Abstract Careful tailoring the quantum state of probes offers the capability of investigating matter at unprecedented precisions. Rarely, however, the interaction with the sample is fully encompassed by a single parameter, and the information contained in the probe needs to be partitioned on multiple parameters. There exist, then, practical bounds on the ultimate joint-estimation precision set by the unavailability of a single optimal measurement for all parameters. Here, we discuss how these considerations are modified for two-level quantum probes — qubits — by the use of two copies and entangling measurements. We find that the joint estimation of phase and phase diffusion benefits from such collective measurement, while for multiple phases no enhancement can be observed. We demonstrate this in a proof-of-principle photonics setup.
Abstract Careful tailoring the quantum state of probes offers the capability of investigating matter at unprecedented precisions. Rarely, however, the interaction with the sample is fully encompassed by a single parameter, and the information contained in the probe needs to be partitioned on multiple parameters. There exist, then, practical bounds on the ultimate joint-estimation precision set by the unavailability of a single optimal measurement for all parameters. Here, we discuss how these considerations are modified for two-level quantum probes — qubits — by the use of two copies and entangling measurements. We find that the joint estimation of phase and phase diffusion benefits from such collective measurement, while for multiple phases no enhancement can be observed. We demonstrate this in a proof-of-principle photonics setup.
This work compares the performance of single- and two-qubit probes for estimating several phase rotations simultaneously under the action of different noisy channels. We compute the quantum limits for this simultaneous estimation using collective and individual measurements by evaluating the Holevo and Nagaoka-Hayashi Cramér-Rao bounds, respectively. Several quantum noise channels are considered, namely the decohering channel, the amplitude damping channel, and the phase damping channel. For each channel, we find the optimal single- and two-qubit probes. Where possible we demonstrate an explicit measurement strategy that saturates the appropriate bound and we investigate how closely the Holevo bound can be approached through collective measurements on multiple copies of the same probe. We find that under the action of the considered channels, two-qubit probes show enhanced parameter estimation capabilities over single-qubit probes for almost all non-identity channels, i.e., the achievable precision with a single-qubit probe degrades faster with increasing exposure to the noisy environment than that of the two-qubit probe. However, in sufficiently noisy channels, we show that it is possible for single-qubit probes to outperform maximally entangled two-qubit probes. This work shows that, in order to reach the ultimate precision limits allowed by quantum mechanics, entanglement is required in both the state preparation and state measurement stages. It is hoped the tutorial-esque nature of this paper will make it easily accessible.
Accurate phase estimation plays a pivotal role in quantum metrology, yet its precision is significantly affected by noise, particularly phase-diffusive noise caused by phase drift. To address this challenge, the joint estimation of phase and phase diffusion has emerged as an effective approach, transforming the problem into a multi-parameter estimation task. However, the incompatibility between optimal measurements for different parameters prevents single-copy measurements from reaching the fundamental precision limits defined by the quantum Cramer-Rao bound. Meanwhile, collective measurements performed on multiple identical copies can mitigate this incompatibility and thus enhance the precision of joint parameter estimation. This work experimentally demonstrates joint phase and phase-diffusion estimation using deterministic Bell measurements on a two-qubit system. A linear optical network is employed to implement both parameter encoding and deterministic Bell measurements, achieving improved estimation precision compared to any separable measurement strategy. This work proposes a new framework for phase estimation under phase-diffusive noise and underscores the substantial advantages of collective measurements in multi-parameter quantum metrology.
In multi-parameter quantum metrology, the resource of entanglement can lead to an increase in efficiency of the estimation process. Entanglement can be used in the state preparation stage, or the measurement stage, or both, to harness this advantage; here we focus on the role of entangling measurements. Specifically, entangling or collective measurements over multiple identical copies of a probe state are known to be superior to measuring each probe individually, but the extent of this improvement is an open problem. It is also known that such entangling measurements, though resource-intensive, are required to attain the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks. In this work we investigate the maximum precision improvement that collective quantum measurements can offer over individual measurements for estimating parameters of qudit states, calling this the 'collective quantum enhancement'. We show that, whereas the maximum enhancement can, in principle, be a factor of n for estimating n parameters, this bound is not tight for large n . Instead, our results prove an enhancement linear in dimension of the qudit is possible using collective measurements and lead us to conjecture that this is the maximum collective quantum enhancement in any local estimation scenario.
Entanglement is a fundamental feature of quantum mechanics and holds great promise for enhancing metrology and communications. Much of the focus of quantum metrology so far has been on generating highly entangled quantum states that offer better sensitivity, per resource, than what can be achieved classically. However, to reach the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks, collective measurements, which generate entanglement between multiple copies of the quantum state, are necessary. Here, we experimentally demonstrate theoretically optimal single- and two-copy collective measurements for simultaneously estimating two non-commuting qubit rotations. This allows us to implement quantum-enhanced sensing, for which the metrological gain persists for high levels of decoherence, and to draw fundamental insights about the interpretation of the uncertainty principle. We implement our optimal measurements on superconducting, trapped-ion and photonic systems, providing an indication of how future quantum-enhanced sensing networks may look.
Abstract Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of two parameters that are respectively associate with the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy—first, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values j , then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show that when the state is nearly pure, for sufficiently but not arbitrarily large N , most information will be captured in the largest three j -subspaces. Then, we study how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N = 2, 3, or approximately when the qubits are nearly pure for other N values. We also obtain numerical results that suggest that using a Bell multiport setup, one can distinguish between projection onto the j = N /2 and j = N /2 − 1 subspaces from their respective interference signatures at the output.
Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known, but is given by the solution of a non-trivial minimisation problem. We present a new expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.