Detection of arbitrary quantum correlations via synthesized quantum channels

Correlations of physical quantities are key to understanding quantum many-body physics[1, 2, 3], nonlinear optics[4], solid-state nuclear magnetic resonance (NMR)[5, 6] and open quantum systems[7, 8, 9, 10], and are relevant to some quantum-enhanced technologies[11, 12, 13]. Second-order correlations[14, 15] are the underlying physical quantities measured in a broad range of fields such as linear optics[16], transport[17], thermodynamics[18], neutron scattering[19], and are directly extracted in various quantum systems such as single solid impurities[20, 21, 22, 23, 24], quantum dots[25, 26] and superconductor qubits[27]. Recently, it is indicated that high-order correlations are relevant to mesoscopic quantum many-body systems[28, 2, 29]. How to systematically characterize these correlations then plays a central role in investigation of various quantum systems[4, 30].

In general, the dynamics of a quantum system is determined by the correlations $C^{\eta_{N}\cdots\eta_{1}}=\Tr_{\mathrm{B}}\left(\mathcal{B}_{N}^{\eta_{N}}\cdots\mathcal{B}_{2}^{\eta_{2}}\mathcal{B}_{1}^{\eta_{1}}\hat{\rho}\right)$, where $\hat{\rho}$ is the initial state of the system and $\eta_{k}=\pm 1$. The super-operators of the commutator and anti-commutator at time $t_{k}$ (with $t_{1}\leq t_{2}\leq\cdots t_{N}$) are defined as $\mathcal{B}_{k}^{-}\hat{\rho}\equiv-\mathrm{i}[\hat{B}\left(t_{k}\right)\hat{\rho}-\hat{\rho}\hat{B}\left(t_{k}\right)]/2$ and $\mathcal{B}_{k}^{+}\hat{\rho}\equiv[\hat{B}\left(t_{k}\right)\hat{\rho}+\hat{\rho}\hat{B}\left(t_{k}\right)]/2$, respectively. The quantum quantities $\hat{B}\left(t_{k}\right)$ at different time $t_{k}$ in general do not commute, i.e., $\hat{B}\left(t_{i}\right)\hat{B}\left(t_{j}\right)\neq\hat{B}\left(t_{j}\right)\hat{B}\left(t_{i}\right)$ when $i\neq j$. Therefore, the high-order quantum correlations have a rich structure resulting from different orderings[31, 32, 33]. There are $2^{N-1}$ inequivalent correlations corresponding to different nesting of commutators and anti-commutators in time order[10, 30, 34]. Among these numerous correlations, several special forms have been widely investigated and play significant roles in many subjects. For example, the nonlinear spectroscopy[13], in which the system is detected by a classical sensor, measures only one type of time-ordered correlations, namely, $C^{+--\cdots-}=\Tr_{\mathrm{B}}\left(\mathcal{B}_{N}^{+}\cdots\mathcal{B}_{2}^{-}\mathcal{B}_{1}^{-}\hat{\rho}_{\mathrm{B}}\right)$. Another widely used tool is the noise spectroscopy[35, 9], in which the target system is approximated as a classical stochastic noise field, and it usually extracts the symmetric correlations like $C^{++\cdots+}=\Tr_{\mathrm{B}}\left(\mathcal{B}_{N}^{+}\cdots\mathcal{B}_{2}^{+}\mathcal{B}_{1}^{+}\hat{\rho}_{\mathrm{B}}\right)$. As shown in Fig. 1a, using the quantum sensors to detect the target systems is necessary for the full extraction of the quantum correlations, but few studies on this subject are carried out. Recently, it is proposed theoretically that the sequential weak measurements via a quantum sensor can extract arbitrary-order correlations of a quantum bath[30]. By preparing the initial state and choosing a certain measurement basis of the sensor, one can pre- and post-select the coupling between the sensor and the target system to access arbitrary types and orders of correlations of the target system. A very recent experimental work also used the sequential weak measurement to obtain the mixed signals of the fourth-order correlations of single nuclear spin[36]. Until now, selective detection of arbitrary types of correlations has not yet been realized in experiments.

