Sisi Zhou

The optimal quantum measurements for estimating different unknown parameters in a parameterized quantum state are usually incompatible with each other. Traditional approaches to addressing the measurement incompatibility issue, such as the Holevo Cram\'{e}r--Rao bound, suffer from multiple difficulties towards practical applicability, as the optimal measurement strategies are usually state-dependent, difficult to implement and also take complex analyses to determine. Here we study randomized measurements as a new approach for multi-parameter quantum metrology. We show quantum measurements on single copies of quantum states given by $3$-designs perform near-optimally when estimating an arbitrary number of parameters in pure states and more generally, {approximately low-rank well-conditioned states}, whose metrological information is largely concentrated in a low-dimensional subspace. The near-optimality is also shown in estimating the maximal number of parameters for three types of mixed states that are well-conditioned on their supports. Examples of fidelity estimation and Hamiltonian estimation are explicitly provided to demonstrate the power and limitation of randomized measurements in multi-parameter quantum metrology.

The Heisenberg limit (HL, with estimation error scales as 1/n) and the standard quantum limit (SQL, 1/sqrt(n)) are two fundamental limits in estimating an unknown parameter in n copies of quantum channels and are achievable with full quantum controls, e.g., quantum error correction (QEC). It is unknown though, whether these limits are still achievable in restricted quantum devices when QEC is unavailable, e.g., with only unitary controls or bounded system sizes. In this talk, I will discuss various new limits for estimating qubit channels under restrictive controls. The HL is proven to be unachievable in various cases, indicating the necessity of QEC in achieving the HL. Furthermore, a necessary and sufficient condition to achieve the SQL is determined, where a novel unitary control protocol is identified to achieve the SQL for certain types of noisy channels, and a constant floor on the estimation error is proven for other cases.

Abstract The quantum Fisher information (QFI), as a function of quantum states, measures the amount of information that a quantum state carries about an unknown parameter. The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla. In quantum metrology, people are interested in calculating the QFI of N identical copies of a quantum channel when N\rightarrow\infty, which we call the asymptotic QFI. It was known that the asymptotic QFI grows either linearly or quadratically with N. Here we obtain a simple criterion that determines whether the scaling is linear or quadratic. In both cases, the asymptotic QFI and a quantum error correction protocol to achieve it are solvable via a semidefinite program. When the scaling is quadratic, the Heisenberg limit, a feature of noiseless quantum channels, is recovered. When the scaling is linear, we show the asymptotic QFI is still in general larger than N times the single-channel QFI and furthermore, sequential estimation strategies provide no advantage over parallel ones. For details, see arXiv: 2003.10559.

Imaging thermal sources naturally yields Gaussian states at the receiver, raising the question of whether Gaussian measurements can perform optimally in quantum imaging. In this work, we establish no-go theorems on the performance of Gaussian measurements for imaging thermal sources in the limit of mean photon number per temporal mode $\epsilon \to 0$ or source size $L \to 0$. We show that non-Gaussian measurements can outperform any Gaussian measurement in the scaling of the estimation variance with $\epsilon$ (or $L$). We also present several examples to illustrate the no-go results.

Quantum effect enables enhanced estimation precision in metrology, with the Heisenberg limit (HL) representing the ultimate limit allowed by quantum mechanics. Although the HL is generally unattainable in the presence of noise, quantum error correction (QEC) can recover the HL in various scenarios. A notable example is estimating a Pauli-$Z$ signal under bit-flip noise using the repetition code, which is both optimal for metrology and robust against noise. However, previous protocols often assume noise affects only the signal accumulation step, while the QEC operations---including state preparation and measurement---are noiseless. To overcome this limitation, we study fault-tolerant quantum metrology where all qubit operations are subject to noise. We focus on estimating a Pauli-$Z$ signal under bit-flip noise, together with state preparation and measurement errors in all QEC operations. We propose a fault-tolerant metrological protocol where a repetition code is prepared via repeated syndrome measurements, followed by a fault-tolerant logical measurement. We demonstrate the existence of an error threshold, below which errors are effectively suppressed and the HL is attained.