trureturing — Discovering truth

bibkey: seshadreesan2015renyi
authors: K. P. Seshadreesan, M. Berta, M. M. Wilde
year: 2015
title: "Rényi squashed entanglement, discord, and relative entropy differences"
doi: 10.1088/1751-8113/48/39/395303
url: https://arxiv.org/abs/1410.1443v3
claim: "Conjecture 1 The Rényi conditional quantum mutual information obeys monotonicity with respect to CPTP maps on system A, i.e., for ρABE ∈ S(HABE) and α ∈ (0, 1) ∪ (1, 2], Iα(A; B|E)ρ ≥ Iα(A′; B|E)τ, where τA′BE ≡ MA→A′(ρABE) and MA→A′ is a CPTP map acting on system A alone."
strata_touched:
  - D5/S3/Quantum/Information/RenyiCMILocalChannelRefutation
license: citation-only
triage: anchor

Locator

Author text: https://arxiv.org/abs/1410.1443v3.

• DOI: https://doi.org/10.1088/1751-8113/48/39/395303

J. Phys. A: Math. Theor. 48 (2015) 395303, §3, equation (3.18):

Throughout this paper, we take our definition of Rényi conditional quantum mutual information for α ∈ (0, 1) ∪ (1, ∞) to be as follows [BSW15a, Definition 7]:

$$I_{\alpha }(A;B|E{)}_{\rho }=\frac{\alpha }{\alpha -1}\log \mathrm{Tr}\left\{\right({\rho }_E^{(\alpha -1)/2}{\mathrm{Tr}}_A\{{\rho }_{AE}^{(1-\alpha )/2}{\rho }_{ABE}^{\alpha }{\rho }_{AE}^{(1-\alpha )/2}\}{\rho }_E^{(\alpha -1)/2}{)}^{1/\alpha }\}$$

§3, introduction to Conjecture 1:

Some results in this paper rely on the following conjecture:

§3, Conjecture 1:

Conjecture 1 The Rényi conditional quantum mutual information obeys monotonicity with respect to CPTP maps on system A, i.e., for ρABE ∈ S(HABE) and α ∈ (0, 1) ∪ (1, 2], Iα(A; B|E)ρ ≥ Iα(A′; B|E)τ, where τA′BE ≡ MA→A′(ρABE) and MA→A′ is a CPTP map acting on system A alone.

§2, discussion of local quantum operations:

it was left as an open question to determine if the property holds for local quantum operations on the other system (with numerical evidence supporting a positive answer). Here, we assume the latter to hold.

Mathematical scope

The definition is equation (3.18), with generalized inverse powers on the support. Conjecture 1 concerns channels on $A$ for $0<\alpha <1$ and $1<\alpha \le 2$. The finite-dimensional reading uses positive semidefinite trace-one matrices and finite Kraus families with ${\sum }_j{K}_j^{†}{K}_j=1$. The settlement fixes three qubits, a faithful state with classical $B$, complete dephasing of $A$, and $\alpha =2$. It does not concern monotonicity on $B$.