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Between quantum mutual information and coherent information, which one is more similar to classical mutual information? I understand that both measures have some similarities to classical mutual information, but I'm wondering which one captures these similarities more closely. Additionally, how do these measures differ in their respective contexts?

The Wikipedia page for coherent information states:

[coherent information] attempts to describe how much of the quantum information in the state will remain after the state goes through the channel. In this sense, it is intuitively similar to the mutual information of classical information theory

Is the quantum mutual information not the quantum analogue of the classical mutual information? and if so, is it less similar to the classical mutual information than the coherent information is to the classical mutual information?

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  • $\begingroup$ Well the quantum mutual information is a direct generalisation of the mutual information. If you use classical states to encode your probability distributions then it will give you back exactly the mutual information. I don't see how much `closer' you can get. $\endgroup$
    – Rammus
    Feb 26, 2023 at 18:46
  • $\begingroup$ @Rammus well, one could argue that the QMI isn't that close to the mutual information between classical random variables, except for the case of classical states where the two notions coincide. It quantifies information rates achievable with entangled resources. I'd say quantities such as the Holevo information, or even better the accessible information, are closer to the purely classical notion (albeit they're all just different quantities really, so the question isn't entirely well-defined) $\endgroup$
    – glS
    Feb 26, 2023 at 21:41
  • $\begingroup$ But what does it even mean for it to be close to the MI when you plug in quantum states? It's just a very poorly-defined question. $\endgroup$
    – Rammus
    Feb 26, 2023 at 22:51
  • $\begingroup$ On the contrary, I think this is a well-defined question. In the context of quantum information, we constantly refer to the classical setting to look for inspiration for introducing new concepts, while acknowledging that sometimes there are concepts in quantum information that do not have counterparts in the classical setting. We constantly compare and contrast quantum vs classical, and say things like "unlike the classical setting, the quantum conditional entropy can be negative". In this question, I am interested in understanding as to why or why not QMI or CI is similar to CMI. $\endgroup$
    – Josh
    Feb 26, 2023 at 23:11

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You are asking about the quantum analogue of a classical quantity, but quantum information contains features that have no classical analogue and so you won't find an objective answer. Here are some various ideas though:

  1. QMI reduces to CMI: A plain answer is that QMI is more similar to classical MI because QMI reduces to MI if you restrict yourself to diagonal (classical) density matrices. On the other hand, the coherent information is defined with respect to a purifying system, and therefore requires a notion of entanglement that does not exist in classical Shannon theory.

  2. Communication: A more operational answer is that QMI is analogous to MI in the context of transmitting classical data over quantum channels, while coherent information is more analogous (by design) for transmitting quantum data over quantum channels.

    • If Alice wants to transmit some classical information to Bob over a quantum channel by sending him the state $\rho_X$ instead of the classical symbol $X$, then Holevo's theorem says that the QMI between these classical and quantum variables upper bounds the classical MI between $X$ and any choice of measurement for Bob; this in turn bounds the classical capacity of quantum channel. In this way, the QMI is being used as a stand-in for the classical MI that appears in Shannon's noisy channel theorem.

    • Meanwhile, coherent information plays a similar role to classical MI for bounding transmission of quantum information over quantum channels. Here the definition of quantum capacity is very different from classical capacity because it involves entanglement, but it is otherwise upper bounded by a term involving coherent information in a similar way that classical capacity is upper bounded by classical MI. I suspect this is what the Wikipedia article was hinting at.

  3. Hypothesis testing: In the setting where you have variables $X$ and $Y$ where the goal is to determine $X$ given $Y$, the QMI plays a similar role to classical MI if we think of $X$ as labeling the states in a quantum ensemble and $Y$ as measurements. Then, our knowledge of $X$ given $Y$ is bounded by $I(X:Y)$ (e.g. Fano's inequality). This relationship is used to derive sample complexity for some tomography tasks, in a similar way that classical MI is sometimes used to derive sample complexity for some compressed sensing tasks.

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