Efficient computation of the Nagaoka--Hayashi bound for multi-parameter estimation with separable measurements (2008.02612v2)

Abstract: Finding the optimal attainable precisions in quantum multiparameter metrology is a non trivial problem. One approach to tackling this problem involves the computation of bounds which impose limits on how accurately we can estimate certain physical quantities. One such bound is the Holevo Cramer Rao bound on the trace of the mean squared error matrix. The Holevo bound is an asymptotically achievable bound when one allows for any measurement strategy, including collective measurements on many copies of the probe. In this work we introduce a tighter bound for estimating multiple parameters simultaneously when performing separable measurements on finite copies of the probe. This makes it more relevant in terms of experimental accessibility. We show that this bound can be efficiently computed by casting it as a semidefinite program. We illustrate our bound with several examples of collective measurements on finite copies of the probe. These results have implications for the necessary requirements to saturate the Holevo bound.

• The paper introduces a semidefinite programming reformulation for the Nagaoka–Hayashi bound, achieving tighter precision limits for separable measurements.
• It compares performance on quantum systems, demonstrating that finite-copy separable measurements can approach the precision of collective strategies in realistic cases.
• The method offers a practical tool for experimental quantum metrology, guiding optimal measurement strategies in high-dimensional and complex scenarios.

Efficient Computation of the Nagaoka–Hayashi Bound for Multi-Parameter Estimation with Separable Measurements

This paper addresses a significant problem in quantum multiparameter metrology: determining the optimal attainable precisions when restrictively employing separable measurements instead of collective ones. The paper introduces the Nagaoka–Hayashi bound, which extends the Nagaoka bound to situations involving more than two parameters, making it applicable for separable measurements on finite copies of quantum probes. This is in contrast to the Holevo Cramér–Rao bound, typically only achievable asymptotically with collective measurements on infinitely many copies.

Key Contributions and Results

1. Nagaoka–Hayashi Bound:
   • Provides a methodology to calculate a tighter precision bound than the Holevo bound in scenarios where only separable measurements are feasible.
   • The bound applies to finite copies of the quantum probe, thus being experimentally more attainable than the collective measurements required for the Holevo bound.
2. Semidefinite Programming Approach:
   • The authors reformulate the minimization process of the Nagaoka–Hayashi bound as a semidefinite program (SDP). This reformulation enables efficient computation, enhancing practical applicability.
   • The worst-case computational complexity of the SDP indicates feasibility for high-dimensional problems, showing favorable performance compared to typical expectations.
3. Comparative Analysis:
   • The research features a comparative analysis on specific quantum systems:
     • Quantum Metrology for Qubit Rotations: In a context involving a two-qubit state under phase damping, the authors demonstrate the gap between separable measurement precision and the Holevo bound. They illustrate the ability of the Nagaoka–Hayashi bound to highlight cases where collective measurement's additional complexity is necessary.
     • Interferometry: The Nagaoka–Hayashi bound coincides with the Holevo bound when estimating phase and transmissivity in interferometers, revealing that separable measurements are as optimal as collective ones in some realistic scenarios.

Implications and Future Directions

• Experimental Quantum Metrology: This framework allows experimentalists to efficiently gauge the necessary measurement strategies and resources, providing insight into whether the complexities of collective measurements are warranted.
• Quantum Multi-Parameter Estimation: The generalizability facilitated by the SDP suggests broad utility across quantum technologies, potentially extending to quantum imaging, sensing, and communications.
• Nuisance Parameters and Robust Estimation: While current results assume known noise parameters (e.g., phase damping), challenges remain in estimating quantum systems entangled with nuisance parameters. The methodologies discussed could inform strategies in developing bounds and protocols resilient to such uncertainties.

Lastly, by providing an accessible computational tool, this paper sets the groundwork for further exploration in quantum measurement theory, with the potential to refine and tailor quantum systems measurement strategies. Whether the Nagaoka–Hayashi bound consistently holds across all scenarios or if separable measurements universally suffice remains an open question for future inquiry. This work provides significant steps towards a comprehensive understanding of quantum measurement optimality in realistic settings.