Completely Positive Maps and Normality

Complete positivity requires positivity after every finite matrix amplification. In infinite-dimensional quantum mechanics, this algebraic property must be paired with normality on the bounded-observable side to obtain an operation on trace-class states. The canonical Completely Positive Maps article explains physical admissibility with reference systems; this page identifies the spaces, duality, and continuity hypotheses.

Helpful background. Normal states supplies ultraweak continuity; trace-norm duality pairs bounded observables and trace-class operators.

Matrix amplification defines complete positivity

For Hilbert spaces $\mathcal H_{\mathrm{in}},\mathcal H_{\mathrm{out}}$, consider a bounded linear Heisenberg map

$$\Psi:\mathcal B(\mathcal H_{\mathrm{out}}) \longrightarrow\mathcal B(\mathcal H_{\mathrm{in}}).$$

It is completely positive (CP) if, for every $n\geq1$, entrywise application $[A_{ij}]\mapsto[\Psi(A_{ij})]$ preserves positivity on operator matrices. Such matrices act on $\mathbb C^n\otimes\mathcal H$. Positivity for $n=1$ is ordinary positivity; requiring all finite $n$ expresses compatibility with arbitrary finite reference systems.

For a normal CP map, these finite tests extend to infinite separable references by finite-rank approximation of positive trace-class states. The topology is essential: one does not insert an infinite, nonnormalizable maximally entangled vector into a finite-dimensional Choi formula.

Every map $\Psi(A)=K^*AK$ is CP, since an amplified positive operator is conjugated by $I_n\otimes K$. Sums and suitable normal limits of these maps retain complete positivity.

Normality and the Schrödinger predual

A normal bounded map is ultraweakly continuous. Equivalently here, it has a unique bounded preadjoint

$$\Phi:\mathcal S_1(\mathcal H_{\mathrm{in}}) \longrightarrow\mathcal S_1(\mathcal H_{\mathrm{out}})$$

such that

$$\operatorname{Tr}(\Phi(T)A)=\operatorname{Tr}(T\Psi(A)).$$

Complete positivity passes between this predual pair. The condition $\Psi(I)\leq I$ means that $\Phi$ is trace nonincreasing on positive inputs; equality means trace preservation. Hence normal CP subunital maps describe measurement branches, and normal CP unital maps describe channels in the Heisenberg picture.

The arrow direction reverses because an output observable is pulled back to an input observable, while an input state is pushed forward to an output state. This prevents a common dimension mismatch when input and output Hilbert spaces differ.

Normality cannot be omitted. Let $\omega_s$ be a nonnormal state on $\mathcal B(\mathcal H)$ and set $\Psi(A)=\omega_s(A)I$. This map is unital and CP: positivity of each scalar matrix $[\omega_s(A_{ij})]$ follows from positivity of the functional. But it is not normal. If it had a trace-class predual, applying any input state would produce a density operator representing $\omega_s$, contradicting nonnormality.

A positive map that fails the reference test

Matrix transposition $T(A)=A^{\mathsf T}$ in a fixed qubit basis is positive and unital. For the normalized Bell projector $P_\Omega$, however,

$$(T\otimes\mathrm{id})(P_\Omega)=\tfrac12F,$$

where $F$ swaps the two qubits. The antisymmetric vector has swap eigenvalue $-1$, so the output has eigenvalue $-1/2$. Transposition fails $2$-positivity.

This finite-dimensional corner already witnesses failure on any larger Hilbert space containing the qubit subspace. Enlarging the ambient space cannot repair a violation detected by one allowed reference. Complete positivity is stronger than testing output positivity on a list of unentangled input states.

The Schwarz inequality for unital CP maps

The block operator

$$\begin{pmatrix} A^*A&A^*\\ A&I \end{pmatrix}$$

is positive, being the product of a column operator and its adjoint. Apply $2$-positivity and use $\Psi(I)=I$. The Schur-complement condition gives

$$\Psi(A)^*\Psi(A)\leq\Psi(A^*A).$$

Thus unital CP maps obey an operator version of the classical inequality between the square of a mean and the mean square. For an effect, positivity and unitality also ensure that its image remains an effect.

These facts imply trace-distance contraction of the predual channel: its adjoint maps every optimizing effect into the allowed effect interval. This connects analytic structure with the operational trace-distance bound.

Exercises

1. For a trace-nonincreasing operation $\Phi$ with dual $\Psi$, show that its success effect is $E=\Psi(I)$.

Solution

The probability of the branch is $\operatorname{Tr}\Phi(\rho)=\operatorname{Tr}(\rho\Psi(I))$. Complete positivity gives $E\geq0$, while trace nonincrease gives $E\leq I$. This is the effect associated with an outcome of a quantum instrument.

2. Does the Schwarz inequality alone prove that a positive map is completely positive?

Solution

No. The argument establishes a consequence of $2$-positivity for unital maps. It does not reconstruct positivity of every matrix amplification. Complete positivity remains an all-matrix-level requirement unless an additional theorem, such as the finite-dimensional Choi criterion with its dimension assumptions, applies.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987 — normal maps and duality of operator algebras. Publisher.
• Busch, Paul, Pekka Lahti, Juha-Pekka Pellonpää, and Kari Ylinen, Quantum Measurement, Springer, 2016 — completely positive operations and instruments. Publisher.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018 — finite-dimensional CP maps and reference-system tests. Author manuscript.