Entanglement Assisted Metrology

We propose a new approach to the measurement of a single spin state, based on nuclear magnetic resonance (NMR) techniques and inspired by the coherent control over many-body systems envisaged by quantum information processing. A single target spin is coupled via the magnetic dipolar interaction to a large ensemble of spins. Applying radio frequency pulses, we can control the evolution so that the spin ensemble reaches one of two orthogonal states whose collective properties differ depending on the state of the target spin and are easily measured. We first describe this measurement process using quantum gates; then we show how equivalent schemes can be defined in terms of the Hamiltonian and thus implemented under conditions of real control, using well established NMR techniques. We demonstrate this method with a proof of principle experiment in ensemble liquid state NMR and simulations for small spin systems.

Figure 1

Scheme 1: series of cnot gates between the target and amplifier spins. A collective measurement is sufficient to detect the state of the target spin.

Figure 2

Entanglement permits us to use only one local action of the target spin. The entangling operator $U_{\mathrm{m}\mathrm{a}\mathrm{p}}$ can be created with gates on single spins (above) or with a phase modulated sequence, using only the internal dipolar Hamiltonian and collective rf pulses to create the grade raising operator (below).

Figure 3

Experimental results: (a) spectrum of the first carbon at thermal equilibrium, showing the coupling with the proton and the other two carbons (the methyl group produces the multiplet splitting). (b) Initial state: pseudopure state of the three carbons. (c) Final state: the polarization of the carbons has been inverted (left) indicating a coupling to the target spin in state $|1⟩$, while it is unchanged for spins coupled to target spin in the $|0⟩$ state (right).

Figure 4

Entanglement (ten spins) and contrast for different numbers of spins. Inset: number of repetitions to reach contrast $\approx 1$ as a function of the Hilbert space size, showing a logarithmic dependence. The (perturbed) map applied to the initially fully polarized amplifier is $U=e^{-it{H}_{\mathrm{G}\mathrm{R}1}^{(2)}}{e}^{-iT{H}_{\mathrm{d}\mathrm{i}\mathrm{p}}}$, with $t{b}_{1,2}=1/2$ and $T{b}_{1,2}=\pi /\sqrt{2}$.

Figure 5

Comparison of the four proposed methods.