Here we demonstrate the extraction of arbitrary types of quantum correlations. In stead of the original weak measurement scheme[30], we propose a more general protocol using synthesized quantum channels to select the coupled evolution of the quantum sensor and target system along a specific path, leading to a desired correlation detected by the final measurement. This quantum channel scheme is universal, applicable to both single and ensemble quantum systems. Using the versatility of NMR techniques, which are a powerful tool for studying quantum many-body physics[6, 37] and correlation measurements[38, 39, 40], we demonstrate the scheme using nuclear-spin targets and sensors. We extract the second-order correlation $C^{+-}$ and fourth-order correlation $C^{+--+}$. It is expected that the measurement protocol for arbitrary quantum correlations will provide an essential tool for studying quantum many-body physics and finding the applications in quantum technologies.

Consider a quantum sensor S transiently coupled to a quantum many-body target B at time $t_{k}$ for a short time $\delta t$, the state evolution is governed by the interaction Hamiltonian $\hat{V}=\hat{S}\otimes\hat{B}(t_{k})$. Then the corresponding Liouville equation $\partial_{t}\hat{\rho}\left(t\right)=-\mathrm{i}\left[\hat{V},\hat{\rho}\left(t\right)\right]$ has the first-order approximation[41]

where $\mathcal{S}^{\pm}/\mathcal{B}_{k}^{\pm}$ are the super-operators as defined before and $\hat{B}_{k}=\hat{B}\left(t_{k}\right)$. Suppose the sensor and target are initially separable, i.e., $\hat{\rho}\left(0\right)=\hat{\rho}_{\mathrm{S}}\otimes\hat{\rho}_{\mathrm{B}}$. After passing through the $N$ transient interactions successively at $t_{1}\leq t_{2}\leq\cdots\leq t_{N}$, the final state after $t_{N}$ becomes

where $\mathcal{T}$ is the time-ordering operator and $\Theta=\sum_{k=1}^{N}\left|\eta_{k}\right|$. Here $\eta_{k}\in\left\{\pm,0\right\}$ and we define $\mathcal{S}^{0}=\mathcal{B}_{k}^{0}\equiv\mathds{1},k=1,\cdots,N$. $\overline{\eta}_{k}=-\eta_{k}$, which means $\mathcal{S}^{\pm}$ are always adjoint with $\mathcal{B}_{k}^{\mp}$. Then by taking the partial trace over the target, one obtains the reduced density of the quantum sensor:

which is completely determined by all types of quantum correlations $C^{\eta_{N}\cdots\eta_{1}}=\Tr_{\mathrm{B}}\left(\mathcal{B}_{N}^{\eta_{N}}\cdots\mathcal{B}_{2}^{\eta_{2}}\mathcal{B}_{1}^{\eta_{1}}\hat{\rho}_{\mathrm{B}}\right)$ and $\Theta$ defines the order number of the correlation $C^{\eta_{N}\cdots\eta_{1}}$. In general, direct measurement on the quantum sensor would involve all possible kinds of quantum correlations.

To selectively extract arbitrary types of quantum correlations $C^{\eta_{N}\cdots\eta_{1}}$ via the quantum sensor, we insert a general ‘quantum channel’ (denoted by a super-operator $\mathcal{P}_{k}$) before each short-time ($\delta t$)-coupling evolution, as shown in Fig. 1b. Such a set of quantum channels can be realized by some unitary or non-unitary operations applied on the quantum sensor, such as rotation, measurement or polarization (see Supplementary Note 2). Then the final state of the quantum sensor turns into

After measuring the observable $\hat{O}$ on the final state of the quantum sensor, the obtained signal is

where the coefficient is

In order to selectively extract arbitrary types of quantum correlations, we can design a set of quantum channels $\left\{\mathcal{P}_{k}\right\}(k=1\cdots,N$) together with an observable $\hat{O}$, so that only one term of equation (5) is reserved, e.g.,that associated with the correlation $C^{\gamma_{N}\cdots\gamma_{1}}$. To achieve this goal, the channel sequence $\left\{\mathcal{P}_{k}\right\}$ and observable $\hat{O}$ have to make sure the coefficient $A^{\overline{\gamma}_{N}\cdots\overline{\gamma}_{1}}\neq 0$, while all other $A^{\overline{\eta}_{N}\cdots\overline{\eta}_{1}}$ vanish for $\eta_{N}\cdots\eta_{1}\neq\gamma_{N}\cdots\gamma_{1}$. Fig. 1b visualizes this idea. When passing through each $\delta t$ slice, the quantum target has three evolution options: $\mathcal{B}^{+}_{k}$, $\mathcal{B}^{-}_{k}$ and $\mathds{1}$. If the designed quantum channels $\mathcal{P}_{k}$ are inserted between the neighbouring $\delta t$ slices, only one connected path that leads to the desired correlation is reserved while other evolving paths of the quantum target are blocked. The extraction of arbitrary correlations in the case of more general coupling interaction ($\hat{V}=\sum_{\alpha=1}^{3}\hat{S}_{\alpha}\otimes\hat{B}_{\alpha}$) can be achieved by a generalized method, where the problem is reduced to solve the indefinite linear equations. It can be proved that it is always possible to find a solution because that the number of the linear equations is always less than the control elements (see Supplementary Note 2).

Next we will take the second-order correlation $C^{+-}$ and the fourth-order correlation $C^{+--+}$ as examples, where a spin-1/2 system is chosen as the quantum sensor coupled to a quantum target by pure dephasing spin model $\hat{V}(t)=\frac{1}{2}\hat{\sigma}_{z}\otimes\hat{B}(t)$. Here $\frac{1}{2}\hat{\sigma}_{z}$ is the $z$ component of the sensor’s spin operator, which corresponds to the super-operators of commutator $\mathcal{S}_{z}^{-}$ and anti-commutator $\mathcal{S}_{z}^{+}$. The initial state of the quantum sensor is $\hat{\rho}_{\mathrm{S}}=\left(\mathds{1}+p\hat{\sigma}_{z}\right)/2$ ($p$ is the polarization of the sensor) and the final observable is $\hat{O}=\hat{\sigma}_{y}$. The quantum channels are constructed by synthesizing the spin rotations of the quantum sensor.

For measuring $C^{+-}$, two channels $\mathcal{P}_{1}^{\rm 2nd},\mathcal{P}_{2}^{\rm 2nd}$ are designed to reserve the coefficient $A^{-+}$ while making other coefficients like $A^{+-}$ and $A^{--}$ vanish (see Methods). The scheme can be clearly illustrated by the channel diagram as shown in Fig. 1c. Since $\{\mathds{1},\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\}$ constitutes a complete basis for the Liouville space of spin-$1/2$ system, the initial state $\hat{\rho}_{\mathrm{S}}$ and observable $\hat{O}$ can be represented by four-vectors. The super-operators like $\mathcal{S}_{z}^{\eta_{k}}$ and $\mathcal{P}_{k}$ are the $4\times 4$ matrices in this representation. $\mathcal{S}_{z}^{\pm}$ are always adjoint with $\mathcal{B}_{k}^{\mp}$. Consequently, the only non-vanishing coefficient $A^{-+}$ associated with the correlation $C^{+-}$ corresponds to the connected path (in solid line) that starts from $\mathds{1}$ of the initial state and ends at $\hat{\sigma}_{y}$ measurement of the final state (see Fig. 1c). As given in Methods, the coefficient $A^{-+}=1$, thus the measurement signals of $\left<\hat{\sigma}_{y}\right>$ on the quantum sensor give the information of second-order correlation $S_{2}$ for $n=2$ in equation (5)].

The measurement of $C^{+--+}$ can be realized by four quantum channels $\mathcal{P}_{1}^{\rm 4nd},\mathcal{P}_{2}^{\rm 4nd},\mathcal{P}_{3}^{\rm 4nd},\mathcal{P}_{4}^{\rm 4nd}$ (see Methods). As shown in Fig. 1d, the non-vanishing coefficient $A^{-++-}$ associated with $C^{+--+}$ corresponds to the connected path (in solid line) that starts from $\hat{\sigma}_{z}$ of the initial state and ends at $\hat{\sigma}_{y}$ measurement of the final state. The other irrelevant paths are blocked because the coefficients related to them all vanish. Since $A^{-++-}=p$ (see Methods), the final measurement signals of $\left<\hat{\sigma}_{y}\right>$ on the quantum sensor give the information of fourth-order correlation $S_{4}$ for $N=4$ in equation (5).

