The theory of entanglement-assisted metrology for quantum channels

Abstract

The quantum Fisher information (QFI) measures the amount of information that a quantum state carries about an unknown parameter. The (entanglement-assisted) QFI of a quantum channel is defined to be the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla. Both the channel QFI and the optimal input state could be solved via a semidefinite program (SDP). In quantum metrology, people are interested in calculating the QFI of $N$ identical copies of a quantum channel when $N \rightarrow \infty$, which we call the asymptotic QFI. It was known that the asymptotic QFI grows either linearly or quadratically with $N$. Here we obtain a simple criterion that determines whether the scaling is linear or quadratic. In both cases, we found a quantum error correction protocol achieving the asymptotic QFI and an SDP to solve the optimal code. When the asymptotic QFI is quadratic, the Heisenberg limit, a feature once thought unique to unitary quantum channels, is recovered. When the asymptotic QFI is linear, we show it is still in general larger than $N$ times the channel QFI, showing the non-additivity of the channel QFI of general quantum channels.