Topological tensor products and quantum entanglement

The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.The state of two or more particle systems in quantum mechanics is generally expressed as linear superpositions of tensor products of discrete states; and entanglement is connected to the superpositions that cannot be expressed by a single tensor product. On the other hand a discussion of nonlocality is always present in quantum entanglement and furthermore this nonlocality may be revealed via entanglement in the cases of the topological effects in quantum mechanics. In this work, aiming a more systematic and a complementary framework we focus on a comparison of the topological effects of Aharonov-Bohm type whose roots seem to be common. In this context, we will discuss the roles of three topological effects on an entangled quantum mechanical system.

AIP conference proceedingsJan 1, 20184 citations

Topological order and long range quantum entanglements

Xiao-Gang Wen

Bulletin of the American Physical SocietyMar 1, 20120 citations

Tensor-product representations for string-net condensed states

Zheng‐Cheng Gu, Michael Levin, Brian G. Swingle +1 more

We show that general string-net condensed states have a natural representation in terms of tensor product states (TPSs). These TPSs are built from local tensors. They can describe both states with short-range entanglement (such as the symmetry-breaking states) and states with long-range entanglement (such as string-net condensed states with topological/quantum order). The tensor product representation provides a kind of ``mean-field'' description for topologically ordered states and could be a powerful way to study quantum phase transitions between such states. As an attempt in this direction, we show that the constructed TPSs are fixed points under a certain wave-function renormalization-group transformation for quantum states.We show that general string-net condensed states have a natural representation in terms of tensor product states (TPSs). These TPSs are built from local tensors. They can describe both states with short-range entanglement (such as the symmetry-breaking states) and states with long-range entanglement (such as string-net condensed states with topological/quantum order). The tensor product representation provides a kind of ``mean-field'' description for topologically ordered states and could be a powerful way to study quantum phase transitions between such states. As an attempt in this direction, we show that the constructed TPSs are fixed points under a certain wave-function renormalization-group transformation for quantum states.

arXiv:0809.2821Physical Review BFeb 23, 2009153 citations

The entanglement structure of topological quantum phases

Michael Mariën

Jan 1, 20160 citations

Tensor product representation of a topological ordered phase: Necessary symmetry conditions

Xie Chen, Bei Zeng, Zheng‐Cheng Gu +2 more

The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long-range entanglement. However, an important issue arises when we use tensor product states (TPSs) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with ${\mathbb{Z}}_{2}$ topological order, we show that, surprisingly, not all variations in the tensors correspond to the variation in the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations in the tensors to be physical is that they respect certain ${\mathbb{Z}}_{2}$ symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order.The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long-range entanglement. However, an important issue arises when we use tensor product states (TPSs) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with ${\mathbb{Z}}_{2}$ topological order, we show that, surprisingly, not all variations in the tensors correspond to the variation in the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations in the tensors to be physical is that they respect certain ${\mathbb{Z}}_{2}$ symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order.

arXiv:1003.1774Physical Review BOct 20, 201053 citations

{\mathbb{Z}}_{2}

topological order, we show that, surprisingly, not all variations in the tensors correspond to the variation in the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations in the tensors to be physical is that they respect certain

{\mathbb{Z}}_{2}

symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order.