Quantum Metrology with Indefinite Causal Order

We address the study of quantum metrology enhanced by indefinite causal order, demonstrating a quadratic advantage in the estimation of the product of two average displacements in a continuous variable system. We prove that no setup where the displacements are used in a fixed order can have root-mean-square error vanishing faster than the Heisenberg limit $1/N$, where $N$ is the number of displacements contributing to the average. In stark contrast, we show that a setup that probes the displacements in a superposition of two alternative orders yields a root-mean-square error vanishing with super-Heisenberg scaling $1/N^2$, which we prove to be optimal among all superpositions of setups with definite causal order. Our result opens up the study of new measurement setups where quantum processes are probed in an indefinite order, and suggests enhanced tests of the canonical commutation relations, with potential applications to quantum gravity.

Figure 1

Two causally ordered schemes. (a) Parallel scheme with measurements of individual displacements. $2N$ independent probes, each with average energy bounded by $E$, are used to estimate the $2N$ displacements $(x_i{)}_{i=1}^N$ and $(p_j{)}_{j=1}^N$. The average displacements $\overline{x}={\sum }_i{x}_i/N$ and $\overline{p}={\sum }_j{p}_j/N$, and their product $A=\overline{x}\overline{p}$ are then computed by classical postprocessing. The RMSE of the scheme has the standard quantum limit scaling $1/\sqrt{N}$. (b) Sequential scheme with independent $x$ and $p$ measurements. The average displacements $\overline{x}$ and $\overline{p}$ are measured directly by applying the total $x$ displacement $D_{x_1}{D}_{x_2}\cdots D_{x_N}$ and the total $p$ displacement $D_{p_1}{D}_{p_2}\cdots D_{p_N}$ to two independent probes, each with average energy bounded by $E$. The product $A=\overline{x}\overline{p}$ is then computed by classical postprocessing. The RMSE of this scheme has the Heisenberg scaling $1/N$.

Figure 2

Definite vs indefinite order in a quantum metrology setup. (a) Estimation scheme using the quantum switch. The total $x$ displacements $D_{x_1}{D}_{x_2}\cdots D_{x_N}$ and $p$ displacements $D_{p_1}{D}_{p_2}\cdots D_{p_N}$ act in a coherent superposition of two alternative orders, controlled by the state of a control qubit. If the control is prepared in the state $|0⟩$ ($|1⟩$), the probe will experience the displacements in the order corresponding to the blue (orange) path. By preparing the probe in the minimum-energy state $|0⟩$ and the control qubit in the state $|+⟩$, this scheme achieves the super-Heisenberg scaling $1/N^2$ of the RMSE. (b) Generic causally ordered scheme. A probe and an auxiliary system are prepared in a generic state, with average energy of the probe bounded by $E$. Then, the probe undergoes a sequence of displacements, arranged in a fixed order $(z_1,\dots ,z_{2N})$, where $(z_1,\dots ,z_{2N})$ is an arbitrary permutation of the sequence $(x_1,\dots ,x_N,p_1,\dots ,p_N)$. Each displacement operation $z_i$ is followed by a unitary gate $V_i$, acting jointly on the probe and the auxiliary system. Finally, a joint measurement is performed on the probe and the auxiliary system. Every estimation scheme of this form, including the schemes in Figs. 1 and 1, must have the RMSE vanishing no faster than $1/N$.

Figure 3

Definite vs indefinite order in the nonasymptotic regime. The RMSE achievable with the quantum switch is plotted against the lower bound to the RMSE for every estimation scheme with definite causal order. The four plots correspond to the parameter values $|\overline{x}|=|\overline{p}|=\overline{z}>0$, $\nu =10$, and (a) $E=0.5$, $N=5$; (b) $E=1$, $N=5$; (c) $E=0.5$, $N=15$; (d) $E=1$, $N=15$. The $y$ axis shows the RMSE $\mathrm{\Delta }A$ in units of $2\pi /N^2$. The solid red lines show the RMSE $\mathrm{\Delta }{A}_{\mathrm{SWITCH}}^{\prime }$, achievable by measuring the probe and the control [Eq. (7)]. The dashed lines show the RMSE $\mathrm{\Delta }{A}_{\mathrm{SWITCH}}$, achievable by measuring the control alone [Eq. (6)]. The blue lines show the lower bound of the RMSE $\mathrm{\Delta }{A}_{\text{fixed}}$ [Eq. (9)].