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arXiv:2409.16515

Quantum error correction-inspired multiparameter quantum metrology

Sivaprasad Omanakuttan, Jonathan A. Gross, Tyler J. Volkoff

Published Sep 25, 2024• 0 citations
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Abstract

We present a novel strategy for obtaining optimal probe states and measurement schemes in a class of noiseless multiparameter estimation problems with symmetry among the generators. The key to the framework is the introduction of a set of quantum metrology conditions, analogous to the quantum error correction conditions of Knill and Laflamme, which are utilized to identify probe states that saturate the multiparameter quantum Cramér-Rao bound. Similar to finding two-dimensional irreps for encoding a logical qubit in error correction, we identify trivial irreps of finite groups that guarantee the satisfaction of the quantum metrology conditions. To demonstrate our framework, we analyze the SU(2) estimation with symmetric states in which three parameters define a global rotation of an ensemble of $N$ qubits. For even $N$, we find that tetrahedral symmetry and, with fine-tuning, $S_{3}$ symmetry, are minimal symmetry groups providing optimal probe states for SU(2) estimation, but that the quantum metrology conditions can also be satisfied in an entanglement-assisted setting by using a maximally entangled state of two spin-$N/2$ representations for any $N$. By extending the multiparameter method of moments to non-commuting observables, we use the quantum metrology conditions to construct a measurement scheme that saturates the multiparameter quantum Cramér-Rao bound for small rotation angles.

DOI
10.48550/arxiv.2409.16515

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We present a novel strategy for obtaining optimal probe states and measurement schemes in a class of noiseless multiparameter estimation problems with symmetry among the generators. The key to the framework is the introduction of a set of quantum metrology conditions, analogous to the quantum error correction conditions of Knill and Laflamme, which are utilized to identify probe states that saturate the multiparameter quantum Cramér-Rao bound. Similar to finding two-dimensional irreps for encoding a logical qubit in error correction, we identify trivial irreps of finite groups that guarantee the satisfaction of the quantum metrology conditions. To demonstrate our framework, we analyze the SU(2) estimation with symmetric states in which three parameters define a global rotation of an ensemble of NNN qubits. For even NNN, we find that tetrahedral symmetry and, with fine-tuning, S3S_{3}S3​ symmetry, are minimal symmetry groups providing optimal probe states for SU(2) estimation, but that the quantum metrology conditions can also be satisfied in an entanglement-assisted setting by using a maximally entangled state of two spin-N/2N/2N/2 representations for any NNN. By extending the multiparameter method of moments to non-commuting observables, we use the quantum metrology conditions to construct a measurement scheme that saturates the multiparameter quantum Cramér-Rao bound for small rotation angles.
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.
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Quantum error correction (QEC) codes have emerged as a powerful tool to protect quantum-enhanced metrology against noise. However, the ability to correct errors alone does not guarantee high metrological sensitivity, as the encoded states may become insensitive to the parameter of interest. Here we show that this limitation can be overcome by exploiting an intrinsic freedom of QEC codes: for a fixed set of correctable errors, the Knill-Laflamme conditions admit an equivalence class of encodings. When the correctable noise possesses unitary symmetries, these symmetries generate continuous transformations within this class, allowing systematic optimization of the encoding to increase the quantum Fisher information while preserving the correctable set of noise. Based on this observation, we develop a symmetry-based optimization approach and derive criteria identifying when such optimization can enhance metrological sensitivity. In particular, for stabilizer-sum Hamiltonians, symmetry optimization can convert a code with vanishing QFI into one achieving the standard quantum limit in general or even Heisenberg scaling in specific cases, illustrating the power of symmetry optimization for QEC-assisted quantum metrology.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
Estimating multiple unknown parameters simultaneously is essential for practical quantum sensing. However, it faces a fundamental challenge: the optimal strategy for estimating one parameter is often incompatible with that for another, making it impossible to simultaneously achieve the ultimate precision limits for all parameters. Here we develop a general and efficient computational framework that jointly optimizes probe states, control operations, and measurements across different strategy families, including parallel, sequential, and those with indefinite causal order. Our approach provides exact semidefinite-program formulations for several precision bounds, including the Holevo, Nagaoka-Hayashi, and quantum Cramér-Rao bounds. We demonstrate the capabilities of the framework in multiparameter magnetometry and frequency estimation, identifying optimal protocols within each class and revealing a strict hierarchy among the achievable performances of different classes in the multiparameter regime. The framework also directly incorporates resource constraints, such as energy budgets, enabling systematic investigation of experimentally realistic sensing scenarios. Furthermore, we develop a finite-memory optimization method for sequential strategies with restricted ancillary-memory dimension. By decomposing the protocol into initial probe preparation and intermediate control operations, this method provides a practical route to designing resource-constrained sequential sensing schemes. Our work establishes a versatile computational tool for determining fundamental precision limits and designing optimal quantum-sensing protocols in complex multiparameter settings.
Determining when the multiparameter quantum Cramér--Rao bound (QCRB) is saturable with experimentally relevant single-copy measurements is a central open problem in quantum metrology. Here we establish an equivalence between QCRB saturation and the simultaneous hollowization of a set of traceless operators associated with the estimation model, i.e., the existence of complete (generally nonorthogonal) bases in which all corresponding diagonal matrix elements vanish. This formulation yields a geometric characterization: optimal rank-one measurement vectors are confined to a subspace orthogonal to a state-determined Hermitian span. This provides a direct criterion to construct optimal Positive Operator-Valued Measures(POVMs). We then identify conditions under which the partial commutativity condition proposed in [Phys. Rev. A 100, 032104(2019)] becomes necessary and sufficient for the saturation of the QCRB, demonstrate that this condition is not always sufficient, and prove the counter-intuitive uselessness of informationally-complete POVMs.