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Bipartite time-energy uncertainty relation for quantum metrology with noise



Philippe Faist, Mischa Woods, Victor Albert, Joseph Renes, Jens Eisert, John Preskill


Noise in quantum metrology reduces the sensitivity to which one can determine an unknown parameter in the evolution of a quantum state, such as time. Here, we consider a system prepared in a pure state that evolves according to a given Hamiltonian. We study the resulting local sensitivity to time, measured in terms of the quantum Fisher information, of the system after the application of a given noise channel. We show that the decrease in the quantum Fisher information due to the noise is equal to the quantum Fisher information that the environment gains with respect to the energy of the noiseless system. We generalize this Fisher information tradeoff to any two parameters whose evolutions are generated by arbitrary Hermitian operators. In this case, the tradeoff is quantified in terms of the commutator of the associated generators. Upper bounds on the quantum Fisher information of the noisy system are obtained via our tradeoff relation, by applying known sensitivity lower bounds on the environment's system. We discuss conditions under which our setting can model a quantum system subject to continuous noise. We furthermore obtain necessary and sufficient conditions for when the system does not suffer any sensitivity loss; these conditions are analogous to, but weaker than, the Knill-Laflamme quantum error correction conditions. We find states for many-body systems with Ising and Heisenberg interactions on any graph that do not lose any sensitivity to time after any single located error. For a one-dimensional spin chain with nearest-neighbor Ising interactions subject to amplitude damping noise on each site, we verify numerically that in our setting the state does not lose any sensitivity to first order in the noise parameter.
Physics Arxiv


Faist, P. , Woods, M. , Albert, V. , Renes, J. , Eisert, J. and Preskill, J. (2023), Bipartite time-energy uncertainty relation for quantum metrology with noise, Physics Arxiv, [online],, (Accessed May 26, 2024)


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Created December 5, 2023, Updated April 11, 2024