What are the prominent visualizations used to depict large, entangled states and in what context are they most commonly applied?
What are their advantages and disadvantages?
In Verifying Genuine High-Order Entanglement the following graphs represent entangled qudits
In an answer to 'Alternative to Bloch sphere to represent a single qubit' @Rob references Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates which states
The Majorana representation for spin−$s$ systems has found widespread applications such as determining geometric phase of spins, representing $N$ spinors by $N$ points, geometrical representation of multi-qubit entangled states, statistics of chaotic quantum dynamical systems and characterizing polarized light.
The paper also includes this style of representation for qudits
An n-cube can be projected inside a regular 2n-gonal polygon by a skew orthogonal projection
This method seems to allow for the complexity of entanglement to be visualized in a scalable fashion.
The ZX-calculus is a graphical language for dealing with linear maps of qubits, and it can in particular represent any state of qubits. Basically, ZX-diagrams are tensor networks, but there is an additional set of rewrite rules that allows you to manipulate them graphically. On the Wikipedia page you can find an example of how to prove that a certain quantum circuit indeed implements a GHZ-state. It has also been used to reason about Measurement-Based Quantum Computing, because it allows you to straightforwardly reason about graph states.
In PyZX (disclaimer: I am a lead developer) we use automated graph rewriting to reason and prove results with ZX-diagrams involving thousands of vertices, and we can visualize circuits and states on dozens of qubits.
My personal view:
Yes, large entangled states can be visualized using quantum bayesian networks. See
Factorization of Quantum Density Matrices According to Bayesian and Markov Networks, by Robert R. Tucci (obviously I am the author here)
Python tools for analyzing both classical and quantum Bayesian Networks (Disclaimer: artiste-qb.net is my company)
Other people will probably advise using Tensor Networks instead of quantum Bayesian nets. This begs the question: How do Quantum Bayesian Networks and Tensor Networks compare? I have thought about that and gathered my thoughts in this blog post.
First lines of blog post:
A question I am often asked is what is the difference between tensor networks and quantum Bayesian networks, and is there any advantage to using one over the other.
When dealing with probabilities, I prefer quantum Bayesian networks because b nets are a more natural way of expressing probabilities (and probability amplitudes) whereas tensor nets can be used to denote many physical quantities other than probabilities so they are not tailor made for the job as b nets are. Let me explain in more detail for the technically inclined.
One can consider bipartite entanglement for the two sides of a partition, of a quantum bayesian network. One can write nice inequalities for such bipartite entanglements. See, for example, Entanglement Polygon Inequality in Qubit Systems, Xiao-Feng Qian, Miguel A. Alonso, Joseph H. Eberly.
One can also try to define a measure of n-partite entanglement for n>2, where n is the number of nodes of a quantum Bayesian net. See, for example, Verifying Genuine High-Order Entanglement, Che-Ming Li, Kai Chen, Andreas Reingruber, Yueh-Nan Chen, Jian-Wei Pan.