We study the effect of local decoherence on arbitrary quantum states. Adapting techniques developed in quantum metrology, we show that the action of generic local noise processes --though arbitrarily small-- always yields a state whose Quantum Fisher Information (QFI) with respect to local observables is linear in system size $N$, independent of the initial state. This implies that all macroscopic quantum states, which are characterized by a QFI that is quadratic in $N$, are fragile under decoherence, and cannot be maintained if the system is not perfectly isolated. We also provide analytical bounds on the effective system size, and show that the effective system size scales as the inverse of the noise parameter $p$ for small $p$ for all the noise channels considered, making it increasingly difficult to generate macroscopic or even mesoscopic quantum states. In turn, we also show that the preparation of a macroscopic quantum state, with respect to a conserved quantity, requires a device whose QFI is already at least as large as the one of the desired state. Given that the preparation device itself is classical and not a perfectly isolated macroscopic quantum state, the preparation device needs to be quadratically bigger than the macroscopic target state.
In this work, we consider the preparation and maintenance of macroscopic quantum states, and study how such states are affected by local decoherence. By means of the Quantum Fisher Information, we characterize the macroscopicity of arbitrary quantum states and we show that all quantum states of macroscopic sizes are fragile and cannot be maintained if the system is not perfectly isolated. In addition, we find that, in order to prepare such states, one needs the classical preparation device to be quadratically bigger than the target state.
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