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WORLD SCIENTIFIC eBooks

Quantum Paths

Hui Khoon Ng, Rui Han

Published Jan 8, 2015• 1 citation
Publisher

Abstract

Quantum Optics and Open Quantum Systems Wave-Particle Duality Mutually Unbiased Bases and the Mean King's Problem Foundations of Quantum Mechanics Cold Atoms and Density Functionals Information About Quantum Systems Personal Anecdotes
DOI
10.1142/9485

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The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension ddd is investigated. It is shown that a solution of the problem exists if and only if the maximal number (d+1)(d+1)(d+1) of orthogonal Latin squares exists. This implies that there is no solution in d=6d=6d=6 or d=10d=10d=10 dimensions even if the maximal number of MUB's exists in these dimensions.
In quantum information, complementarity of quantum mechanical observables plays a key role. The eigenstates of two complementary observables form a pair of mutually unbiased bases (MUBs). More generally, a set of MUBs consists of bases that are all pairwise unbiased. Except for specific dimensions of the Hilbert space, the maximal sets of MUBs are unknown in general. Even for a dimension as low as six, the identification of a maximal set of MUBs remains an open problem, although there is strong numerical evidence that no more than three simultaneous MUBs do exist. Here, by exploiting a newly developed holographic technique, we implement and test different sets of three MUBs for a single photon six-dimensional quantum state (a "qusix"), encoded exploiting polarization and orbital angular momentum of photons. A close agreement is observed between theory and experiments. Our results can find applications in state tomography, quantitative wave-particle duality, quantum key distribution.
The mean king problem is a conditional retrodiction problem. In this problem Alice prepares a two prime-dimensional particles state and avails one of the particles to the king who measures its state in one of mutually unbiased bases of his choice. The king tells Alice his choice of basis after she completes a control measurement on his particle. Conditioned on this knowledge, she now infers the state observed by the king by utilizing the outcome of her control measurement. In the extended mean king problem, studied in this paper, the king does not tell Alice his measurement basis, but instead both the king and Alice repeat their measurements. Proper ordering of these allows Alice to deduce both the basis used by the king and the outcome of his first measurement, with the king reticent throughout, i.e., this protocol effects a (almost) complete retrodiction of the king's first measurement.