It is well known that, despite the misleading imagery conjured by the name, entanglement in a multipartite system cannot be understood in terms of pair-wise entanglement of the parts. Indeed, there are only pairs of systems, but the number of qualitatively distinct types of entanglement scales exponentially in . A good way to think about this is to recognize that a quantum state of a multipartite system is, in terms of parameters, much more akin to a classical probability distribution than a classical state. When we ask about the information stored in a probability distributions, there are lots and lots of “types” of information, and correlations can be much more complex than just knowing all the pairwise correlations. (“It’s not just that A knows something about B, it’s that A knows something about B conditional on a state of C, and that information can only be unlocked by knowing information from either D or E, depending on the state of F…”).
However, Gaussian distributions (both quantum and classical) are described by a number of parameters that grows on quadratically with the number of variables. The pairwise correlations really do tell you everything there is to know about the quantum state or classical distribution. The above paper makes me wonder to what extent we can understand multipartite Gaussian entanglement in terms of pairs of modes. They have shown that this works at a single level, that entanglement across a bipartition can be decomposed into modewise entangled pairs. But since this doesn’t work for mixed states, it’s not clear how to proceed in understanding the remain entanglement within a partition. My intuition is that there is a canonical decomposition of the Gaussian state that, in some sense, lays bare all the multipartite entanglement it has in any possible partitioning, in much the same way that the eigendecomposition of a matrix exposes its the inner workings.
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