Entanglement Entropy

Entanglement entropy is the primary quantitative tool for measuring how entangled two parts of a quantum system are. Given a composite quantum system divided into subsystems A and B, the entanglement entropy S(A) is the von Neumann entropy of the reduced density matrix obtained by tracing out subsystem B: S(A) = -Tr(rho_A log rho_A). For a pure global state, S(A) equals S(B), even when A and B have different sizes, making it a well-defined, symmetric measure of the entanglement across any bipartition.

Computing entanglement entropy

The most direct route to entanglement entropy is the Schmidt decomposition. Any pure bipartite state can be written as a sum over orthonormal pairs: |psi> = sum_i sqrt(lambda_i) |a_i>|b_i>, where the lambda_i are non-negative real numbers summing to 1. These Schmidt coefficients are the eigenvalues of rho_A. The entanglement entropy is then S(A) = -sum_i lambda_i log(lambda_i). For a product state, only one Schmidt coefficient is nonzero, giving S = 0. For a maximally entangled state of local dimension d, all d coefficients equal 1/d and S reaches its maximum value of log(d).

Area laws and volume laws

One of the most important structural results in quantum many-body physics concerns how entanglement entropy scales with subsystem size. For gapped ground states of local Hamiltonians in one dimension, the entanglement entropy across any bipartition is bounded by a constant independent of system size; this is the area law. The boundary between A and B in one dimension is a single point, so the entropy scales with the boundary area (a constant) rather than the volume of A. Critical systems at a phase transition and systems with long-range interactions can exhibit logarithmic corrections or full volume-law scaling, where S(A) grows proportionally to the size of A.

This distinction has direct computational consequences. States obeying area laws can be efficiently represented as matrix product states (MPS) or, in higher dimensions, projected entangled pair states (PEPS). These tensor network representations have bond dimensions that grow polynomially rather than exponentially, enabling classical simulation. Volume-law states require exponentially large tensors and are generically hard to simulate classically.

Connections to quantum simulation advantage

Entanglement entropy provides a quantitative explanation for why quantum simulation is hard for classical computers and where quantum computers are expected to provide genuine advantage. Strongly correlated systems such as high-temperature superconductors, frustrated magnets, and quantum spin liquids exhibit high entanglement entropy across large bipartitions, especially during time evolution or near phase transitions. This high entanglement is precisely the regime where tensor network methods break down and exact diagonalization becomes intractable due to exponential Hilbert space growth.

Role in quantum error correction

Entanglement entropy plays an inverted role in quantum error correction. A good quantum error-correcting code deliberately encodes logical information in highly entangled states, so that any local measurement reveals nothing about the logical content. This is quantified by the requirement that the reduced density matrix of any correctable subsystem is maximally mixed, meaning the entanglement entropy of that subsystem is maximal. The threshold theorems of fault-tolerant quantum computing rely on this structure: the logical information is spread nonlocally enough that local errors cannot destroy it.

See also

• Von Neumann Entropy
• Entanglement
• Density Matrix
• Tensor Network
• Quantum Error Correction