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

Le résumé fourni par la source

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.

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