Abstract Multipartite entanglement has a much more complex structure than bipartite entanglement. A state that lacks generic multipartite entanglement is 2-producible, i.e. it can be written as a tensor product of at most 2-partite entangled states. Recently, it has been proved that a tripartite pure state is 2-producible if and only if the gap between the entanglement of purification and its lower bound vanishes. Here, we show that the entanglement of purification gap is insufficient to detect more than tripartite entanglement in 4-partite stabilizer states. We then generalize entanglement of purification to the multipartite cases, and demonstrate that a multipartite pure state is 2-producible if and only if all the generalized entanglement of purification gaps vanish. The generalized entanglement of purification gap quantifies the quantum communication cost for redistributing one part of the system to the others, and also relates to the local recoverability of a multipartite state and the relative entropy between that state and 2-producible states. Moreover, we calculate the generalized entanglement of purification for states satisfying the general Schmidt decomposition, which implies that 4-partite stabilizer states do not necessarily have a general Schmidt decomposition. Our results provide a quantitative characterization of multipartite entanglement in multipartite system, which will promote further investigations and understanding of multipartite entanglement.
Abstract Multipartite entanglement has a much more complex structure than bipartite entanglement. A state that lacks generic multipartite entanglement is 2-producible, i.e. it can be written as a tensor product of at most 2-partite entangled states. Recently, it has been proved that a tripartite pure state is 2-producible if and only if the gap between the entanglement of purification and its lower bound vanishes. Here, we show that the entanglement of purification gap is insufficient to detect more than tripartite entanglement in 4-partite stabilizer states. We then generalize entanglement of purification to the multipartite cases, and demonstrate that a multipartite pure state is 2-producible if and only if all the generalized entanglement of purification gaps vanish. The generalized entanglement of purification gap quantifies the quantum communication cost for redistributing one part of the system to the others, and also relates to the local recoverability of a multipartite state and the relative entropy between that state and 2-producible states. Moreover, we calculate the generalized entanglement of purification for states satisfying the general Schmidt decomposition, which implies that 4-partite stabilizer states do not necessarily have a general Schmidt decomposition. Our results provide a quantitative characterization of multipartite entanglement in multipartite system, which will promote further investigations and understanding of multipartite entanglement.
Projective norms are capable of measuring entanglement of multipartite quantum states. However, typically, the explicit computation of these distance-based geometric entanglement monotones is very difficult even for finite dimensional systems. Motivated by the significance of Schmidt decompositions for our quantitative understanding of bipartite quantum entanglement, a generalization of this concept to multipartite scenarios is proposed, in the sense that generalized Schmidt decomposability of a multipartite pure state implies that its projective norm can be calculated in a simple way analogous to the bipartite case. Thus, this concept of generalized Schmidt decomposability of multipartite quantum states is linked in a natural way to projective norms as entanglement monotones. Therefore, it may not only be a convenient tool for calculations, but may also shed new light onto the intricate features of multipartite entanglement in an analogous way as the 'classical' Schmidt decomposition does for bipartite quantum systems.
Abstract Quantum entanglement and quantum entropy are crucial concepts in the study of multipartite quantum systems. In this work, we show how the notion of concurrence vector, re-expressed in a particularly useful form, provides new insights and computational tools for the analysis of both. In particular, using this approach for a general multipartite pure state, one can easily prove known relations in an easy way and to build up new relations between the concurrences associated with the different bipartitions. The approach is also useful to derive sufficient conditions for genuine entanglement in generic multipartite systems that are computable in polynomial time. From an entropy-of-entanglement perspective, the approach is powerful to prove properties of the Tsallis-2 entropy, such as the subadditivity, and to derive new ones, e.g., a modified version of the strong subadditivity which is always fulfilled; thanks to the purification theorem these results hold for any multipartite state, whether pure or mixed.
A bstract We introduce a new information-theoretic measure of multipartite correlations Δ P , by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjecture a holographic dual of multipartite entanglement of purification Δ W , as a sum of minimal areas of codimension-2 surfaces which divide the entanglement wedge into multi-pieces. We prove that this geometrical quantity satisfies all properties we proved for the multipartite entanglement of purification. These agreements strongly support the Δ P = Δ W conjecture. We also show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement. We discuss potential saturation of multipartite squashed entanglement onto multipartite mutual information in holographic CFTs and its applications.
For bipartite pure and mixed quantum states, in addition to the quantum mutual information, there is another measure of total correlation, namely, the entanglement of purification. We study the monogamy, polygamy, and additivity properties of the entanglement of purification for pure and mixed states. In this paper, we show that, in contrast to the quantum mutual information which is strictly monogamous for any tripartite pure states, the entanglement of purification is polygamous for the same. This shows that there can be genuinely two types of total correlation across any bipartite cross in a pure tripartite state. Furthermore, we find the lower bound and actual values of the entanglement of purification for different classes of tripartite and higher-dimensional bipartite mixed states. Thereafter, we show that if entanglement of purification is not additive on tensor product states, it is actually subadditive. Using these results, we identify some states which are additive on tensor products for entanglement of purification. The implications of these findings on the quantum advantage of dense coding are briefly discussed, whereby we show that for tripartite pure states, it is strictly monogamous and if it is nonadditive, then it is superadditive on tensor product states.
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.
Despite the fundamental importance of quantum entanglement in many-body systems, our understanding is mostly limited to bipartite situations. Indeed, even defining appropriate notions of multipartite entanglement is a significant challenge for general quantum systems. In this work, we initiate the study of multipartite entanglement in a rich, yet tractable class of quantum states called stabilizer tensor networks. We demonstrate that, for generic stabilizer tensor networks, the geometry of the tensor network informs the multipartite entanglement structure of the state. In particular, we show that the average number of Greenberger-Horne-Zeilinger (GHZ) triples that can be extracted from a stabilizer tensor network is small, implying that tripartite entanglement is scarce. This, in turn, restricts the higher-partite entanglement structure of the states. Recent research in quantum gravity found that stabilizer tensor networks reproduce important structural features of the AdS/CFT correspondence, including the Ryu-Takayanagi formula for the entanglement entropy and certain quantum error correction properties. Our results imply a new operational interpretation of the monogamy of the Ryu-Takayanagi mutual information and an entropic diagnostic for higher-partite entanglement. Our technical contributions include a spin model for evaluating the average GHZ content of stabilizer tensor networks, as well as a novel formula for the third moment of random stabilizer states, which we expect to find further applications in quantum information.