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Photonics

Experimental Direct Measurement of the Relative Entropy of Coherence

Xufeng Huang, Yuan Yuan, Yueping Niu, Shangqing Gong

Published Sep 2, 2023• 3 citations
Publisher

Abstract

Quantum coherence is the most distinguished feature of quantum mechanics, which characterizes the superposition properties of quantum states. It plays a critical role in various fields, ranging from quantum information technology to quantum biology. Although various coherence quantifiers have been proposed since the resource theory of coherence was established, there are a lack of experimental methods to estimate them efficiently, which restricts the applications of coherence. Relative entropy of coherence is one of the main quantifiers of coherence, and is frequently used in quantum information science. Such nonlinear properties of quantum states are usually calculated from full descriptions of the quantum state, although they are not related to all parameters that specify the state. Here, we experimentally measure the relative entropy of coherence for the arbitrary qubit states directly in the photonic system without using standard state tomography. In the experiment, we directly measure the von Neumann entropy of the quantum states through interference, thus obtaining the relative entropy of coherence, and finding that the experimental results are in good agreement with the theory. Our work provides a nice alternative experimental scheme for measuring the relative entropy of coherence.

DOI
10.3390/photonics10091004

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Quantum coherence is the most distinguished feature of quantum mechanics, which characterizes the superposition properties of quantum states. It plays a critical role in various fields, ranging from quantum information technology to quantum biology. Although various coherence quantifiers have been proposed since the resource theory of coherence was established, there are a lack of experimental methods to estimate them efficiently, which restricts the applications of coherence. Relative entropy of coherence is one of the main quantifiers of coherence, and is frequently used in quantum information science. Such nonlinear properties of quantum states are usually calculated from full descriptions of the quantum state, although they are not related to all parameters that specify the state. Here, we experimentally measure the relative entropy of coherence for the arbitrary qubit states directly in the photonic system without using standard state tomography. In the experiment, we directly measure the von Neumann entropy of the quantum states through interference, thus obtaining the relative entropy of coherence, and finding that the experimental results are in good agreement with the theory. Our work provides a nice alternative experimental scheme for measuring the relative entropy of coherence.
Abstract Quantum coherence is the most distinguished signature of quantum mechanics, also recognized to be an essential resource for many promising quantum technologies, playing a central role in phenomena related to quantum information science, quantum thermodynamics and quantum biology, just to mention few examples. However, the resource theory of coherence is in the preliminary stage of its development, being still limited to the study of the manipulation and quantification of coherence, mostly from a theoretical viewpoint. Here, we propose an experimental method to directly measure the relative entropy of coherence, which according to the resource theory is one of the main quantifiers of coherence. To achieve this aim, we discuss a procedure to measure the von Neumann entropy of a generic quantum state directly in terms of the Shannon entropy of the probability distribution of outcomes in a defined measurement basis. In both cases, by direct we mean that tomographic methods are not required. Two quantum-optical applications of our method are discussed in order to give support to our predictions.
Quantum coherence is the most distinguished feature of quantum mechanics. It lies at the heart of the quantum-information technologies as the fundamental resource and is also related to other quantum resources, including entanglement. It plays a critical role in various fields, even in biology. Nevertheless, the rigorous and systematic resource-theoretic framework of coherence has just been developed recently, and several coherence measures are proposed. Experimentally, the usual method to measure coherence is to perform state tomography and use mathematical expressions. Here, we alternatively develop a method to measure coherence directly using its most essential behavior-the interference fringes. The ancilla states are mixed into the target state with various ratios, and the minimal ratio that makes the interference fringes of the "mixed state" vanish is taken as the quantity of coherence. We also use the witness observable to witness coherence, and the optimal witness constitutes another direct method to measure coherence. For comparison, we perform tomography and calculate l_{1} norm of coherence, which coincides with the results of the other two methods in our situation. Our methods are explicit and robust, providing a nice alternative to the tomographic technique.
Abstract The recently established resource theory of quantum coherence allows for a quantitative understanding of the superposition principle, with applications reaching from quantum computing to quantum biology. While different quantifiers of coherence have been proposed in the literature, their efficient estimation in today’s experiments remains a challenge. Here, we introduce a collective measurement scheme for estimating the amount of coherence in quantum states, which requires entangled measurements on two copies of the state. As we show by numerical simulations, our scheme outperforms other estimation methods based on tomography or adaptive measurements, leading to a higher precision in a large parameter range for estimating established coherence quantifiers of qubit and qutrit states. We show that our method is accessible with today’s technology by implementing it experimentally with photons, finding a good agreement between experiment and theory.
The ability of quantum states to be in superposition is one of the key features that sets them apart from the classical world. This “coherence” is rigorously quantified by resource theories, which aim to understand how such properties may be exploited in quantum technologies. There has been much research on what the resource theory of coherence can reveal about quantum metrology, almost all of which has been from the viewpoint of Fisher information. We prove, however, that the relative entropy of coherence, and its recent generalization to positive operator-valued measures (POVMs), naturally quantify the performance of Bayesian metrology. In particular, we show how a coherence measure can be applied to an ensemble of states. We then prove that during parameter estimation, the ensemble relative entropy of coherence (C) is equal to the difference between the optimal Holevo information (X), and the mutual information attained by a measurement (I). We call this relation the CXI equality. The ensemble coherence lets us visualize how much information is locked away in superposition and hence is inaccessible with a given measurement scheme and quantifies the advantage that would be gained by using a joint measurement on multiple states. Our results hold regardless of how the parameter is encoded in the state, encompassing unitary, dissipative, and discrete settings. We consider both projective measurements and general POVMs. This work suggests new directions for research in coherence, provides a novel operation interpretation for the relative entropy of coherence and its POVM generalization, and introduces a new tool to study the role of quantum features in metrology. Published by the American Physical Society 2024
Coherence and entanglement are fundamental properties of quantum systems, promising to power the near future quantum computers, sensors and simulators. Yet, their experimental detection is challenging, usually requiring full reconstruction of the system state. We show that one can extract quantitative bounds to the relative entropy of coherence and the coherent information, coherence and entanglement quantifiers respectively, by a limited number of purity measurements. The scheme is readily implementable with current technology to verify quantum computations in large scale registers, without carrying out expensive state tomography.
As one of the most essential properties of quantum mechanics, quantum coherence is also considered a key physical resource which can be used to accomplish certain tasks of quantum information processing. Significant efforts are involved in the quantification of quantum coherence using measures such as the l1{l}_{1}l1​-norm and the relative entropy of coherence. The quantum coherence as a physical resource is usually defined with respect to a given basis, which also arouses an issue of interest concerning the variation of quantum coherence for a quantum state in different reference bases. In this paper, using an all optical setup, we experimentally investigate a trade-off relation between the coherence measures quantified by relative entropy of coherence in two noncommuting reference bases. Our result shows that the sum of quantum coherence under these bases is bounded in a region defined by its lower bound and upper bound.