Locally Purified Maximally Mixed States At Scale: Entanglement Pruning and Symmetries

TL;DR

The paper tackles the challenge that Locally Purified Density Operators (LPDOs) can represent mixed states with sub-optimal resource use, especially near the maximally mixed fixed point $\rho_{\mathds{1}}$. It proposes three complementary methods—fidelity-preserving truncation, Riemannian optimization on the Stiefel manifold acting in the $\\kappa$-space, and symmetry/injectivity analysis—to prune coherent correlations and reduce bond-dimension complexity, showing that in the maximally mixed case the optimal $\chi_i=1$ and $\kappa_i=2$. The authors derive analytic expressions for the disentangler via injectivity and demonstrate numerically that $\chi$ can be driven to 1 while preserving fidelity, significantly improving the scalability of LPDO-based simulations. These results advance tensor-network methods for mixed states on near-term quantum devices, with implications for error mitigation, noise-extrapolation, and exploration of fundamental scalability limits for tensor networks in open quantum systems.

Abstract

Locally Purified Density Operators (LPDOs) are state-of-the-art tensor network ansatze candidates that efficiently represent mixed quantum states at scale. However, given their non-uniqueness, their representational complexity is generally sub-optimal in practical computations. In this work we perform a comprehensive numerical and analytical analysis and resolve this issue in the experimentally relevant limit where noise depolarizes the density operator into a maximally mixed state. To resolve the sub-optimality issue, we analyze two numerical tools, one analytic method, and detail the relations between them. The numerical tools used are fidelity-preserving truncations and isometric gauge transformations leveraging Riemannian optimizations over entropic objective functions. In addition, by invoking the injectivity and symmetry constraints of the maximally mixed LPDO, we also present analytical closed-form expressions for the disentangler and discuss their relation to numerical optimizers. Our work shows how, by minimizing the resources required to represent key states of practical interest in experiment, the efficiency of tensor network algorithms can be substantially increased. This paves the path for uncovering tensor network's fundamental scalability limits and latent potential in representing the wide locus of mixed quantum states that are accessible on near-term quantum devices.

Figures (13)

• Figure 1: Optimal representation of LPMM$\rho$. (a) MPO representation of $\rho_{\mathds{1}}$. The system is completely separable, therefore $\chi=1$ between each of the neighoring tensors. (b) Performing an SVD operation on the individual tensors results in the singular values $\Sigma$, as in Eq. \ref{['eq:A_j']}. The square root of $\Sigma$ is then merged into $A^{(i)}$ and ${A^{(i)}}^{\dag}$ at all sites resulting in the canonical LPDO. (c) This generates the purification index $\kappa$ that encodes the mixture correlations. In the context of the optimal LPMM$\rho$, we note that $\chi_{i}=1$ and $\kappa_{i}=2$ at all sites.
• Figure 2: Sub-optimal representation of LPMM$\rho$. (a) Initializing a pure random MPS of $N$ sites with $\chi_{i}$'s bounded by $\chi_{\text{max}}$ by employing the randomMPS function call in ITensors.jl. We decorate the above with the mixture indices, $\kappa_{i}=1$ (denoted by dashed orange line) at each $i$ as the state remains pure. (b) The action of bitflip and dephasing single qubit noise channels at each of the sites, for a more detailed description of the above, see App. \ref{['appa:channel_application']}. The noise channels completely decohere the initial entangled state, resulting in a LPMM$\rho$. (c) Representationally, $\kappa_{i}$'s are inflated due to the noise channels (denoted by solid orange line), however $\chi$ remains unchanged leading to a sub-optimal representation of LPMM$\rho$.
• Figure 3: Fidelity-preserving truncation protocol. (a) Choose two sites, $s_{i}$ and $s_{i+1}$ whose corresponding $\chi_{i}$, the shared virtual index, needs to be truncated. Fix the orthogonality center on one of the above sites. (b) Perform a contraction over $\chi_{i}$. (c) Further perform an SVD on the above with a cutoff, $\Lambda$ resulting in (d) where the corresponding singular values are truncated and renormalized in the $\chi$-subspace (the approximation arises out of the truncation). (e) Retrieving the LPDO form by merging the singular values into either the left or right tensor depending on the orthogonality center.
• Figure 4: (a) To regauge into optimal representation we sweep multiple times across the LPMM$\rho$, with each sweep involving serial two-body updates. Averaged bond dimension, $\chi_\text{mean}$, as a function of sweeps for different cutoff, $\Lambda$ for an initial sub-optimal LPMM$\rho$ of size $N=100$ and $\chi_{\text{max}}=16$. We note that $\chi_{\text{mean}}$ flattens out after a few number of sweeps for a given cutoff as the distribution associated with the singular values at each $\chi$ becomes fixed. Scaling of $\chi_{\text{mean}}$ post fidelity-preserving truncation as a function of the cutoff, $\Lambda$ for (b) different $\chi_{\text{max}}$ with fixed $N=100$, (c) different system sizes $N$ for fixed $\chi_{\text{max}}=16$, with number of sweeps fixed at 20. We observe exponential scaling of $\chi_{\text{mean}}$ as a a function (a) sweeps, and (b, c) cutoff $\Lambda$. To quantify the scaling, in App. \ref{['sec:scaling_at']}, we perform a fit to an exponential function, $f(x) = \alpha + \beta e^{-\gamma x}$ and obtain the fit coefficients, $\alpha, \beta, \gamma$ as in Fig. \ref{['fig:at_fit_coeffs']}.
• Figure 5: Riemannian manifold based optimization protocol. We initialize the optimization by choosing a random isometry, labeled as $P_{i}$, on the Stiefel manifold, $\mathcal{M}$. To generate the next update in the manifold, $P_{i+1}$, we compute the numerical derivative of the objective function, $D$ with respect to $P_{i}$. We then estimate the gradient, $G$ by projecting $D$ onto the tangent space of $P_{i}$, $\mathcal{T}_{P_{i}}(\mathcal{M})$. Finally, we perform a retraction of the gradient onto the manifold to obtain $P_{i+1}$. In each iteration the optimizer performs a search on the manifold that minimizes the value of an entropic objective function.