Fig. 3: Variance and bias of a linear spin chain Gibbs state expansion.
We illustrate in this figure the different sampling probabilities of the terms of the expansion truncating after the Δ4 term. Moreover, we calculate the value of the negativity λ in this case. When truncating the expansion, one must take into account the variance-bias tradeoff that arises. We define \({\sigma }_{0}^{2}\) as the variance of measuring observable O with respect to the full Gibbs state at inverse temperature β. The variance scales as λ2 in the case of the approximated Gibbs state using the linked cluster expansion, where λ increases with the more terms included in the expansion. On the other hand, the bias increases when including a smaller number of terms in the expansion. Hence, it is important to optimize the number of expansion terms in the approximation.