It is worth noting that some quantum correlations are inaccessible to the synthesized spin rotations of a spin-1/2 quantum sensor. For instance, the extraction of the third-order correlation $C^{+-+}$ or any correlation like $\mathcal{C}^{+-\cdots-+}$ with an odd number of commutators ($\mathcal{B}_{k}^{-}$) requires the quantum channels that connect $\mathds{1}$ with $x/y/z$, which can’t be synthesized by the rotations of a single spin-1/2 system. To measure them then, quantum channels beyond synthesized spin rotations or multi-spin quantum sensors are required.

The scheme above is experimentally demonstrated by using nuclear spins at room temperature on a Bruker Avance III 400 MHz nuclear magnetic resonance (NMR) spectrometer. The sample is carbon-13 labeled acetic acid (¹³CH₃COOH) dissolved in heavy water (D₂O). The methyl group (-¹³CH₃) in acetic acid can be seen as a central spin system, where the ¹³C nucleus is the central spin as the sensor while three ¹H nuclei constitute the quantum many-body target. The natural Hamiltonian of the system in doubly rotating frame is the coupling term $\hat{H}_{\rm NMR}=\frac{\pi}{2}J_{\rm CH}\hat{\sigma}_{z}^{\rm C}\otimes\sum_{i=1}^{3}\hat{\sigma}_{i,z}^{\mathrm{H}}$ with the coupling constant $J_{\rm CH}=129.6$ Hz, and directly generates the pure dephasing Hamiltonian $\hat{V}=\frac{1}{2}\hat{\sigma}_{z}\otimes\hat{B}$ between the sensor and the target, where the target operator is $\hat{B}=\frac{1}{2}J\sum_{i=1}^{3}\hat{\sigma}_{i,z}^{\mathrm{H}}$ with $J=2\pi J_{\rm CH}$.

Figure 2a and Figure 3a show the experimental procedures for measuring the second-order correlation $C^{+-}$ and the fourth-order correlation $C^{+--+}$, respectively. The system is initially prepared in a separable equilibrium state $\hat{\rho}\left(0\right)=\hat{\rho}_{\mathrm{S}}\otimes\hat{\rho}_{\mathrm{B}}$ with $\hat{\rho}_{\mathrm{S}}=\left(\mathds{1}+p_{\mathrm{C}}\hat{\sigma}_{z}^{\rm C}\right)/2$ and $\hat{\rho}_{\mathrm{B}}=\left(\mathds{1}+p_{\mathrm{H}}\sum_{i=1}^{3}\hat{\sigma}_{i,x}^{\rm H}\right)/8$, where $\mathds{1}$ is the unit operator and $p_{\mathrm{C}},p_{\mathrm{H}}$ are the thermal polarizations ($\sim 10^{-5}$) for the ¹³C and ¹H spins, respectively. A continuous radio frequency (RF) field on resonance along $x$ axis is applied on the ¹H spins to create the local Hamiltonian of the target: $\hat{H}_{\mathrm{B}}=\pi\nu\sum_{i=1}^{3}\hat{\sigma}_{i,x}^{\rm H}$, where $\nu\approx 24000$ Hz is the nutation frequency of ¹H spins. The quantum channels are constructed by the synthesis of $\pi/2$ pulses $\mathcal{R}_{\alpha}\left(\pi/2\right)$ with different phases $\alpha$. In experiments, they can be well realized by the phase cycling technology in NMR[42]. Finally, the signals are recorded by measuring the magnetizations of central spins (¹³C) along $y$ axis, i.e., $\left<\hat{\sigma}_{y}^{C}\right>$, from which arbitrary correlations can be extracted by different quantum circuits with suitable phase cycling schemes.

