Experimental Test of the Quantum No-Hiding Theorem

The no-hiding theorem says that if any physical process leads to bleaching of quantum information from the original system, then it must reside in the rest of the Universe with no information being hidden in the correlation between these two subsystems. Here, we report an experimental test of the no-hiding theorem with the technique of nuclear magnetic resonance. We use the quantum state randomization of a qubit as one example of the bleaching process and show that the missing information can be fully recovered up to local unitary transformations in the ancilla qubits.

Figure 1

Circuit diagram for testing the no-hiding theorem using the state randomization. $U$ is the unitary operator for randomization and $H$ represents the Hadamard gate. Dots and circles represent the CNOT gates. After randomization, the state $|\psi ⟩$ on the first qubit has been transferred to the ancilla qubit.

Figure 2

(a) The equilibrium spectra of ${}^{13}\mathrm{CHFBr}_2$ dissolved in acetone-d6 at 300 K on a AV500 NMR spectrometer. Labeling of the transition is given on top of each spectral line. The scalar couplings between different spins are measured as $J_{\mathrm{HC}}=224.5$, $J_{\mathrm{FC}}=-310.9$, $J_{\mathrm{HF}}=49.7$. For $\theta =\pi /2$ and $\varphi =0$, (b) and (c) are, respectively, the experimental spectra for the input ($|\psi {⟩}_1|00{⟩}_{23}$) and output ($|{\varphi }^{+}{⟩}_{12}|\psi {⟩}_3$) state for the 3 spins. The receiver phase is set using a separate experiment so that $y$ magnetization appears as positive absorption mode. For $\theta =\varphi =\pi /2$, (d) and (e) are the experimental spectra showing, respectively, the input and output states for the 3 spins.

Figure 3

NMR pulse sequence for the implementation of the no-hiding theorem. The filled and the empty boxes, respectively, represent $\pi /2$ and $\pi$ pulses while the gray boxes represent pulses with flip angles on the top. The phase of a pulse is given on top of the pulse for $\pi /2$ and $\pi$ pulses and are subscripts of angles for other flip angle pulses. $G_z$ is the $z$-gradient pulse used to destroy all unwanted transverse magnetization. For each ($\theta ,\varphi$) pair three identical experiments are performed to observe each qubit independently.

Figure 4

Summarized experimental results for the no-hiding theorem using the state randomization. The integrals of the real part of the NMR signal from spins ${}^1\mathrm{H}$, ${}^{19}\mathrm{F}$, and ${}^{13}\mathrm{C}$ are shown as mesh and contour plots as a function of $\theta$ and $\varphi$. The plots show the expected sine and cosine behavior. (a) Input state $|\psi {⟩}_1|00{⟩}_{23}$. The information about $\theta$ and $\varphi$ is encoded in the first spin, and second and third spins are in $|00⟩$ state. (b) Quantum information has been transferred from the first qubit to the third qubit.

Figure 5

The output density matrix for the first two qubits showing that they are in the Bell state $|{\varphi }^{+}⟩$. (a) $\theta =\varphi =\pi /2$. (b) $\theta =\pi /2$ and $\varphi =\pi$.