| entropy of entanglement | |
|---|---|
| Name | Entropy of entanglement |
| Quantity | Information-theoretic measure |
| Unit | bit (when log base 2) |
| Used in | Quantum information science |
entropy of entanglement
The entropy of entanglement is a quantitative measure of quantum correlations between parts of a composite quantum system. For a bipartite pure state it equals the von Neumann entropy of either subsystem's reduced density matrix and plays a central role in Quantum information theory and the study of quantum entanglement. It underpins resource theories of entanglement, informs the performance of protocols such as quantum teleportation and entanglement distillation, and connects to many-body phenomena in condensed matter physics.
The entropy of entanglement is defined for a bipartite pure state |ψ⟩ ∈ H_A ⊗ H_B by S(ρ_A), where ρ_A = Tr_B(|ψ⟩⟨ψ|) is the reduced density operator on subsystem A and S(ρ) = −Tr(ρ log ρ) is the von Neumann entropy. This definition is invariant under local unitary operations and symmetric: S(ρ_A) = S(ρ_B). The measure arises from the formalism of density matrixs and the partial trace operation introduced in foundational work by John von Neumann and later formalized in information-theoretic contexts by researchers such as Nicolas Cerf and Charles H. Bennett.
Alternative formulations express the entropy of entanglement in terms of the eigenvalue spectrum {λ_i} of ρ_A: S = −∑_i λ_i log λ_i. For two-qubit pure states the Schmidt decomposition yields a single nontrivial Schmidt coefficient, enabling closed-form expressions. The measure uses a logarithm base choice to set the unit (bits for base 2, nats for base e).
Entropy of entanglement is a special case of entanglement measures tied directly to the von Neumann entropy; it differs from other measures like entanglement of formation, relative entropy of entanglement, negativity (quantum) and concurrence which generalize to mixed states or provide operationally distinct monotones. In resource-theoretic terms it is the unique asymptotic entanglement measure for bipartite pure states, coinciding with the distillable entanglement and entanglement cost in the asymptotic limit as shown in works by Bennett et al. Connections to Shannon entropy appear in classical-quantum correspondences; however, quantum nonlocal correlations captured by entanglement entropy do not have a simple classical analogue.
In many-body physics the entropy of entanglement is compared to measures like Rényi entropys, which are parametrized generalizations S_α(ρ) = (1/(1−α)) log Tr(ρ^α) used to probe entanglement spectra in models studied by groups at institutions such as Institute for Quantum Information and Matter and in landmark papers by Holzhey, Larsen, and Wilczek and Calabrese and Cardy relating entanglement scaling to conformal field theory.
For a pure state with Schmidt decomposition |ψ⟩ = ∑_i √λ_i |u_i⟩_A |v_i⟩_B, the reduced density ρ_A = ∑_i λ_i |u_i⟩⟨u_i| and the entropy of entanglement is S = −∑_i λ_i log λ_i. Practical computation therefore reduces to finding Schmidt coefficients via singular value decomposition of the state coefficients with respect to product bases, a technique used in numerical methods such as density matrix renormalization group (DMRG).
Special cases include maximally entangled states (e.g., Bell states of Bell pair form) where all nonzero λ_i are equal and S = log d for local dimension d, and product states where one λ_i = 1 and S = 0. For two-qubit states the concurrence C relates to Schmidt coefficients and can be used to derive S via eigenvalues of the reduced density matrix.
Entropy of entanglement satisfies key properties: invariance under local unitaries, nonincreasing under local operations and classical communication (LOCC), additivity on tensor-product pure states, and continuity. It serves as an operational currency: asymptotically, one ebit (entropy 1 bit) corresponds to a Bell pair, and the entropy quantifies how many Bell pairs can be distilled or are required for state formation under LOCC as formalized in entanglement concentration and entanglement dilution protocols by Bennett et al.
The measure also provides insight into phenomena such as area laws in quantum many-body systems: ground states of gapped local Hamiltonians often exhibit entanglement entropy scaling with the boundary area rather than volume, a concept developed in studies involving Xiao-Gang Wen, Matthew Hastings, and others. Violations of area laws characterize critical systems with logarithmic corrections predicted by conformal field theory.
Entropy of entanglement is central to protocols including quantum teleportation, superdense coding, and entanglement-assisted classical capacity proofs. It quantifies resource requirements in quantum error correction codes (e.g., stabilizer codes developed by Daniel Gottesman), and bounds performance in quantum key distribution analyses by groups such as those around Artur Ekert and Charles H. Bennett.
In quantum computing models, entanglement entropy serves as a diagnostic for the hardness of classical simulation and for identifying quantum phase transitions in simulator platforms like ion trap quantum computers and superconducting qubits experiments at institutions such as IBM and Google Quantum AI.
For mixed bipartite states, entropy of entanglement does not uniquely extend; several measures exist including entanglement of formation (E_F), distillable entanglement (E_D), and relative entropy of entanglement (E_R). For multipartite systems, various entropic quantities (e.g., global entanglement, entanglement entropy across bipartitions) and concepts like multipartite entanglement classes (GHZ, W states) are used. The entanglement spectrum—eigenvalues of reduced density matrices—provides richer structure than a single scalar entropy and has been employed in topological order studies by researchers at Perimeter Institute and MIT.
Experimentally, entanglement entropy is estimated via state tomography for small systems, using projective measurements to reconstruct ρ_A and compute S. For larger systems, randomized measurement protocols such as ``classical shadows'' and randomized benchmarking permit estimation of Rényi entropies; techniques pioneered by groups at Harvard and University of Innsbruck leverage randomized unitary ensembles and statistical reconstruction. Interferometric methods can directly access second Rényi entropy via swap tests implemented in photonic, cold-atom, and superconducting platforms. Practical challenges include scaling tomography, mitigating noise, and distinguishing classical correlations from genuine entanglement in mixed states.
Category:Quantum information theory Category:Quantum entanglement