This repository provides MATLAB code to evaluate the bounds derived in the paper F. Albarelli and R. Demkowicz-Dobrzański, "Probe incompatibility in multiparameter noisy quantum channel estimation"
In the paper the general scenario of bounding the sum of the single-parameter QFI of different channels is considered (i.e. the random sensing scenario described in the paper). However, the functions provided here work only in the multiparameter channel estimation setting, where there is only one channel.
We provide the function
[trFIbound,hvec ] = totalQFI_SDP(KrausOps,KrausOpsDeriv,npar,nKraus,variant,dimIn,dimOut,wInv)
that evaluates the optimal total QFI trFIbound, either for a single use of the channel variant='single' or it gives the asymptotic standard quantum limit bound with variant='asymptotic'.
Some possibly nontrivial conventions for using the function are that the Kraus operators K1,K2,... are stacked in column as KrausOps=[ K1 ; K2 ; K3; ... ] and their partial derivatives are also stacked in column, one parameter after the other, as KrausOpsDeriv=[ d1K1 ; d1K2 ; d1K3; ... d2K1 ; d2K2; d2K3 ... ].
The purification matrices are returned in a single tensor of dimension [ nKraus, nKraus, npar ] and are individually accessed as hvec(:,:,i).
The other input arguments should be intuitive.
The functions do not have sanity checks, so make sure to provide consistent input data.
A detailed description of the input and output arguments can be obtained by typing help totalQFI_SDP in the MATLAB Command Window.
The files hamiltonian_erasure.m, multiphase_loss.m, phase_loss.m, phase_dephasing.m, GADchannel.m and grover_dephasing.m are the examples considered in the paper and can be inspected for further instruction on how to use the main function and to check numerically the analytical results reported in the paper.