# Advantages of Density in Tensor Network Geometries for Gradient-Based Training
## 2 Tensor Network Structure  
### Bond Dimension and Correlations  
Entanglement is a crucial property of quantum systems that distinguishes them from systems with only classical properties [26]. It is related to the correlations between different parts of the system, and while it has been extensively studied in the case of bipartitions, characterizing and quantifying multipartite entanglement in general is still a profoundly complex topic [27; 28]. For a quantum state \(\ket{\psi}\) with density matrix \(\rho=\ket{\psi}\bra{\psi}\), subsystems A and B of a physical state \(\rho\) are not entangled (and are thus completely uncorrelated) when they can be written down as a tensor product \(\rho=\rho_{A}\otimes\rho_{B}\)[29]. Otherwise, this decomposition requires a sum of different product states, and \(A\) exhibits entanglement with \(B\). This can be related to the Schmidt decomposition of a quantum state [29], which for a bipartition \(A/B\) is also  
\[\ket{\psi}=\sum_{i}^{\chi_{S}}\lambda_{i}\ket{\phi_{i}}_{A}\otimes\ket{\phi_{ i}}_{B}. \tag{1}\]  
When looking for the tensor product decomposition on \(A/B\) of the density matrix of such a state, \(\rho=\ket{\psi}\bra{\psi}\), it is clear that the number of terms in the Schmidt decomposition dictates how many tensor product terms are needed. We call the number of states needed in this sum the Schmidt number \(\chi_{S}\). Sometimes, we do not need many elements in the decomposition, so \(\chi_{S}\) is small, which can happen, for example, when the entanglement between \(A\) and \(B\) is small. Tensor networks [30] can encode high-dimensional tensors into the product of smaller, low-dimensional ones. This can be generally useful for data encoding [31; 32] in the classical setting, where they are sometimes known as tensor trains [33], although they are more commonly used in the simulation of quantum systems [15]. In the quantum setting, the high-dimensional tensor is commonly a quantum state of \(n\) sites, which, before decomposing, needs to store \(d^{n}\) coefficients, with \(d\) representing the dimensionality of each site. For example, quantum digital computing uses qubits with \(d=2\). Such decomposition is advantageous whenever correlations between tensors that are far apart in the network are not too high, as it means that those sites present a small amount of entanglement. Then, a bipartition separating these tensors, following Equation (1), does not need a large dimension \(\chi_{S}\) to encode the full state, and the sum of memory required by all the small tensors can be smaller than that required by the large, contracted tensor. The low-dimensional tensors can be connected through bonds in various geometries, which can be useful depending on the entanglement structure of the system they encode, although networks with more complexity and expressive power will entail higher performance costs. One of the advantages of tensor networks is the flexibility of their graphic notation. Each tensor is represented by a small node, usually round-shaped, and the bonds between tensors are lines that connect these nodes. These are called virtual bonds, or virtual indices, as they depend on the geometry of the tensor network and not necessarily the properties of the system they encode. If their dimension is as large as the Schmidt number \(\chi_{S}\) for the bipartition of the two parts of the system they connect, the tensor network is a faithful encoding of the system. We denote the maximum of the dimensions among all virtual bonds as the bond dimension \(\chi\). The dimension of the system, on the other hand, is encoded in the physical bonds (indices), which are connected to a node on one side and free on the other. A few examples can be seen in Figure 1. The original high-dimensional tensor can be recovered by contracting all virtual bonds. A tensor network diagram is directly equivalent to an equation between the original tensor and the small tensors, which makes the contractions explicit.