Improving sum uncertainty relations with the quantum Fisher information

Abstract

We show how preparation uncertainty relations that are formulated as sums of variances may be tightened by using the quantum Fisher information to quantify quantum fluctuations. We apply this to derive stronger angular momentum uncertainty relations, which in the case of spin-$1/2$ turn into equalities involving the purity. Using an analogy between pure-state decompositions in the Bloch sphere and the moment of inertia of rigid bodies, we identify optimal decompositions that achieve the convex- and concave-roof decomposition of the variance. Finally, we illustrate how these results may be used to identify the classical and quantum limits on phase estimation precision with an unknown rotation axis.

11 pages, 5 figures. See also the related work by G. Tóth and F. Fröwis, Phys. Rev. Research 4, 013075 (2022)

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• Entanglement detection
• Reference frames, superselection rules, and quantum information
• Quantum metrology from a quantum information science perspective
• Fisher information under decoherence in Bloch representation
• Stronger uncertainty relations for the sum of variances
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• Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices
• Hierarchies of Frequentist Bounds for Quantum Metrology: From Cramér-Rao to Barankin
• Quantum Wasserstein distance based on an optimization over separable states
• Entanglement Criteria Based on Quantum Fisher Information
• Alternatives of entanglement depth and metrological entanglement criteria
• Standard symmetrized variance with applications to coherence, uncertainty and entanglement
• Wigner-Yanase skew information-based uncertainty relations for quantum channels
• Uncertainty relations based on state-dependent norm of commutator
• Uncertainty from the Aharonov-Vaidman Identity
• Parameterized Multi-observable Sum Uncertainty Relations
• Uncertainty relations between quantum Fisher information and entanglement monotones
• Uncertainty relation and the constrained quadratic programming