Coherence-breaking channels and coherence sudden death

Quantum noise is ubiquitous to quantum systems as they incessantly interact with their surroundings and results in degrading useful resources such as coherence for single quantum systems and quantum correlations for multipartite systems. Given the importance of these resources in various quantum information processing protocols, it is of the utmost importance to characterize how deteriorating a particular noise scenario (quantum channel) is in reference to a certain resource. Here we develop a theory of coherence-breaking channels for single quantum systems. Any quantum channel on a single quantum system will be called a coherence-breaking channel if it is an incoherent channel and maps any state to an incoherent state. We explicitly and exhaustively characterize these coherence-breaking channels. Moreover, we define the coherence-breaking indices for incoherent quantum channels and present various examples to elucidate this concept. We further introduce the concept of coherence sudden death under noisy evolutions and make an explicit connection of the phenomenon of coherence sudden death with the coherence-breaking channels and the coherence-breaking indices together with various suggestive examples. Furthermore, for higher-dimensional Hilbert spaces, we establish the typicality of the dynamics of coherence under any incoherent quantum channel exploiting the concentration of measure phenomenon.

Figure 1

The relationship between the sets of entanglement-breaking channels (EBTs), quantum-classical channels (QCs), and coherence-breaking channels (CBCs).

Figure 2

Consider a qubit state $\rho =\frac{1}{2}\left(\mathbb{I}+\vec{r}·\vec{\sigma }\right)$ with $\vec{r}={(0.3,0.5,0.2)}^T$. Also, take $J=10$ in the stroboscopic Markovian process of both Examples 3 and 4. In Example 3, take $\alpha =0.5$. We have $C_{l_1}\left(\rho \right)=0.5830,C_{l_1}\left(\mathrm{\Phi }\left[\rho \right]\right)=0.25$, and $C_{l_1}\left({\mathrm{\Phi }}^J\left[\rho \right]\right)=0$ for $J\ge 2$. The solid line in above figure plots $C_{l_1}\left({\mathrm{\Phi }}^J\left[\rho \right]\right)$ as a function of $J$ and shows the coherence sudden death at the second iteration of the channel. In Example 4, take $p=0.7$ and $t=1$. Here $C_{l_1}\left(D_{p,t}^J\left[\rho \right]\right)=p^{\frac{J}{2}}{C}_{l_1}\left(\rho \right)$. The dotted line in the above figure plots $C_{l_1}\left(D_{p,t}^J\left[\rho \right]\right)$ as a function of $J$ and shows no sudden death of coherence as the coherence is nonzero at every iteration of the channel $D_{p,t}$.

Figure 3

The figure plots $S_0=\frac{S\left(\frac{1+{\lambda }_1}{2},\frac{1-{\lambda }_1}{2}\right)}{S\left(\frac{1+z}{2},\frac{1-z}{2}\right)}-S\left(\frac{1+c}{2},\frac{1-c}{2}\right)$ as a function of $c$ for a fixed value of $z=0.6$. $S_0\ne 0$ implies that the factorization relation of the form $C_r(\mathrm{\Phi }\left(\rho \right))=C_r\left(\rho \right)C_r(\mathrm{\Phi }\left(|\psi \rangle \langle \psi |\right))$ does not hold in general.