Tensor Network Efficiently Representing Schmidt Decomposition of Quantum Many-Body States.

Area of Science:

• Quantum Many-Body Physics
• Quantum Information Science
• Computational Physics

Background:

• Characterizing entanglement in quantum many-body states is computationally challenging due to exponential complexity scaling with system size (N).
• Efficient access to the entanglement structure, particularly the Schmidt decomposition, is crucial for understanding and simulating complex quantum systems.

Purpose of the Study:

• To develop an efficient method for representing the Schmidt decomposition of quantum many-body states.
• To introduce the Schmidt tensor network state (Schmidt TNS) for linear complexity scaling with system size.
• To enable efficient full-state sampling and analysis of both finite and infinite quantum systems.

Main Methods:

• Representing Schmidt coefficients (entanglement spectrum) and transformations using tensor networks (TNs).
• Encoding Schmidt coefficients in a positive-definite matrix product state (MPS).
• Utilizing local unitary tensors to form TNs for transformations and imposing translational invariance for infinite systems.

Main Results:

• Demonstrated the validity of Schmidt TNS by simulating a frustrated quasi-one-dimensional spin model.
• Showed that the MPS encoding Schmidt coefficients exhibits weak entanglement, even for highly entangled states.
• Achieved linear complexity scaling with system size for Schmidt decomposition representation.

Conclusions:

• The Schmidt TNS provides an efficient representation of quantum state entanglement, overcoming exponential complexity.
• The weak entanglement of the MPS for Schmidt coefficients justifies the method's efficiency.
• This approach promises an exponential speedup for full-state sampling tasks in quantum many-body systems.