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arXiv:quant-ph/0306119 · Zeitschrift für Naturforschung A

Best Conventional Solutions to the King’s Problem

P. K. Aravind

Published Dec 1, 2003• 10 citations
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Abstract

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.

DOI
10.1515/zna-2003-1204

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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.

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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.
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.
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%.
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.
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.
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.