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arXiv:1912.08457 · Physical Review A

Experimental investigation of entropic uncertainty relations and coherence uncertainty relations

Zhiyong Ding, Huan Yang, Dong Wang, Hao Yuan, Jie Yang, Liu Ye

Published Mar 4, 2020• 44 citations
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Abstract

The uncertainty relation usually is one of the most important features in quantum mechanics and is the backbone of quantum theory, which distinguishes it from the rule in its classical counterpart. Specifically, entropy-based uncertainty relations are of fundamental importance in the region of quantum information theory, offering one nontrivial bound of key rate towards quantum key distribution. In this work, we experimentally demonstrate the entropic uncertainty relations and coherence-based uncertainty relations in an all-optics platform. By means of preparing two kinds of bipartite initial states with high fidelity, i.e., Bell-like states and Bell-like diagonal states, we carry out local projective measurements over a complete set of mutually unbiased bases on the measured subsystem. In terms of quantum tomography, the density matrices of the initial states and the postmeasurement states are reconstructed. It shows that our experimental results coincide with the theoretical predictions very well. Additionally, we also verify that the lower bounds of both the entropy-based and coherence-based uncertainty can be tightened by imposing the Holevo quantity and mutual information, and the entropic uncertainty is inversely correlated with the coherence. Our demonstrations might offer an insight into their uncertainty relations and their connection to quantum coherence in quantum information science, which might be applicable to the security analysis of quantum key distributions.

DOI
10.1103/physreva.101.032101

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The uncertainty relation usually is one of the most important features in quantum mechanics and is the backbone of quantum theory, which distinguishes it from the rule in its classical counterpart. Specifically, entropy-based uncertainty relations are of fundamental importance in the region of quantum information theory, offering one nontrivial bound of key rate towards quantum key distribution. In this work, we experimentally demonstrate the entropic uncertainty relations and coherence-based uncertainty relations in an all-optics platform. By means of preparing two kinds of bipartite initial states with high fidelity, i.e., Bell-like states and Bell-like diagonal states, we carry out local projective measurements over a complete set of mutually unbiased bases on the measured subsystem. In terms of quantum tomography, the density matrices of the initial states and the postmeasurement states are reconstructed. It shows that our experimental results coincide with the theoretical predictions very well. Additionally, we also verify that the lower bounds of both the entropy-based and coherence-based uncertainty can be tightened by imposing the Holevo quantity and mutual information, and the entropic uncertainty is inversely correlated with the coherence. Our demonstrations might offer an insight into their uncertainty relations and their connection to quantum coherence in quantum information science, which might be applicable to the security analysis of quantum key distributions.
We report an all-optical experiment to investigate the uncertainty relations based on linear entropy. First we prepare two kinds of biphoton states initially, i.e., Bell-like states and Werner states. Then we perform projective measurements on one subsystem over a complete or incomplete set of mutually unbiased bases; the corresponding measurement probabilities can be directly determined by the photon coincidence counting. Together with the initial state and the postmeasurement states of the system available by quantum state tomography, linear-entropy-based uncertainty relations can be fully verified. It is shown that the experimental results agree very well with the theoretical predictions. Also, it is revealed that entanglement of the initial biphoton states is anticorrelated with the total of uncertainties. Our experimental results could not only deepen the understanding of uncertainty relations, but also facilitate potential applications to quantum technologies.
The Heisenberg uncertainty principle is one of the characteristics of quantum mechanics. With the vigorous development of quantum information theory, uncertain relations have gradually played an important role in it. In particular, in order to solved the shortcomings of the concept in the initial formulation of the uncertainty principle, we brought entropy into the uncertainty relation, after that, the entropic uncertainty relation has exploited the advantages to the full in various applications. As we all know the entropic uncertainty relation has became the core element of the security analysis of almost all quantum cryptographic protocols. This review mainly introduces development history and latest progress of uncertain relations. After Heisenberg's argument that incompatible measurement results are impossible to predict, many scholars, inspired by this viewpoint, have made further relevant investigations. They combined the quantum correlation between the observable object and its environment, and carried out various generalizations of the uncertainty relation to obtain more general formulas. In addition, it also focuses on the entropy uncertainty relationship and quantum-memory-assisted entropic uncertainty relation, and the dynamic characteristics of uncertainty in some physical systems. Finally, various applications of the entropy uncertainty relationship in the field of quantum information are discussed, from randomnesss to wave-particle duality to quantum key distribution.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
Abstract As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations.
Recently, an entropic uncertainty relation for multiple measurements was proposed by Liu et al. [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements. Herein, we improve the lower bound of the entropic uncertainty relation for multiple measurements, referred to as a simply constructed bound (SCB). We verify that the SCB is tighter than Liu et al.'s result for arbitrary mutually unbiased basis measurements, which might play a fundamental and crucial role in practical quantum information processing. Moreover, we optimize the SCB by considering mutual information and the Holevo quantity, and propose an optimized SCB. Notably, the proposed bounds are extrapolations of the behavior of two measurements to a larger collection of measurements. It is believed that our findings would shed light on entropy-based uncertainty relations in the multiple measurement scenario and will be beneficial for security analysis in quantum key distributions.
The quantum component in uncertainty relation can be naturally characterized by the quantum coherence of a quantum state, which is of paramount importance in quantum information science. Here, we experimentally investigate quantum uncertainty relations construed with relative entropy of coherence, l1 norm of coherence, and coherence of formation. Instead of quantum state tomographic technology, we employ the classical shadow algorithm for the detection of lower bounds in quantum uncertainty relations. With an all-optical setup, we prepare a family of quantum states whose purity can be fully controlled. We experimentally explore the tightness of various lower bounds in different reference bases on the prepared states. Our results indicate that the tightness of quantum coherence lower bounds depends on the reference bases and the purity of the quantum state.