Quantum

Upper and lower bounds for few-copy Gaussian tomography, and the absence of a fermionic warm regime

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Briefings

Chen, Mele, Mele, and Preskill proved that few-copy Gaussian tomography requires cubic scaling in copies for fermionic states, unlike the quadratic scaling for bosons. This note establishes that no fermionic warm regime exists, showing a single-copy rate of Theta(m^3/epsilon^2) across all fixed beta_0 values. The research extends these bounds to blocks of k copies and identifies the exact maximal Fisher information budget for fermionic states, though the true asymptotic rate for intermediate temperatures remains open.

Key points

• Chen, Mele, Mele, and Preskill proved that few-copy Gaussian tomography requires cubic scaling in copies for fermionic states, unlike the quadratic scaling for bosons.
• The research establishes that no fermionic warm regime exists, showing a single-copy rate of Theta(m^3/epsilon^2) across all fixed beta_0 values.

Context

This note extends previous results by proving the absence of a fermionic warm regime and establishing exact bounds for block tomography, addressing a gap in understanding fermionic Gaussian states.

Limitations

The true asymptotic rate for intermediate temperatures remains open, and the exact formula for the maximal block Fisher information is known only for specific block sizes.

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