The tightness of multipartite coherence from spectrum estimation

Quantum coherence, as a fundamental characteristic of quantum mechanics, describes the ability of a quantum state to present quantum interference phenomena Nielsen and Chuang (2010). It also plays a central role in many emerging areas, including quantum metrology Giovannetti et al. (2004, 2011), nanoscale thermodynamics Lostaglio et al. (2015); Narasimhachar and Gour (2015); Åberg (2014); Gour et al. (2015), and energy transportation in the biological system Huelga and Plenio (2013); Lloyd (2011); Lambert et al. (2013); Romero et al. (2014). Recently, a rigorous framework for quantifying coherence as a quantum resource was introduced Baumgratz et al. (2014); Streltsov et al. (2017a); Hu et al. (2018). Meanwhile, the framework of resource theory of coherence has been extended from a single party to the multipartite scenario Bromley et al. (2015); Radhakrishnan et al. (2016); Streltsov et al. (2015); Yao et al. (2015).

Based on the general framework, several coherence measures have been proposed, such as the $l_{1}$ norm of coherence, the relative entropy of coherence Baumgratz et al. (2014), the geometric measure of coherence Streltsov et al. (2015), the robustness of coherence Napoli et al. (2016); Piani et al. (2016), some convex roof quantifiers of coherence Yuan et al. (2015); Winter and Yang (2016); Zhu et al. (2017); Liu et al. (2017); Qi et al. (2017), and others Shao et al. (2015); Chin (2017); Rana et al. (2016); Zhou et al. (2017); Xi and Yuwen (2019a, b); Cui et al. (2020). These coherence measures make it possible to quantify the role of coherence in different quantum information processing tasks, especially in the multipartite scenario, such as quantum state merging Streltsov et al. (2016), coherence of assistance Chitambar et al. (2016), incoherent teleportation Streltsov et al. (2017b), coherence localization Styliaris et al. (2019), and anti-noise quantum metrology Zhang et al. (2019). However, detecting or estimating most coherence measures requires the reconstruction of quantum states, which is inefficient for large-scale quantum systems.

Efficient protocols for detecting quantum coherence without quantum state tomography have been recently investigated Smith et al. (2017); Wang et al. (2017); Yuan et al. (2020); Zhang et al. (2018); Yu and Gühne (2019); Ding et al. (2021); Dai et al. (2020); Ma et al. (2021). However, the initial proposals require either complicated experiment settings for multipartite quantum systems Smith et al. (2017); Wang et al. (2017); Yuan et al. (2020) or complex numerical optimizations Zhang et al. (2018). An experiment-friendly tool, the so-called spectrum-estimation-based method, requires local measurements and simple post-processing Yu and Gühne (2019), and has been experimentally demonstrated to measure the relative entropy of coherence Ding et al. (2021). Other experiment-friendly tools, such as the fidelity-based estimation method Dai et al. (2020) and the witness-based estimation method Ma et al. (2021), have been successively proposed very recently. The fidelity-based estimation method delivers lower bounds for coherence concurrence Qi et al. (2017), the geometric measure of coherence Streltsov et al. (2015), and the coherence of formation Winter and Yang (2016), and the witness-based estimation method can be used to estimate the robustness of coherence Napoli et al. (2016).

Still, there are several unexplored matters along this line of research. First, on the theoretical side, although it has been studied that the spectrum-estimation-based method is capable to detect coherence of several coherence measures Yu and Gühne (2019), there still exists some coherence measures unexplored. On the experimental side, the realization is focused on the detection of relative entropy of coherence Ding et al. (2021), and its feasibility for other coherence measures has not been tested. Second, the tightness of estimated bounds on multipartite states with spectrum-estimation-based method has not been extensively discussed. Third, while the efficient schemes have been studied either theoretically or experimentally, their feasibility and comparison with the same realistic hardware are under exploration. In particular, implementing efficient measurement schemes and analysing how the noise in realistic hardware affects the measurement accuracy are critical for studying their practical performance with realistic devices.

The goal of this work is to investigate the spectrum-estimation-based method in three directions: First, we generalize the spectrum-estimation-based method to detect the geometric measure of coherence, which has not been investigated yet. Second, we investigate the tightness of the estimated bound with the spectrum-estimation-based method on multipartite Greenberger-Horne-Zeilinger(GHZ) states and linear cluster states. Finally, we present the comparison of the efficient methods with the same experimental data.

The article is organized as follows. In Section II, we briefly introduce the theoretical background, including the review of definitions of well-explored coherence measures, the present results of coherence estimation with the spectrum-estimation-based method and the construction of constraint in the spectrum-estimation-based method. In Section III, we provide the generalization of the spectrum-estimation-based method for the geometric measure of coherence. In Section IV, we discuss the tightness of estimated bounds on multipartite states. In Section V, we present the results of comparison for three estimation methods. Finally, we conclude in Section VI.

