Genuine multipartite entanglement is crucial for quantum information and related technologies, but quantifying it has been a long-standing challenge. Most proposed measures do not meet the "genuine" requirement, making them unsuitable for many applications. In this work, we propose a journey toward addressing this issue by introducing an unexpected relation between multipartite entanglement and hypervolume of geometric simplices, leading to a tetrahedron measure of quadripartite entanglement. By comparing the entanglement ranking of two highly entangled four-qubit states, we show that the tetrahedron measure relies on the degree of permutation invariance among parties within the quantum system. We demonstrate potential future applications of our measure in the context of quantum information scrambling within many-body systems.
Genuine multipartite entanglement is crucial for quantum information and related technologies, but quantifying it has been a long-standing challenge. Most proposed measures do not meet the "genuine" requirement, making them unsuitable for many applications. In this work, we propose a journey toward addressing this issue by introducing an unexpected relation between multipartite entanglement and hypervolume of geometric simplices, leading to a tetrahedron measure of quadripartite entanglement. By comparing the entanglement ranking of two highly entangled four-qubit states, we show that the tetrahedron measure relies on the degree of permutation invariance among parties within the quantum system. We demonstrate potential future applications of our measure in the context of quantum information scrambling within many-body systems.
Genuine multipartite entanglement is crucial for quantum information and quantum technologies but quantifying it has been a long-standing challenge. Most existing measures do not meet a “genuine” requirement, making them unsuitable for many applications. Here, we present a surprising triangle measure for tripartite entanglement, and introduce the extension to four-qubit systems. We discuss potential avenues for further generalizations along this geometric path. This indicates that our work may pave the way for further advancements.
Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, we give an approach of constituting genuine m -partite entanglement measures from any bipartite entanglement and any k -partite entanglement measure, 3 ⩽ k < m . In addition, as a complement to the three-qubit concurrence triangle proposed in ( Phys. Rev. Lett. 127 040403), we show that the triangle relation is also valid for any continuous entanglement measure and system with any dimension. We also discuss the tetrahedron structure for the four-partite system via the triangle relation associated with tripartite and bipartite entanglement respectively. For multipartite system that contains more than four parties, there is no symmetric geometric structure as that of tri- and four-partite cases.
Abstract Quantifying genuine entanglement is a key task in quantum information theory. We study the quantification of genuine multipartite entanglement for four-qubit systems. Based on the concurrence of nine different classes of four-qubit states, with each class being closed under stochastic local operation and classical communication, we construct a concurrence tetrahedron. Proper genuine four-qubit entanglement measure is presented by using the volume of the concurrence tetrahedron. For non genuine entangled pure states, the four-qubit entanglement measure classifies the bi-separable entanglement. We show that the concurrence tetrahedron based measure of genuine four-qubit entanglement is not equivalent to the genuine four-partite entanglement concurrence. We illustrate the advantages of the concurrence tetrahedron by detailed examples.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
Quantifying entanglement is vital to understand entanglement as a resource in quantum information processing, and many entanglement measures have been suggested for this purpose. When mathematically defining an entanglement measure, we should consider the distinguishability between entangled and separable states, the invariance under local transformation, the monotonicity under local operations and classical communication, and the convexity. These are reasonable requirements but may be insufficient, in particular when taking into account the usefulness of quantum states in multi-party quantum information processing. Therefore, if we want to investigate multipartite entanglement as a resource, then it can be necessary to consider the usefulness of quantum states in multi-party quantum information processing when we define a multipartite entanglement measure. In this paper, we define new multipartite entanglement measures for three-qubit systems based on the three-party teleportation capability, and show that these entanglement measures satisfy the requirements for being genuine multipartite entanglement measures. We also generalize our entanglement measures for N-qubit systems, where [Formula: see text], and discuss that these quantities may be good candidates to measure genuine multipartite entanglement.
The space Inv(j1,j2,j3,j4) of SU(2)-invariant four-valent tensors, also known as intertwiners, can be understood as the quantum states of a tetrahedron in Euclidean space with fixed areas. In loop quantum gravity, they are states of the smallest ``atom of space" with non-zero volume. At the same time they correspond to four-party tensor product states invariant under global rotations. We consider the multipartite entanglement of states in Inv(j1,j2,j3,j4) using the recently proposed entropic fill.
Numerically evaluating entropic fill in the case of equal spins between 1/2 and 11, we find that the distributions of entanglement are very different for intertwiners as compared to generic tensors, and for coherent intertwiners as compared to generic ones. The peak in the distribution seems to be at the highest entanglement for generic intertwiners and at the lowest for generic tensors, but in terms of average entanglement, the roles are switched: average entanglement is highest in arbitrary tensors and lower in intertwiners, at least in the regime of large j.
We also consider the dependence of entanglement on the geometric data of coherent intertwiners. Coherent intertwiners labeled by a symmetric tetrahedron have maximal entanglement, while minimal entanglement implies a coherent state labeled by a degenerate tetrahedron. Away from these cases, entanglement depends on the geometry in an intricate way.