Steering-enhanced quantum metrology using superpositions of noisy phase shifts

Quantum steering is an important correlation in quantum information theory. A recent work [Nat. Commun. 12, 2410 (2021)] showed that quantum steering is also useful for quantum metrology. Here, we extend the exploration of steering-enhanced quantum metrology from single noiseless phase shifts to superpositions of noisy phase shifts. As concrete examples, we consider a control system that manipulates a target system to pass through a superposition of either dephased or depolarized phase shifts channels. We show that using such superpositions of noisy phase shifts can suppress the effects of noise and improve metrology. Furthermore, we also implemented proof-of-principle experiments for a superposition of dephased phase shifts on the IBM quantum experience, demonstrating a clear improvement on metrology.

Figure 1

Illustration of steering-enhanced quantum metrology with a superposition of quantum channels. Alice ($\mathrm{A}$) and Bob ($\mathrm{B}$) share a bipartite state ${\rho }_{\text{A}\text{B}}$. Alice performs a measurement $A$ and obtains the corresponding outcome $a$. Then, Alice sends her information $(a,A)$ to Bob through a classical communication. A local phase shift $\theta$ on Bob's side is generated by a Hamiltonian $H$. Different from Ref. [29], in which a phase shift is generated noiseless, we use a system $\mathrm{C}$ to control the evolution of system $\mathrm{B}$ and create a superposition of noisy phase shifts. When $\mathrm{C}$ is in state $|0\rangle$ ($|1\rangle$), represented by the white (black) dot on the left, system $\mathrm{B}$ interacts with the environment ${\mathrm{E}}_1$ (${\mathrm{E}}_2$). After creating the superposition of noisy phase shifts, Bob collects the conditional state and measure on $\mathrm{C}$. According to Alice's information, Bob can decide to either measure $H$ or estimate the phase-shift $\theta$ through the measurement $M$. Then, he can obtain the optimal variance $\mathrm{\Delta }{H}_{\text{opt}}$ and the optimal quantum Fisher information $F_{Q,\text{opt}}$.

Figure 2

Average violations $\mathcal{V}$ of the metrological steering inequality. We set $\theta =0$ and plot the average violations of two examples: (a) dephased and (b) depolarized phase shifts with the visibilities $w$ and $v$, respectively. Note that ${\rho }_{\mathrm{C}}={\mathbb{1}}_{\mathrm{C}}/2$ and ${\rho }_{\mathrm{C}}=|+\rangle {\langle +|}_{\mathrm{C}}$ represent different choices of the initial states of the control systems. One can observe in example (a) that the control ${\rho }_{\mathrm{C}}=|+\rangle {\langle +|}_{\mathrm{C}}$ can enhance the violation for a given visibility; although the system is fully dephased, we can still witness $\mathcal{V}\approx 0.33$ with ${\rho }_{\mathrm{C}}=|+\rangle {\langle +|}_{\mathrm{C}}$. For example, (b), the depolarized noise (with ${\rho }_{\mathrm{C}}={\mathbb{1}}_{\mathrm{C}}/2$) causes a sudden vanishing of the violation when $v\approx 0.29$; however, if ${\rho }_{\mathrm{C}}=|+\rangle {\langle +|}_{\mathrm{C}}$, it can enhance the violation and extend the sudden-vanishing effect to $v\approx 0.42$.

Figure 3

Circuit model for steering-enhanced quantum metrology with a superposition of dephased phase shifts. Schematics of our circuit model (a) without and (c) with details. (b) In the topology of the four qubits that we chose in the IBM-Cairo device, the numbers label the qubits No. 24, No. 25, No. 26, and No. 22, representing system $\mathrm{C}$, $\mathrm{B}$, ${\mathrm{E}}_1$, ${\mathrm{E}}_2$, respectively. Here, the visibility $w$ is tuned by the angle $\varphi$, such that $w=2{sin}^2(\varphi /2)$. We use the standard symbols for quantum gates (see the Appendix).

Figure 4

Experimental results and noise simulations of the metrological tasks with the dephased phase shifts: the control $\mathrm{C}$ is prepared in (a) ${\rho }_{\mathrm{C}}={\mathbb{1}}_{\mathrm{C}}/2$ (without a superposition of phase shifts) and (b) ${\rho }_{\mathrm{C}}=|+\rangle {\langle +|}_{\mathrm{C}}$ (with a superposition of phase shifts). Specifically, the red-cross (blue-circle) data points are the experimental results of the average optimal Fisher information (the optimal variance) with respect to the visibility $w$. Note that $\theta =0$. Here, the error bars are obtained from 40 individual rounds of experiments; each experimental data point consists of 10 000 individual runs performed on about ten different dates. Therefore, the error bars represent the variance of the IBM-Cairo device to conduct these experiments in long timescales. The solid curves represent the noise simulations for the dephased phase shifts with ${\rho }_{\mathrm{C}}=|+\rangle {\langle +|}_{\mathrm{C}}$; the dashed curves represent the noise simulations with ${\rho }_{\mathrm{C}}={\mathbb{1}}_{\mathrm{C}}/2$. Although we consider only several common noisy resources, the tendencies and magnitudes of noise simulations approach the actual experiment in both cases (a) and (b). We clearly observe that the control in a superposition state, i.e., $|+\rangle {\langle +|}_{\mathrm{C}}$, can enhance the optimal Fisher information and decrease the optimal variance; thus, it extends the violations of the metrological steering inequality from $w\approx 0.38$ to $w\approx 0.69$.

Figure 5

Model of noise simulations. We model the qubit relaxation and qubit dephasing effects that occur after performing a total unitary evolution. We apply depolarizing and bit-flip channels to describe gate errors and read-out errors (on the system $\mathrm{B}$ and $\mathrm{C}$), respectively. Here, qubit relaxation and qubit dephasing (QRQD) represents the qubit relaxation and qubit dephasing channel which action is given by the master equation in Eq. (22); the cnot error ${\mathcal{G}}_{\mathrm{err}}$ is modeled by the depolarizing noise in Eq. (23); the readout error ${\mathcal{R}}_{\mathrm{err}}$ is described by the bit-flip channel in Eq. (24).

Figure 6

Circuit implementation of a superposition of depolarized phase shifts. We assume that the control system $\mathrm{C}$ and all the environments ${\mathrm{E}}_i(i=1,...,4)$ are initialized to $|0\rangle$.

Figure 7

Circuit implementing the unitary operation $U(\zeta ,\xi )$, which maps $|0\rangle {|0\rangle }_{\mathrm{E}}$ onto the state in Eq. (A3).