Pathways to optimize the representation of Locally Purified Density Operators

Locally Purified Density Operators (LPDOs) have emerged as one among several tensor network ansatzes that efficiently represent mixed quantum states at scale. However, the LPDO representation of some states that appear in practical computations remain sub-optimal. One such example is the representation of a maximally mixed state obtained by the action of depolarizing noise on an initially entangled state. To optimize the representation, we present several numerical and analytical methods while also outlining the relationship between them. The numerical protocols involve truncation strategies that preserve the fidelity, in addition to isometric Riemannian optimization routines that minimize entropy based objective functions. Further, by invoking the injectivity and symmetry properties of the maximally mixed LPDO we present analytical closed form expressions involving isometries that effectively prune the correlations thereby mapping the sub-optimal representation to its optimal counterpart. The methods discussed have a broad spectrum of applications ranging from efficiently simulating noisy quantum processes to effectively estimating the scalability of tensor networks.