# Identifying families of multipartite states with non-trivial local entanglement transformations

^{1}Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21A, 6020 Innsbruck, Austria^{2}Department of Physics, QAA, Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany^{3}Current address: Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria^{4}Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, E-28911, Leganés (Madrid), Spain^{5}Instituto de Ciencias Matemáticas (ICMAT), E-28049 Madrid, Spain

Published: | 2024-02-29, volume 8, page 1270 |

Eprint: | arXiv:2302.03139v2 |

Doi: | https://doi.org/10.22331/q-2024-02-29-1270 |

Citation: | Quantum 8, 1270 (2024). |

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### Abstract

The study of state transformations by spatially separated parties with local operations assisted by classical communication (LOCC) plays a crucial role in entanglement theory and its applications in quantum information processing. Transformations of this type among pure bipartite states were characterized long ago and have a revealing theoretical structure. However, it turns out that generic fully entangled pure multipartite states cannot be obtained from nor transformed to any inequivalent fully entangled state under LOCC. States with this property are referred to as isolated. Nevertheless, multipartite states are classified into families, the so-called SLOCC classes, which possess very different properties. Thus, the above result does not forbid the existence of particular SLOCC classes that are free of isolation, and therefore, display a rich structure regarding LOCC convertibility. In fact, it is known that the celebrated $n$-qubit GHZ and W states give particular examples of such classes and in this work, we investigate this question in general. One of our main results is to show that the SLOCC class of the 3-qutrit totally antisymmetric state is isolation-free as well. Actually, all states in this class can be converted to inequivalent states by LOCC protocols with just one round of classical communication (as in the GHZ and W cases). Thus, we consider next whether there are other classes with this property and we find a large set of negative answers. Indeed, we prove weak isolation (i.e., states that cannot be obtained with finite-round LOCC nor transformed by one-round LOCC) for very general classes, including all SLOCC families with compact stabilizers and many with non-compact stabilizers, such as the classes corresponding to the $n$-qunit totally antisymmetric states for $n\geq4$. Finally, given the pleasant feature found in the family corresponding to the 3-qutrit totally antisymmetric state, we explore in more detail the structure induced by LOCC and the entanglement properties within this class.

### Popular summary

So far, only two classes of states [the stochastic LOCC (SLOCC) classes of the GHZ and the W states] have been shown to contain no isolated states (isolation-free). Here, we discover a new isolation-free class, containing the 3-qutrit totally antisymmetric state, which turns out to have some fascinating entanglement properties. Additionally, we found evidence that many other classes of fully entangled pure states contain isolated states.

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[71] Since the perturbation series of eigenvector $|e_p\rangle $ converges in $\epsilon$, one can choose $\epsilon$ small enough such that the absolute value of the sum of the $\mathcal{O}(\epsilon^2)$ terms for $\langle0|e_p\rangle$ is strictly smaller than 1 for $|e_0\rangle $ and $|\frac{\epsilon\sqrt{r}^{p}}{(1-r^p)(1-\omega^{-p})}|$ for every $|e_p\rangle $ where $p\in\{1,\ldots,d-1\}$, while keeping $\{E_p\}$ non-degenerate footnote:pert.

[72] It is easy to see the following: If $S\in SL(d,\mathbb{C})$ quasi-commutes with two $d\times d$ positive definite diagonal matrices $\Lambda$ and $D$ such that $\Lambda\not\propto D$, $S$ must be a direct sum of block matrices that act on the (degenerate) eigenspaces of $\Lambda^{-1}D$. Moreover, for each block in $S$ of which the range lies within the (degenerate) eigenspace of a single eigenvalue of $\Lambda$ or $D$, the block is unitary.

[73] When multiplying Eq. (1) by $|A_3\rangle $ (which is the seed state $|\Psi_s\rangle $ here) where $g=\sqrt{\Delta'}\otimes \sqrt{D'}\otimes {1}$ and $h=\sqrt{\Delta}\otimes \sqrt{D}\otimes {1}$, the term $g^\dagger\sum_q N_q^\dagger N_q g|A_3\rangle =0$ because all $N_q\in\mathcal{N}_{g\Psi_s}$ satisfy $N_q g|A_3\rangle =0$ by definition.

[74] Alternatively, one can see this by showing that $|A_3\rangle $ is the only state among all the MES candidates in Observation 11 that has a completely mixed single qutrit reduced density matrix for all 3 bipartite splittings. Applying Nielsen's theorem Nielsen to all 3 bipartitions proves that $|A_3\rangle $ is indeed not LOCC-reachable.

[75] The preparation procedure above does not work for $|\psi(\alpha_1,\alpha_2,\beta_1,\beta_2)\rangle $ with $\beta_1=\beta_2$ because one of the columns in $U_2$ and $U_3$ becomes all zeros when $\beta_1=\beta_2$.

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[1] Moisés Bermejo Morán, Alejandro Pozas-Kerstjens, and Felix Huber, "Bell Inequalities with Overlapping Measurements", Physical Review Letters 131 8, 080201 (2023).

[2] Anubhav Kumar Srivastava, Guillem Müller-Rigat, Maciej Lewenstein, and Grzegorz Rajchel-Mieldzioć, "Introduction to quantum entanglement in many-body systems", arXiv:2402.09523, (2024).

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