Seminar: Philippe Faist (Freie Universität Berlin): "Time-energy uncertainty relation and quantum error correction for noisy quantum metrology"

Date: 

Tuesday, October 25, 2022, 9:30am

Location: 

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09
Philippe Faist
Speaker:
  Philippe Faist (Freie Universität Berlin)
Title:  Time-energy uncertainty relation and quantum error correction for noisy quantum metrology
Abstract:  Detection of weak forces and precise measurement of time are two of the many applications of quantum metrology to science and technology. We consider a quantum system initialized in a pure state and whose evolution is governed by a Hamiltonian H. If H is known, we can estimate t by this method; if t is known, we can estimate classical parameters on which H depends. If the system is subject to noise, its ability to accurately reveal the unknown parameter deteriorates. In this work, we introduce and study a fundamental trade-off which relates the amount by which noise reduces the accuracy of a quantum clock to the amount of information about the energy of the clock that leaks to the environment. Specifically, we consider an idealized scenario in which Alice prepares an initial pure state of the clock, allows the clock to evolve for a time t that is not precisely known, and then transmits the clock through a noisy channel to Bob. The environment (Eve) receives any information that is lost. The sensitivity to which t can be determined is quantified in terms of the quantum Fisher information (QFI). We prove that Bob's loss of QFI about t is equal to Eve's gain of QFI about a complementary energy parameter. We also prove a more general trade-off that applies when Bob and Eve wish to estimate the values of parameters associated with two non-commuting observables. We then connect our results to the notion of quantum error correction: We show that a weaker variant of the standard quantum conditions for error correction form necessary and sufficient conditions for the accuracy of the clock to be unaffected by the noise. We call states satisfying this weaker notion of error correction a "metrological code," and discuss its relation with existing schemes using quantum error correction for metrology. We show that there are metrological codes that cannot be written as a quantum error-correcting code with similar distance in which the Hamiltonian acts as a logical operator, potentially offering new schemes for constructing states that do not lose any sensitivity upon application of a noisy channel. We discuss applications of our results to sensing using a many-body state subject to erasure or amplitude-damping noise.
Joint work with: Mischa P. Woods, Victor V. Albert, Joseph M. Renes, Jens Eisert, and John Preskill.
Reference: https://arxiv.org/abs/2207.13707
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