Finding maximal quantum resources

TL;DR

The paper presents an iterative, universal method to maximize resourceful multipartite quantum states across diverse quantifiers, anchored by the geometric measure $G$. By alternating closest-product-state searches with controlled perturbations and rank-1 re-optimizations, the algorithm monotonically increases the chosen resource, and it is implemented with gradient-based optimizers and flexible unitary parametrizations. Demonstrated results for qubits reveal known maximizers (Bell, W) and novel AME-adjacent states, as well as maximally entangled subspaces, with extensions to higher dimensions, stabilizer ranks, and MPS-based analyses. The approach provides a powerful toolkit for identifying and classifying highly entangled states, with implications for AME existence, quantum networks, and tensor-space geometry.

Abstract

For many applications the presence of a quantum advantage crucially depends on the availability of resourceful states. Although the resource typically depends on the particular task, in the context of multipartite systems entangled quantum states are often regarded as resourceful. We propose an algorithmic method to find maximally resourceful states of several particles for various applications and quantifiers. We discuss in detail the case of the geometric measure, identifying physically interesting states and delivering insights to the problem of absolutely maximally entangled states. Moreover, we demonstrate the universality of our approach by applying it to maximally entangled subspaces, the Schmidt-rank, the stabilizer rank as well as the preparability in triangle networks.

Figures (9)

• Figure 1: Schematic illustration of the iteration step of the algorithm. The set of all states is represented by the half sphere and the set of product states by the lower dimensional manifold $\mathcal{P}$. If the algorithm is initialized in state $\vert \varphi \rangle$ (red arrow), we first compute the best approximation within $\mathcal{P}$, denoted by $\vert \pi \rangle$ (blue arrow). Then, we compute the projector into the orthocomplement of $\vert \pi \rangle$, which is here given by the $xy$-plane. The portion of $\vert \varphi \rangle$ within the $xy$-plane is given by $\vert \eta \rangle$ (gray arrow). The new state $\vert \tilde{\varphi}\rangle \sim \vert \varphi \rangle + \epsilon \vert \eta \rangle$ is then the normalized version of $\vert \varphi \rangle$ shifted by a small amount $\epsilon>0$ into the direction $\vert \eta \rangle$.
• Figure 2: The probability distribution of the geometric measure for different multi-qubit systems. It can be easily seen that the maximum of the distribution is shifted to higher values when the number of parties increases. From the approximation to the distribution, we can also calculate the first and second moments, see Tab. \ref{['tab:expected_entanglement']}.
• Figure 3: Trajectory of the geometric measure of the iterates of the algorithm for different staring points and different numbers of qubits. While for the case $n=3$ the different trajectories differ, this deviation becomes smaller if the number of parties increase. However, it should be noted that after $300$ iterations each state displays roughly the same amount of entanglement, showing that at least for the case of $n=3,4,5,6$ the algorithm is robust w.r.t. to local optima. It can be seen that for increasing $n$ also the geometric measure of the initial state increases, as the expected geometric measure increases. Further, the different trajectories are getting more narrow with increasing $n$, reflecting the concentration property of the geometric measure.
• Figure 4: Graph of the known $\text{AME}(4,4)$ state AME_qudit_helwig_2013.
• Figure 5: Highly entangled qubit graph states. The first four graphs yielding the known AME states on two/three/five/six qubits respectively. The last graph state, the Fano graph state huber2017, is a $2$-uniform state on seven qubits where $32$ of its $35$ three-body marginals are maximally mixed.

Theorems & Definitions (8)

• Lemma 1
• proof
• Corollary 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof