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A zero-temperature energy-relaxation channel transfers the excited population of a qubit to its ground state. Its Kraus operators are $A_0=\mathrm{diag}(1,\sqrt{1-p})$ and $A_1=\sqrt{p}|0⟩⟨1|$ for $0\le p\le 1$. They satisfy ${\sum }_j{A}_j^{†}{A}_j=I$. The excited population is multiplied by $1-p$ and coherences by $\sqrt{1-p}$. For nonzero $p$, it is not a unital quantum channel.

Choose input eigenvectors $u_i$ and output eigenvectors $v_j$ for a unital quantum channel. Positivity makes the displayed entries nonnegative. Trace preservation makes every column sum one; unitality makes every row sum one. Thus $A$ is a doubly stochastic matrix and the output spectrum satisfies $s=Ar$. The mixing matrix need not be a matrix of squared overlaps of a single unitary, even though its row and column constraints are the same.

A unital quantum channel preserves the identity operator: $\Phi (I)=I$. In equal input and output dimensions, it therefore preserves the maximally mixed state. For amplitude damping channel,Hence the channel is not unital for $p>0$; it is unital only in the identity case $p=0$. Trace preservation involves the opposite products $K_i^{†}{K}_i$ and does not imply unitality.

A unital quantum channel preserves the identity operator: $\Phi (I)=I$. For equal input and output dimensions, this is equivalent to preserving the maximally mixed density operator.

For the amplitude damping channel,Thus the channel is not unital for $p>0$; it is unital only at $p=0$, when it is the identity channel. In particular, it biases the maximally mixed state toward the atomic ground state. The trace-preservation condition involves ${\sum }_j{A}_j^{†}{A}_j$, whereas unitality involves the opposite products ${\sum }_j{A}_j{A}_j^{†}$.

The final printed direction is reversed under the majorization convention stated in the same part. A completely depolarizing unital quantum channel sends a pure qubit state to $I/2$, giving spectra $r=(1,0)$ and $s=(1/2,1/2)$. The proposed $r\prec s$ fails already at the first partial sum, since $1>1/2$. The valid conclusion isHere is its general proof. Choose eigenbases $\rho ={\sum }_i{r}_i|u_i⟩⟨u_i|$ and $\sigma ={\sum }_j{s}_j|v_j⟩⟨v_j|$, and define the eigenvalue mixing matrix of a unital quantum channelIts entries are nonnegative by positivity. Trace preservation gives ${\sum }_j{A}_{ji}=1$ for every column. Unitality gives ${\sum }_i{A}_{ji}=⟨v_j|\Lambda (I)|v_j⟩=1$ for every row. Thus $A$ is a doubly stochastic matrix, and linearity givesThe supplied doubly stochastic characterization of majorization then yields $s\prec r$, completing the intended corrected result. This proves spectral majorization under a unital quantum channel. A unital quantum channel can make a spectrum more uniform; it cannot generally make the output spectrum majorize the input. The counterexample resolves the false printed assertion rather than treating the reversed inequality as a convention change.

Every random unitary channel satisfiesIt is therefore a unital quantum channel.

For mutually orthogonal projectors summing to the identity, a pinching map discards coherences between their ranges while preserving each diagonal block. It is a quantum channel with Kraus operators $P_i$, and a unital quantum channel. A state is fixed by pinching exactly when it commutes with every projector.

A unital quantum channel mixes its input spectrum with a doubly stochastic matrix, so its output spectrum satisfies the majorization relation with its input spectrum. The reverse inequality is generally false: a pure state can be sent to a maximally mixed state. The direction expresses the fact that unital noise can make eigenvalues more uniform. Equality of the two spectra is possible, for example under a unitary channel.