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Characterizing resources for multiparameter estimation of SU(2) and SU(1,1) unitaries

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Researchers identified optimal quantum states for estimating three non-commuting parameters in SU(2) and SU(1,1) systems, achieving Heisenberg-limited precision scaling simultaneously for all parameters using eigenstates of Jz² and balanced particle-number states. In practical scenarios with measurements limited to first/second moments of generators, twin-Fock states emerged as the only resource enabling Heisenberg scaling for two of three parameters in both SU(2) and SU(1,1) cases. For states with fluctuating particle numbers and quadratic measurements, simultaneous Heisenberg scaling becomes largely impossible except for two-mode squeezed states estimating two-parameter SU(2) transformations. The work bridges theoretical bounds and experimental feasibility, showing that while ideal states allow perfect scaling, practical measurement constraints severely restrict achievable precision in multiparameter estimation. This extends prior findings that SU(1,1) operations can enhance SU(2) interferometry, providing a unified framework for quantum-enhanced sensing with non-commuting observables.

Characterizing resources for multiparameter estimation of SU(2) and SU(1,1) unitaries

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AbstractWe analyze the task of estimating a multi-parameter unitary belonging to the $SU(2)$ or $SU(1,1)$ groups, in a two-bosonic-mode scenario and investigate the scaling of the precision in terms of the total particle number. For the $SU(2)$ case, the total particle number is conserved by the evolution and we discuss optimal states in fixed-$n$ subspaces, identifying eigenstates of $J_z^2$ as useful resources, even allowing simultaneous Heisenberg precision scaling for all three parameters. In the $SU(1,1)$ case instead, the conserved quantity is the particle number difference between the two modes, and we identify useful probe states in the sector with an equal number of particles in the two modes. These states are analogous to the $SU(2)$ case and would also allow simultaneous Heisenberg precision scaling for all three parameters. We then consider the more pragmatic scenario of an estimation via expectation values of time-evolved observables, which we restrict to be the first two moments of the generators. We analyze the maximal precision achievable in this setting and we find that the twin-Fock state emerges in both the $SU(2)$ and the $SU(1,1)$ cases as the only one potentially allowing Heisenberg scaling for the estimation of two out of the three parameters. As a complement, we also consider other probe states with fluctuating number of particles, with measurements restricted to quadratic expressions in the mode operators. In this scenario, simultaneous Heisenberg scaling in multiple parameters seems mostly forbidden, with the only exception being an input two-mode squeezed state for the estimation of a two-parameter $SU(2)$. This extends to the multiparameter scenario the well-established intuition that the performance of a $SU(2)$ interferometer can be enhanced by a prior $SU(1,1)$ operation.Featured image: Three-parameter $SU(2)$ estimation as an example. An input two-mode twin Fock state $\left|\Psi_{\mathbf{0}}\right\rangle$ undergoes a $SU(2)$ rotation, evolving into the state $U(\boldsymbol{\theta})\left|\Psi_{\mathbf{0}}\right\rangle$, and then detected at two output ports.Popular summaryQuantum technologies promise sensors that can measure physical quantities with a precision beyond what is possible with ordinary classical resources. A central example is quantum interferometry, where particles such as photons or atoms are used to estimate phases. With uncorrelated particles, the precision improves only with the square root of the number of particles used. Quantum theory, however, allows more strongly correlated probe states that can in principle reach the so-called Heisenberg scaling, where the improvement is proportional to the total number of particles. Similar enhancements are known to be possible also when several parameters are estimated at the same time, but multiparameter estimation brings an additional difficulty: the formal quantum limits are not always attainable in practice, especially when the different parameters are generated by non-commuting observables. In this work, we study this question in an abstract but widely applicable setting. We