Theorem: Quantum information is not destroyed by complete unitary evolution

Let $J:C\to {\mathcal{H}}_{\mathrm{in}}$ be an isometry, let $|\Omega ⟩$ be a fixed normalized ancillary state, and let $U$ be unitary on the complete physical input space. After any fixed unitary regrouping of the output, set

Then:

1. $V^{†}V=I_C$;

2. the complete channel $\mathcal{V}(X)=VX{V}^{†}$ has a CPTP left inverse;

3. for every trace-class operator $X$, including operators on the code and an arbitrary reference,

   $${\Vert (V\otimes I)X(V^{†}\otimes I)\Vert }_1={\Vert X\Vert }_1;$$

4. every logical coherence and every code-state trace distance are preserved;

5. for a Bell input, the reference and the complete output have mutual information $2{\log }_{2}d$.

These assertions are conditional on the stated unitary dynamics and complete output space.

Proof. The composition of an isometry, a normalized pure preparation, and a unitary is an isometry, proving the first item.

Let $P=V{V}^{†}$, and choose any code density operator $\tau$. Define

Both terms are completely positive. Their traces sum to $\mathrm{Tr}Y$. Thus $\mathcal{L}$ is CPTP, and

An isometric embedding leaves the nonzero singular values of an operator unchanged. Equivalently, its modulus is the isometric embedding of the original modulus on the range, with zero on the orthogonal complement. Taking its trace proves the third item, also after tensoring with a reference. Applying this identity to differences of density operators proves trace-distance preservation. Applying the defining isometry to $E_{ij}$ proves preservation of all matrix units.

Finally,

is pure, and both reduced states have nonzero eigenvalues $1/d$. Therefore

All three entropies in this equality are finite, even if the complete output space is infinite dimensional.