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arXiv:2607.22195

Maximal Fisher Budget Forces Blind Directions in Bipartite Collective SU(d) Metrology

Tariq Aziz, Naeem Akhtar, Dong Wang, Augusto Smerzi

quant-phSubmitted Jul 24, 2026• Announced Jul 27, 2026• 0 citations
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Abstract

For pure two-qudit probes used in a single fixed setting to estimate a common SU(d) transformation, maximal total Fisher sensitivity and full local identifiability are mutually exclusive. We derive exact identities that determine the trace of the quantum Fisher information matrix from two measurable probe properties: exchange symmetry and collective polarization. Every maximizer is exchange symmetric and maximally entangled. Its Fisher matrix has equal sensitivity in every visible direction, while $d(d-1)/2$ generator directions are blind. More generally, weak compatibility, defined by vanishing mean generator commutators, forces at least $\lfloor d/2\rfloor$ blind directions for every pure bipartite probe. A probe-defined antiunitary invariant classifies the complete Fisher spectrum at maximal entanglement, and approaches to the optimum that preserve exchange symmetry have a divergent Holevo cost. The bipartite obstruction is sharp in particle number: for $N\geq 3$, generalized GHZ probes can satisfy weak compatibility, attain the corresponding $N$-partite Fisher trace bound, and retain full local identifiability. Hence, Fisher sensitivity, weak compatibility, identifiability, and attainable precision are distinct resources in quantum metrology with noncommuting generators.

