Dephasing channel

A dephasing channel suppresses off-diagonal entries of a density matrix in a preferred basis while preserving quantum state populations. Complete dephasing is identical to a projective measurement in that basis with its outcome discarded.

For the Bell correlations versus a classical mixture example, completely dephasing Alice's qubit in the computational basis maps ${\rho }_{\mathrm{Bell}}$ to ${\rho }_{\mathrm{mix}}$. It preserves their perfect $Z$ agreement and each reduced density matrix, yet destroys the $X$ correlation. This is why testing only populations can miss damage to a quantum memory: the stored quantum coherence may have disappeared while every preferred-basis measurement still looks correct. The joint Von Neumann entropy rises from zero to one bit, although each local entropy stays fixed.

Correlated dephasing caused by the same fluctuating $z$ rotation on all qubits. Averaging $U_{\varphi }\rho U_{\varphi }^{†}$ over the unknown angle suppresses quantum coherence between sectors with different total $Z$ eigenvalues. Within one eigenspace, however, every state acquires the same global phase, so its density matrix is preserved. Such an eigenspace is a decoherence-free subspace for this noise model. The assumption concerns the shared rotation; independent local phase noise generally destroys this protection.

For a qubit, computational basis dephasing with complex overlap $c$ maps ${\rho }_{01}$ to $\overline{c}{\rho }_{01}$ and preserves the diagonal entries. Conjugation by the phase gate $P_{\phi }=\mathrm{diag}(1,e^{i\phi })$ maps the same entry to $e^{-i\phi }{\rho }_{01}$. Since these scalar factors commute, their quantum channels commute on every density matrix. More directly, an interaction $|j⟩|e⟩\mapsto |j⟩|e_j⟩$ and a diagonal phase gate give the same joint quantum state in either order on the specified initial environment state. This need not hold for an interaction that changes the quantum state populations.

Different ensemble members accumulating different deterministic phases can shorten their averaged Bloch vector even when each member remains in a pure state. Averaging static frequency offsets gives an effective dephasing channel. It need not represent irreversible phase loss in each individual system; an appropriate refocusing pulse can reverse static offsets. Changing offsets during the sequence requires a temporal noise model.

An environment interaction preserving the computational populations multiplies a qubit's upper off-diagonal density entry by $\overline{c}$. A real phase-flip channel representation exists for every input precisely when $c$ is real, with probability $(1-c)/2$. For $c=r{e}^{i\phi }$, the channel is a phase-flip channel of probability $(1-r)/2$ followed by the diagonal unitary operator $\mathrm{diag}(1,e^{i\phi })$. The coherent phase cannot be discarded just because the populations are unchanged.

In a Hadamard gate, phase gate, Hadamard gate sequence starting from $|0⟩$, let a population-preserving interaction correlate $|j⟩$ with a normalized environment state $|e_j⟩$ between the two Hadamard gates. Use $P_{\phi }=\mathrm{diag}(1,e^{i\phi })$ and $⟨e_0|e_1⟩=v{e}^{i\alpha }$. The environment-valued output-zero coefficient is $(|e_0⟩+e^{i\phi }|e_1⟩)/2$, whose squared norm is the displayed probability. The modulus $v$ controls contrast, while $\alpha$ shifts the quantum interference fringe. For $v=1$ there is no mixing; a deterministic relative quantum phase accounts for the shift.