Quantum operations and CPTP maps

Setup. The system starts in $\hat\rho$, the environment in a pure state $\ket{e_0}$ (if the environment is mixed, purify it first using section 10.1 — no loss of generality). The two start uncorrelated, so the whole is $\hat\rho\otimes\ket{e_0}\bra{e_0}$. Let the whole evolve unitarily under $\hat U$, then trace out the environment:

Expand the trace. Take an orthonormal basis $\{\ket{e_k}\}$ for the environment:

Look at the object $\bra{e_k}\hat U\ket{e_0}$: $\hat U$ acts on “system $\otimes$ environment”, and once the environment states are sandwiched off both ends, what remains is an operator acting on the system alone. Give it a name:

(The intuitive reading: $\hat E_k$ is “the operation the system undergoes in the branch where the environment jumps from $\ket{e_0}$ to $\ket{e_k}$”.) Then

This is the Kraus representation (also called the operator-sum representation), and the $\hat E_k$ are the Kraus operators.

The completeness condition. Unitarity of $\hat U$ gives a constraint:

(the middle step used the completeness of the environment basis, $\sum_k\ket{e_k}\bra{e_k}=\hat 1$).

All three properties are automatically saved. Verify that $\mathcal{E}(\hat\rho)$ is a legitimate density matrix:

• Hermitian: each term $\hat E_k\hat\rho\hat E_k^\dagger$ is its own Hermitian conjugate.
• Unit trace: $\operatorname{Tr}\sum_k\hat E_k\hat\rho\hat E_k^\dagger =\operatorname{Tr}\big(\sum_k\hat E_k^\dagger\hat E_k\,\hat\rho\big)=\operatorname{Tr}\hat\rho=1$ (using cyclic invariance of the trace and the completeness condition — the entire responsibility for trace preservation rests on the completeness condition).
• Positive semi-definite: $\bra{\phi}\hat E_k\hat\rho\hat E_k^\dagger\ket{\phi} =\bra{\phi'}\hat\rho\ket{\phi'}\ge0$, where $\ket{\phi'}=\hat E_k^\dagger\ket{\phi}$.

Unitary evolution is the special case with a single Kraus operator ($\hat E_0=\hat U$); with more than one operator, pure states are generally sent to mixed states — precisely the “shrinking” evolution we were looking for.

On its own, nothing wrong. The transpose $\mathcal{T}(\hat\rho)=\hat\rho^{\mathsf T}$ preserves Hermiticity, preserves the trace, and preserves eigenvalues (a matrix and its transpose have the same spectrum), so positive semi-definite in, positive semi-definite out — it is a positive map. Acting alone on any single qubit, its output is a legitimate density matrix.

Put it inside entanglement and disaster strikes at once. Readers of chapter 9 know a real system may be entangled with someone else. Take the Bell state $\ket{\Phi^+}=\frac{1}{\sqrt2}(\ket{00}+\ket{11})$ and let the transpose act on B only (doing nothing to A); write the combined operation as $\mathcal{I}\otimes\mathcal{T}$.

The Bell state’s density matrix (basis order $\ket{00},\ket{01},\ket{10},\ket{11}$):

Transposing on B swaps B’s indices: matrix elements go $\rho_{(a b),(a'b')}\to\rho_{(a b'),(a' b)}$. The two corner $\frac12$‘s (that is, $\ket{00}\bra{11}$ and $\ket{11}\bra{00}$) get moved into the middle (becoming $\ket{01}\bra{10}$ and $\ket{10}\bra{01}$):

The middle $2\times2$ block $\frac12\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has eigenvalues $\pm\frac12$. So the full spectrum is

A negative eigenvalue has appeared: the output is not a density matrix, and it would predict probability $-\frac12$ for some measurement.

Conclusion. “Preserving positivity for every individual state” is not enough to guarantee “still legitimate when acting on half of an entangled system”. A physical evolution must survive the test “my system might be entangled with anything in the world” — that is what the word “completely” protects. Maps in Kraus form pass the test automatically (repeat the positivity check in the derivation above for $\mathcal{I}\otimes\mathcal{E}$), and the converse holds too (Choi’s theorem, stated here without proof): completely positive + trace-preserving $\Leftrightarrow$ a Kraus representation exists.

A footnote: the transpose map’s “illegitimacy” did not get it thrown away — it became a tool. It flips out negative eigenvalues only on entangled states, so “partial transpose yields a negative spectrum” became a standard entanglement test (the PPT criterion).