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arXiv:0710.2952

The "mean king's problem" with continuous variables

Alonso Botero, Yakir Aharonov

Published Oct 16, 2007• 2 citations
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Abstract

We present the solution to the "mean king's problem" in the continuous variable setting. We show that in this setting, the outcome of a randomly-selected projective measurement of any linear combination of the canonical variables x and p can be ascertained with arbitrary precision. Moreover, we show that the solution is in turn a solution to an associated "conjunctive" version of the problem, unique to continuous variables, where the inference task is to ascertain all the joint outcomes of a simultaneous measurement of any number of linear combinations of x and p.

DOI
10.48550/arxiv.0710.2952

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We present the solution to the "mean king's problem" in the continuous variable setting. We show that in this setting, the outcome of a randomly-selected projective measurement of any linear combination of the canonical variables x and p can be ascertained with arbitrary precision. Moreover, we show that the solution is in turn a solution to an associated "conjunctive" version of the problem, unique to continuous variables, where the inference task is to ascertain all the joint outcomes of a simultaneous measurement of any number of linear combinations of x and p.
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 the Mean King's Problem a ship-wrecked physicist is challenged to infer the outcome of an intermediate spin measurement by a suitable preparation in conjunction with an appropriate control measurement -- the 1987 spin retrodiction puzzle of Vaidman, Aharonov, and Albert. We report the experimental realization of a quantum-optical version in which the outcome of an intermediate polarization projection is inferred by exploiting single-photon two-qubit states. The experimental success probability is consistently above the 90.2% threshold of the optimal one-qubit strategy, with an average success probability of 95.6%.
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 with mutually unbiased bases is reconsidered for arbitrary ddd-level systems. Hayashi et al. [Phys. Rev. A 71, 052331 (2005)] related the problem to the existence of a maximal set of $d\ensuremath{-}1$ mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the problem when, e.g., d=6d=6d=6 or d=10d=10d=10. In contrast to their result, we show that the king's problem always has a solution for arbitrary levels if we also allow positive operator-valued measures. In constructing the solution, we use orthogonal arrays in combinatorial design theory.