consider multiparameter quantum estimation problems in which the unknown transformation is generated by three non-commuting observables. This situation appears, for instance, when estimating the three angles of a rotation, or the parameters of optical transformations related to squeezing. Mathematically, these two paradigmatic cases are described by the groups SU(2) and SU(1,1), which play a central role in quantum optics, interferometry, and many-body physics. Our first goal is to identify the ultimate quantum limits: which probe states, in principle, allow simultaneous Heisenberg scaling for all three parameters? We show that such states can indeed be characterized in both the SU(2) and SU(1,1) settings, and we highlight a close analogy between the two cases. We then move to a more practical scenario, where the information is extracted from expectation values of experimentally natural observables, namely first and second moments of the relevant generators. In this restricted setting, the situation becomes much more selective: only special states, such as twin-Fock states, retain the possibility of Heisenberg scaling for more than one parameter. Overall, our results clarify which quantum resources are useful for enhanced multiparameter estimation when the parameters are generated by non-commuting transformations. They also distinguish between formal precision bounds and precision that can actually be achieved with concrete measurements, thereby identifying when simultaneous Heisenberg scaling is genuinely attainable.► BibTeX data@article{Du2026characterizing, doi = {10.22331/q-2026-06-08-2130}, url = {https://doi.org/10.22331/q-2026-06-08-2130}, title = {Characterizing resources for multiparameter estimation of {SU}(2) and {SU}(1,1) unitaries}, author = {Du, Shaowei and Liu, Shuheng and Steinhoff, Frank E. S. and Vitagliano, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2130}, month = jun, year = {2026} }► References [1] Gregg Jaeger, Abner Shimony, and Lev Vaidman. ``Two interferometric complementarities''. Phys. Rev. A 51, 54 (1995). https://doi.org/10.1103/PhysRevA.51.54 [2] Berthold-Georg Englert. ``Fringe visibility and which-way information: An inequality''. Phys. Rev. Lett. 77, 2154 (1996). https://doi.org/10.1103/PhysRevLett.77.2154 [3] Patrick J Coles, Jedrzej Kaniewski, and Stephanie Wehner. ``Equivalence of wave–particle duality to entropic uncertainty''. Nat. Commun. 5, 5814 (2014). https://doi.org/10.1038/ncomms6814 [4] Frank E. S. Steinhoff and Marcos César de Oliveira. ``Limitations imposed by complementarity''. Quantum Inf. Process. 19, 1–17 (2020). https://doi.org/10.1007/s11128-020-02869-1 [5] R. Hanbury Brown and Richard Q. 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Journal of Physics A: Mathematical and Theoretical 50, 323001 (2017). https://doi.org/10.1088/1751-8121/50/32/323001Cited by[1] Giuseppe Vitagliano, Otfried Gühne, and Géza Tóth, "su(d)-squeezing and many-body entanglement geometry in finite-dimensional systems", Quantum 9, 1844 (2025). [2] F. E. S. Steinhoff, "Implementation and representation of qudit multi-controlled unitaries and hypergraph states by N-body angular momentum couplings", arXiv:2506.21831, (2025). [3] F. E. S. Steinhoff, "Qutrit Clifford+T gates by two-body angular momentum couplings, rotations and one-axis-twistings", arXiv:2604.23007, (2026). [4] T. J. Volkoff, "Relative phase and dynamical phase sensing in a Hamiltonian model of the optical SU(1,1) interferometer", arXiv:2505.15635, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-06-08 14:35:46). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-08 14:35:45: Could not fetch cited-by data for 10.22331/q-2026-06-08-2130 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe analyze the task of estimating a multi-parameter unitary belonging to the $SU(2)$ or $SU(1,1)$ groups, in a two-bosonic-mode scenario and investigate the scaling of the precision in terms of the total particle number. For the $SU(2)$ case, the total particle number is conserved by the evolution and we discuss optimal states in fixed-$n$ subspaces, identifying eigenstates of $J_z^2$ as useful resources, even allowing simultaneous Heisenberg precision scaling for all three parameters. In the $SU(1,1)$ case instead, the conserved quantity is the particle number difference between the two modes, and we identify useful probe states in the sector with an equal number of particles in the two modes. These states are analogous to the $SU(2)$ case and would also allow simultaneous Heisenberg precision