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arXiv:quant-ph/0502138

New Tales of the Mean King

Andreas Klappenecker, Martin Roetteler

Published Feb 22, 2005• 11 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0502138

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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.
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 problem is a quantum mechanical retrodiction problem, in which Alice has to name the outcome of an ideal measurement made in one of several different orthonormal bases. Alice is allowed to prepare the state of the system and to do a final measurement, possibly including an entangled copy. However, Alice gains knowledge about which basis was measured only after she no longer has access to the quantum system or its copy. We give a necessary and sufficient condition on the bases, for Alice to have a strategy to solve this problem, without assuming that the bases are mutually unbiased. The condition requires the existence of an overall joint probability distribution for random variables, whose marginal pair distributions are fixed as the transition probability matrices of the given bases. In particular, in the qubit case the problem is decided by Bell's original three variable inequality. In the standard setting of mutually unbiased bases, when they do exist, Alice can always succeed. However, for randomly chosen bases her success probability rapidly goes to zero with increasing dimension.
In the King’s Problem, a physicist is asked to prepare a d-state quantum system in any state of her choosing and give it to a king who measures one of (d+1) sets of mutually unbiased observables on it. The physicist is then allowed to make a control measurement on the system, following which the king reveals which set of observables he measured and challenges the physicist to predict correctly all the eigenvalues he found. This paper obtains an upper bound on the physicist’s probability of success at this task if she is allowed to make measurements only on the system itself (the “conventional” solution) and not on the system as well as any ancillary systems it may have been coupled to in the preparation phase, as in the perfect solutions proposed recently. An optimal conventional solution, with a success probability of 0.7, is constructed in d = 4; this is to be contrasted with the success probability of 0.902 for the optimal conventional solution in d = 2. The gap between the best conventional solution and the perfect solution grows quite rapidly with increasing d.
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.
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.