Determining an $n$-qubit state by a single apparatus through a pairwise interaction

This paper shows how one can reconstruct an unknown $n$-qubit state by only a single apparatus. Its core is a redistribution of the information within an extended Hilbert space by coupling the unknown system with an assistant system through only a pairwise interaction, which results in a one-to-one mapping between the unknown density matrix elements and the probabilities of the occurrence of the eigenvalues of a single, factorized observable of the composite system. In such an interaction configuration, which is more feasible in experiments, the quality of the measurement only depends on that of the subsystem consisting a of one-qubit system coupled to a one-qubit assistant. We analyze in detail the performance of this procedure and experimentally implement it for reconstructing an unknown two-qubit state using a four-nuclear-spin system.

Figure 1

(Color online) Schematic diagram for measuring an $n$-qubit unknown state by a single observable through a pairwise interaction. The upper spin chain consisting of qubits $1,2,\cdots ,n$ is the unknown system $S$, while the lower chain consisting of qubits $1_a,2_a,\cdots ,n_a$ is the assistant $A$. The solid lines represent the pairwise interactions between qubit $k$ and $k_a$.

Figure 2

(Color online) Molecular structure and relevant parameters of iodotrifluoroethylene. ${}^{13}$C and three ${}^{19}$F nuclei are used as a four-qubit quantum system, and the labels in brackets near the nuclei correspond to the qubits in Fig. 1. The diagonal and nondiagonal elements represent the chemical shifts and the coupling constants (in units of Hz) respectively. The spin-lattice relaxation times $T_1$ are 21 s for ${}^{13}$C and $12.5$ s for ${}^{19}$F. These parameters are measured at the temperature $T=303$ K.

Figure 3

(Color online) Flowchart of the experiments. The system $S$ is initially in an unknown state ${\widehat{\rho }}_S$ and the assistant $A$ is initially in a completely disordered state $\frac{1}{4}\widehat{1}$. Then we drive the composite system $S+A$ to evolve under an optimal Hamiltonian of Eq. (19) for a time $\tau$ which is simulated by NMR techniques. Finally we measure a single, factorable observable $\widehat{P}$, i.e., their $x$ components of Eq. (4).

Figure 4

Pulse sequence for implementing the propagator ${\widehat{U}}_{ap}^{\left(2\right)}\left(\tau \right)$. Here, ${\widehat{U}}_{zz}\left({\theta }_i\right)=e^{-i{\theta }_i({\widehat{I}}_z^1{\widehat{I}}_z^{1_a}+{\widehat{I}}_z^2{\widehat{I}}_z^{2_a})}$ with ${\theta }_3=2{\theta }_1=2\delta t,{\theta }_2=4\sqrt{2}\delta t$ and $t_1=|{\theta }_i/\left(2\pi J_{11_a}\right)|,t_2=\left|{\theta }_i/\left(2\pi J_{22_a}\right)-{\theta }_i/\left(2\pi J_{11_a}\right)\right|$. The local operators ${\widehat{U}}_1=e^{-i\frac{\sqrt{2}}{2}\delta t({\widehat{I}}_z^1+{\widehat{I}}_z^2+{\widehat{I}}_z^{1_a}+{\widehat{I}}_z^{2_a})}{e}^{-i\frac{\pi }{2}({\widehat{I}}_y^1+{\widehat{I}}_y^2+{\widehat{I}}_y^{1_a}+{\widehat{I}}_y^{2_a})}$ and ${\widehat{U}}_2=e^{i\frac{\pi }{2}({\widehat{I}}_y^1+{\widehat{I}}_y^2+{\widehat{I}}_y^{1_a}+{\widehat{I}}_y^{2_a})}{e}^{-i\frac{\sqrt{2}}{2}\delta t({\widehat{I}}_z^1+{\widehat{I}}_z^2+{\widehat{I}}_z^{1_a}+{\widehat{I}}_z^{2_a})}$.

Figure 5

Experimental ${}^{13}$C NMR spectra (denoted by thick blue lines) for reconstructing the three different unknown states: the thermal equilibrium state ${\widehat{\rho }}_{S1}$, the pseudo-pure state ${\widehat{\rho }}_{S2}$, and the pseudo-entangled state ${\widehat{\rho }}_{S3}$ from left to right. From top to bottom are the final information of ${}^{13}$C, F${}_1$, F${}_2$, and F${}_3$ sequently. The intensities of different resonance lines correspond directly to population differences between two related energy levels. Fluorine's information is observed through the ${}^{13}$C NMR spectra by SWAP gates acting on ${}^{13}$C and ${}^{19}$F. The theoretical expectations are denoted by thin red lines obtained by a NMR simulation program (nmrsim) in Bruker's TopSpin${}^{\mathrm{TM}}$ software package. The numbers of repeated scans for ${\widehat{\rho }}_{S1}$, ${\widehat{\rho }}_{S2}$ and ${\widehat{\rho }}_{S3}$ are, respectively, 32, 32, and 64.

Figure 6

Experimentally reconstructed coefficients $c_s$ from the experimentally measured joint probabilities $p_q$ by the inverse mapping ${\mathcal{M}}^{-1}$ for three initial states ${\widehat{\rho }}_{S1}$, ${\widehat{\rho }}_{S2}$, and ${\widehat{\rho }}_{S3}$. $c_s$ is from 1 to 15 where $c_0$ is the coefficient of identity.