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

Nicky Kai Hong Li1,2,3, Cornelia Spee1, Martin Hebenstreit1, Julio I. de Vicente4,5, and Barbara Kraus1,2

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

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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.

Multipartite entanglement is a type of correlation that is stronger than any classical correlation for multiple parties. Many quantum technologies that have an advantage over their classical counterparts need multipartite entanglement which is thereby considered to be a resource for quantum information processing. The resource theory of entanglement aims at characterizing and quantifying entanglement, providing protocols to harness this resource as well as ways to quantify the efficiency of these protocols. The free operations in this theory are local operations assisted by classical communication (LOCC), which naturally describe state manipulation protocols carried out by multiple spatially separated parties. As applying LOCC to any quantum state cannot increase its entanglement, we can identify states that are more useful for certain quantum information processing tasks by characterizing states that can be transformed into many other less entangled states. For bipartite pure states, there exists one maximally entangled state that can be LOCC transformed to any other bipartite states with compatible local dimensions. However, for $n$-qudit states, it has been shown that the counterpart of this single maximally entangled state, the maximally entangled set, is almost the whole Hilbert space. In fact, it has been shown that there is almost no LOCC transformation possible among pure, fully entangled multipartite states. That is, almost all states are isolated, i.e., cannot be obtained from nor transformed to any inequivalent fully entangled state under LOCC.

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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[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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