Tensor rank and entanglement of pure quantum states

Abstract

The rank of a tensor is analyzed in context of description of entanglement of pure states of multipartite quantum systems. We discuss the notions of the generic rank of a tensor with d indices and n levels in each mode and the maximal rank of a tensor of these dimensions. Other variant of this notion, called border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. As entanglement of a given quantum state depends on the way the total system is divided into subsystems, we introduce a notion of 'partitioning rank' of a tensor, which depends on a way how the entries forming the tensor are treated. In particular, we analyze the tensor product of several copies of the n-qubit state |Wn and analyze its partitioning rank for various splittings of the entire space. Some results concerning the generic rank of a tensor are also provided.

An Algebraic-Geometric Characterization of Tripartite Entanglement

“…Interested reader can learn more about the tensor rank in the context of quantum entanglement in Ref. [32].…”

On the complexity of finding tensor ranks

“…In the last fifty years it became clear that multiarrays with more than two indices, known as tensors, are vital tools in data processing [19], mathematical biology [1], numerical linear algebra [28], quantum physics [4], theoretical computer science [5] and theoretical mathematics [18]. Formally, we define a d-tensor, as a multiarray with d 3 indices.…”