Quantum Channels and Dilations

2.1 Complete positivity

2.1.1 An entangled input is a stronger test

A channel \(\mathcal N:A\to B\) is linear, completely positive, and trace preserving. Linearity respects preparation mixtures. Trace preservation keeps total probability one. Complete positivity requires \(\operatorname{id}_R\otimes\mathcal N\) to preserve positivity for every reference system, because a device must work when its input is part of a larger experiment. The reference need not be available to the device.

Ordinary transposition is a useful counterexample. It preserves the eigenvalues and trace of an isolated density matrix. Apply it to the second half of \(|\Phi\rangle=(|00\rangle+|11\rangle)/\sqrt2\), however. The resulting operator is one half of the swap operator. Its antisymmetric eigenvector \((|01\rangle-|10\rangle)/\sqrt2\) has eigenvalue \(-1/2\). This is not a state. Positivity alone therefore fails the physical consistency test.

We fix the Choi convention explicitly. Let \(|\Omega\rangle=\sum_i|i\rangle_R|i\rangle_A\), without normalization, and define \(J_{\mathcal N}=(\operatorname{id}_R\otimes\mathcal N)(|\Omega\rangle\langle\Omega|)\). Complete positivity is equivalent to \(J_{\mathcal N}\ge0\); trace preservation is equivalent to \(\tr_BJ_{\mathcal N}=I_R\). Thus \(\tr J_{\mathcal N}=d_A\), not one. The normalized Choi state is \(J/d_A\). The reconstruction formula is \(\mathcal N(X)=\tr_R[(X^T\otimes I)J]\), in the basis used for \(|\Omega\rangle\).

2.2 Kraus operators and environments

2.2.1 Where the missing coherence goes

A Kraus representation writes \(\mathcal N(X)=\sum_kK_kXK_k^\dagger\), with \(\sum_kK_k^\dagger K_k=I\). It is completely positive because each term stays positive even after adjoining a reference. Trace preservation follows by moving factors around the trace. Conversely, a positive Choi matrix can be spectrally decomposed and its eigenvectors reshaped into Kraus matrices; this proves the finite-dimensional representation theorem.

For amplitude damping, use \(K_0=\operatorname{diag}(1,\sqrt{1-\gamma})\) and \(K_1=\sqrt\gamma|0\rangle\langle1|\), where \(0\le\gamma\le1\). An input with entries \(\rho_{00}=a,\rho_{01}=b,\rho_{11}=1-a\) becomes

Define an isometry \(V|\psi\rangle=\sum_kK_k|\psi\rangle|k\rangle_E\). Completeness gives \(V^\dagger V=I\); discarding \(E\) yields the channel. Discarding \(B\) instead defines a complementary channel, which records what the environment learns. Kraus representations are not unique, so their labels are not automatically physical detector outcomes. An actual measurement of the environment supplies an instrument and a classical record.

The equivalence of these representations is treated in Watrous, Chapter 2. We use the isometry later to compare information reaching the receiver and environment.

2.3 Comparing devices

2.3.1 A reference can expose hidden errors

Comparing channels on a few chosen states does not certify their behavior on every input. A stronger distance allows a reference: \(\|\mathcal N-\mathcal M\|_\diamond=\max_{\rho_{RA}}\|(\operatorname{id}_R\otimes(\mathcal N-\mathcal M))(\rho_{RA})\|_1\), where for channel differences a reference of input dimension suffices. Equal-prior single-use discrimination succeeds with probability \(1/2+\|\mathcal N-\mathcal M\|_\diamond/4\). This characterization is a theorem, not a conclusion from a small test set.

Consider identity versus complete dephasing on a qubit. Both act identically on \(|0\rangle\) and \(|1\rangle\); basis-state testing misses the error entirely. On \(|+\rangle\), the outputs are \(|+\rangle\langle+|\) and \(I/2\), whose trace distance is one half. This already permits success 0.75. It proves a lower bound on the optimized device-discrimination success. To establish optimality one still needs an upper bound.

For a sequence of devices, the triangle inequality and contraction by channels imply a useful telescoping bound: replacing \(m\) devices whose diamond distances are at most \(\delta_i\) changes the overall channel by at most \(\sum_i\delta_i\). This is conservative but applies to entangled and adaptive inputs when the surrounding operations are channels. State-specific accuracy cannot simply be substituted into this worst-case statement.