Comments
43 pages, 4 figures

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For pure two-qudit probes used in a single fixed setting to estimate a common SU(d) transformation, maximal total Fisher sensitivity and full local identifiability are mutually exclusive. We derive exact identities that determine the trace of the quantum Fisher information matrix from two measurable probe properties: exchange symmetry and collective polarization. Every maximizer is exchange symmetric and maximally entangled. Its Fisher matrix has equal sensitivity in every visible direction, while d(d−1)/2d(d-1)/2d(d−1)/2 generator directions are blind. More generally, weak compatibility, defined by vanishing mean generator commutators, forces at least ⌊d/2⌋\lfloor d/2\rfloor⌊d/2⌋ blind directions for every pure bipartite probe. A probe-defined antiunitary invariant classifies the complete Fisher spectrum at maximal entanglement, and approaches to the optimum that preserve exchange symmetry have a divergent Holevo cost. The bipartite obstruction is sharp in particle number: for N≥3N\geq 3N≥3, generalized GHZ probes can satisfy weak compatibility, attain the corresponding NNN-partite Fisher trace bound, and retain full local identifiability. Hence, Fisher sensitivity, weak compatibility, identifiability, and attainable precision are distinct resources in quantum metrology with noncommuting generators.
Achieving joint quantum-enhanced precision in multiparameter sensing requires both high sensitivity and measurement compatibility. These two aspects are characterized by the quantum Fisher information matrix (QFIM) and the Uhlmann curvature matrix (UCM), respectively, with weak compatibility corresponding to the vanishing of the relevant UCM elements. Here, we develop a symmetry-projection framework that classifies phase generators into subspace-preserving and subspace-changing sectors. For probe states confined to a symmetry subspace, symmetry projection imposes a common block-diagonal structure on the QFIM and UCM, rendering cross-sector parameters simultaneously free from information cross-talk and measurement incompatibility. When the subspace-changing generators act as scalars within the occupied subspace, the corresponding QFIM block reduces to four times the symmetrized covariance matrix, even for mixed probe states. For parity-protected collective SU(2)\mathrm{SU}(2)SU(2) systems, this structure singles out the transverse anti-squeezed quadrature and the longitudinal mean-spin direction as natural optimal sensing axes. Applied to a dissipative one-axis twisting model, the dynamically generated probe state exhibits identically vanishing UCM elements for transverse--longitudinal parameter pairs, while maintaining nearly balanced, Heisenberg-scaled QFIM components over a broad transient window. Our work opens a route to symmetry-protected, weakly compatible multiparameter sensing in interacting quantum many-body systems.
Fisher information appears in quantum theory in two mathematically distinct settings: as a classical probability functional in variational reconstructions of Schrodinger dynamics and as the quantum Fisher metric of parameterized density operators. Here we examine how far these roles can be connected without identifying the two objects. Starting from a Hamilton-Jacobi ensemble action, we recover the Fisher term that generates the quantum potential and isolate its contribution to the kinetic-energy expectation. We then analyze the quantum Fisher information matrix (QFIM) of bipartite states. For any pure two-qubit state, the optimized magnitude of the off-diagonal QFIM element for normalized local generators equals the concurrence, while a collective phase generator yields FQ = 4 C^2. Extending the optimized pure-state quantity by a convex roof gives an exact identity with Wootters concurrence for arbitrary mixed two-qubit states. This differs from evaluating the QFIM directly on a mixed density operator: for Werner states the raw optimized cross-QFIM is 2 p^2/(1 + p) and remains nonzero inside the separable region. The gap between the raw and convex-roof quantities therefore isolates, for this family, correlation geometry not captured by entanglement. Finally, in a two-path interferometer with a which-way marker, the optimal phase Fisher information available from local output statistics is I_phi^opt = V^2. These results place probability-gradient energy, bipartite correlation geometry, entanglement, and decoherence-induced loss of local phase sensitivity in a common mathematical setting while preserving the distinctions between classical Fisher information, QFIM geometry, Bell nonlocality, and global phase quantization.
Quantum metrology achieves Heisenberg-limited precision in single-parameter estimation, but its multiparameter extension is fundamentally constrained by both parameter-encoding and measurement incompatibility. Noncommuting signal generators may cause incompatible parameter-encoding, preventing the quantum Fisher information matrix from simultaneously achieving the Heisenberg scale for all parameters. Due to incompatible optimal measurements, the classical Fisher information matrix represents the practical attainable precision. Here, we introduce a multiparameter metrology framework based on indefinite evolution (IE), in which different control operations and signal reversal are placed in a coherent superposition. For a single-qubit probe with mutually orthogonal signal generators, IE enables compatible parameter encoding and optimal measurement without the signal reversal. For parallel generators, where only signal reversal realized by its generator is available, IE can achieve the same performance. We further extend this mechanism to noisy, many-body, and high-dimensional probes, and establish general conditions for achieving the simultaneous Heisenberg-limit. In contrast, definite evolution cannot achieve the same performance under compatible optimal measurements, even when signal reversal is available. Our results identify IE as an operational resource for overcoming multiparameter incompatibility and open a route toward attainable Heisenberg-limited sensing in interferometric platforms.
We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length ηηη. Every fixed-local architecture, including recorded classical randomization, heterogeneous single-copy measurements, and arbitrary outcome spaces, reduces exactly to a zero-barycenter probability measure on the Bloch ball. For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid: the unique optimal aggregate design is the spin-coherent Haar positive-operator-valued measure (POVM). For NNN copies, inverse-Fisher AAA loss has the exact value 2/[Nf(η)]2/[Nf(η)]2/[Nf(η)], where $ f(η)= \frac{2η-(1-η^2)\log[(1+η)/(1-η)]}{4η}$. This uniqueness has an immediate finite-readout consequence. No finite-support measurement attains the unrestricted mixed-state optimum, while at the smallest globally regular support the tetrahedral symmetric informationally complete (SIC) measurement is uniquely AAA- and DDD-minimax, with exact worst-direction values. Relaxing the fixed-readout constraint separates the asymptotic resources: one-way local operations and classical communication (LOCC), unrestricted LOCC, and separable measurements have AAA-loss coefficient 4/η24/η^24/η2, whereas collective measurements attain 2(1+η)/η22(1+η)/η^22(1+η)/η2. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.
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.
The choice of probe state is essential to achieving ultimate precision in quantum metrology. For a single parameter, an equal superposition of eigenstates associated with the largest and smallest generator eigenvalues is optimal. This simple construction does not directly extend to multiple parameters, since different parameters can favor different probes. We address this difficulty through a geometric construction of optimal probes for multiparameter estimation in distributed quantum sensing. In this setting, the generators commute and share a common eigenbasis, so each common eigenstate defines a vector of generator eigenvalues. We show that the minimal enclosing balls and ellipsoids of these vectors determine the maximal total quantum Fisher information and the minimum weighted covariance, respectively. In both cases, the boundary of the convex body identifies the common eigenstates that can contribute to an optimal probe, and its center and shape specify the conditions that their populations must satisfy. Our results provide a systematic approach to designing optimal probe states, linking the structure of the parameter generators directly to the attainable precision in multiparameter quantum metrology.