Indefinite Causal Orderings for Quantum Depolarizing Channel Identification

Quantum channel identification (QCI) is the statistical estimation of channel parameters, accomplished by passing probe systems in prepared states through the quantum channel. Indefinite causal ordering (ICO) is a channel probing scheme in which copies of a quantum channel are physically or logically arranged so that the path through them is a superposition of different possible paths. Channel probing is known to be aided in a strong sense by ICO in the case of the qudit (d-dimensional) depolarizing channel. Specifically, ICO-assisted probing increases the quantum Fisher information (QFI) in the processed probe state about the unknown parameter(s), with increasing order indefiniteness yielding correspondingly greater QFI. Investigations up until now have involved just superpositions of forward and reverse channel orderings; for example, three channel copies, A, B, and C, in a superposition of orderings A-B-C and C-B-A. Alternatively, these channels copies can be cyclically indefinitely ordered: a superposition of A-B-C, B-C-A, and C-A-B. The present work compares the effectiveness (QFI) of disparate competing ICOs for probing the qudit depolarizing channel. The QFIs associated with these competing ICOs are expressed analytically.