scaling for all three parameters. We then consider the more pragmatic scenario of an estimation via expectation values of time-evolved observables, which we restrict to be the first two moments of the generators. We analyze the maximal precision achievable in this setting and we find that the twin-Fock state emerges in both the $SU(2)$ and the $SU(1,1)$ cases as the only one potentially allowing Heisenberg scaling for the estimation of two out of the three parameters. As a complement, we also consider other probe states with fluctuating number of particles, with measurements restricted to quadratic expressions in the mode operators. In this scenario, simultaneous Heisenberg scaling in multiple parameters seems mostly forbidden, with the only exception being an input two-mode squeezed state for the estimation of a two-parameter $SU(2)$. This extends to the multiparameter scenario the well-established intuition that the performance of a $SU(2)$ interferometer can be enhanced by a prior $SU(1,1)$ operation.Featured image: Three-parameter $SU(2)$ estimation as an example. An input two-mode twin Fock state $\left|\Psi_{\mathbf{0}}\right\rangle$ undergoes a $SU(2)$ rotation, evolving into the state $U(\boldsymbol{\theta})\left|\Psi_{\mathbf{0}}\right\rangle$, and then detected at two output ports.Popular summaryQuantum technologies promise sensors that can measure physical quantities with a precision beyond what is possible with ordinary classical resources. A central example is quantum interferometry, where particles such as photons or atoms are used to estimate phases. With uncorrelated particles, the precision improves only with the square root of the number of particles used. Quantum theory, however, allows more strongly correlated probe states that can in principle reach the so-called Heisenberg scaling, where the improvement is proportional to the total number of particles. Similar enhancements are known to be possible also when several parameters are estimated at the same time, but multiparameter estimation brings an additional difficulty: the formal quantum limits are not always attainable in practice, especially when the different parameters are generated by non-commuting observables. In this work, we study this question in an abstract but widely applicable setting. We consider multiparameter quantum estimation problems in which the unknown transformation is generated by three non-commuting observables. This situation appears, for instance, when estimating the three angles of a rotation, or the parameters of optical transformations related to squeezing. Mathematically, these two paradigmatic cases are described by the groups SU(2) and SU(1,1), which play a central role in quantum optics, interferometry, and many-body physics. Our first goal is to identify the ultimate quantum limits: which probe states, in principle, allow simultaneous Heisenberg scaling for all three parameters? We show that such states can indeed be characterized in both the SU(2) and SU(1,1) settings, and we highlight a close analogy between the two cases. We then move to a more practical scenario, where the information is extracted from expectation values of experimentally natural observables, namely first and second moments of the relevant generators. In this restricted setting, the situation becomes much more selective: only special states, such as twin-Fock states, retain the possibility of Heisenberg scaling for more than one parameter. Overall, our results clarify which quantum resources are useful for enhanced multiparameter estimation when the parameters are generated by non-commuting transformations. They also distinguish between formal precision bounds and precision that can actually be achieved with concrete measurements, thereby identifying when simultaneous Heisenberg scaling is genuinely attainable.► BibTeX data@article{Du2026characterizing, doi = {10.22331/q-2026-06-08-2130}, url = {https://doi.org/10.22331/q-2026-06-08-2130}, title = {Characterizing resources for multiparameter estimation of {SU}(2) and {SU}(1,1) unitaries}, author = {Du, Shaowei and Liu, Shuheng and Steinhoff, Frank E. S. and Vitagliano, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2130}, month = jun, year = {2026} }► References [1] Gregg Jaeger, Abner Shimony, and Lev Vaidman. ``Two interferometric complementarities''. Phys. Rev. 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Characterizing resources for multiparameter estimation of SU(2) and SU(1,1) unitaries

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