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arXiv:1812.07601 · Research

Experimental Test of Tracking the King Problem

Cheng-Qiu Hu, Jun Gao, Lu-Feng Qiao, Ruo-Jing Ren, Zhu Cao, Zeng-Quan Yan, Zhi-Qiang Jiao, Hao Long Tang, Zhihao Ma, Xian-Min Jin

Published Jan 1, 2019• 4 citations
arXivHTMLPDF

Abstract

In quantum theory, the retrodiction problem is not as clear as its classical counterpart because of the uncertainty principle of quantum mechanics. In classical physics, the measurement outcomes of the present state can be used directly for predicting the future events and inferring the past events which is known as retrodiction. However, as a probabilistic theory, quantum-mechanical retrodiction is a nontrivial problem that has been investigated for a long time, of which the Mean King Problem is one of the most extensively studied issues. Here, we present the first experimental test of a variant of the Mean King Problem, which has a more stringent regulation and is termed "Tracking the King." We demonstrate that Alice, by harnessing the shared entanglement and controlled-not gate, can successfully retrodict the choice of King's measurement without knowing any measurement outcome. Our results also provide a counterintuitive quantum communication to deliver information hidden in the choice of measurement.

DOI
10.34133/2019/3474305

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In quantum theory, the retrodiction problem is not as clear as its classical counterpart because of the uncertainty principle of quantum mechanics. In classical physics, the measurement outcomes of the present state can be used directly for predicting the future events and inferring the past events which is known as retrodiction. However, as a probabilistic theory, quantum-mechanical retrodiction is a nontrivial problem that has been investigated for a long time, of which the Mean King Problem is one of the most extensively studied issues. Here, we present the first experimental test of a variant of the Mean King Problem, which has a more stringent regulation and is termed "Tracking the King." We demonstrate that Alice, by harnessing the shared entanglement and controlled-not gate, can successfully retrodict the choice of King's measurement without knowing any measurement outcome. Our results also provide a counterintuitive quantum communication to deliver information hidden in the choice of measurement.
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 complete (almost) retrodiction of the King's first measurement.
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.
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.
When a photon passes through an interferometer, quantum mechanics does not provide a clear answer as to its past. Quantum retrodiction is a quantitative theory, which endeavors to make statements about the past of a system based on present knowledge. Quantum retrodiction may be used to analyze the past of a photon, that is, its trajectory. Here we experimentally retrodict the trajectories of single photons in double interferometers by measuring the final state of the photon. A sequence of measurements is made on a photon to determine which path the photon followed, so a series of retrodiction of measurement results can be regarded as a photon trajectory. We obtain information about the partial trajectory and the entire trajectory of the photon. Furthermore, we also observe the effect of different measurements in the extraction of trajectory information. Our experiment highlights the application of retrodiction theory to the study of the photon's past, and provides potential application in quantum communications.
Experimental realization of Gedanken experiments on Quantum Mechanics continue to suggest inferences of action at a distance, action into the future or past as reported in Delayed choice Quantum entanglement experiments recently by Ma et al (2012). Is it possible to read through this and investigate through simple logic and experiments whether control of the past or future is possible? A set of experiments are suggested here to do the same whose results may be interesting. Gedanken experiments on Quantum Mechanics principles Einstein, Podolsky and Rosen (Einstein et al, 1935) were the first to suggest a Gedanken experiment which would suggest instant communication over a distance between particles which was against the principle of Special