Estimating the entanglement of random multipartite quantum states

TL;DR

The paper addresses estimating the injective norm, equivalently the geometric measure of entanglement for random multipartite pure states, by introducing a normalized gradient descent (NGD) algorithm and comparing it to ALS, PIM, and symmetrized variants. It systematically benchmarks these methods on real and complex Gaussian tensors (both symmetrized and non-symmetrized) and extends the approach to random matrix product states (MPS), reporting first numerical estimates of genuine multipartite entanglement in these models. The authors formulate two conjectures on the asymptotics of the injective norms for random Gaussian tensors and Gaussian MPS, supported by extensive numerical evidence. They demonstrate that NGD consistently outperforms or matches existing methods, especially for high-dimensional or non-normalized inputs, and provide open-source code to enable further exploration and application across quantum information and many-body physics. Overall, the work delivers a practical, scalable tool for quantifying multipartite entanglement in complex tensor models and offers new numerical insights into entanglement structure in random states and tensor networks.

Abstract

Genuine multipartite entanglement of a given multipartite pure quantum state can be quantified through its geometric measure of entanglement, which, up to logarithms, is simply the maximum overlap of the corresponding unit tensor with product unit tensors, a quantity that is also known as the injective norm of the tensor. Our general goal in this work is to estimate this injective norm of randomly sampled tensors. To this end, we study and compare various algorithms, based either on the widely used alternating least squares method or on a novel normalized gradient descent approach, and suited to either symmetrized or non-symmetrized random tensors. We first benchmark their respective performances on the case of symmetrized real Gaussian tensors, whose asymptotic average injective norm is known analytically. Having established that our proposed normalized gradient descent algorithm generally performs best, we then use it to obtain numerical estimates for the average injective norm of complex Gaussian tensors (i.e., up to normalization, uniformly distributed multipartite pure quantum states), with or without permutation-invariance. We also estimate the average injective norm of random matrix product states constructed from Gaussian local tensors, with or without translation-invariance. All these results constitute the first numerical estimates on the amount of genuinely multipartite entanglement typically present in various models of random multipartite pure states. Finally, motivated by our numerical results, we posit two conjectures on the injective norms of random Gaussian tensors (real and complex) and Gaussian MPS in the asymptotic limit of the physical dimension.

Figures (16)

• Figure 1: Illustration of our NGD algorithm for finding a rank-1 approximation of a tensor $\Psi$.
• Figure 2: The average injective norm of symmetrized real Gaussian tensors approximated using the ALS, PIM, NGD and SGD algorithms. All the algorithms perform equally well on order $2$ tensors. NGD and SGD perform substantially better than ALS and PIM on order $3$ tensors. The analytical bounds are labeled "ANA".
• Figure 3: The average injective norm of non-symmetrized real Gaussian tensors approximated using the ALS and NGD algorithms. Both algorithms perform equally well on order $2$ tensors. NGD performs substantially better than ALS on order $3$ tensors since both algorithms estimate lower bounds on the unknown analytical value.
• Figure 4: All algorithms perform equally well on inputs that are normalized real Gaussian tensors, either symmetrized or non-symmetrized. However, it is preferable in this case to work with non-normalized tensors because otherwise the injective norm goes to $0$ as $d \to \infty$.
• Figure 5: We fit functions of the form $C_0 + C_1 d^\alpha$ on the data points obtained using our NGD algorithm for symmetrized and non-symmetrized real Gaussian tensors of order $2$ and $3$. The asymptotic constant $C_0$ is fixed to the known analytical value while $C_1$ and $\alpha$ are estimated by minimizing the squared error.

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2