We present that the geometric measure of coherence $C_{g}(\rho)$ is related to $l_{C_{l_{2}}}(\rho)$.

The lower bounds $l_{C_{r}}$, $l_{C_{l_{2}}}$, $l_{C_{l_{1}}}$ and $l_{C_{R}}$ are tight for pure states Yu and Gühne (2019). $l_{\tilde{C_{l_{1}}}}$ and $l_{C_{f}}$ related to $l_{C_{l_{1}}}$ and $l_{C_{r}}$ as shown in Eq. 11 are tight for pure states as well. However, the tightness of $l_{C_{g}}$ is quite different. As shown in Eq. 14, $l_{C_{g}}$ is related to the dimension of quantum system $d$ as well as $l_{C_{l_{2}}}$. Although $l_{C_{l_{2}}}$ is tight for stabilizer states, $l_{C_{g}}$ is generally not due to the fact of $\frac{d-1}{d}$. The equality in Eq. 14 holds for a special family of states, i.e., the maximal coherent states $|\Psi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{\alpha=0}^{d-1}e^{i\theta_{\alpha}}|\alpha\rangle$ Zhang et al. (2017).

To investigate the tightness of the estimated bounds $l_{C}$ on multipartite states, we consider the graph states

For a target graph $G$ with $n$ qubits (vertices), the initial states are the tensor product of $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$. An edge $(i,j)\in E$ corresponds to a two-qubit controlled Z gate $CZ_{(i,j)}$ acting on two qubits $i$ and $j$. Particularly, we investigate two types of graphs. The first one is star graph, and the corresponding state is $n$-qubit GHZ states $|\text{GHZ}_{n}\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})$ with local unitary transformations acting on one or more of the qubits. The second one is linear graph, and the corresponding state is $n$-qubit linear cluster state $|\text{C}_{n}\rangle$ Briegel and Raussendorf (2001), which is the ground state of the Hamiltonian

where $X^{(i)}$, $Y^{(i)}$ and $Z^{(i)}$ denote the Pauli matrices acting on qubit $i$.

For $|\text{GHZ}_{n}\rangle$ and $|\text{C}_{n}\rangle$ with $n$ up to 10, we calculate $\mathcal{P}_{C}=l_{C}/C$ to indicate the tightness (accuracy) of estimations, and the results are shown in Fig. 1(a) and Fig. 1(b), respectively. For $|\text{GHZ}_{n}\rangle$, we observe that $\mathcal{P}_{C}$ is 1 in the estimation of $C_{f}(C_{r}),\tilde{C_{l_{1}}}(C_{l_{1}})$ and $C_{R}$, which indicates the corresponding bounds are tight as the target states are pure state. The reason is that $l_{C_{g}}$ is determined by $d$ and $l_{C_{l_{2}}}$ as shown in Eq. 14. For $|\text{GHZ}_{n}\rangle$, $l_{C_{l_{2}}}=1/2$ regardless of $n$. Then, we take the partial derivative of $l_{C_{g}}$ in Eq. 14 with respect to $d$, and obtain

It is clear that $l_{C_{g}}$ is monotonically decreasing with respect to $d$. Note that $l_{C_{g}}\to(\sqrt{2}-1)/\sqrt{2}\approx 0.2929$ and $\mathcal{P}_{C}\to 2(\sqrt{2}-1)/\sqrt{2}\approx 0.5858$ when $d\to\infty$.

The results of $|\text{C}_{n}\rangle$ is quite different as shown in Fig. 1(b). $\mathcal{P}_{C}$ is 1 in the estimation of $C_{g}$ as $|\text{C}_{n}\rangle$ is in the form of maximally coherent state $|\Psi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{\alpha=0}^{d-1}e^{i\theta_{\alpha}}|\alpha\rangle$. For example, $|\text{C}_{3}\rangle=(|+0+\rangle+|-1-\rangle)/\sqrt{2}$ and we rewrite it in the computational basis

By re-encoding $|\alpha_{1}\alpha_{2}\alpha_{3}\rangle$ to $|\alpha\rangle$ by $\alpha=\alpha_{1}2^{2}+\alpha_{2}2^{1}+\alpha_{3}2^{0}$, $|\text{C}_{3}\rangle$ can be represented in the form of maximally coherent state $\frac{1}{\sqrt{d}}\sum_{\alpha=0}^{d-1}e^{i\theta_{\alpha}}|\alpha\rangle$ with $\bm{\theta_{\alpha}}=(0,0,0,\pi,0,0,\pi,0)$.

