Von Neumann entropy and purification

Let the nonnegative eigenvalues of a normalized finite-dimensional density operator be lambda values summing to one. Define entropy by the negative sum of each eigenvalue times its base-two logarithm, with zero times log zero assigned its limiting value zero. The result is measured in bits. It vanishes precisely for a pure state and reaches log base two of dimension for the maximally mixed state. Multiplication by Boltzmann's constant and conversion to natural logarithms gives a thermodynamic entropy convention, which we do not use here.

A projective measurement in the eigenbasis has outcome Shannon entropy equal to this spectral entropy. Other complete rank-one projective bases can produce larger outcome entropy. For example, a pure plus state has zero von Neumann entropy but yields one bit in the zero–one basis. This distinction prevents a common mistake: inserting measurement frequencies from a single arbitrary basis into the spectral formula does not reconstruct the entropy of an unknown quantum state.

Write a spectral decomposition on system A and introduce orthonormal reference states on factor R. Weight each product eigenvector by the square root of the corresponding eigenvalue. The resulting normalized joint vector purifies the original density operator. In the outer product, cross terms carry reference operators with different labels. Taking their reference trace gives zero because the reference basis is orthonormal, leaving exactly the original weighted projectors on A.

The partial trace is defined by preserving every expectation of an A-only observable. In components, sum matrix elements whose two reference indices agree; do not merely select the upper block of the joint matrix. A reference dimension at least the rank of the state is sufficient. Purifications are not unique: a unitary on the reference leaves the A marginal unchanged. More generally the corresponding statement uses an isometry when reference spaces have different dimensions.

For a pure bipartite state, both reduced states share the same nonzero Schmidt eigenvalues. Their equal entropy quantifies pure-state entanglement across that specified partition. A product pure state gives zero; a maximally entangled pair of qubits gives one bit locally while the joint entropy remains zero. Entropy here measures inaccessible correlations with the other factor rather than uncertainty about which global pure vector was prepared.

For a mixed joint state, the local entropy alone is not an entanglement measure. A fair classical mixture of the product states zero-zero and one-one has the same one-bit local entropies as a Bell pair, but its joint entropy is one bit and it is separable. Even an uncorrelated product of maximally mixed qubits has those local entropies. Additional global information is therefore necessary to distinguish entanglement, classical correlation, and independent noise. The Hilbert-space partition and allowed operations are part of any entanglement claim.