| von Neumann entropy | |
|---|---|
| Name | von Neumann entropy |
| Unit | nat (dimensionless) |
| Field | Quantum mechanics |
| Introduced by | John von Neumann |
| Introduced in | 1932 |
von Neumann entropy
The von Neumann entropy is a quantum generalization of Shannon entropy that quantifies the information content or mixedness of a quantum state expressed by a density operator. It plays a central role in quantum information theory, quantum statistical mechanics, and the study of entanglement and decoherence, providing a bridge between microscopic quantum descriptions and thermodynamic properties. The quantity is fundamental to characterizing irreversibility, resource theories, and the limits of quantum communication.
For a quantum system described by a density matrix ρ acting on a Hilbert space H, the von Neumann entropy is defined as S(ρ) = −Tr(ρ log ρ). If ρ has spectral decomposition ρ = ∑_i p_i |ψ_i⟩⟨ψ_i| with eigenvalues p_i ≥ 0, then S(ρ) = −∑_i p_i log p_i, directly paralleling the Shannon entropy of the eigenvalue distribution. Typical logarithm bases are natural (nats) or base 2 (bits). The definition relies on the trace and the functional calculus for Hermitian operators. For pure states (ρ = |ψ⟩⟨ψ|) S(ρ)=0, while the maximum entropy for a d-dimensional system is log d, attained by the maximally mixed state I/d.
Von Neumann entropy satisfies numerous important properties used in formal proofs and operational tasks. It is nonnegative and unitary invariant: S(UρU†)=S(ρ) for any unitary U. The entropy is concave in ρ, reflecting mixing increases entropy. Key inequalities include subadditivity S(ρ_AB) ≤ S(ρ_A) + S(ρ_B) and strong subadditivity S(ρ_ABC) + S(ρ_B) ≤ S(ρ_AB) + S(ρ_BC), proved by Elliott H. Lieb and Mary Beth Ruskai. The equality conditions relate to product and conditional independence structures and underpin the quantum analogues of classical information inequalities used in quantum coding theory and entanglement theory. The entropy is continuous with bounds provided by Fannes' inequality and refined continuity estimates by Audenaert.
Von Neumann entropy reduces to classical Shannon entropy when ρ is diagonal in a fixed measurement basis, thereby recovering classical information content. The quantum relative entropy D(ρ||σ)=Tr[ρ(log ρ − log σ)] generalizes the Kullback–Leibler divergence and is related to distinguishability and hypothesis testing results such as the Quantum Stein's lemma. Conditional entropy in the quantum setting can be negative, a phenomenon directly tied to entanglement as exemplified by the Bell state and other maximally entangled states. Von Neumann entropy underlies capacities of quantum channels (e.g., Holevo bound, quantum channel capacity) and concepts like coherent information and entanglement measures including entanglement entropy and entanglement of formation.
Operationally, von Neumann entropy quantifies compressibility of quantum ensembles via Schumacher compression and bounds rates in quantum source coding. In thermodynamics, it connects to the von Neumann formulation of quantum statistical mechanics where equilibrium states maximize entropy subject to constraints, recovering Gibbs and canonical ensembles via the maximum entropy principle. It measures decoherence and information loss when systems interact with environments modeled by open quantum dynamics such as Lindblad equation evolutions. In quantum thermodynamic protocols, it governs work extraction limits and the formulation of quantum versions of Landauer's principle.
In quantum thermodynamics von Neumann entropy appears in fluctuation theorems, resource theory formulations of thermal operations, and studies of equilibration and thermalization in closed quantum systems (e.g., eigenstate thermalization hypothesis). In condensed matter physics and many-body theory, entanglement entropy (von Neumann entropy of reduced density matrices) is a diagnostic for quantum phase transitions, area laws, and topological order, used in tensor network algorithms such as density matrix renormalization group (DMRG) and matrix product states. It is central to studies of topological entanglement entropy and critical scaling in conformal field theories described by Virasoro algebra methods.
Computing S(ρ) often reduces to diagonalizing ρ to obtain eigenvalues p_i. For finite-dimensional systems this is done numerically via linear algebra packages (e.g., routines in LAPACK or Eigen). Analytic results exist for simple systems: two-level systems (qubits), thermal states of harmonic oscillators, and free fermion/boson lattices where correlation matrix methods yield entanglement spectra. In many-body systems, entanglement entropy is computed using exact diagonalization, DMRG, or quantum Monte Carlo with replica-trick methods. Examples include entropy of a single qubit mixed state, thermal Gibbs states S(ρ)=β⟨H⟩+log Z for Hamiltonian H, and the logarithmic scaling S∼(c/3) log L in 1D critical systems with central charge c from conformal field theory.
Extensions include quantum Rényi entropies S_α(ρ)= (1/(1−α)) log Tr[ρ^α], which interpolate between min- and max-entropies and converge to von Neumann entropy as α→1; these are used in one-shot information theory and quantum cryptography. The quantum conditional entropy S(A|B)=S(AB)−S(B) and quantum mutual information I(A:B)=S(A)+S(B)−S(AB) quantify correlations and are central to entanglement theory and channel capacities. Relative entropy measures such as Umegaki relative entropy and variants like sandwiched Rényi relative entropy underpin operational tasks in hypothesis testing and resource theories. These generalizations link to mathematical results by Araki, Umegaki, Petz, and researchers in quantum information such as Mark M. Wilde.
Category:Quantum information theory Category:Entropy