Tensor Networks for Entanglement Evolution

Meznaric, Sebastian; Biamonte, Jacob

doi:10.1002/9781118742631.ch17

Cited by 2 publications

(3 citation statements)

References 20 publications

(36 reference statements)

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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]. In the string diagram notation, a tensor is a graphical shape with a number of input legs (or 'arms') pointing up, and output legs pointing down 6 .…”

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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Section: Penrose Graphical Notation For Tensor Networkmentioning

confidence: 99%

“…This has also been understood as a diagrammatic form of map-state duality underlying bipartite entanglement evolution[29].…”

mentioning

confidence: 99%

Tensor network methods for invariant theory

Biamonte¹,

Bergholm²,

Lanzagorta³

2013

J. Phys. A: Math. Theor.

Self Cite

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“…Here an entangled state |ψ acted on by a map A can instead be viewed as a map ψ acting on a state |A . 6 This is a diagrammatic form of map-state duality underlying bipartite entanglement evolution [29,30]. See e.g.…”

Section: The Fully Antisymmetric Tensor From Example 1 Has An Interes...mentioning

confidence: 99%

Tensor Networks in a Nutshell

Biamonte,

Bergholm

2017

Preprint

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104

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Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of 3-edge-colorings of 3-regular planar graphs.

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Tensor Networks for Entanglement Evolution

Cited by 2 publications

References 20 publications

Tensor network methods for invariant theory

Tensor network methods for invariant theory

Tensor Networks in a Nutshell

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