Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 37 / 3 / b

Expand an initial bipartite pure state in local orthonormal bases asBy the Schmidt decomposition, or equivalently the singular value decomposition of $C$, its Schmidt rank is $\mathrm{rank}C$. In a specified local measurement branch, the product Kraus operator $A\otimes B$ changes the coefficient matrix toIndeed the coefficient of $|k⟩|l⟩$ is ${\sum }_{i,j}{A}_{ki}{C}_{ij}{B}_{lj}$. The elementary matrix rank inequality givesNormalizing a nonzero branch multiplies $C^{\prime }$ by a scalar and therefore leaves its matrix rank unchanged.

Now refine a complete LOCC transcript to record every individual Kraus operator. Although later choices depend on the earlier classical outcomes, fixing a transcript fixes all those choices. Composing the local operations therefore gives a single product Kraus operator $A_{\tau }\otimes B_{\tau }$ in that branch. Each nonzero branch obeys the same Schmidt-rank contraction under product operators. Local auxiliary systems prepared independently are included by local isometries and do not add entanglement. Discarding a local auxiliary system is included by resolving its partial trace in an orthonormal basis; this again yields refined product Kraus operators.

If the protocol produces a specified pure state deterministically, its output density operator is a sum of positive rank-one branch density operators. Every nonzero branch vector must be proportional to that output vector: its squared overlap with any vector orthogonal to the output sums to zero, so each such overlap is separately zero. Thus the output Schmidt rank equals that of each surviving branch, andThis proves monotonicity of Schmidt rank under LOCC, even for a nonzero branch selected by postselection. If the classical transcript is instead discarded and the result is mixed, the refined branch vectors supply an ensemble with Schmidt ranks at most $r_{\mathrm{in}}$; by definition its Schmidt number is at most $r_{\mathrm{in}}$. This is the correct mixed-state extension, rather than a bound on the ranks of its reduced density matrices. In finite dimensions, any limiting exact protocol obeys the same pure-state bound because the set of coefficient matrices of rank at most $r_{\mathrm{in}}$ is closed, being defined by vanishing minors.