Minimal decomposition entropy and optimal representations of absolutely maximally entangled states

TL;DR

This work introduces the minimal decomposition entropy $\mathop{\mathrm{S_q}}$ as a LU-invariant tool to analyze multipartite entanglement, with a focus on absolutely maximally entangled (AME) states. It derives general lower bounds $\mathop{\mathrm{S^{min}_q}} \ge \lfloor N/2\rfloor\log(d)$ and develops a complex $\mathcal{L}_p$-PCA–based algorithm to compute $\mathop{\mathrm{S^{min}_q}}$ for finite $q>1$, as well as a seesaw-style method for $q=\infty$ linked to the geometric measure of entanglement. Numerical results compare Haar-random and AME states across qubits, qutrits, and ququads, revealing cases where AME states achieve more localized optimal representations (lower $\mathop{\mathrm{S^{min}_2}}$) and higher geometric entanglement (higher $\mathop{\mathrm{S^{min}_\infty}}$), and identifying new extremal states. The paper also demonstrates how the algorithm yields sparser, more interpretable AME representations and uses these to distinguish genuinely quantum AME states from those constructible from classical combinatorial designs. Together, these results provide both a practical method for characterizing AME entanglement structures and insights into the role of minimal decomposition entropy in LU classification and quantum-design distinctions.

Abstract

Understanding and classifying multipartite entanglement is fundamental to quantum information processing. A useful measure of multipartite entanglement is the minimal decomposition entropy, defined as the minimum of the Rényi entropy $ S_q $ associated with the state's decomposition over all local product bases. This quantity identifies the product bases in which the state is maximally localized, thereby yielding optimal representations for analyzing local-unitary equivalence and structural properties of multipartite states. We investigate the minimal decomposition entropy for absolutely maximally entangled (AME) states, a class of highly entangled states characterized by their maximal entanglement across any bipartitions. We present a numerical algorithm for computing the minimal decomposition entropy for finite $ q>1 $. Entropy distributions for AME and Haar random states are obtained for $ q=2 $ and $ q=\infty $ in qubit, qutrit, and ququad systems. For $ q=2 $, AME states of four qutrits and four ququads exhibit smaller minimal decomposition entropy than Haar random states, indicating more localized optimal representations. For $ q=\infty $, corresponding to the geometric measure of entanglement, AME states display higher entanglement than Haar random states. The algorithm additionally produces simpler and sparser decompositions of known AME states, aiding in distinguishing genuinely quantum AME states from those associated with classical combinatorial designs.

Figures (9)

• Figure 1: Distribution of $\mathop{\mathrm{S_2}}\nolimits$ and $\mathop{\mathrm{S^{min}_2}}\nolimits$ for Haar random states and random AME states in (a) three qubits, (b) four qubits, (c) three qutrits, (d) four qutrits, and (e) four ququads. For the ensembles of AME states, the lower bound of $\mathop{\mathrm{S_2}}\nolimits$ for $\mathop{\mathrm{AME}}\nolimits(N,d)$ states, given by $\log\left(d^{\lfloor N/2 \rfloor}\right)$, is indicated. The upper bound for $\mathop{\mathrm{S_2}}\nolimits$, $\log(d^{N})$, and the known upper bound for $\mathop{\mathrm{S^{min}_2}}\nolimits$, $\log\left(d^{N} - \tfrac{1}{2} N d(d-1)\right)$, are included wherever possible. The ensembles of $\mathop{\mathrm{AME}}\nolimits(3,3)$, $\mathop{\mathrm{AME}}\nolimits(4,3)$, and $\mathop{\mathrm{AME}}\nolimits(4,4)$ consist of $2.5 \times 10^4$ states, while all other Haar random and AME state ensembles contain $10^5$ states.
• Figure 2: Distribution of $\mathop{\mathrm{S_\infty}}\nolimits$ and $\mathop{\mathrm{S^{min}_\infty}}\nolimits$ for Haar random states and random AME states in (a) three qubits, (b) four qubits, (c) three qutrits, (d) four qutrits, and (e) four ququads. For the ensembles of AME states, the lower bound of $\mathop{\mathrm{S_2}}\nolimits$ for $\mathop{\mathrm{AME}}\nolimits(N,d)$ states, given by $\log\left(d^{\lfloor N/2 \rfloor}\right)$, is indicated. The upper bound for $\mathop{\mathrm{S_2}}\nolimits$, $\log(d^{N})$, and the known upper bound for $\mathop{\mathrm{S^{min}_2}}\nolimits$, $\log\left(d^{N} - \tfrac{1}{2} N d(d-1)\right)$, are included wherever possible. The ensembles of $\mathop{\mathrm{AME}}\nolimits(3,3)$, $\mathop{\mathrm{AME}}\nolimits(4,3)$, and $\mathop{\mathrm{AME}}\nolimits(4,4)$ consist of $2.5 \times 10^4$ states, while all other Haar random and AME state ensembles contain $10^5$ states.
• Figure 3: The convergence of the algorithm for twenty random $\mathop{\mathrm{AME}}\nolimits(4,3)$ states is shown, with each state represented by a different color. The decomposition entropy at each iteration step is plotted. The plot demonstrates convergence of $\mathop{\mathrm{S_2}}\nolimits$ to the theoretical value $\log(9)$.
• Figure 4: The minimal decomposition entropy $\mathop{\mathrm{S_q}}\nolimits$ and $\mathop{\mathrm{S^{min}_q}}\nolimits$ as a function of $q$ for a Haar random three qutrit state and a random $\mathop{\mathrm{AME}}\nolimits(3,3)$ state plotted for $q>1$ with a step size 0.5. The values at $q=\infty$ are marked on the right side.
• Figure 5: (a) 2-unitary matrix corresponding to a random $\mathop{\mathrm{AME}}\nolimits(4,3)$ state. (b) 2-unitary matrix corresponding the state obtained by applying the algorithm. Only the absolute values are shown in both cases. The values of entropy are given, and the $\mathop{\mathrm{S^{min}_2}}\nolimits$ value matches the lower bound $\log(9)$ up to five decimal places for $\mathop{\mathrm{AME}}\nolimits(4,3)$ states.

Theorems & Definitions (3)

• Definition 1
• Proposition 1
• proof