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arXiv:quant-ph/0101065 · Zeitschrift für Naturforschung A

The Mean King's Problem: Spin

Yakir Aharonov, Berthold‐Georg Englert

Published Feb 1, 2001• 40 citations
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Abstract

Abstract We show how one can ascertain the values of four mutually complementary observables of a spin-1 degree of freedom. -Pacs: 03.65.Bz
DOI
10.1515/zna-2001-0104

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We investigate the king's problem of the measurement of operators ${\stackrel{P\vec}{n}}_{k}∙\stackrel{P\vec}{\ensuremath{\sigma}}$ (k=1k=1k=1, 2, 3) instead of the three Cartesian components ${\ensuremath{\sigma}}_{x},{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}$ of the spin operator $\stackrel{P\vec}{\ensuremath{\sigma}}$. Here, ${\stackrel{P\vec}{n}}_{k}$ are three-dimensional real unit vectors. We show the condition over three vectors ${\stackrel{P\vec}{n}}_{k}$ to ascertain the result for measurement of any one of these operators.
The Mean King's problem asks to determine the outcome of a measurement that is randomly selected from a set of complementary observables. We review this problem and offer a combinatorial solution. More generally, we show that whenever an affine resolvable design exists, then a state reconstruction problem similar to the Mean King's problem can be defined and solved. As an application of this general framework we consider a problem involving three qubits in which the outcome of nine different measurements can be determined without using ancillary qubits. The solution is based on a measurement derived from Hadamard designs.
In this thesis, we study several variants of the Mean King problem from the perspective of quantum information and optimization. The Mean King problem is a quantum guessing task, where one party prepares a quantum state, another party performs a measurement chosen from a given set of bases, and the first party tries to guess the measurement outcome. We consider four variants of this problem, including both non-interactive and interactive settings. We analyze their optimal guessing performance in both the average-case and the worst-case settings. We formulate the success probabilities of these Mean King games as optimization problems over quantum states and measurement strategies. This approach gives a unified way to study the different variants. For the non-interactive games, we derive the winning operators and show that some of the optimization problems can be reduced to eigenvalue computations. For the interactive games, we formulate them as semidefinite programs. These programs include extra constraints that come from returned information and partial trace conditions. These formulations support both theoretical analysis and numerical computation. We use MATLAB and CVX to compute optimal success probabilities for several choices of measurement bases. These bases include mutually unbiased bases and rotated bases. The results show how the choice of measurement bases affects the guessing probability. The results also show the difference between average-case and worst-case performance across different Mean King variants. In particular, in one variant, the numerical results for two and three mutually unbiased bases are close to the known success probabilities of the 2 → 1 and 3 → 1 quantum random access code tasks. These numerical similarities suggest that Mean King problems and random access codes may share a deeper connection. This observation gives a possible direction for future work.