The set of eigenvalues describing entanglement in quantum systems, often related to geometry and topology.The quantum entanglement spectrum is the set of eigenvalues of the reduced density matrix that describes a subsystem of a larger quantum system. Unlike the entanglement entropy, which is a single number, the full spectrum provides a much richer description of how quantum information is distributed. In many-body physics, the entanglement spectrum often reveals hidden structure, such as topological order or edge states, that is not visible in the entropy alone. It connects quantum information theory to geometry and topology, particularly in the context of the AdS/CFT correspondence, where the spectrum is related to the spectrum of a gravitational theory in a higher-dimensional space.
The concept was introduced by Li and Haldane in 2008, who showed that the entanglement spectrum of a fractional quantum Hall state contains information about the edge physics, a discovery that sparked widespread application in condensed matter and high-energy physics.
To obtain the entanglement spectrum, one divides a quantum system into two parts, A and B, and traces out the degrees of freedom of B to obtain a reduced density matrix for A. The eigenvalues of this matrix, when normalized, form a probability distribution that encodes how the quantum state is shared. The spectrum can be visualized as a set of 'pseudo-energies' that often resemble the energy levels of a boundary theory. In topological phases, the low-lying part of the spectrum mirrors the edge excitations, providing a fingerprint of the phase. The spectrum is computed using techniques like exact diagonalization, tensor networks, or quantum Monte Carlo, and it is a key tool in diagnosing quantum phases.
The entanglement spectrum is used in quantum simulation and materials science to identify topological phases, detect quantum phase transitions, and characterize the entanglement structure of many-body systems, which is essential for designing quantum computers and understanding exotic materials.
In the AdS/CFT correspondence, the entanglement spectrum of a boundary quantum field theory is conjectured to be related to the spectrum of a bulk gravitational theory, providing a concrete link between quantum entanglement and the geometry of spacetime.
| C_{lm l'm'} | variable | Angular power spectrum coefficients — angular power spectrum |
| a_{lm} | variable | Spherical harmonic coefficients of temperature map — spherical harmonic coefficient kelvin |
| ⟨ ⟩ | operator | Ensemble average over realizations — ensemble average |
| * | operator | Complex conjugation — complex conjugate |
| f | variable | frequency of the quantum field oscillation — frequency hertz |
| m | variable | mass of the particle associated with the field — mass kilogram |
| c | constant | universal constant relating space and time in relativity — speed of light = 299792458 metre per second |
| h | constant | fundamental constant relating energy to frequency in quantum mechanics — planck constant = 6.62607015e-34 joule second |
