Experimental verification of many-body entanglement using thermodynamic quantities

The phenomenon of quantum entanglement underlies several important protocols that enable emerging quantum technologies. Entangled states, however, are extremely delicate and often get perturbed by tiny fluctuations in their external environment. Certification of entanglement is therefore immensely crucial for the successful implementation of protocols involving this resource. In this Letter, we propose a set of entanglement criteria for multiqubit systems that can be easily verified by measuring certain thermodynamic quantities. In particular, the criteria depend on the difference in optimal global and local works extractable from an isolated quantum system under global and local interactions, respectively. As a proof of principle, we demonstrate the proposed scheme on nuclear spin registers of up to 10 qubits using the nuclear magnetic resonance architecture. We prepare a noisy Bell diagonal state and noisy Greenberger-Horne-Zeilinger class of states in star-topology systems and certify their entanglement through our thermodynamic criteria. Along the same line, we also propose an entanglement certification scheme in many-body systems when only partial or even no knowledge about the state is available.

Figure 1

(a) Two-qubit system sodium fluorophosphate (NAFP, with ${}^{19}\mathrm{F}$ and ${}^{31}\mathrm{P}$ being first and second qubits) used in experiment I. (b) The star-topology configuration, wherein each ancillary spin A interacts with the central spin ${}^{13}\mathrm{C}$. (c), (d) Three-qubit star-system fluoroacetonitril (FAN) and ten-qubit star-system trimethylphosphate (TMP) used in experiment II. (e) Dibromofluoromethane (DBFM, with ${}^1\mathrm{H}$, ${}^{13}\mathrm{C}$, and ${}^{19}\mathrm{F}$ being first, second, and third qubits) used in experiment III.

Figure 2

(a) The NMR pulse sequence to prepare the Bell diagonal state with control parameter $\beta$ and $\gamma$. Here, PFG is the pulsed-field gradient and $\mathrm{\Delta }\nu$ is the resonance offset of both ${}^{19}\mathrm{F}$ and ${}^{31}\mathrm{P}$. One of the operations ($U_{\mathrm{per}}\circ H\circ \mathrm{CNOT}$) takes the Bell diagonal state to its passive state. A detailed explanation of this sequence is given in the Supplemental Material [48]. (b) Gradient-color plot for theoretical values of ${\mathrm{\Delta }}_{1|2}$ (in units of ${\omega }_P$) vs the control parameters $\beta$ and $\gamma$. States outside the white line (inner perimeter) are entangled as ${\mathrm{\Delta }}_{1|2}>{\delta }_{1|2}^G$. For the states outside the black line (outer perimeter) ${\mathrm{\Delta }}_{1|2}>{\delta }_{1|2}^I$ and hence they are also entangled. (c) Gradient-color plot for the experimental values of ${\mathrm{\Delta }}_{1|2}^{\text{Expt}}$ with estimated errors of $\pm 0.1{\omega }_P$. Here, the error originates both from the spin system as well as the NMR hardware, so accordingly we have estimated the random error from the experimental NMR spectrum corresponding to the least signal-to-noise ratio providing a useful upper bound for errors. (d) $11\times 11$ pixel theoretical gradient-color plot of ${\mathrm{\Delta }}_{1|2}$: Evidently the state-independent certification scheme is weaker than the state-dependent scheme.

Figure 3

(a) ${}^{19}\mathrm{F}$ spectra of FAN corresponding to the one-pulse experiment on a thermal state (front) and to the three-qubit GHZ state (back). (b) Plot ${\mathrm{\Delta }}_{1|23}$ (in the unit of ${\omega }_H$) vs purity $\lambda$ for three-qubit noisy GHZ states. Comparing the values of ${\mathrm{\Delta }}_{1|23}$ and ${\delta }_{1|23}^G,{\delta }_{1|23}^I$, we identify the threshold values marked by the dotted lines: $\lambda =3/7$ and $\lambda =0.68$, respectively. Above these thresholds, the state exhibits entanglement. (c) ${}^{31}\mathrm{P}$ spectra of TMP corresponding to the one pulse experiment (front), and to GHZ (back). (d) ${\mathrm{\Delta }}_{1|1^c}$ vs purity $\lambda$ for the ten-qubit GHZ class. Here, $\lambda =0.499$ and $\lambda =0.957$ for ${\delta }_{1|1^c}^G$ and ${\delta }_{1|1^c}^I$ marks the entanglement threshold boundary. The error bar represents the estimated random error due to noise.

Figure 4

Plot of ${\mathrm{\Delta }}_{12|3}$ (experimental as well as theoretical), ${\delta }_{12|3}^{GL}$, ${\delta }_{12|3}^G$, and ${\delta }_{12|3}^I$ against the noise parameter $\lambda$. The state-independent, the global, and the global-local separability conditions certify entanglement for the parameter ranges $\lambda >0.91$, $\lambda >0.71$, and $\lambda >1/5$, respectively, establishing hierarchy among these conditions.