a–c, Experimental measurements of NV–NV couplings. NVi+(−) denotes that the ith NV is prepared in the state m s = +1(−1). The angular parameters of the magnetic field θ and ϕ in spherical coordinates are defined through the coordinate transformation relationships from the Cartesian system (x, y, z), in which θ ∈ [0, π] denotes the polar angle and φ ∈ [0, 2π] specifies the azimuthal angle. d, The possible spatial positions of NV2 determined from the hyperfine coupling obtained in a according to ({r|[|A(r, B) − A NV1,NV2+ | < ΔA] ∩ [|A(r, B) − A NV1,NV2− | < ΔA]}) with a coupling error of ΔA = 20 kHz used as illustration. e, Correlated determination of NV2 location according to hyperfine coupling of a (red band), b (blue band) and c (green band). The position of NV2 is listed in the upper-left corner. f, The spatial positioning schematic diagram of DS1 by the sensing of the entangled state (|ψ 2 ⟩). Each band in the figure is determined by the effective coupling of the entangled states and DS1 under two magnetic fields (\(\{{\bf{r}}| [| A({\bf{r}},{{\bf{B}}}_{1})-{A}_{{\psi }_{2},{\rm{DS1}}}({{\bf{B}}}_{1})| < \Delta A]\cap [| A({\bf{r}},{{\bf{B}}}_{2})-{A}_{{\psi }_{2},{\rm{DS1}}}({{\bf{B}}}_{2})| < \Delta A]\}\)). For the red band, in the magnetic field (B 0 , θ, φ) = (100.6 G, 0.62π, 0.68π) and (B 0 , θ, φ) = (151.84 G, 0.696π, 0.5π), the effective couplings are 585 and 907 kHz, respectively. For the cyan band, in the magnetic field (B 0 , θ, φ) = (147.4 G, 0.7π, 1.58π) and (B 0 , θ, φ) = (66 G, 0.85π, 0.37π), the effective couplings are 702 and 359 kHz, respectively.