Schmidt spectrum (${\mathrm{spec}}_{>0}({\rho }_A)={\mathrm{spec}}_{>0}({\rho }_B)=\{p_k>0\}$)

The nonzero eigenvalues, including multiplicities, of either reduced density operator of a bipartite pure state. A Schmidt decomposition $|\psi ⟩={\sum }_k\sqrt{p_k}|a_k{b}_k⟩$ gives ${\rho }_A={\sum }_k{p}_k|a_k⟩⟨a_k|$ and the corresponding expression for ${\rho }_B$ by partial trace. Thus unequal marginal spectra rule out a pure joint state. The amplitudes $\sqrt{p_k}$ are often called Schmidt coefficients; the spectrum here uses the probabilities $p_k$.