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arXiv:quant-ph/0612053 · Zeitschrift für Naturforschung A

Comments on “Best Conventional Solutions to the King’s Problem”

Gen Kimura, Hajime Tanaka, Masanao Ozawa

Published Apr 1, 2007• 4 citations
arXivHTMLPDF

Abstract

Conventional solutions of the (Mean) King’s problem without using entanglement have been investigated by P. K. Aravind, Z. Naturforsch. 58a, 682 (2003). We report that the upper bound for the success probability claimed is not valid in general but we give a condition for the claim to be justified.
DOI
10.1515/zna-2007-3-406

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Comment on “Quantum Controlled Teleportation of Bell State Using Seven-Qubit Entangled State”

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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.
Vaidman, Aharanov, and Albert [ Phys. Rev. Lett. 58 ( 14 ), 1385 ( 1987 ) 10.1103/PhysRevLett.58.1385 ] put forward a puzzle called the mean king’s problem (MKP) that can be solved only by harnessing quantum entanglement. Prime-powered solutions to the problem have been shown to exist, but they have not yet been experimentally realized for any dimension beyond two. We propose a general first-of-its-kind experimental scheme for solving the MKP in prime dimensions ( D ). Our search is guided by the digital discovery framework Pytheus , which finds highly interpretable graph-based representations of quantum optical experimental setups; using it, we find specific solutions and generalize to higher dimensions through human insight. As proof of principle, we present a detailed investigation of our solution for the three-, five-, and seven-dimensional cases. We obtain maximum success probabilities of 82.3%, 56.2%, and 35.5%, respectively. We therefore posit that our computer-inspired scheme yields solutions that implement Alice’s strategy with quantum advantage, demonstrating its promise for experimental implementation in quantum communication tasks.
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.