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We describe an approach to fix the gauge degrees of freedom in tensornetworks, including those with closed loops, which allows a canonical form forarbitrary tensor networks to be realized. Additionally, a measure for theinternal correlations present in a tensor network is proposed, which quantifiesthe extent of resonances around closed loops in the network. Finally wedescribe an algorithm for the optimal truncation of an internal index from atensor network, based upon proper removal of the redundant internalcorrelations. These results, which offer a unified theoretical framework forthe manipulation of tensor networks with closed loops, can be applied toimprove existing tensor network methods for the study of many-body systems andmay also constitute key algorithmic components of sophisticated new tensormethods.

Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops