Tensor Networks for Entanglement Evolution

Meznaric, Sebastian; Biamonte, Jacob

doi:10.48550/arxiv.1204.3599

Cited by 1 publication

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“…Contributions on the topic we found influential can be found in [23][24][25][26][27]. In our previous work, we have adapted the graphical notation and surrounding methods to describe generalized quantum circuits [11], tensor network states [10][11][12], open quantum systems [28,29] as well as decidability in algorithms based on tensor contractions [30].…”

Section: Penrose Graphical Notation For Tensor Networkmentioning

confidence: 99%

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Tensor network methods for invariant theory

Biamonte¹,

Bergholm²,

Lanzagorta³

2013

J. Phys. A: Math. Theor.

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Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state factorization) when studying polynomial invariants. As our main example, we consider invariants of matrix product states. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the Rényi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as matrix product states, contain excellent tools that can be applied in the study of invariants.

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