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arXiv:1206.3884

Maximal Entanglement via Collective Coordinates

M. Revzen

Published Jun 18, 2012• 0 citations
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Abstract

Maximal entangled states (MES) provide a basis to 2d-dimensional particles Hilbert space, d=prime $\ne2$. 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.

DOI
10.48550/arxiv.1206.3884

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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.
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.
Every Maximally Entangled State (MES) of two d-dimensional particles is shown to be a product state of suitably chosen collective coordinates. The state may be viewed as defining a "point" in a "phase space" like d^2 array representing d^2 orthonormal Maximally Entangled States basis for the Hilbert space. A finite geometry view of MES is presented and its relation with the afore mentioned "phase space" is outlined: "straight lines" in the space depict product of single particle mutually unbiased basis (MUB) states, inverting thereby Schmidt's diagonalization scheme in giving a product single particle states as a d-terms sum of maximally entangled states. To assure self sufficiency the essential mathematical results are summarized in the appendices.
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.
Relative and center-of-mass coordinates are used to generalize mutually unbiased bases (MUB) and define mutually unbiased collective bases (MUCB). Maximally entangled states are given as product states in the collective variables. These states are analyzed in terms of the Wigner representative function of the states and shown to display a discontinuous attribute of the entanglement. Finite Hilbert space dimensionality collective coordinates are introduced and provide a framework for the analysis.
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.
A few simply stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs) and constrain the combinations of such patterns that can coexist in full complements of MUBs. We consider Hilbert spaces of prime power dimensions (D=pND={p}^{N}D=pN), as realized by systems of NNN prime-state particles, where full complements of D+1D+1D+1 MUBs are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using any particular construction. The general rules include the following: (1) In any MUB, a given particle appears either in a pure state or totally entangled and (2) in any full MUB complement, each particle is pure in (p+1p+1p+1) bases (not necessarily the same ones) and totally entangled in the remaining (${p}^{N}\ensuremath{-}p$). It follows that the maximum number of product bases is p+1p+1p+1 and, when this number is realized, all remaining (${p}^{N}\ensuremath{-}p$) bases in the complement are characterized by the total entanglement of every particle. This ``standard distribution'' is inescapable for two-particle systems (of any ppp), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits [Phys. Rev. A 65. 032320 (2002); Phys. Rev. A 72, 062310 (2005)] and qutrits [Phys. Rev. A 70, 012302 (2004)], drawing particularly upon [Phys. Rev. A 72, 062310 (2005)]. With three particles there are three MUB types, and these may be combined in (p+2p+2p+2) different ways to form full complements. With N=4N=4N=4, there are 6 MUB types for p=2p=2p=2, but new MUB types become possible with larger ppp, and these are essential to realizing full complements. With this example, we argue that new MUB types that show new entanglement patterns should enter with every step in NNN and, also, when NNN is a prime plus 1, at a critical ppp value, $p=N\ensuremath{-}1$. Such MUBs should play critical roles in filling complements.