Experimental determination of tripartite quantum discord

Abstract

Quantum discord is a measure of nonclassical correlations in quantum systems. While the bipartite version of quantum discord is experimentally well-studied, the multipartite version has never been convincingly measured. In this study, we experimentally investigate tripartite quantum discord using an NMR quantum information processor. Building on a theoretical framework for conditional projective measurements and quantum conditional mutual information, we quantify the tripartite quantum discord and its contributions in different three-qubit states such as the Greenberger-Horne-Zeilinger (GHZ) and Werner (W) states as well as classical mixtures of biseparable Bell states, and classical mixtures of product states. The experiments employed full quantum state tomography and temporal averaging to prepare mixed states, achieving fidelities exceeding 95%. Our results confirm that quantum discord persists even in the absence of entanglement, highlighting its utility as a broader indicator of quantum correlations. Furthermore, we validate the nonconvexity of discord, confirming that classical mixtures of zero-discord states can exhibit nonzero discord. This experimentally confirms that quantum discord does not fit into the framework of resource theory. This work establishes a robust methodology for measuring quantum discord, illuminating the structure and distribution of quantum correlations in multipartite systems.

Keywords: NMR; multipartite quantum correlations; quantum conditional mutual information; quantum discord; quantum information.

Fig. 1.

The three-qubit system considered in this work. (A) Molecular structure of ¹³C-labeled diethylfluoromalonate, highlighting three qubits represented by ¹H, ¹⁹F, and ¹³C nuclei. We denote the qubit labeling $A,B,C$ of each nuclei. (B) Relaxation parameters: spin-lattice relaxation time (T₁, in seconds) and spin–spin relaxation time (T2, in seconds) for each nucleus. (C) Spectroscopic details: chemical shifts (ν_i, in Hz) of the nuclear spins and spin–spin coupling constants (J_ij, in Hz) between them. (D) Decompositions of the tripartite quantum discord into the three bipartite components ${\mathrm{\Delta }}_{A;B|C},{\mathrm{\Delta }}_{A;C|B},{\mathrm{\Delta }}_{B;C|{\mathrm{\Pi }}^A}$ and the genuine tripartite discord ${\mathrm{\Delta }}_{A:B:C}$.

Fig. 2.

NMR pulse sequences for the preparation of various quantum states and gate operations. The pink rectangles represent pulses that induce rotation about the y-axis, while the blue rectangles correspond to rotations about the x-axis. The angle of rotation for each pulse is indicated above its respective rectangle. (A) Preparation of the completely mixed state with gradient pulse shown in green. (B) GHZ state preparation sequence. The NMR pulse sequence consisting of single-qubit gate and controlled-NOT (CNOT) gates is shown with similar sequences for other CNOT gates. (C) W state preparation using a combination of CNOT and controlled-rotation (CROT) gates. One of the rectangles in figure (C) is labeled with a rotation angle of $2\beta$, where $\beta =0.195\pi$. (D and E) Preparation of the Bell states $|{\varphi }_{\mathit{AB}}^{+}\rangle$ and $|{\varphi }_{\mathit{AC}}^{+}\rangle$, respectively. (F) Pulse sequence for the CNOT_AB gate used in state preparation. (G) Pulse sequence for the CROT gate employed in the W state preparation, with one rectangle indicating a pulse with angle $\gamma =\frac{\pi }{4}$. The time interval τ is defined as $\frac{1}{2J}$, where J represents the scalar coupling strength between the corresponding qubits.

Fig. 3.

Tripartite discord as a function of mixedness for the states (A) $\mu |\text{GHZ}\rangle \langle \text{GHZ}|+(1-\mu )\frac{I_8}{8}$, (B) $\mu |\text{W}\rangle \langle \text{W}|+(1-\mu )\frac{I_8}{8}$, (C) $\mu |{\varphi }_{\mathit{AB}}^{+}\rangle \langle {\varphi }_{\mathit{AB}}^{+}|+(1-\mu )|{\varphi }_{\mathit{AC}}^{+}\rangle \langle {\varphi }_{\mathit{AC}}^{+}|$, and (D) $\mu |000\rangle \langle 000|+(1-\mu )|+++\rangle \langle +++|$. Solid lines represent the theoretical values, while the dots with error bars correspond to the experimental values for total discord and its decompositions. While we have included the error bars corresponding to the experimental value, the range of the error bar is within $\pm 0.06$. Hence, in (A–C), the error bars are too small in comparison with the values of the discord along the vertical axis and hence are not visible. In the case of (D) the smaller range of the vertical-axis makes the error bar more discernible.