Symmetry-Protected Topological Entanglement

Abstract

We propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the "SPT-entanglement", is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A, B and the boundaries of the system. In the case of 1-dimensional systems we prove that in the limit where A and B are large and far from each other compared to the correlation length, the SPT-entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the non-trivial phases. Moreover, we show that the SPT-entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT-entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to a new algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.

V4: close to the published version. V3: The paper is rewritten (24 pages, 8 Figures). Several new sections are added to the paper: (i) An example about (perturbed) cluster Hamiltonian, (ii) A discussion on the implications of our results in the context of measurement-based quantum computation. (iii) A preliminary section reviewing basic concepts (Entanglement, symmetry, SPT order). in PRB (2017)

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• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Symmetry protected topological order at nonzero temperature
• Quantum circuit complexity of one-dimensional topological phases
• Chiral central charge from a single bulk wave function
• Symmetry-protected sign problem and magic in quantum phases of matter
• Probing sign structure using measurement-induced entanglement
• Detecting subsystem symmetry protected topological order via entanglement entropy
• Quantum computational advantage with string order parameters of 1D symmetry-protected topological order
• Visualizing Quasiparticles from Quantum Entanglement for general 1D phases
• Disentangling quantum matter with measurements
• Inaccessible entanglement in symmetry protected topological phases
• Stable Symmetry-Protected Topological Phases in Systems with Heralded Noise
• Jones index, secret sharing and total quantum dimension
• Universal resources for quantum computing
• Symmetry protected entanglement in random mixed states
• String orders in the Luttinger liquid phase of one-dimensional spin-1/2 systems
• Quantum teleportation implies symmetry-protected topological order
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Detection of 2D SPT Order with Partial Symmetries