# Twisting the noise away

This is a Perspective on "Ramsey interferometry with generalized one-axis twisting echoes" by Marius Schulte, Victor J. Martínez-Lahuerta, Maja S. Scharnagl, and Klemens Hammerer, published in Quantum 4, 268 (2020).

By Luca Pezzè (QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy).

### Entanglement-enhanced sensors

Improving the performance of sensors by engineering specific quantum correlations between probe particles is a research frontier in quantum technology. Outstanding experimental results have been obtained in the recent years [1], clearly demonstrating phase sensitivities overcoming the standard quantum limit, $\Delta \theta = 1/\sqrt{N}$, that is achieved with $N$ uncorrelated particles. However, noise sources pose notable challenges [1,2,3]. For instance, poor detection efficiency is one of the main limitations in many current experiments exploring entanglement-enhanced phase sensitivity.

Interestingly, the requirement of high detection efficiency can be relaxed when interaction between the probe particles is applied after the phase imprinting and before the readout measurement. This strategy — generally called interaction-based readout (IBR) [4,5,6,7,8] — can make phase estimation robust against detection noise, removing the requirement of single-particle resolution to reach the Heisenberg limit $\Delta \theta = 1/N$. The advantage offered by IBR has been demonstrated experimentally in atomic [9,10] and trapped ions [11,12] ensembles.

### Encoding phase in spin parity

The basic working principle of IBR is nicely illustrated by considering a Greenberger-Horne-Zeilinger (GHZ) state of $N$ qubits, $\vert {\rm GHZ} \rangle = (\vert \uparrow \rangle^{\otimes N} + \vert \downarrow \rangle^{\otimes N})/\sqrt{2}$ [11]. Following the standard readout procedure [13], after phase encoding, described by the transformation $e^{-i \hat{J}_z \theta}$ (where $\theta$ is the unknown phase to be determined), the state undergoes a collective rotation by $\pi/2$ around the $x$ axis, $e^{-i \tfrac{\pi}{2} \hat{J}_x}$, which converts the rotation angle $\theta$ into a measurable signal given by the spin population [14]. The probability to measure $n_\uparrow$ spin-up qubits is
\label{P1}
P(n_\uparrow \vert \theta) = \big\vert \langle n_\uparrow \vert e^{-i \tfrac{\pi}{2} \hat{J}_x} e^{-i \hat{J}_z \theta} \vert {\rm GHZ} \rangle \big\vert^2 =
f_{n_{\uparrow},+}\cos^2 \frac{N\theta}{2} + f_{n_{\uparrow},-} \sin^2 \frac{N\theta}{2},

where $f_{n_{\uparrow},\pm} = \binom{N}{n_{\uparrow}} \tfrac{1 \pm (-1)^{n_{\uparrow}}}{2^N}$ are binomial distributions in which odd (in the case of $f_{n_{\uparrow},+}$) or even (in the case of $f_{n_{\uparrow},-}$) values of $n_{\uparrow}$ are suppressed.

The information about $\theta$ is thus contained in the parity of $n_{\uparrow}$: the probability to detect an even [odd] value of $n_{\uparrow}$ is $\cos^2 (N\theta/2)$ [$\sin^2 (N\theta/2)$]. This fast-oscillating signal, with period $2\pi/N$, allows for an estimate of $\theta$ with a sensitivity at the Heisenberg limit [13].

### Twisting to amplify spin population differences

However, measuring the parity of $n_{\uparrow}$ requires a detection system with single-particle resolution in order to distinguish $n_{\uparrow}$ from $n_{\uparrow} \pm 1$: this is experimentally prohibitive, especially when $N$ is large. IBR overcomes this limitation by letting the qubits interact before readout. When replacing the the local transformation $e^{-i \tfrac{\pi}{2} \hat{J}_x}$ by the non-local one-axis twisting (OAT) evolution $e^{-i \tfrac{\pi}{2} \hat{J}_x^2}$ [15], the probability to measure $n_\uparrow$ spin-up qubits becomes
\label{P2}
P(n_{\uparrow} \vert \theta) = \big\vert \langle n_\uparrow \vert e^{-i \tfrac{\pi}{2} \hat{J}_x^2} e^{-i \hat{J}_z \theta} \vert {\rm GHZ} \rangle \big\vert^2 = \delta_{n_{\uparrow}, N} \cos^2 \frac{N\theta}{2} + \delta_{n_{\uparrow}, 0} \sin^2 \frac{N\theta}{2},

where $\delta_{n_{\uparrow},\nu}$ is the Kronecker delta.

Only two detection events are possible: the $N$ qubits are measured all spin-up ($n_{\uparrow} = N$) or all spin-down ($n_{\uparrow} = 0$), with probabilities $\cos^2(N\theta/2)$ and $\sin^2(N\theta/2)$, respectively. Two nearby values of $\theta$, such as $0$ and $\pi/N$, for instance, are “magnified” by the OAT evolution, leading to two detection events that can be distinguished without requiring single-particle resolution. Notice that OAT generates the state $\vert {\rm GHZ} \rangle$ from $N$ uncorrelated qubits, namely $\vert {\rm GHZ} \rangle = e^{i \tfrac{\pi}{2}\hat{J}_x^2} \vert \downarrow \rangle^{\otimes N}$ [11,16]: therefore, IBR realizes an echo protocol [4,6,10] where the unitary operation $e^{i \tfrac{\pi}{2}\hat{J}_x^2}$ that
entangles the $N$ qubits is inverted, by $e^{-i \tfrac{\pi}{2}\hat{J}_x^2}$, before readout.

### Optimizing that twist

The paper of Schulte at al. [17] investigates the optimal use of IBR based on twisting dynamics, where the optimization involves interaction strength as well as rotation and spin-projection directions, and addresses the different regimes of OAT evolution. This provides a comprehensive overview on the possibilities offered by OAT echo protocols, which apply to a variety of experimental systems [1,9]. An interesting prediction is that over-squeezed (or non-Gaussian) states produced with OAT can be detected using IBR, with error propagation predicting a sensitivity with Heisenberg scaling $\Delta \theta \sim 1/N$. Non-Gaussian spin states have been only partially explored experimentally [18], and Ref. [17] provides a feasible and robust approach to the optimal detection and use of these states [19].

In a broader context, the work of Schulte at al. [17], belongs to a significant research direction aimed at strengthening entanglement-enhanced sensing against experimental noise and imperfections, making the Heisenberg limit less elusive. In addition to IBR, interesting possibilities are also offered, for instance, by decoherence-free protocols [21,22], error correction techniques [23,24,25] and the use of robust probes states [20].

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[14] Here, $\hat{J}_{x,yz} = \sum_{k=1}^N \hat{\sigma}_{x,y,z}^{(k)}/​2$ are collective spin operators, $\hat{\sigma}_{x,y,z}^{(k)}$ are Pauli matrices for particle $k$ and we consider, without loss of generality, even values of $N$.

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### Cited by

[1] Johannes Jakob Meyer, Johannes Borregaard, and Jens Eisert, "A variational toolbox for quantum multi-parameter estimation", npj Quantum Information 7 1, 89 (2021).

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