Trace (in)fidelity

[relugzosiraba]

What does “trace fidelity” show? The fidelity of the matrix consisting of only essential levels or the fidelity of the whole matrix including guard levels?

[andersp]

Trace fidelity = |Tr(U^dag V)/E|^2. Infidelity = 1 - fidelity Here U and V are (N x E) matrices, so all levels are accounted for. However, the target matrix (V) is always zero for the guard levels. Look in our paper on arXiv for further details.

[relugzosiraba]

By "the target matrix (V) is always zero for the guard levels", do you mean that the corresponding elements are zero? It shouldn't be the matrix itself...

[andersp]

The corresponding rows of the matrix V are always zero

[relugzosiraba]

Why do V and U in Eq. (13) here are N-by-N? No guard levels considered? But it seems that leakage is taken into account.

[andersp]

There are several ways of writing the fidelity. In that report we use a matrix P of size NxE that inflates a state vector with E essential levels to size N, by inserting zeros at all guard levels. As a result U-tilde = P-dagger U P is of size ExE. Note that there is a typo in Eqn. (13). The factor in front of the trace should be 1/ E^2. In this case, E=4.

[relugzosiraba]

There is also typo in Eq. (11) there, right? It should be C^{4 x 4}

[andersp]

Obviously