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arXiv:quant-ph/0201017

Unspeakable quantum information

Asher Peres, Petra F. Scudo

Published Jan 6, 2002• 9 citations
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Abstract

No verbal explanation can indicate a direction in space or the orientation of a coordinate system. Only material objects can do it. In this article we consider the use of a set of spin-\half particles in an entangled state for indicating a direction, or a hydrogen atom in a Rydberg state for transmitting a Cartesian frame. Optimal strategies are derived for the emission and detection of the quantum signals.

DOI
10.48550/arxiv.quant-ph/0201017

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No verbal explanation can indicate a direction in space or the orientation of a coordinate system. Only material objects can do it. In this article we consider the use of a set of spin-\half particles in an entangled state for indicating a direction, or a hydrogen atom in a Rydberg state for transmitting a Cartesian frame. Optimal strategies are derived for the emission and detection of the quantum signals.
We consider the use of NNN spin- 12\frac{1}{2}21​ particles for indicating a direction in space. If N>2N>2N>2, their optimal state is entangled. For large NNN, the mean square error decreases as ${N}^{\ensuremath{-}2}$ (rather than ${N}^{\ensuremath{-}1}$ for parallel spins).
The orientation in space of a Cartesian coordinate system can be indicated by the two vectorial constants of motion of a classical Keplerian orbit: the angular momentum and the Laplace-Runge-Lenz vector. In quantum mechanics, the states of a hydrogen atom that mimic classical elliptic orbits are the coherent states of the SO(4) rotation group. It is known how to produce these states experimentally. They have minimal dispersions of the two conserved vectors and can be used as direction indicators. We compare the fidelity of this transmission method with that of the idealized optimal method.
We construct the optimal strategy for the estimation of an unknown unitary transformation $U\ensuremath{\in}\mathrm{SU}(d).$ This includes, in addition to a convenient measurement on a probe system, finding which is the best initial state on which U is to act. When $U\ensuremath{\in}\mathrm{SU}(2),$ such an optimal strategy can be applied to simultaneously estimate both the direction and the strength of a magnetic field, and shows how to use a spin 1/2 particle to transmit information about a whole coordinate system instead of only a direction in space.