Tensor products and completely positive maps

The algebraic tensor product combines finite sums of elementary products, but completing arbitrary C*-algebras requires a C*-norm. The minimal tensor norm comes from faithful Hilbert-space representations and their spatial tensor product. The maximal norm allows commuting representations on a common Hilbert space. These completions need not agree for arbitrary algebras. Matrix algebras and commutative C*-algebras are nuclear, so their minimal and maximal tensor norms agree with any other C*-algebra; no such claim is made for every algebra.

A linear map Phi from A to B is completely positive when the entrywise map on matrices over A preserves positivity for every positive integer matrix size. Size one is ordinary positivity, so complete positivity implies positivity. The converse fails because matrix-level positivity detects correlations with an auxiliary system. The transpose on M₂ is positive by ordinary spectral reasoning but fails at amplification size two. Testing only separate positive elementary tensors would miss the entangled input causing the failure.

Stinespring's theorem says a completely positive map from a unital C*-algebra A to B(H) can be written as V-star pi(a)V for a *-representation pi on a larger Hilbert space and a bounded V from H into that space. When the map is unital, V is an isometry. This theorem is stated rather than fully proved; its positivity-based quotient construction resembles GNS with matrix-valued rather than scalar-valued averages. A compressed representation need not remain multiplicative after compression.

In finite dimensions, an observable map of the form sum of K_j-star a K_j is completely positive. It is unital exactly when the sum of K_j-star K_j equals the identity. The dual state map sends rho to the sum of K_j rho K_j-star, and that same identity makes it trace-preserving. Distinguish the two map directions before checking a formula. A map may erase coherence while preserving probabilities and positivity, which is useful precisely because it need not be an algebra homomorphism.

Let Omega be the normalized two-qubit vector with equal coefficients on 00 and 11. Its rank-one density operator is positive. Transposing only the second subsystem turns it into one half of the flip operator, which exchanges the two tensor factors. The antisymmetric unit vector is an eigenvector with eigenvalue minus one half. Thus the amplified transpose sends a positive input to a nonpositive output, proving that ordinary transpose is not completely positive.

This counterexample does not imply transposing a single positive matrix creates negative eigenvalues; it does not. The obstruction appears only with the untouched auxiliary factor and its correlated input. It also does not say every positive map has a Kraus form: complete positivity is what supports that representation in the matrix setting. Tensor products, channels, and entangled tests therefore need to be developed together rather than assuming scalar positivity automatically survives adjoining another system.