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arXiv:1205.5406

Geometrical view of the Mean King Problem

M. Revzen

Published May 24, 2012• 2 citations
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Abstract

Finite geometry is used to underpin finite, $d^2$, dimensional Hilbert space accommodating two particles, d dimensional each. d=prime $\ne2$. 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.

DOI
10.48550/arxiv.1205.5406

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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.
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Maximal entangled states (MES) provide a basis to two d-dimensional particles Hilbert space, d=prime ≠2\ne 2=2. The MES forming this basis are product states in the collective, center of mass and relative, coordinates. These states are associated (underpinned) with lines of finite geometry whose constituent points are associated with product states carrying Mutual Unbiased Bases (MUB) labels. This representation is shown to be convenient for the study of the Mean King Problem and a variant thereof, termed Tracking the King which proves to be a novel quantum communication channel. The main topics, notions used are reviewed in an attempt to have the paper self contained.
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.
Maximal entangled states (MES) provide a basis to 2d-dimensional particles Hilbert space, d=prime ≠2\ne2=2. These states allow generalization of the Mean King Problem. The states may be viewed as build of points each underpins a product state carrying a mutual unbiased bases (MUB) label or, alternatively, as product states labeled with center of mass and relative coordinates. The coordinate-like label of the center of mass and the momentum-like of the relative coordinates provides a MES account of the Hilbert space in close analogy with the single particle phase space coordinates.
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.