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Noise is resource-contextual in quantum communication

Aditya Nema, Ananda G. Maity, Sergii Strelchuk, David Elkouss

TL;DR

A one-parameter family of channels for which as the parameter increases its one-way quantum and private capacities increase while its two-way capacities decrease, and constructions demonstrate that noise is context dependent in quantum communication.

Abstract

Estimating the information transmission capability of a quantum channel remains one of the fundamental problems in quantum information processing. In contrast to classical channels, the information-carrying capability of quantum channels is contextual. One of the most significant manifestations of this is the superadditivity of the channel capacity: the capacity of two quantum channels used together can be larger than the sum of the individual capacities. Here, we present a one-parameter family of channels for which as the parameter increases its one-way quantum and private capacities increase while its two-way capacities decrease. We also exhibit a one-parameter family of states with analogous behavior with respect to the one- and two-way distillable entanglement and secret key. Our constructions demonstrate that noise is context dependent in quantum communication.

Noise is resource-contextual in quantum communication

TL;DR

A one-parameter family of channels for which as the parameter increases its one-way quantum and private capacities increase while its two-way capacities decrease, and constructions demonstrate that noise is context dependent in quantum communication.

Abstract

Estimating the information transmission capability of a quantum channel remains one of the fundamental problems in quantum information processing. In contrast to classical channels, the information-carrying capability of quantum channels is contextual. One of the most significant manifestations of this is the superadditivity of the channel capacity: the capacity of two quantum channels used together can be larger than the sum of the individual capacities. Here, we present a one-parameter family of channels for which as the parameter increases its one-way quantum and private capacities increase while its two-way capacities decrease. We also exhibit a one-parameter family of states with analogous behavior with respect to the one- and two-way distillable entanglement and secret key. Our constructions demonstrate that noise is context dependent in quantum communication.
Paper Structure (11 sections, 6 theorems, 49 equations, 6 figures)

This paper contains 11 sections, 6 theorems, 49 equations, 6 figures.

Table of Contents

  1. XZ invariance of the $I_c$ of $\mathcal{N}_{\lambda, \; p}$
  2. Proof that $\mathcal{N}_{\lambda, \; p}$ is degradable
  3. Variation of $\mathcal{Q}(\mathcal{N}_{\lambda, \; p})$ for $\lambda \leq 1/2$
  4. An upper bound for $E_{r}(\overline{\mathcal{N}_{\lambda, \; p}})$
  5. Proof of the one-way capacity of $\mathcal{N}_{\lambda, \; p}$ for $\lambda\in[0,1/2]$ and two-way capacity of $\mathcal{N}_{\lambda, \; p}$ for $\lambda\in[0,1]$
  6. Characterizing the one-way capacity using continuity when $\lambda \in [1/2,1]$
  7. Lower bound on $\mathcal{Q}(\mathcal{N}_{\lambda, \; p})$ for $\lambda \in [1/2,1]$
  8. Upper bound on $\mathcal{Q}(\mathcal{N}_{\lambda, \; p})$ for $\lambda \in [1/2,1]$
  9. A discrete family of channels for which the one-way capacity increases and the two-way capacity decreases with $\lambda \in [1/2,1]$
  10. One-way and two-way assisted capacity of the complementary channel
  11. One-way distillable entanglement and secret key of degradable states

Key Result

Proposition 1

The channel $\mathcal{N}_{\lambda, \; p}$ given in Equation eq:N(rho) is degradable for $\lambda \in [0,1/2]$.

Figures (6)

  • Figure 1: Noise resource-contextuality: capacity of a family of channels for transmitting quantum information as a function of the noise parameter $\lambda$. The top scenario corresponds to the capacity in the absence of feedback (one-way capacity) and the bottom scenario corresponds to the capacity with feedback (two-way capacity). Intuitively, one would expect both capacities to be quantitatively different but have similar qualitative behavior; that is, either both increase or both decrease as a function of the noise parameter. Here we show that the very meaning of noise can depend on the resources. We exhibit families of channels for which as the two-way capacity decreases, the one-way capacity increases. Increasing $\lambda$ represents noise for two-way communications while decreasing $\lambda$ represents noise for one-way communications.
  • Figure 2: Weighted direct sum construction. $\overline{\mathcal{D}_p}$ is the complement of a qubit dephasing channel with parameter $p$, $\mathcal{I}$ is the noiseless channel, and $\lambda,\;p \in [0,1]$. This way of combining quantum channels is also referred to as 'gluing' Vikesh_superadditivity.
  • Figure 3: One- vs two-way capacity of $\mathcal{N}_{\lambda, \; p}$ as a function of $\lambda$ when $p(\lambda)=4\lambda-1$. In the range $\lambda\in[0.25,0.3125]$ the one-way quantum (and private) capacity monotonically increases while the two-way quantum (and private) capacity decreases.
  • Figure 4: One- vs two-way quantum (and private) capacity of $\mathcal{N}_{\lambda, \; p}$ as a function of $p$ when $\lambda(p)=p/\log p$. In the range $p\in[0.35,0.5]$ the one-way quantum (and private) capacity monotonically increases while the two-way quantum (and private) capacity decreases.
  • Figure 5: Weighted direct sum classical wiretap channel. With probability $1-\lambda$ the channel gives Bob Alice's input and Eve a random outcome. With probability $\lambda$ it gives Eve Alice's input and Bob a noisy version of Alice's input.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Definition 2
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Definition 6
  • proof : Proof of Equation \ref{['eq:upper_bound_superadd']}
  • Proposition 8
  • ...and 6 more