Researchers Unveil Quantum Sensor Network’s Optimal Function Estimation, Enhancing Quantum Sensing

A team of researchers from the Joint Center for Quantum Information and Computer Science, Joint Quantum Institute, and the Department of Physics at MIT have made significant strides in the field of quantum sensing. They have resolved open questions related to Mach-Zehnder interferometers and quadrature displacement sensing, deriving lower bounds on the achievable mean square error in estimating a linear function of either local phase shifts or quadrature displacements. The team also provided optimal protocols for achieving these bounds and proved necessary conditions for the amount of entanglement needed for any optimal protocol. This research could have far-reaching implications for quantum computing and communication.

What is the Optimal Function Estimation with Photonic Quantum Sensor Networks?

The study, conducted by Jacob Bringewatt, Adam Ehrenberg, Tarushii Goel, and Alexey V Gorshkov from the Joint Center for Quantum Information and Computer Science, Joint Quantum Institute, and the Department of Physics at the Massachusetts Institute of Technology, focuses on the optimal measurement of an analytic function of unknown local parameters each linearly coupled to a qubit sensor. This problem is well understood with applications ranging from field interpolation to noise characterization.

The researchers have resolved a number of open questions that arise when extending this framework to Mach-Zehnder interferometers and quadrature displacement sensing. They have derived lower bounds on the achievable mean square error in estimating a linear function of either local phase shifts or quadrature displacements. In the case of local phase shifts, these results prove and somewhat generalize a conjecture by Proctor et al. For quadrature displacements, they extend proofs of lower bounds to the case of arbitrary linear functions.

The team provides optimal protocols achieving these bounds up to small multiplicative constants and describes an algebraic approach to deriving new optimal protocols possibly subject to additional constraints. Using this approach, they prove necessary conditions for the amount of entanglement needed for any optimal protocol for both local phase and displacement sensing.

How Does Quantum Metrology Work?

In quantum metrology, entangled states of quantum sensors are used to try to obtain a performance advantage in estimating an unknown parameter or parameters, such as field amplitudes coupled to the sensors. In addition to this practical advantage of quantum sensing, the theory of the ultimate performance limits for parameter estimation tasks is deeply related to a number of topics of theoretical interest in quantum information science such as resource theories, the geometry of quantum state space, quantum speed limits, and quantum control theory.

Initial experimental and theoretical work on quantum sensing focused on optimizing the estimation of a single unknown parameter. More recently, the problem of distributed quantum sensing has become an area of particular interest. Here one considers a network of quantum sensors, each coupled to a local unknown parameter. The prototypical task in this setting is to measure some function or functions of these parameters.

For qubit sensors, the asymptotic limits on performance for these function estimation tasks are rigorously understood and techniques for generating optimal protocols subject to various constraints such as limited entanglement between sensors are known. However, despite extensive theoretical and experimental research on distributed quantum sensing for photonic quantum sensors, the asymptotic performance limits for function estimation are not yet rigorously established.

What are the Performance Limits for Function Estimation?

The researchers have closed this gap, proving an ultimate bound on performance as measured by the mean square error of the estimator for measuring a linear function of unknown parameters each coupled to a different photonic mode via either the number operator or a field quadrature operator chosen without loss of generality to be the momentum quadrature.

That is, they are interested in determining a function of either unknown local phase shifts or unknown quadrature displacements. For case one, their primary focus, they derive this bound subject to a strict constraint on photon number, proving a longstanding conjecture appearing in Ref 8. In case two, they derive their bound subject to a constraint on the average photon number, which is more natural in this setting as quadrature displacements are not photon number conserving.

Their results are consistent with existing bounds in the literature, but for completeness, they include derivations in this setting using an equivalent mathematical framework to the number operator case and the qubit sensor case. This allows for a natural comparison of the various performance limits and resource requirements of function estimation in quantum sensor networks and opens the door to designing new information-theoretically optimal protocols.

How is the Problem Set Up?

The problem is set up by considering a sensor network of d optical modes, each coupled to an unknown parameter θj for j = 1, d via a local coupling Hamiltonian and boldface denotes vectors. Here they consider the following two cases: where the local coupling Hamiltonian is the number operator acting on mode j and the momentum quadrature on mode j.

The choice of quadrature is arbitrary. All results apply equally well for coupling to any quadrature. The θ-independent time-dependent Hamiltonian is a control Hamiltonian, possibly including coupling to an arbitrary number of ancilla modes. Here s = 0, t where t is the total sensing time. In either case, their task is to measure a linear function of the unknown parameters.

What are the Implications of this Research?

This research has significant implications for the field of quantum sensing. By proving an ultimate bound on performance for function estimation tasks, the researchers have provided a valuable tool for future studies in this area. Their work also opens the door to the design of new information-theoretically optimal protocols, which could further enhance the performance of quantum sensor networks.

Furthermore, their research has deepened our understanding of the theoretical underpinnings of quantum sensing, shedding light on topics such as resource theories, the geometry of quantum state space, quantum speed limits, and quantum control theory. This could have far-reaching implications for a range of fields, from quantum computing to quantum communication and beyond.

In conclusion, this study represents a significant step forward in our understanding of quantum sensing and its potential applications. By rigorously establishing the performance limits for function estimation tasks and providing a framework for the design of optimal protocols, the researchers have laid a solid foundation for future advancements in this exciting field.