Question

I know how to calculate EE for a contiguous sub-system using Schmidt decomposition aka SVD. Algorithm 9 on page 11 of this paper. Is there any similar SVD algorithm of restructuring the wave function matrix when the subsystem is not contiguous? One could calculate the partial trace but is there any other way that can be used for many-body numerical calculations?

In a 1D system of L degrees of freedom(e.g. bosons) , the above SVD method assumes a subsystem A=1≤j≤l_A. The subsystem I am interested in is A=j.

Answer 1

The question is about finding a numerical method to calculate the entanglement entropy for a non-contiguous subsystem in a quantum many-body system, potentially using an approach like SVD used for contiguous subsystems, but the student sought alternatives to the partial trace method. Techniques like Matrix Product States (MPS) might be relevant, and understanding the underlying quantum mechanics through the Schrödinger equation and wave functions is important for comprehending the entanglement properties in such systems.

Step-by-step explanation:

The student is asking about a method to calculate the entanglement entropy (EE) for a non-contiguous subsystem within a quantum many-body system, similar to the way it is done using the Schmidt decomposition or Singular Value Decomposition (SVD) for contiguous subsystems. When dealing with a non-contiguous subsystem in a one-dimensional system of L degrees of freedom, a direct application of the SVD may not be straightforward due to the spatial separation of the subsystem parts. Partial tracing over the degrees of freedom not in the subsystem is a standard method, but the student is inquiring about alternative numerical techniques that might be utilized in situations such as this.

While a direct method analogous to the contiguous case SVD is not specified, it's possible that techniques such as Matrix Product States (MPS) or other tensor network methods could be used to represent the state of a non-contiguous subsystem to calculate its entanglement properties. These methods have been shown to be efficient for many numerical calculations in one-dimensional quantum systems. Nonetheless, the specific algorithm or method would depend on the details of the system being studied and the nature of the wavefunction.

In a broader context of quantum mechanics, solving the Schrödinger equation and discussing wave functions such as in the stationary states of a particle in a potential well or box, can provide insights into the quantization of energy and other properties that are relevant in understanding the entanglement in quantum systems.