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arXiv:2407.20142

Precision in estimating independent local fields: attainable bound and indispensability of genuine multiparty entanglement

Aparajita Bhattacharyya, Ujjwal Sen

Published Jul 29, 2024• 0 citations
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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 genuine multipartite entangled state as the input probe. The feature depends on the choice of the weight matrix considered, which is full-rank and contains non-vanishing off-diagonal terms. We obtain this result by providing a lower bound on the precision of multiparameter estimation, optimized over input probes, for an arbitrary positive semi-definite weight matrix. We show that there exists a weight matrix for which this bound is always attainable by the Greenberger-Horne-Zeilinger (GHZ) state, chosen in a certain basis. Furthermore, we find the parametric form of the most general optimal state for three parties. We also show that no pure product state can achieve the lower bound. Finally, for an arbitrary weight matrix and an arbitrary multiparty local encoding Hamiltonian, we prove that using a probe that is in any mixed state provides a precision lower than that obtainable using pure states. To emphasize the importance of the weight matrix considered, we also prove that the choice of identity operator as the same - thereby ignoring the "off-diagonal" covariances in the precision matrix - does not require the use of genuine multiparty entanglement in input probes for attaining the best precision, and the optimal probe can be a pure product.

DOI
10.48550/arxiv.2407.20142

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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 genuine multipartite entangled state as the input probe. The feature depends on the choice of the weight matrix considered, which is full-rank and contains non-vanishing off-diagonal terms. We obtain this result by providing a lower bound on the precision of multiparameter estimation, optimized over input probes, for an arbitrary positive semi-definite weight matrix. We show that there exists a weight matrix for which this bound is always attainable by the Greenberger-Horne-Zeilinger (GHZ) state, chosen in a certain basis. Furthermore, we find the parametric form of the most general optimal state for three parties. We also show that no pure product state can achieve the lower bound. Finally, for an arbitrary weight matrix and an arbitrary multiparty local encoding Hamiltonian, we prove that using a probe that is in any mixed state provides a precision lower than that obtainable using pure states. To emphasize the importance of the weight matrix considered, we also prove that the choice of identity operator as the same - thereby ignoring the "off-diagonal" covariances in the precision matrix - does not require the use of genuine multiparty entanglement in input probes for attaining the best precision, and the optimal probe can be a pure product.
We ask whether the optimal probe is entangled, and if so, what is its character and amount, for estimating the noise parameter of a large class of local quantum encoding processes that we refer to as vector encoding, examples of which include the local depolarizing and bit-flip channels. We first establish that vector encoding is invariably ``continuously commutative'' for optimal probes. We utilize this result to deal with the queries about entanglement in the optimal probe. We show that for estimating noise extent of the two-party arbitrary-dimensional local depolarizing channel, there is a descending staircase of optimal-probe entanglement for increasing depolarizing strength. For the multi-qubit case, the analysis again leads to a staircase, but which can now be monotonic or not, depending on the multiparty entanglement measure used. We also find that when sufficiently high depolarizing noise is to be estimated, fully product multiparty states are the only choice for being optimal probes. In many cases, for even moderately high depolarizing noise, fully product states are optimal. For two-qubit local bit-flip channels, the continuous commutativity of the channel and optimal probe implies that a product state suffices for obtaining the optimal precision.
Quantum multiparameter estimation offers a framework for the simultaneous estimation of multiple parameters, pertaining to possibly noncommutating observables. While the optimal probe for estimating a single unitary phase is well understood - being a pure state that is an equal superposition of the eigenvectors of the encoding Hamiltonian corresponding to its maximum and minimum eigenvalues - the structure of optimal probes in the multiparameter setting remains more intricate. We investigate the simultaneous estimation of two phases, each encoded through arbitrary qubit Hamiltonians, using arbitrary weight matrices, and considering single-qubit probes. We also consider single-qutrit probes, for which the encoding Hamiltonians are chosen as SU(2) generators. We find that in both the qubit and qutrit scenarios, the optimal probe is a coherent superposition of the eigenstates corresponding to the largest and smallest eigenvalues of the total encoding Hamiltonian, with a fixed - and not arbitrary - relative phase. Remarkably, this optimal probe is independent of the specific choice of weight matrix, making it a universally optimal input state for the estimation of any pair of SU(2) parameters, which can be reparameterized to the phase estimation problem. Furthermore, we show that this probe also maximizes the determinant of the quantum Fisher information matrix, providing a handy tool for identifying the optimal probe state. We also examine the role of commutativity between the generators of the unitary encodings. Our results demonstrate that a high degree of commutativity degrades the achievable precision, rendering estimation infeasible in the extreme case. However, maximal noncommutativity does not necessarily yield optimal precision.
In a recent paper, Bancal et al. [Phys. Rev. Lett. 106, 250404 (2011)] put forward the concept of device-independent witnesses of genuine multipartite entanglement. These witnesses are capable of verifying genuine multipartite entanglement produced in a laboratory without resorting to any knowledge of the dimension of the state space or of the specific form of the measurement operators. As a by-product they found a multiparty three-setting Bell inequality which makes it possible to detect genuine nnn-partite entanglement in a noisy nnn-qubit Greenberger-Horne-Zeilinger (GHZ) state for visibilities as low as 2/32/32/3 in a device-independent way. In this paper, we generalize this inequality to an arbitrary number of settings, demonstrating a threshold visibility of $2/\ensuremath{\pi}~0.6366$ for number of settings going to infinity. We also present a pseudotelepathy Bell inequality achieving the same threshold value. We argue that our device-independent witnesses are optimal in the sense that for nnn odd the above value cannot be beaten with nnn-party-correlation Bell inequalities.
Abstract Quantum multiparameter estimation involves estimating multiple parameters simultaneously and can be more precise than estimating them individually. Our interest here is to determine fundamental quantum limits to the achievable multiparameter estimation precision in the presence of noise. We first present a lower bound to the estimation error covariance for a noisy initial probe state evolving through a noiseless quantum channel. We then present a lower bound to the estimation error covariance in the most general form for a noisy initial probe state evolving through a noisy quantum channel. We show conditions and accordingly measurements to attain these estimation precision limits for noisy systems. We see that the Heisenberg precision scaling of 1/N can be achieved with a probe comprising N particles even in the presence of noise. In fact, some noise in the initial probe state or the quantum channel can serve as a feature rather than a bug, since the estimation precision scaling achievable in the presence of noise in the initial state or the channel in some situations is impossible in the absence of noise in the initial state or the channel. However, a lot of noise harms the quantum advantage achievable with N parallel resources, and allows for a best precision scaling of 1/N{\bf{1}}/\sqrt{{\boldsymbol{N}}}1/N​ 1 / N . Moreover, the Heisenberg precision limit can be beaten with noise in the channel, and we present a super-Heisenberg precision limit with scaling of 1/N2 for optimal amount of noise in the channel, characterized by one-particle evolution operators. Furthermore, using γ-particle evolution operators for the noisy channel, where γ > 1, the best precision scaling attainable is 1/N2γ, which is otherwise known to be only possible using 2γ-particle evolution operators for a noiseless channel.
Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cramér-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.
Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.