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arXiv:2012.13038 · Physical Review A

Detecting and estimating coherence based on coherence witnesses

Zhao Ma, Zhou Zhang, Yue Dai, Yuli Dong, Chengjie Zhang

Published Jan 13, 2021• 28 citations
arXivHTMLPDF

Abstract

Quantum coherence has wide-ranging applications from quantum thermodynamics to quantum metrology, quantum channel discrimination, and even quantum biology. Thus, detecting and quantifying coherence are two fundamental problems in quantum resource theory. Here, we introduce feasible methods to detect and estimate the coherence by constructing coherence witnesses for any finite-dimensional states. Our coherence witnesses detect coherent states by testing whether the expectation value of the witness is negative or not. Two typical coherence witnesses are proposed and discussed based on our witness-constructing method, which are also used to estimate the robustness of coherence, the ${l}_{1}$ norm, and the ${l}_{2}$ norm of coherence measures. Furthermore, we compare one of our coherence witnesses with a previously introduced witness by proving that our witness is strictly stronger than that previous witness. We also present an application of coherence in a quantum metrology task, in which we estimate an unknown parameter by measuring our coherence witness.

DOI
10.1103/physreva.103.012409

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Quantum coherence has wide-ranging applications from quantum thermodynamics to quantum metrology, quantum channel discrimination, and even quantum biology. Thus, detecting and quantifying coherence are two fundamental problems in quantum resource theory. Here, we introduce feasible methods to detect and estimate the coherence by constructing coherence witnesses for any finite-dimensional states. Our coherence witnesses detect coherent states by testing whether the expectation value of the witness is negative or not. Two typical coherence witnesses are proposed and discussed based on our witness-constructing method, which are also used to estimate the robustness of coherence, the l1{l}_{1}l1​ norm, and the l2{l}_{2}l2​ norm of coherence measures. Furthermore, we compare one of our coherence witnesses with a previously introduced witness by proving that our witness is strictly stronger than that previous witness. We also present an application of coherence in a quantum metrology task, in which we estimate an unknown parameter by measuring our coherence witness.
Quantum states may exhibit asymmetry with respect to the action of a given group. Such an asymmetry of states can be considered as a resource in applications such as quantum metrology, and it is a concept that encompasses quantum coherence as a special case. We introduce explicitly and study the robustness of asymmetry, a quantifier of asymmetry of states that we prove to have many attractive properties, including efficient numerical computability via semidefinite programming, and an operational interpretation in a channel discrimination context. We also introduce the notion of asymmetry witnesses, whose measurement in a laboratory detects the presence of asymmetry. We prove that properly constrained asymmetry witnesses provide lower bounds to the robustness of asymmetry, which is shown to be a directly measurable quantity itself. We then focus our attention on coherence witnesses and the robustness of coherence, for which we prove a number of additional results; these include an analysis of its specific relevance in phase discrimination and quantum metrology, an analytical calculation of its value for a relevant class of quantum states, and tight bounds that relate it to another previously defined coherence monotone.
The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state ∣ψ⟩|ψ\rangle∣ψ⟩. Having access to mmm copies of an unknown pure state ∣ψ⟩|ψ\rangle∣ψ⟩ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success ≥2/3\ge 2/3≥2/3 is Θ(c(∣ψ⟩)−1)Θ(c(|ψ\rangle)^{-1})Θ(c(∣ψ⟩)−1), where c(∣ψ⟩)c(|ψ\rangle)c(∣ψ⟩) is the geometric measure of coherence of ∣ψ⟩|ψ\rangle∣ψ⟩. However, assuming access to the unitary UψU_ψUψ​ which prepares the unknown state ∣ψ⟩|ψ\rangle∣ψ⟩, and its inverse Uψ†U_ψ^\daggerUψ†​, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number O(c(∣ψ⟩)−1/2)O(c(|ψ\rangle)^{-1/2})O(c(∣ψ⟩)−1/2) of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error ε\varepsilonε with a sample complexity that scales as O(1/ε)O(1/\varepsilon)O(1/ε) as compared to the O(1/ε2)O(1/\varepsilon^2)O(1/ε2) sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.
Quantifying coherence is an essential endeavor for both quantum foundations and quantum technologies. Here, the robustness of coherence is defined and proven to be a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task.
Imaginarity has shown to be an important resource in quantum information. The witness theory of quantum resource, such as entanglement witness, coherence witness, and imaginarity witness, has been established, in particular entanglement witness and coherence witness have been extensively explored. We present here another class of imaginarity witnesses beyond the one in [{Phys. Lett. A} \textbf{530}, 130135 (2025)]. Within our framework, any nonreal Hermitian operator under the reference basis is an imaginarity witness and only finite of them can detect all the imaginarity states. As the common approaches that were explored in the witness theories of entanglement and coherence, we then explore the relations between these imaginarity witnesses in different cases: (i) when they can detect common imaginary states, (ii) when they can detect the same sets of imaginary states, and (ii) when they obey the finer relation. Finally, we define an imaginarity measure in terms of witnesses, termed witnessed imaginarity, and prove that it coincides with both the trace norm of imaginarity and the robustness of imaginarity.
Abstract Quantum coherence is one of the most basic characteristics of quantum mechanics. Here we give some methods to detect and measure quantum coherence. Firstly, we propose a coherence criterion without full quantum state tomography based on partial transposition. Moreover, we present a coherent nonlinear detection strategy from witnesses, in which we find that for some coherent states, normal witness detection fails but our nonlinear detection succeeds. In addition, we prove that when the nonlinear detection on the two copies of the coherent state fails, the nonlinear detection on the three copies may be successful. Finally, due to the difficulty in calculating robustness of coherence for general states, we introduce a lower bound for coherent robustness based on the witness operator, and after comparing our lower bound with the currently known lower bound, one show that our lower bound is better. Coherence is believed to play a crucial role in quantum information tasks, making the detection and quantization of coherence particularly significant. Therefore, these results help to open up new avenues for advancement in quantum theory.
Abstract Coherence is a fundamental resource in quantum information processing, which can be certified by a coherence witness. In order to detect all the coherent states, we introduce a useful concept of coherence witness and structure the set of coherence witnesses C [ m , M ] d . We present necessary and sufficient conditions of detecting quantum coherence of d − dimensional quantum states based on C [ m , M ] d . Moreover, we show that each coherent state can be detected by one of the coherence witnesses in a finite set. The corresponding finite set of coherence witnesses is presented explicitly, which detects all the coherent states.