= Schmidt spectrum
{c}
{title2=$\operatorname{spec}_{>0}(\rho_A)=\operatorname{spec}_{>0}(\rho_B)=\{p_k>0\}$}
The nonzero eigenvalues, including multiplicities, of either reduced density operator of a bipartite pure state. A <Schmidt decomposition> $|\psi\rangle=\sum_k\sqrt{p_k}|a_kb_k\rangle$ gives $\rho_A=\sum_kp_k|a_k\rangle\langle a_k|$ 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 $p_k$.
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