Theory of entanglement-assisted metrology for quantum channels*

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 by the maximum QFI of the output state assuming an entangled input state over a single probe and an ancilla. In quantum metrology, people are interested in computing the QFI of N identical copies of a quantum channel when N→∞, 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, the asymptotic QFI and a quantum error correction protocol to achieve it are solvable via a semidefinite program. When the scaling is quadratic, the Heisenberg limit is recovered. When the scaling is linear, the asymptotic QFI is still in general larger than N times the single-channel QFI and furthermore, sequential estimation strategies provide no advantage over parallel ones. For details, see arXiv: 2003.10559.