Thread by @fgrosshans

Nows at #MCQS, Rafał Demkowicz-Dobrzański from (@UniWarsawski ) on the grand unified theory of quantum metrology.
#LTQI

Rafał Demkowicz-Dobrzański seez quantum metrology as a channel estimation problem. The channel U_φ is given, one optimizes the state sent and the measurement.
#LTQI #MCQS

Rafał Demkowicz-Dobrzański: The Cramer-Rao bount gives limit on the variance Δ²φ≥1/F of an unbiased estimator., where F is the (Quantum or classical) Fisher information, defined in term of derivatives of probability/kets over φ
#LTQI #MCQS

Rafał Demkowicz-Dobrzańsk: e.g. phase estimation with N uses of a channel. We just have to maximize the Fisher information. It’s N for uncorrelated states, but some statest (N00N) have N²
#LTQI #MCQS

Rafał Demkowicz-Dobrzańsk: But the most general scheme is an adaptive scheme, where U_φ is applied N successive times, with interleaved wide N qubit unitaries.
F_Q≤N² : Adaptiveness is useless !

#LTQI #MCQS

Rafał Demkowicz-Dobrzańsk: it can be applied to frequency aevaluation over time T, showing Δω≥1/T
#LTQI #MCQS

Rafał Demkowicz-Dobrzańsk now looks at the impact of decoherence. The problem, there is no explicit quantum Fisher information for mixed states, or at least no one which is easy to analyze.
Looking at purifications of mixed states is helpful
#LTQI #MCQI

Rafał Demkowicz-Dobrzańsk: Actually, the Fisher information of the worse purification is the Fisher information of the mixed state. If one fails to find the optimal purification, one nevertheless have a bound #LTQI #MCQS

Rafał Demkowicz-Dobrzańsk: Here we work with channel, so we have to look at “pourification of channels”: we minimize over equivalent Kraus representations of the channels.
#LTQI #MCQS

Rafał Demkowicz-Dobrzańsk: We arrive to a single channel bound
max F_Q≤4 min N||α|| + N(N-1)||α||(||α||+||β||+1), where ||α|| and ||β|| are norms defined in term of K_i and their derivatives, min is taken over Kraus representations
#LTQI #MCQS

Rafał Demkowicz-Dobrzańsk: Depending on ||α|| and ||β||, F_Q is either linear (||β|| vanishes) or quadratic.
Translating this in time, with infinetesimal time steps gives a condition on the Lindblad operators to have F_Q∝T.
#LTQI #MCQS

Rafał Demkowicz-Dobrzańsk: this conditions on H and Ls are typically fulfilled and Heisenberg scalling is then lost. E.g. losses in interferometer Δφ≥√((1-η)/ηN) . Classical scaling, but still a quantum gain #LTQI #MCQS

Rafał Demkowicz-Dorzański: When H is not in the span specified by the condition, Heisenberg T² scaling can be recovered, as with error correction. #LTQI #MCQS

Rafał Demkowicz-Dorzański also applied his tools to metrology with condensed matter many-body HAmiltonians. #LTQI #MCQS

Rafał Demkowicz-Dorzański now works on genralizing this to correlated noises #LTQI #MCQS

Rafał Demkowicz-Dorzański:eHeisenberg scalling iff H∈span{I, L_j^Harmitian, i×L_j^(antihermitian), (L_j^†L₁j')^hermitian and i×(L_j^†L₁j')^antihermitian}

#LTQI #MCQS

Rafał Demkowicz-Dobrzański’s talk from (@UniWarsawski ) on the grand unified theory of quantum metrology is available at
#LTQI

In the talk mentioned in the #previoustweet, Rafał Demkowicz-Dobrzański refers to a “future” talk by Liang Jiang from @Yale_QI on the Acievability of the eisenberg limit. This talk is online at