Entanglement entropy ($S_A=-\mathrm{Tr}({\rho }_A\log {\rho }_A)$)

For a bipartite pure state, the entanglement entropy is the Von Neumann entropy of either reduced density matrix. The two reduced states have the same nonzero eigenvalues, so their entropies agree.

For a unit-trace two-dimensional reduced state of a bipartite pure state, determinant $D$ gives eigenvalues $(1\pm \sqrt{1-4D})/2$. Their binary entropy quantifies entanglement. Natural logarithms give a maximum $\ln 2$; base-two logarithms give one bit.

For a bipartite pure state, entanglement entropy is the Von Neumann entropy of either reduced density matrix. When Alice makes a local measurement with pure conditional branches, Bob's reductions average to his original reduction. Concavity of Von Neumann entropy therefore gives ${\sum }_r{p}_{r}S({\rho }_{B,r})\le S({\rho }_B)$. When Bob measures, apply concavity to Alice's reductions. In each pure branch the two reduced entropies coincide by the Schmidt decomposition. Refining measurement outcomes to individual Kraus operators and iterating proves the inequality for arbitrary finite-round LOCC. Discarded local systems can be measured and their outcomes refined as well, so discarding does not evade the bound.

Every finite-dimensional bipartite pure state has a decomposition $|\psi ⟩={\sum }_{j=1}^r{s}_j|a_j⟩|b_j⟩$ with positive Schmidt coefficients $s_j$ and orthonormal families on each subsystem.

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 $p_k$.

An orthonormal subsystem basis appearing in a Schmidt decomposition. Its vectors diagonalize the corresponding reduced density matrix. Degenerate Schmidt coefficients permit a freedom to rotate the paired bases together; zero-weight basis vectors complete the subsystem space without contributing to the state.

The operator Schmidt rank of a bipartite linear operator is its shortest expansion as a sum of product linear operators. Expanding in local orthonormal operator bases gives a coefficient matrix; its matrix rank is the operator Schmidt rank, by a singular value decomposition. It is unchanged by invertible local operator-basis changes. A rank-one projector onto a bipartite pure state of Schmidt rank $d$ has operator Schmidt rank $d^2$: its expansion contains all $d^2$ independent products of local matrix units.

If two parties share a pure resource of Schmidt rank $R$, a fully refined branch of their local operations has Kraus operator $K_{\mu \nu }={\sum }_{j=1}^R\sqrt{p_j}{A}_{\mu j}\otimes B_{\nu j}$. Hence its operator Schmidt rank is at most $R$. Local ancillas, locally adaptive readouts and refined discarded environments are included in the branch operators. Shared classical randomness only mixes such branches. Classical comparison of records does not increase their operator Schmidt rank.

For the pure state ${\lambda }_0|00⟩+{\lambda }_1|11⟩$ with real normalized Schmidt coefficients, the two-body Pauli correlators form $T=\mathrm{diag}(s,-s,1)$, $s=2{\lambda }_0{\lambda }_1$. Thus arbitrary local axes give $⟨(\mathbf{a}\cdot \sigma )\otimes (\mathbf{b}\cdot \sigma )⟩=s(a_x{b}_x-a_y{b}_y)+a_z{b}_z$. The negative $yy$ component follows from the phases in the Pauli operators.

The Schmidt coefficients of a normalized bipartite pure state are the nonnegative numbers $s_j$ in its Schmidt decomposition; they satisfy ${\sum }_j{s}_j^2=1$ and their squares are the nonzero eigenvalues of either reduced density matrix.

The Schmidt rank is the number of nonzero terms in a Schmidt decomposition, equivalently the rank of either reduced density matrix of a bipartite pure state.

The coefficient matrix of a bipartite pure state has rank equal to its Schmidt rank, by singular value decomposition. A local product operation $A\otimes B$ sends it to $AM{B}^{\mathsf{T}}$, whose rank cannot increase. A complete classical transcript of an LOCC protocol still selects one product Kraus operator, so every nonzero branch obeys the bound. Discarding the transcript gives a mixed output with Schmidt number at most the initial rank.

The Schmidt rank of a bipartite pure state cannot increase under local operations, even in a nonzero selected branch. A coefficient matrix $C$ changes under a local product operator to $AC{B}^T$, whose rank is at most that of $C$. For deterministic exact conversions the same conclusion follows from Nielsen's pure-state conversion theorem: at the input rank $r$, the majorization inequality forces all output coefficients after $r$ to vanish.

A normalized bipartite pure state of Schmidt rank $r$ has quantum fidelity at most $\sqrt{r/d}$ with a fixed maximally entangled state on $d\otimes d$. Its coefficient matrix has matrix rank $r$ and Hilbert-Schmidt norm one. Its overlap is the trace of that matrix divided by $\sqrt{d}$; trace duality and the Cauchy-Schwarz inequality bound this by $\sqrt{r/d}$. Uniform Schmidt coefficients on $r$ matching basis pairs attain equality.

The Schmidt number of a mixed bipartite state is the smallest $k$ for which it has a pure-state ensemble whose members all have Schmidt rank at most $k$. Schmidt number one is equivalent to separability.