In quantum information theory, the distance between two quantum channels is often measured using the diamond norm. There are also a number of ways to measure distance between two quantum states, such as the trace distance, fidelity, etc. The Jamiokowski isomorphism provides a duality between quantum channels and quantum states.This is interesting, to me at least, because the diamond norm is notoriously hard to calculate, and the Jamiokowski isomorphism would seem to imply some correlation between distance measures of quantum channels and quantum states. So, my question is this: Is there any known relation between the distance in the diamond norm and the distance between the associated states (in some measure)?
Is there any connection between the diamond norm and the distance of the associated states?
quantum computing;it.information theory;quantum information
For a quantum channel $\Phi$, let us write $J(\Phi)$ to denote the associated state:$$ J(\Phi) = \frac{1}{n} \sum_{1\leq i,j \leq n} \Phi(|i \rangle \langle j|) \otimes |i \rangle \langle j|. $$Here we are assuming that the channel maps $M_n(\mathbb{C})$ (i.e., $n\times n$ complex matrices) to $M_m(\mathbb{C})$ for whatever choice of positive integers $n$ and $m$ you like.  The matrix $J(\Phi)$ is sometimes called the Choi matrix or Choi-Jamiolkowski representation of $\Phi$, but it is more frequent that those terms are used when the $\frac{1}{n}$ normalization is omitted.Now, suppose that $\Phi_0$ and $\Phi_1$ are quantum channels.  We may define the diamond norm distance between them as$$ \| \Phi_0 - \Phi_1 \|_{\Diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 \otimes \operatorname{Id}_k)(\rho) \|_1 $$where $\operatorname{Id}_k$ denotes the identity channel from $M_k(\mathbb{C})$ to itself, $\| \cdot \|_1$ denotes the trace norm, and the supremum is taken over all $k \geq 1$ and all density matrices $\rho$ chosen from $M_{nk}(\mathbb{C}) = M_n(\mathbb{C}) \otimes M_{k}(\mathbb{C})$.  The supremum always happens to be achieved for some choice of $k\leq n$ and some rank 1 density matrix $\rho$.(Note that the above definition does not work for arbitrary mappings, only those of the form $\Phi = \Phi_0 - \Phi_1$ for completely positive maps $\Phi_0$ and $\Phi_1$.  For general mappings, the supremum is taken over all matrices with trace norm 1, as opposed to just density matrices.)If you don't have any additional assumptions on the channels, you cannot say too much about how these norms relate aside from these coarse bounds:$$ \frac{1}{n} \| \Phi_0 - \Phi_1 \|_{\Diamond} \leq \| J(\Phi_0) - J(\Phi_1) \|_1 \leq \| \Phi_0 - \Phi_1 \|_{\Diamond}. $$For the second inequality, one is essentially settling for the specific choice$$ \rho =  \frac{1}{n} \sum_{1\leq i,j \leq n} |i \rangle \langle j| \otimes |i \rangle \langle j| $$rather than taking the supremum over all $\rho$.  The first inequality is a bid tougher, but it would be a reasonable assignment question for a graduate course on quantum information.  (At this point I should thank you for your question, because I fully intend to use this question in the Fall offering of my quantum information theory course.)You can achieve either inequality for an appropriate choice of channels $\Phi_0$ and $\Phi_1$, even under the additional assumption that the channels are perfectly distinguishable (meaning $\| \Phi_0 - \Phi_1 \|_{\Diamond} = 2$).