SchmidtZahl

SchmidtZahl, commonly known in English as the Schmidt number, is a nonnegative integer that characterizes the amount of entanglement in a bipartite quantum state. In quantum information theory it serves as a measure of the complexity of correlations between two subsystems and is used to distinguish different levels of entanglement in mixed states. The concept is named after Erhard Schmidt, whose Schmidt decomposition provides a foundational framework for bipartite pure states. The German term Schmidt-Zahl is sometimes used as well.

For a bipartite pure state |ψ⟩ in a Hilbert space H_A ⊗ H_B, the Schmidt decomposition expresses |ψ⟩ as a sum of Schmidt coefficients, |ψ⟩ = ∑_i √p_i |u_i⟩ ⊗ |v_i⟩, with nonnegative p_i and orthonormal sets {|u_i⟩}, {|v_i⟩}. The Schmidt rank is the number of nonzero Schmidt coefficients and equals the SchmidtZahl for pure states. A higher Schmidt rank indicates greater entanglement, up to the maximum allowed by the subsystem dimensions.

For a mixed state ρ on H_A ⊗ H_B, the SchmidtZahl is defined as the smallest integer r such that ρ can be written as a convex combination ρ = ∑_i p_i |ψ_i⟩⟨ψ_i| with each |ψ_i⟩ having Schmidt rank at most r. The SchmidtZahl satisfies 1 ≤ SchmidtZahl(ρ) ≤ min(dim(H_A), dim(H_B)). A state has SchmidtZahl equal to 1 precisely when it is separable (can be expressed as a mixture of product states); higher values indicate entanglement of increasing complexity.

The SchmidtZahl is used in entanglement theory, including entanglement detection, dimension witnesses, and the study of entanglement structure in quantum information protocols. See also Schmidt decomposition, Schmidt rank, and entanglement.