Furthermore, we investigate the robustness of $\mathcal{P}$ of GHZ states and linear cluster states in a noisy environment. We consider the following imperfect GHZ state and linear cluster state

with $|\psi\rangle$ being either $|\text{GHZ}_{n}\rangle$ or $|\text{C}_{n}\rangle$ and $0\leq\eta\leq 1$. Note that $\rho_{\text{Noisy}}^{\psi}$ can be written in the form of graph-diagonal state , i.e., $\rho_{\text{Noisy}}^{\psi}=\sum\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$ so that $l_{C_{r}}$ and $l_{C_{l_{2}}}$ are tight for $\rho_{\text{Noisy}}^{\psi}$ Ding et al. (2021). The estimation of $l_{C_{l_{1}}}$ is equivalent to the optimization of

For $\rho_{\text{Noisy}}^{\text{GHZ}_{n}}$ in form of

it is easy to calculate that $u_{1}\geq 1$ and $\hat{v}_{k}=\hat{u}_{k}=(1,0,\cdots,0)$. As $l_{C_{l_{2}}}$ is tight for $\rho_{\text{Noisy}}^{\text{GHZ}_{n}}$ so that we can obtain

which indicates $l_{C_{l_{1}}}$ is tight for $\rho_{\text{Noisy}}^{\text{GHZ}_{n}}$. Following the same way, We can also obtain $l_{C_{R}}=C_{R}$.

For $\rho_{\text{Noisy}}^{\text{C}_{n}}$, as its matrix elemetns satisfy

so that we can calculate the $C_{l_{2}}=l_{C_{l_{2}}}=\frac{d-1}{d}(1-\eta)^{2}$, $C_{R}=C_{l_{1}}=(d-1)(1-\eta)$ and

$M=\lfloor\frac{d(d-1)}{2}(1-\eta)^{2}\rfloor$ is the largest integer satisfying $\sum_{l=1}^{M}u_{l}\leq 1$. With $l_{C_{l_{2}}}$ and $M$, we can calculate $l_{C_{l_{1}}}$ and $l_{C_{R}}$

As $M\approx\frac{d(d-1)}{2}(1-\eta)^{2}$ so we have $l_{C_{l_{1}}}=l_{C_{R}}\approx(d-1)(1-\eta)^{2}$. Therefore, $\mathcal{P}_{C}$ of $l_{C_{l_{1}}}$ and $l_{C_{R}}$ for noisy cluster state is $l_{C_{l_{1}}}/C_{l_{1}}=l_{C_{R}}/C_{R}\approx 1-\eta$.

To give an intuitive illustration of our conclusion about tightness of $l_{C}$ on noisy states, we calculate $\mathcal{P}_{C}$ on 4-qubit noisy GHZ state and linear cluster state, i.e., $|\text{GHZ}_{4}\rangle=(|0000\rangle+|1111\rangle)/\sqrt{2}$ and $|\text{C}_{4}\rangle=(|+0+0\rangle+|+0-1\rangle+|-1-0\rangle+|-1+1\rangle)/2$. The results are shown in Fig. 1(c) and Fig. 1(d), respectively. In Fig. 1(c), $\mathcal{P}_{C}$ of $l_{C_{r}}$, $l_{C_{l_{1}}}$ and $l_{C_{R}}$ are still tight. In Fig. 1(d), $\mathcal{P}_{C}$ of $l_{C_{r}}$ is tight while $\mathcal{P}_{C}$ of $l_{C_{l_{1}}}$ and $l_{C_{R}}$ linearly decrease with $\eta$. $l_{C_{g}}$ also exhibits linear decrease with $\eta$ in Fig. 1(c) and Fig. 1(d) because $l_{C_{l_{2}}}\sim(1-\eta)^{2}$.

Besides the spectrum-estimation-based method, another two efficient coherence estimation methods for multipartite states have been proposed recently, namely the fidelity-based estimation method Dai et al. (2020) and the witness-based estimation method Ma et al. (2021), respectively. Specifically, $C_{f}$,$C_{g}$,$\tilde{C_{l_{1}}}$ can be estimated via the fidelity-based estimation method and $C_{R}$ can be estimated via the witness-based estimation method. In this section, we compare the accuracy of $l_{C}$ with difference estimation method with experimental data of $\rho_{\text{expt}}^{\text{GHZ}_{3}}$ and $\rho_{\text{expt}}^{\text{GHZ}_{4}}$ from Ref. Ding et al. (2021).

