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 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.

arXiv:2506.15080Journal of Physics A Mathematical and TheoreticalJun 3, 20252 citations

Classifying coherence with a finite set of witnesses

Xue‐Na Zhu, Zhi‐Xiang Jin, Gui Bao +1 more

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.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.

arXiv:2410.20023Journal of Physics A Mathematical and TheoreticalOct 16, 20240 citations

Robust Coherence

ScienceOct 20, 20000 citations

Quantum detection of coherent-state signal using least-square estimation

Wenbin Yu, Baoyu Zheng, Shengmei Zhao

Coherent-states, representing electromagnetic radiation produced by physical devices such as lasers, are an important class of states for optical communications. For the coherent-state signal detection, a detection method basing on the POVM detection and the least-square estimation is proposed in this paper, concerning the quantum detection theory. This method has been proved to be effective in our paper, and the numerical results indicate that with this method we can fairly improve the performance of coherent-state signal detection.Coherent-states, representing electromagnetic radiation produced by physical devices such as lasers, are an important class of states for optical communications. For the coherent-state signal detection, a detection method basing on the POVM detection and the least-square estimation is proposed in this paper, concerning the quantum detection theory. This method has been proved to be effective in our paper, and the numerical results indicate that with this method we can fairly improve the performance of coherent-state signal detection.

IEEE International Conference on Communication Technology (ICCT)Nov 1, 20100 citations

Amplitude-amplified coherence detection and estimation

Rhea Alexander, Michalis Skotiniotis, Daniel Manzano

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 $|ψ\rangle$ . Having access to $m$ copies of an unknown pure state $|ψ\rangle$ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success $\ge 2/3$ is $Θ(c(|ψ\rangle)^{-1})$ , where $c(|ψ\rangle)$ is the geometric measure of coherence of $|ψ\rangle$ . However, assuming access to the unitary $U_ψ$ which prepares the unknown state $|ψ\rangle$ , and its inverse $U_ψ^\dagger$ , we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number $O(c(|ψ\rangle)^{-1/2})$ 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 $\varepsilon$ with a sample complexity that scales as $O(1/\varepsilon)$ as compared to the $O(1/\varepsilon^2)$ 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.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 $|ψ\rangle$. Having access to $m$ copies of an unknown pure state $|ψ\rangle$ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success $\ge 2/3$ is $Θ(c(|ψ\rangle)^{-1})$, where $c(|ψ\rangle)$ is the geometric measure of coherence of $|ψ\rangle$. However, assuming access to the unitary $U_ψ$ which prepares the unknown state $|ψ\rangle$, and its inverse $U_ψ^\dagger$, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number $O(c(|ψ\rangle)^{-1/2})$ 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 $\varepsilon$ with a sample complexity that scales as $O(1/\varepsilon)$ as compared to the $O(1/\varepsilon^2)$ 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.

quant-pharXiv:2512.15352Dec 18, 20250 citations

|ψ\rangle

. Having access to

m

copies of an unknown pure state

|ψ\rangle

we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success

\ge 2/3

Θ(c(|ψ\rangle)^{-1})

, where

c(|ψ\rangle)

is the geometric measure of coherence of

|ψ\rangle

. However, assuming access to the unitary

U_ψ

which prepares the unknown state

|ψ\rangle

, and its inverse

U_ψ^\dagger

, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number

O(c(|ψ\rangle)^{-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

\varepsilon

with a sample complexity that scales as

O(1/\varepsilon)

as compared to the

O(1/\varepsilon^2)

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.