Figure 2b presents the measured values of $\left<\hat{\sigma}_{y}^{C}\right>$ versus the evolving time $\tau_{21}=t_{2}-t_{1}$ for $\delta t=0.1~{}\&~{}0.5$ ms (black scatters), from which the second-order correlation $C^{+-}$ [equation (15)] is extracted. The target signals $\delta t^{2}C^{+-}(t_{2},t_{1})$ (solid red lines) and the numerical results of $S_{2}$ [equation (13)] (black dashed lines) are also presented. As theoretically expected, the measured signals show oscillatory behavior with $\tau_{21}$. The relative deviation (defined in Methods) between the measured signals and the target signals is $\Delta=32.3\%$ for $\delta t=0.1$ ms and $\Delta=18.8\%$ for $\delta t=0.5$ ms. We also plot the dependence of $\left<\hat{\sigma}_{y}^{C}\right>$ on $J\delta t$ with a fixed interval $\tau_{21}=10$ $\mu$s in Fig. 2c. From this we find that with the increase of $\delta t$, the experimental data show increasing deviation from the target signals (red solid line), but agree well with the numerical simulation (dashed line). The deviation between the numerical simulation and the target signals comes from the theoretical approximation due to the finite value of $\delta t$. Moreover, we fit the measured signals $\left<\hat{\sigma}_{y}^{C}\right>$ with a power law of $\left(J\delta t\right)^{k}$ and find $k\approx 1.856$, which is close to the ideal value $k=2$. A little smaller $k$ is mainly caused by the higher-order contribution in $S_{2}$ due to the finite value of $\delta t$ [see equation (13)]. As expected, we can see from Fig. 2c that the deviation is negligible when $J\delta t\to 0$ while it becomes remarkable when $J\delta t\to 1$. Meanwhile, we can also find from the inset that the fitting increasingly deviates from the linearity and the fitting parameter $k$ becomes unstable due to the lower signal-to-noise ratios (SNRs) when $J\delta t\to 0$. Therefore, there exists a trade-off between the theoretical approximation and the SNR of the measured signals resulting from $\delta t$. The optimal value of $\delta t$ can be obtained with the aid of the numerical simulations (see Methods and Supplementary Note 4), and $\delta t=0.5$ ms is a relatively suitable time to extract the second-order correlations. Besides the errors from the theoretical approximation, other error mechanisms leading to the deviation between the experimental data and the numerical simulation include the control imperfection of the $\pi/2$-pulses, the evolution error of the quantum target caused by the RF inhomogeneity and the readout error from the final measurement. We numerically analyze the contributions of these errors, where the error from the $\pi/2$-pulse imperfection is very small in the measurement of the second-order correlations (see Supplementary Table 3).

It is more difficult to measure the fourth-order correlations since the target signals will be much weaker than those of the second-order correlations. As shown in equation (19), the signal related to $C^{+--+}$ is proportional to $\delta t^{4}$, which leads to higher requirements for the experiments, including the higher sensitivity of the quantum sensor and the higher control accuracy of the quantum channels. While the $\pi/2$-pulse imperfection (about $2\%\sim 3\%$ relative error) is negligible in the measurement of the second-order correlations, it will bring a considerable impact on the measurement of the fourth-order correlations. As analyzed in Methods, the non-ideal channel $\mathcal{P}_{3}^{\rm 4nd}$ will result in a severe leakage of lower-order correlations to the measurement of the four-order correlations and overwhelm the desired signal $C^{+--+}$, which brings a challenge in measurements. Hence the scheme as shown in Fig. 1d is practically infeasible in experiments. To overcome this problem, we design an error-resilient channel by repeating the non-ideal $\mathcal{P}_{3}^{\rm 4th}$ for $n$ times to weaken the unwanted low-order correlations (see Fig. 3a). As demonstrated in equation (23), the robust channel $\left(\mathcal{P}_{3}^{\rm 4th}\right)^{n}$ exponentially approaches to the ideal channel with an error of order $\delta\theta^{n}$ as $n$ increases. Therefore, the unwanted lower-order signals ($\delta t^{2}C^{+00+}$) can be effectively suppressed.

