Abstract
It is well-known that any quantum channel E satisfies the data processing inequality (DPI), with respect to various divergences, e.g., quantum χκ2 divergences and quantum relative entropy. More specifically, the data processing inequality states that the divergence between two arbitrary quantum states ρ and σ does not increase under the action of any quantum channel E. For a fixed channel E and a state σ, the divergence between output states E(ρ) and E(σ) might be strictly smaller than the divergence between input states ρ and σ, which is characterized by the strong data processing inequality (SDPI). Among various input states ρ, the largest value of the rate of contraction is known as the SDPI constant. An important and widely studied property for classical channels is that SDPI constants tensorize. In this paper, we extend the tensorization property to the quantum regime: we establish the tensorization of SDPIs for the quantum χκ1/22 divergence for arbitrary quantum channels and also for a family of χκ2 divergences (with κ≥κ1/2) for arbitrary quantum-classical channels.
Highlights
In information theory, the data processing inequality (DPI) has been an important property for divergence measures to possess operational meaning
The dependence on κ is one major difference between the quantum strong data processing inequality (SDPI) framework and its classical version: all quantum χ2κ divergences coincide for classical states ρ and σ (i.e., ρ and σ commute) and reduce to the classical χ2 divergence; in particular, classical χ2 divergence, as well as the associated classical SDPI constant, does not depend on κ; the SDPI constant for quantum χ2κ divergences might fluctuate significantly between approximately 0 and 1 for various κ, in a special example that we provide below
We provide a partial solution to the problem of the tensorization of SDPIs for quantum channels in Theorem 1
Summary
The data processing inequality (DPI) has been an important property for divergence measures to possess operational meaning. Except for very special cases, in general, obtaining SDPI constants for high-dimensional quantum channels can be rather challenging, even numerically. (ii) κ ≥ κ1/2 and Ej are quantum-classical (QC) channels; we have the tensorization of the SDPI constant for the quantum χ2κ divergence, i.e.,. We notice that there is a particular QC channel E associated with a fixed σ ∈ D+2 such that the largest value of ηχ2κ (E, σ) ≈ 1 for κ = κmin, while ηχ2κ (E, σ) ≈ 0 for κ = κmax (σ is close to a singular matrix); see § 5.1 for details This extreme example shows the high dependence of SDPI constants on the choice of κ, which magnifies the difference between the quantum SDPI constant and its classical analog, because there is only one SDPI constant for the classical χ2 divergence.
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