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arXiv:1111.6446

Geometrical Underpinning of Finite Dimensional Hilbert space

M. Revzen

Published Nov 28, 2011• 2 citations
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Abstract

Finite geometry is employed to underpin operators in finite, d, dimensional Hilbert space. The central role of mutual unbiased bases (MUB) states projectors is exhibited. Interrelation among operators in Hilbert space, revealed through their (finite) dual affine plane geometry (DAPG) underpinning is studied. Transcription to (finite) affine plane geometry (APG) is given and utilized for their interpretation.

DOI
10.48550/arxiv.1111.6446

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Finite geometry is employed to underpin operators in finite, d, dimensional Hilbert space. The central role of mutual unbiased bases (MUB) states projectors is exhibited. Interrelation among operators in Hilbert space, revealed through their (finite) dual affine plane geometry (DAPG) underpinning is studied. Transcription to (finite) affine plane geometry (APG) is given and utilized for their interpretation.
Finite Geometry is used to underpin operators acting in finite, d, dimensional Hilbert space. Quasi distribution and Radon transform underpinned with finite dual affine plane geometry (DAPG) are defined in analogy with the continuous (d→∞d \rightarrow \inftyd→∞) Hilbert space case. An essntial role in these definitions play the projectors of states of mutual unbiased bases (MUB) and their Wigner function-like mapping onto the generalized phase space that lines and points of DAPG constitutes.
A novel analysis of finite-dimensional Hilbert space is outlined. The approach bypasses the general, inherent, difficulties present in handling angular variables in finite-dimensional problems: the finite-dimensional, d , odd prime, Hilbert space operators are underpinned with a finite geometry which provides intuitive perspectives to the physical operators. The analysis emphasizes a central role for projectors of mutual unbiased bases (MUB) states, extending thereby their use in finite-dimensional quantum-mechanics studies. Interrelations among the Hilbert space operators revealed via their (finite) dual affine plane geometry (DAPG) underpinning are displayed and utilized in formulating the finite-dimensional ubiquitous Radon transformation and its inverse illustrating phase-space-like physics encoded in lines and points of the geometry. The finite geometry required for our study is outlined.
Finite geometry is used to underpin finite, d2d^2d2, dimensional Hilbert space accommodating two particles, d dimensional each. d=prime ≠2\ne2=2. Central role is allotted to states with mutual unbiased bases (MUB) labelling underpinned with points of finite dual affine plane geometry (DAPG). The DAPG lines are shown to underpin maximally entangled states which form an orthonormal basis spanning the space and provide a novel, geometrical view to a new solution of the Mean King Problem (MKP). Brief expositions to the topics considered: MUB, DAPG and the MKP are included rendering the paper self contained.
Finite geometry is used to underpin finite, two d-dimensional particles Hilbert space, d=prime 6= 2. A central role is allotted to states with mutual unbiased bases (MUB) labeling. Dual affine plane geometry (DAPG) points underpin single particle, MUB labeled, product states. The DAPG lines are shown to underpin maximally entangled states which form an orthonormal basis spanning the space. The relevance of mutually unbiased collective coordinates bases (MUCB) for dealing with maximally entangled states is discussed and shown to provide an economic alternative mode of study. These maximally entangled, geometrically reasoned states, provide the resource to a transparent solution to what may be termed tracking of the Mean King Problem (MKP): here Alice prepares a state measured by King along some orientation which Alice succeed in identifying with a subsequent measurement. Brief expositions of the topics considered: MUB, DAPG, MUCB and the MKP are included, rendering the paper self contained.
A complete set of N + 1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = pk, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = pk. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.