Greenberger-Horne-Zeilinger state is the best probe for multiparameter estimation of independent local fields

• URL: http://arxiv.org/abs/2407.20142v2
• Date: Thu, 26 Sep 2024 16:02:12 GMT
• Title: Greenberger-Horne-Zeilinger state is the best probe for multiparameter estimation of independent local fields
• Authors: Aparajita Bhattacharyya, Ujjwal Sen,
• Abstract summary: We show that estimation of independent multiple field strengths of a local Hamiltonian requires the utility of a genuine multiparty entangled state. We prove that using a probe that is in any mixed state provides a precision lower than that for the GHZ. To emphasize the importance of the weight matrix considered, we show that choosing the identity operator as the same leads to a product probe attaining the best precision.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Estimation of local quantum fields is a crucial aspect of quantum metrology applications, and often also forms the test-bed to analyze the utility of quantum resources, like entanglement. However, so far, this has been analyzed using the same local field for all the probes, and so, although the encoding process utilizes a local Hamiltonian, there is an inherent ``nonlocality" in the encoding process in the form of a common local field applied on all the probes. We show that estimation of even independent multiple field strengths of a local Hamiltonian, i.e., one formed by a sum of single-party terms, necessitates the utility of a highly symmetric genuine multiparty entangled state, viz. the Greenberger-Horne-Zeilinger (GHZ) state, as the input probe. The feature depends on the choice of the weight matrix considered, to define a figure of merit in the multiparameter estimation. We obtain this result by providing a lower bound on the precision of multiparameter estimation, optimized over input probes, for an arbitrary weight matrix. We find that the bound can be expressed in terms of eigenvalues of the arbitrary multiparty local encoding Hamiltonian. We show that there exists a weight matrix for which this bound is attainable only when the probe is the GHZ state, up to a relative phase. In particular, no pure product state can achieve this lower bound, and the gap in precision with genuinely multiparty entangled states acting as probes increases with increasing number of parties. We finally prove that using a probe that is in any mixed state provides a precision lower than that for the GHZ. To emphasize the importance of the weight matrix considered, we show that choosing the identity operator as the same - thereby ignoring the ``off-diagonal'' covariances in the precision matrix - leads to a product probe attaining the best precision.

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