| Schmidt decomposition | |
|---|---|
| Name | Schmidt decomposition |
| Field | Quantum mechanics |
| Introduced | 1907 |
| Introduced by | Erhard Schmidt |
| Related | Singular value decomposition, Schmidt rank |
Schmidt decomposition
The Schmidt decomposition is a canonical form for expressing a vector in a bipartite Hilbert space as a sum of orthogonal product vectors with nonnegative coefficients. It provides a compact representation of pure states in quantum mechanics that makes structure of bipartite correlations and entanglement manifest, and underpins many methods in quantum information science and linear algebra.
For a pure state |ψ⟩ in the tensor product Hilbert space H_A ⊗ H_B, the Schmidt rank-k decomposition states that there exist orthonormal bases { |u_i⟩ } for H_A and { |v_i⟩ } for H_B and nonnegative real numbers { λ_i } (the Schmidt coefficients) such that |ψ⟩ = ∑_{i=1}^r λ_i |u_i⟩ ⊗ |v_i⟩, where r ≤ min(dim H_A, dim H_B) is the Schmidt rank. The squared coefficients { λ_i^2 } form the nonzero eigenvalues of the reduced density matrices ρ_A = Tr_B(|ψ⟩⟨ψ|) and ρ_B = Tr_A(|ψ⟩⟨ψ|). The decomposition is essentially unique up to phase choices for degenerate coefficients. The Schmidt coefficients are identical to the singular values arising from a matrix representation of |ψ⟩ with respect to chosen bases, linking the construction to the Singular value decomposition.
A standard derivation uses the isomorphism between vectors in H_A ⊗ H_B and linear operators from H_A* to H_B (or between H_A and H_B) and applies the Singular value decomposition to that operator. Diagonalization of the reduced density operator ρ_A (a positive semidefinite operator) yields eigenvectors { |u_i⟩ } and eigenvalues λ_i^2; the corresponding partner vectors { |v_i⟩ } are obtained by projecting |ψ⟩ onto |u_i⟩. Key mathematical properties include: - Orthogonality: |u_i⟩ and |v_i⟩ form orthonormal sets in their respective spaces. - Uniqueness: nonzero Schmidt coefficients are unique; degeneracies permit unitary rotations within degenerate subspaces. - Schmidt rank and entanglement monotones: the rank r classifies separability (r = 1 implies a product state). - Relation to spectra: the nonzero spectrum of ρ_A equals that of ρ_B, so both reduced states share the same nonzero eigenvalues.
The Schmidt decomposition directly quantifies bipartite entanglement for pure states. Measures derived from Schmidt coefficients include: - Entanglement entropy (von Neumann entropy S(ρ_A) = −∑ λ_i^2 log λ_i^2), which connects to thermodynamics and statistical interpretations. - Renyi entropies and purity (Tr ρ_A^2) used in many-body physics and studies of thermalization. - Concurrence and Schmidt number as discrete indicators of quantum correlations. Because the Schmidt basis diagonalizes the reduced states, it is central to understanding local operations and classical communication (LOCC) convertibility via majorization relations (Nielsen's theorem) and to the classification of pure-state transformations under entanglement theory.
Schmidt decomposition is widely used across quantum information theory and algorithmic contexts: - State characterization and tomography: simplifies reconstruction of bipartite pure states and guides compressed sensing protocols. - Quantum communication: analysis of quantum teleportation, superdense coding, and entanglement-assisted channels often assumes Schmidt forms for resource states. - Quantum algorithms and simulation: tensor-network methods such as matrix product states (MPS) and density matrix renormalization group (DMRG) rely on truncating small Schmidt coefficients to approximate ground states of condensed matter physics systems. - Quantum optics and experimental platforms: characterization of entangled photon pairs produced by spontaneous parametric down-conversion and entanglement in ion trap or superconducting qubit experiments exploit Schmidt mode decompositions to quantify mode structure. - Resource theories and entanglement distillation protocols depend on Schmidt spectra to evaluate convertibility and yield.
Several extensions broaden the Schmidt concept beyond finite-dimensional pure bipartite states: - Infinite-dimensional systems: Schmidt decomposition can hold for certain separable Hilbert spaces and continuous-variable systems with appropriate mathematical care (e.g., for two-mode squeezed states in quantum optics). - Multipartite systems: there is no unique Schmidt decomposition for more than two subsystems; instead multipartite generalizations include Schmidt measure, tensor rank, and canonical forms like the Schmidt decomposition for three qubits (e.g., GHZ state vs W state classification). - Operator Schmidt decomposition: applied to linear operators (superoperators) acting on bipartite spaces, useful in characterizing quantum channels and gates; relates to Choi–Jamiołkowski isomorphism. - Schmidt number for mixed states: convex-roof constructions and entanglement witnesses generalize the rank concept to quantify mixed-state entanglement.
Simple illustrative cases demonstrate utility: - Product state: for |ψ⟩ = |a⟩⊗|b⟩ the Schmidt decomposition has r = 1 with λ_1 = 1. - Bell state: the maximally entangled two-qubit singlet |Ψ−⟩ = (|01⟩ − |10⟩)/√2 has Schmidt coefficients (1/√2, 1/√2), Schmidt rank 2, and entanglement entropy log 2. - Two-mode squeezed vacuum: in continuous-variable quantum optics, the Schmidt decomposition of photon-number entanglement yields thermal-like Schmidt spectra parameterized by the squeezing parameter, connecting to squeezed state properties. - Matrix-product truncation: given a chain ground state represented as an MPS, computing Schmidt spectra across a bipartition enables low-rank approximations used in DMRG and time-evolution methods. Worked computations usually begin by expressing |ψ⟩ in product bases, building the coefficient matrix, performing an SVD, and extracting λ_i and orthonormal Schmidt vectors; this aligns with linear-algebra software routines used in numerical linear algebra and scientific libraries.
Category:Quantum mechanics Category:Quantum information theory