# conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open conjectures, and I would like to pose them as open questions to the mathematics community on this forum. Perhaps there is a simple resolution, but we have considered known approaches and it appears that new tools are necessary for solving these problems (or maybe known tools could work?).

## Background

Let $\mathcal{H}$ denote a Hilbert space, and let $\mathcal{D}(\mathcal{H})$ denote the set of density operators acting on this Hilbert space (positive semi-definite operators with trace one). (It suffices for our purposes to consider finite-dimensional Hilbert spaces, but of course the infinite-dimensional case is interesting as well.) We are interested in "three-party" density operators $\rho_{ABC}$ acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$. Let $\rho_A$ denote the "marginal" density operator of $\rho_{ABC}$, obtained by taking a partial trace over the spaces $\mathcal{H}_B \otimes \mathcal{H}_C$: $$\rho_A = \operatorname{Tr}_{BC} \{ \rho_{ABC}\}$$ In a similar way, we can define $\rho_{B}$, $\rho_{C}$, $\rho_{AB}$, $\rho_{BC}$, and $\rho_{AC}$.

We define the "Renyi conditional mutual information" of order $\alpha \geq 0$ as follows: $$I_{\alpha}(A;B|C)_{\rho} \equiv \frac{1}{\alpha - 1} \log \operatorname{Tr} \{ \rho_{ABC}^{\alpha} \rho_{AC}^{(1-\alpha) / 2} \rho_C^{(\alpha-1)/2} \rho_{BC}^{1-\alpha} \rho_C^{(\alpha-1)/2} \rho_{AC}^{(1-\alpha) / 2} \}$$ In the above, identity operators are implicit, so that, e.g., $\rho_C^{(\alpha-1)/2} = I_{AB} \otimes \rho_C^{(\alpha-1)/2}$, where $I_{AB}$ is the identity operator acting on $\mathcal{H}_A \otimes \mathcal{H}_B$. For more motivation for the above quantity, please consult our paper.

We can easily prove that the above quantity is "monotone under a local quantum operation acting on system $B$" for all $\alpha \in [0,2]$, i.e., that the following inequality holds $$I_{\alpha}(A;B|C)_{\rho} \geq I_{\alpha}(A;B|C)_{\omega},$$ where $\omega_{ABC} \equiv (\operatorname{id}_A \otimes \mathcal{M}_B \otimes \operatorname{id}_C)(\rho_{ABC})$, $\operatorname{id}$ denotes the identity map, and $\mathcal{M}_B$ is a completely positive trace preserving linear map acting on system $B$. A proof follows by applying a standard approach with the Lieb concavity theorem and the Ando convexity theorem (proof is detailed in the paper).

## Conjectures/Questions

We have several conjectures which (to fit the MathOverflow format) could be converted into questions in obvious ways (e.g., are the statements true? please provide an argument or counterexample, etc.).

Conjecture 1:

The quantity $I_\alpha(A;B|C)$ is "monotone under a local quantum operation acting on system $A$" for all $\alpha \in [0,2]$, i.e., the following inequality holds $$I_{\alpha}(A;B|C)_{\rho} \geq I_{\alpha}(A;B|C)_{\tau},$$ where $\tau_{ABC} \equiv (\mathcal{N}_A \otimes \operatorname{id}_B \otimes \operatorname{id}_C)(\rho_{ABC})$ and $\mathcal{N}_A$ is a completely positive trace preserving linear map acting on system $A$. (Numerical evidence indicates that this conjecture should be true.)

Conjecture 2:

The Renyi conditional mutual information of order $\alpha$ should be monotone increasing in the Renyi parameter $\alpha$, i.e., if $0 \leq \alpha \leq \beta$, then $$I_{\alpha}(A;B|C)_{\rho} \leq I_{\beta}(A;B|C)_{\rho}.$$ (Numerical evidence indicates that this conjecture should be true as well, and we furthermore have proofs that it is true in some special cases (for $\alpha$ in a neighborhood of one and when $\alpha + \beta = 2$).)

These are the basic conjectures, but please consult our paper for more general forms of them. Solving the first conjecture would be very interesting for us, as it would establish the Renyi conditional mutual as a true "Renyi generalization" of the "von Neumann style" quantum conditional mutual information. Solving the second conjecture would have widespread implications throughout quantum information theory and even for condensed matter physics (topological order - pdf).

• I really like this question --- I hope I manage to get some time to think about it! Jul 1, 2014 at 16:28
• In response to earlier comments (now deleted), I have reformatted the post in order to better highlight the apparent questions. With that, let's please focus on the mathematics. Jul 4, 2014 at 4:06
• This is just to say that there has been what I would consider siginificant progress on Conjecture 2 since it was posted. Fawzi and Renner have proved a variation of the case when $\beta = 1$ and $\alpha = 1/2$, in the paper arxiv.org/abs/1410.0664 . My coauthors and I have also generalized the notion of Renyi conditional mutual information Renyi relative entropy differences in the paper arxiv.org/abs/1410.1443 and we generalized the methods of Fawzi and Renner in the paper arxiv.org/abs/1412.4067 . Of course, the conjectures stated here still remain open. Dec 21, 2014 at 20:54
• There has now been even more progress on Conjecture 2 that is useful enough for applications in quantum information theory. The idea was to make use of the Hadamard three-line theorem (in particular, Riesz-Thorin interpolation). This is detailed in the following paper: arxiv.org/abs/1505.04661 . See also arxiv.org/abs/1506.00981 for follow-up work. Jun 20, 2015 at 3:43