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Threefold Way for Typical Entanglement.

Haruki Yagi1, Ken Mochizuki1,2, Zongping Gong1

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This summary is machine-generated.

Quantum entanglement spectra with time reversal symmetry can now be described using random matrix theory. Fractionalizing symmetry operators reveals the Laguerre symplectic ensemble (LSE) for half-integer spins, unifying entanglement descriptions.

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Area of Science:

  • Quantum Information Theory
  • Condensed Matter Physics
  • Quantum Chaos

Background:

  • Quantum states without symmetry exhibit entanglement spectra following the Laguerre unitary ensemble (LUE).
  • Integer-spin time reversal symmetry leads to entanglement spectra described by the Laguerre orthogonal ensemble (LOE).
  • Kramers' theorem prohibits standard time-reversal symmetric states for half-integer spins, preventing direct application of LSE.

Purpose of the Study:

  • To develop a theoretical framework for describing entanglement spectra in systems with half-integer spin time reversal symmetry.
  • To generalize this framework to arbitrary symmetry fractionalization and its impact on entanglement spectra.
  • To establish the entanglement counterpart of Dyson's threefold way.

Main Methods:

  • Devising a system where the time reversal operator is fractionalized on subsystems.
  • Applying random orthogonal and unitary matrices to study entanglement spectra.
  • Incorporating general symmetry fractionalization and analyzing resulting degeneracies.

Main Results:

  • Demonstrated that fractionalizing the time reversal operator leads to the Laguerre symplectic ensemble (LSE) for half-integer spins.
  • Showed that general symmetry fractionalization decomposes entanglement spectra into a direct sum of LOE, LUE, and LSE.
  • Identified that degeneracies in the entanglement spectrum depend on the symmetry group's non-Abelian nature and cohomology class.

Conclusions:

  • Established a method to realize LSE in quantum entanglement spectra via symmetry fractionalization.
  • Provided a unified description of entanglement spectra statistics for various symmetries, extending Dyson's threefold way.
  • Opened new avenues for understanding quantum entanglement in systems with complex symmetries.