Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 33 / 3

Choose $|0⟩$ as the ground state and $|1⟩$ as the excited state. The relevant quantum channel is the amplitude damping channel. Its Kraus operators areThey satisfy $K_0^{†}{K}_0+K_1^{†}{K}_1=I$. The $K_0$ branch corresponds to no emitted photon: the ground-state amplitude is unchanged and the excited-state amplitude is attenuated. The $K_1$ branch corresponds to photon emission and transition to the ground state. These are unnormalized branch states; the squared norm or matrix trace of each branch gives its probability.

Model the environment's vacuum and one-photon states by $|0_E⟩,|1_E⟩$. A unitary dilation of amplitude damping acts on the initially vacant environment asThese images are orthonormal. One unitary completion takes $U|0,1_E⟩=-\sqrt{p}|1,0_E⟩+\sqrt{1-p}|0,1_E⟩$ and fixes $|1,1_E⟩$. Taking environment matrix elements $⟨k_E|U|0_E⟩$ gives $K_0,K_1$.

Evolve $\rho \otimes |0_E⟩⟨0_E|$ by $U$ and take the partial trace over the environment. Orthogonality of the two environment records removes the cross-branch terms, giving ${\mathcal{A}}_p(\rho )=K_0\rho K_0^{†}+K_1\rho K_1^{†}$. Entrywise,Excited population is transferred to the ground state, while coherence is reduced by the square root of the survival probability.

Successive uses of the same quantum channel mean coupling to a fresh vacuum environment each time, or resetting the environment between uses. They do not mean repeatedly applying the dilation unitary to one unreinitialized two-state environment. Direct composition givesInductively, with $q_n=1-(1-p{)}^n$,For $0<p\le 1$, this tends to the pure ground state $|0⟩⟨0|$, whose Von Neumann entropy is zero. For $p=0$, the channel is the identity: the state remains $\rho$ and its Von Neumann entropy remains $S(\rho )$. Thus the repeated amplitude damping limit must include this no-decay exception.

A zero-Von Neumann entropy output means there is no uncertainty about the final atomic state. It does not mean that the receiver has gained information about the initial preparation: all initial states have the same limiting output, so their distinguishability has been erased from the atom. Under the global unitary evolution, information is carried into the environment. Without observing an emission record, this is a nonselective dissipative process, not an inference about which input was sent. Indeed, starting from the excited pure state, one intermediate output has Von Neumann entropy equal to the binary entropy of $p$, which is positive for $0<p<1$; the atom's Von Neumann entropy need not decrease monotonically during the approach to the final ground state.

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.