Figure 3b presents the measurement values of $\left<\hat{\sigma}_{y}^{C}\right>$ versus the evolving time $\tau_{21}=t_{2}-t_{1}$, including both the cases of $n=1$ and $n=3$. Here $\tau_{32}=t_{3}-t_{2}$ and $\tau_{43}=t_{4}-t_{3}$ are fixed at 10 $\mu$s, and $\delta t=0.5$ ms. The target signal $\delta t^{4}p_{\rm C}C^{+--+}(t_{4},t_{3},t_{1},t_{1})$ (solid red line) and the numerical results of $S^{\rm E}_{4}$ [equation (23)] with (i.e., n = 3, black dashed line) and without (i.e., n=1, blue dashed line) the robust channels for the pulse imperfections, are also presented. For the robust channels ($n=3$), the measured signals (black scatters) agree well with the target signals, and the relative deviation between them is $\Delta=23.5\%$. By contrast, for the non-robust channels ($n=1$), the measured signals (blue scatters) show serious deviation from the target signals ($\Delta=112.2\%$), which makes the data almost untrusted for measuring $C^{+--+}$. For the robust channels, we also plot the dependence of $\left<\hat{\sigma}_{y}^{C}\right>$ on $J\delta t$ with a fixed interval $\tau_{21}=10$ $\mu$s in Fig. 3c. Similar to the results of measuring $C^{+-}$, the experimental data (black scatters) are in good agreement with the numerical simulations (dashed black line), but gradually deviate the target signals (red solid line) when $J\delta t\to 1$. As shown in the inset, the power law fitting of the measured signals $\left<\hat{\sigma}_{y}^{C}\right>\propto\left(J\delta t\right)^{k}$ gives the exponent $k=3.492$, where the deviation from the ideal prediction $k=4$ is also mainly caused by the higher-order contributions to $S_{4}$ due to the finite value of $\delta t$. Besides the approximation errors from $\delta t$, the main experimental errors of measuring the fourth-order correlations by using the robust channels are the evolution error of the quantum target caused by RF inhomogeneity and the readout error, while the control error caused by $\pi/2$-pulse imperfection is almost negligible (see Supplementary Table 4). We also analyze the trade-off between the theoretical approximation and the SNR of the measured signals resulting from $\delta t$, and $\delta t=0.5$ ms corresponds to relatively low errors (see Methods).

To achieve the complete fourth-order quantum correlation $C^{+--+}\left(t_{4},t_{3},t_{2},t_{1}\right)$, we measure the signals of $S_{4}$ versus the evolving time $\tau_{21}$ and $\tau_{43}$ (note that $\mathcal{C}^{+--+}$ doesn’t depend on $\tau_{32}$) using the robust quantum channels (see Fig. 4a). The measured signals show two-dimensional oscillatory behavior along with $\tau_{21}$ and $\tau_{43}$, and the spectral density are obtained from the 2D Fourier transform of $S_{4}$ as shown in Fig. 4b.

We demonstrate selective measurement of arbitrary types and orders of quantum correlations via the synthesized quantum channels with a quantum sensor. The correlation-selection approach based on synthesized quantum channels is more universal than the previously proposed weak measurement scheme in that the former is also applicable to ensemble systems. We successfully extract the second- and fourth-order correlations in a system of nuclear spins with a spin-1/2 sensor. The experiment can be generalized to the quantum sensors of higher or multiple spins, as well as bosonic or fermionic systems. Compared with the conventional nonlinear spectroscopy, this scheme exponentially broadens the accessible correlations. Higher order correlations provide new important information about quantum many-body systems that is not available from conventional nonlinear spectroscopy[2]. Our work offers a new approach to understanding quantum many-body physics (e.g., many-body localization[43, 44] and quantum thermalization[45, 46]), to examining the quantum foundation (e.g., quantum nonlocality[47]), and to providing key information for quantum technologies (e.g., the characterization of quantum computers and quantum simulators, the optimization of quantum control and the improvement of quantum sensing[24]).