Complete Quantum-State Tomography with a Local Random Field

Single-qubit measurements are typically insufficient for inferring arbitrary quantum states of a multiqubit system. We show that, if the system can be fully controlled by driving a single qubit, then utilizing a local random pulse is almost always sufficient for complete quantum-state tomography. Experimental demonstrations of this principle are presented using a nitrogen-vacancy (NV) center in diamond coupled to a nuclear spin, which is not directly accessible. We report the reconstruction of a highly entangled state between the electron and nuclear spin with fidelity above 95% by randomly driving and measuring the NV-center electron spin only. Beyond quantum-state tomography, we outline how this principle can be leveraged to characterize and control quantum processes in cases where the system model is not known.

Figure 1

Random-field quantum-state tomography of a multiqubit system when only a single qubit (dark blue) can be accessed. By randomly driving (light blue) the qubit and measuring the expectation of any single-qubit observable $M$ for a fully controllable system and sufficiently long times of the signal $⟨M{⟩}_t$ (red), any qubit-network state $\rho$ can almost always be reconstructed. We experimentally demonstrate this principle by reconstructing combined states of an electron-nuclear spin system in diamond, in which only the electron spin is accessible.

Figure 2

(a) Schematic illustration of the two-spin system in diamond: NV-center electron spin (dark blue) and nearby carbon nuclear spin (black), along with the level diagram of the electron spin including the fine structure of the ground state, shown below. (b) Sequence of the applied fields: laser for initialization and readout (green), microwave for state preparation and tomography (blue). The central panel shows an example random pulse shape $f(t)$ and the corresponding time trace $⟨M{⟩}_t$ of the electronic ground-state population (bottom panel) when the preparation stage is absent, namely, the system is initially only in the polarized state. The solid black line shows the ideal case obtained by numerical propagation of the initial state. (c) Random-field reconstruction of two different entangled states: both figures show the modulus of the reconstructed density matrix, wherein the solid gray bars depict the tomography results and the blue transparent bars show the exact state obtained numerically. The upper panel shows the reconstruction with fidelity 96.1% of a randomly created state, whereas the lower panel shows the reconstruction with fidelity 94.9% of a highly entangled state (concurrence $C=0.91$) created through an optimized preparation pulse shape.