To this end, we first estimate $l_{C}$ of the coherence measures $C$ introduced in Section II on states $\rho_{\text{expt}}^{\text{GHZ}_{3}}$ and $\rho_{\text{expt}}^{\text{GHZ}_{4}}$ via spectrum-estimation-based method. We employ the experimentally obtained expected values of the stabilizing operators $\mathcal{S}^{\text{GHZ}_{n}}$ and the corresponding statistical errors $\sigma$ to construct constraints in $X$. We denote the lower bound of estimated multipartite coherence as $l_{C,m}^{w}$, where $C$ is the coherence measure $\in\{C_{f}(C_{r}),\tilde{C_{l_{1}}}(C_{l_{1}}),C_{g},C_{R}\}$ and $m\leq 2^{n}-1$ is the stabilizing operators we selected for construction of constraints in $X$. In our estimations, all results are obtained by setting $w=3$. Here, we only consider the case of maximal $l_{C,m}$. In the ideal case, the maximal $l_{C,m}$ is obtained by setting all $m$ stabilizing operators in the constraint. However, a larger $m$ might lead to the case of no feasible solution due to the imperfections in experiments. In practice, the maximal estimated coherence is often obtained with $m\leq 2^{n}-1$ stabilizing operators. Let $l_{C}^{\text{max}}$ be the maximal estimated coherence over all subsets $\{S_{i}\}$, where the number of subset is $\sum_{m=1}^{2^{n}-1}\binom{2^{n}-1}{m}=2^{2^{n}-1}-1$. The results of $l_{C}^{\text{max}}$ are shown in Table 1.

The accuracy estimated bounds is indicated by $\mathcal{P}_{C}=l_{C}^{\text{max}}/C$. Note that $C_{r}$ and $C_{l_{1}}$ of $\rho^{\psi}_{\text{expt}}$ can be calculated directly according to the definition in Eq. 1 and Eq. 2, while the calculations of $C_{g}(\cdot)$ and $C_{R}(\cdot)$ require converting them to the convex optimization problem Napoli et al. (2016); Piani et al. (2016); Zhang et al. (2020) and the corresponding solution Boyd and Vandenberghe (2004); Grant and Boyd (2014, 2008). The calculation of ${C_{f}},\tilde{C}_{l_{1}}$ requires optimizing all pure state decomposition, and there is no general method for analytical and numerical solutions except a few special cases. Therefore, we replace ${C_{f}},\tilde{C}_{l_{1}}$ of these tomographic states by their $C_{r},C_{l_{1}}$ when calculating the estimated accuracy, respectively. The replacement increases $\mathcal{P}_{C}$ when we compare the two estimation methods of spectrum-estimation-based and fidelity-based so that it does not affect our conclusion about the comparison.

We also perform the fidelity-based estimation method and witness-based estimation method on the same experimental data to obtain $l_{C}$ and $\mathcal{P}_{C}$. The results of $l_{C}$ and $\mathcal{P}_{C}$ with these three estimation methods are shown in Table 1. We find that the spectrum-estimation-based and fidelity-based coherence estimation methods have similar performance on the estimation of $\tilde{C_{l_{1}}}$, in which the accuracy is beyond 0.7 for $\rho^{\text{GHZ}_{3}}_{\text{expt}}$ and 0.6 for $\rho^{\text{GHZ}_{4}}_{\text{expt}}$. Importantly, the spectrum-estimation-based method shows a significant enhancement in the estimation of $C_{f}$ and $C_{g}$ compared with the fidelity-based method, as well as in the estimation of $C_{R}$ compared with the witness-based method.

In this work, we first develop the approach to estimating the lower bound of coherence for the geometric measure of coherence via the spectrum-estimation-based method, i.e., we present the relation between the geometric measure of coherence and $l_{2}$ norm of coherence. Then, we investigate the tightness of estimations of various coherence measures on GHZ states and linear cluster states, including the geometric measure of coherence, the relative entropy of coherence, the $l_{1}$-norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence. Finally, we compare the accuracy of the estimated lower bound with the spectrum-estimation-based method, fidelity-based estimation method, and the witness-based estimation method on the same experimental data.

We conclude that the spectrum-estimation-based method is an efficient toolbox to indicate various multipartite coherence measures. For $n$-qubit stabilizer states, it only requires at most $n$ measurements instead of $3^{n}$ measurements required in quantum state tomography. Second, the tightness of the lower bound is not only determined by whether the target state is pure or mixed but also by the coherence measures. We give examples that the lower bound of the geometric measure of coherence is tight for $n$-qubit linear cluster states but is not tight for noisy $n$-qubit GHZ states, and the lower bounds of the robustness of coherence and $l_{1}$-norm of coherence are tight for noisy $n$-qubit GHZ states but is not tight for noisy $n$-qubit linear cluster states. Third, we find that the spectrum-estimation-based method has a significant improvement in coherence estimation compared to fidelity-based method and the witness-based method.