Complete Hierarchies for the Geometric Measure of Entanglement

TL;DR

This work addresses quantifying multiparticle entanglement by computing the geometric measure via a complete set of hierarchies that operate on multiple copies of the state. By inserting symmetric projectors and using multi-copy contractions, the authors establish provable two-sided bounds that converge to the true injective tensor norm, $\Lambda^2(\psi)$, and hence to $E_G(\psi) = 1 - \Lambda^2(\psi)$. The three hierarchies, $\mathfrak{H}_1$, $\mathfrak{H}_2$, and $\mathfrak{H}_3$, combine to provide tight, computable bounds, with strong numerical performance on W/GHZ superpositions and graph states; connections to the product test and extensions to operators and mixed states broaden the method’s applicability. The approach yields practical entanglement witnesses and separability tests, and offers a new perspective on the complexity of separability through a hierarchy of eigenvalue problems, supported by de Finetti arguments and TTN-inspired constructions.

Abstract

In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.

Figures (6)

• Figure 1: Graphical description of the possible estimates in the second hierarchy $\mathfrak{H}_2$. Upper left: a visualization of Eq. \ref{['eq:2nd_hierarchy_example']}. Upper right: relaxation from product states; the dashed lines denote the application of a projector onto the symmetric subspace. Lower left: resulting connectivity graph; the short legs denote the indices on which the symmetric projector acts. Lower right: possible connectivity graphs for six copies of a tripartite state. Note that different labelings are possible which may lead to different results.
• Figure 2: Lower bounds from the three hierarchies for states of the form $| \psi(s) \rangle=\sqrt{s}| W \rangle + \sqrt{1-s}| \mathrm{GHZ} \rangle$. We also show the exact value obtained by solving the one-parameter optimization over symmetric product states wei2003geometric. For the first hierarchy $\mathfrak{H}_1$ we use level $25$ and consider the Schmidt decomposition of $| F_k \rangle$ with respect to the bipartition between the first copy and the rest of the copies for improvement. For $\mathfrak{H}_2$, for which we only consider the norm and level $10$, the used connectivity graph is described in Appendix \ref{['app:trees']}. For $\mathfrak{H}_3$ a level of $60$ was used.
• Figure 3: Bounds on the geometric measure of the $5$-cycle graph state $| C_5 \rangle$, which is the maximally entangled state w.r.t. the geometric measure for five qubits. The horizontal line at height $0.86855$ is the value of the geometric measure of $| C_5 \rangle$steinberg2024finding. For $\mathfrak{H}_1$ and $\mathfrak{H}_2$ we show the upper and lower bounds in solid and dashed, respectively. For $\mathfrak{H}_1$ we also show the lower bound found by considering the largest Schmidt coefficient of $| F_k \rangle$ with respect to a balanced bipartition between the copies, i.e., the bipartition between the first and second half of the copies. Note that the numerical package ENTCALC masajada2025entcalctoolkitcalculatinggeometric provides for this state only a certified bound in the interval $[0.7292, 0.9375]$.
• Figure 4: Examples of graphs corresponding to the second hierarchy for $n=4$ particles. (Left). First non-trivial level $l=1$ of the finite Bethe lattice, where an initial tensor is contracted with $n$ copies. Subsequent levels of the Bethe lattice contract each of the tensors in the 'outer' layer with $n-1$ other copies. (Right) Example tree tensor network of a path graph labeled by $\left[1, 3, 2\right]$, corresponding to the indices connecting the tensors from top to bottom. To calculate the associated bound for a given tree tensor network, we apply symmetric projectors on each of the open legs of a given type, and then contract the tensor with itself. Note that the symmetric projectors can be of different sizes; the tree tensor network on the right requires a symmetric projector on $2$ copies for the leg of type $1$, while requiring a symmetric projector on $4$ copies for the leg of type $4$.
• Figure 5: Lower bounds on the geometric measure of superpositions of Dicke states of the form $\sqrt{s}| D_{1}^{(5)} \rangle + \sqrt{1-s}| D_{2}^{(5)} \rangle$. The lower bounds are found for three path graph TTNs parametrized by $L = \left[1\right]$, $L = \left[1, 2\right]$, $L = \left[1, 2, 3, 4\right]$. We also show a numerical upper bound, found by a brute-force search over a discretization of all symmetric product states.

Theorems & Definitions (3)

• proof
• Theorem 2
• proof