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Erschienen in:
A Brief Overview of the Quantum-Like Formalism in Social Science Kapitel lesen Erstes Kapitel lesen

2023 | OriginalPaper | Buchkapitel

3. Hilbert Space Modelling with Applications in Classical Optics, Human Cognition, and Game Theory

verfasst von : Partha Ghose, Sudip Patra

Erschienen in: Quantum Decision Theory and Complexity Modelling in Economics and Public Policy

Verlag: Springer International Publishing

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Abstract

In this chapter we provide first a basic introduction to Hilbert space modelling, and its applications outside typical quantum mechanics, for example in classical optics, and human cognition. We then present briefly our framework of human cognition model, which we have called, COM: classical optical modelling. Though our chapter is based on the background of and in the wake of quantum-like modelling in cognition, game theory, and different social science areas, our approach differs from the extant models since we follow the Hilbert space modelling in classical optics, with novel features like classical entanglement which can also be exploited in game theory. Hence, we aim to contribute in cognition as well as quantum-like modelling in game theory.

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Vorheriges Kapitel Cooperative Functioning of Unconscious and Consciousness from Theory of Open Quantum Systems
Nächstes Kapitel Remodeling Leadership: Quantum Modeling of Wise Leadership
Fußnoten
1
One anecdotal story goes that David Hilbert, the great German Mathematician, on whose name Neumann named this formulation, was rather oblivious of the fact. In one seminar where Neumann lectured on the properties and applications of Hilbert space, in the end, up went a hand asking but what is a ‘Hilbert’ space? The man was Hilbert himself.
 
2
In modern representations, in case of a product state, the amplitude of the total system is written as a product of amplitudes (outer product to be specific) of its subsystems’ wave functions, but for entangled states, one needs to sum over the proper number of indices to have the amplitude for the total system. A low entanglement state is one where the number of indices to sum over is minimum. For example, if the number is one, then we are dealing with a product state. The number of indices which one has to sum over is often called as bond or auxiliary dimension.
 
3
Here we may think of ‘the Hilbert space’ as an abstract representation, and ‘Hilbert spaces, H’ as concrete Hilbert spaces which are constructed for specific models.
 
4
Any vector ψ ϵ H1 ⊗ H2 such that |ϒ > and |λ > are orthonormal basis in H1 and H2 respectively, can be expressed as ψ = Σcj |ϒj > |λj > , where c s are positive non negative. Again if we have two observables A and B elements of H1 and H2 respectively, then < ψ|(A ⊗ I)|ψ >  = Trace (Aρ1) and < ψ|(I ⊗ B)|ψ >  = Trace (Bρ2), where ρ1 and ρ2 operates on respective Hilbert spaces, and |ϒ > and |λ > s can be eigen vectors for respective density operators, and eigen values of ρ1, say {pj} will be given by cj = √pj in the current example we have two subspaces of same dimensions, but the decomposition formulation holds for any partitions. For example if we have a 5 Qbit space over all, and we partition the space into sub space A of 2 Qbit, such that A is spanned by 4 basis vectors, and sub space B of 3 Qbit space, such that B is spanned by 23 or 8 basis vectors, we will still have the above decomposition formula for ψ, such that the summation index will be over 4, or the no of basis states for the lower dimensional subspace.
Another important concept is of Schimdt number. For example we start with a pure state such as ψ as in the example here, now if we perform a partial trace over it, such that we get back the density matrix of one of the subsystems, then the no of non-zero coefficients in the density matrix representation of that subsystem: ρ = ∑pj |ϒj >  < ϒj|, will be a measure of degree of entanglement in ψ, this is the Schimdt number.
In general we know that from a higher dimensional Hilbert space pure state we can arrive at a lower dimensional mixed state via partial tracing, the converse of this is termed as purification.
 
5
Here we mention that the commutation relations between cognitive variables are not as ‘naturally’ perceived as in standard QM, there is no general theory of commutative relations between cognitive variables, and contextuality may play a deeper role here.
 
6
Poincare sphere is a widely used tool for describing states, like polarization states in Optics as well as in information theory in general. https://​encyclopediaofma​th.​org/​wiki/​Poincar%C3%A9_​sphere.
 
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Metadaten
Titel
Hilbert Space Modelling with Applications in Classical Optics, Human Cognition, and Game Theory
verfasst von
Partha Ghose
Sudip Patra
Copyright-Jahr
2023
Buch
Quantum Decision Theory and Complexity Modelling in Economics and Public Policy

Print ISBN: 978-3-031-38832-3

Electronic ISBN: 978-3-031-38833-0

Copyright-Jahr: 2023

DOI
https://doi.org/10.1007/978-3-031-38833-0_3

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