Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The noisy Landau-Streater(Werner-Holevo) channel in arbitrary dimensions

Vahid Karimipour

TL;DR

This work extends the Landau-Streater/Werner-Holevo channel to arbitrary dimensions by introducing an $SO(d)$-based LS channel and a one-parameter noisy version $\Phi_x(\rho)=(1-x)\rho+\frac{x}{d-1}(\operatorname{tr}(\rho)I-\rho^T)$. It analyzes the spectrum and infinitesimal divisibility, derives the complementary channel, and shows that in even dimensions the channel admits a mixed-unitary decomposition, in contrast to the $d=3$ case. The authors compute closed-form capacities: one-shot classical capacity $C^1(\Phi_x)$ and entanglement-assisted capacity $C_{ea}(\Phi_x)$, and establish a universal lower bound on quantum capacity with a critical noise threshold $x_0\approx 0.4$ below which $Q(\Phi_x)>0$. They also connect the LS channel to the $SU(d)$ framework, showing WH-type covariance and providing avenues for further work on degradability and multi-parameter extensions. The results have implications for high-dimensional quantum communication, offering explicit capacity formulas and structural decompositions across dimensions.

Abstract

Two important classes of quantum channels, namly the Werner-Holevo and the Landau-Streater channels are known to be related only in three dimensions, i.e. when acting on qutrits. In this work, definition of the Landau-Streater channel is extended in such a way which retains its equivalence to the Werner-Holevo channel in all dimensions. This channel is then modified to be representable as a model of noise acting on qudits. We then investigate propeties of the resulting noisy channel and determine the conditions under which it cannot be the result of a Markovian evolution. Furthermore, we investigate its different capacities for transmitting classical and quantum information with or without entanglement. In particular, while the pure (or high noise) Landau-Streater or the Werner-Holevo channel is entanglement breaking and hence has zero capacity, by finding a lower bound for the quantum capacity, we show that when the level of noise is lower than a critical value the quantum capacity will be non-zero. Surprizingly this value turns out to be approximately equal to $0.4$ in all dimensions. Finally we show that, in even dimension, this channel has a decomposition in terms of unitary operations. This is in contrast with the three dimensional case where it has been proved that such a decomposition is impossible, even in terms of other quantum maps.

The noisy Landau-Streater(Werner-Holevo) channel in arbitrary dimensions

TL;DR

This work extends the Landau-Streater/Werner-Holevo channel to arbitrary dimensions by introducing an -based LS channel and a one-parameter noisy version . It analyzes the spectrum and infinitesimal divisibility, derives the complementary channel, and shows that in even dimensions the channel admits a mixed-unitary decomposition, in contrast to the case. The authors compute closed-form capacities: one-shot classical capacity and entanglement-assisted capacity , and establish a universal lower bound on quantum capacity with a critical noise threshold below which . They also connect the LS channel to the framework, showing WH-type covariance and providing avenues for further work on degradability and multi-parameter extensions. The results have implications for high-dimensional quantum communication, offering explicit capacity formulas and structural decompositions across dimensions.

Abstract

Two important classes of quantum channels, namly the Werner-Holevo and the Landau-Streater channels are known to be related only in three dimensions, i.e. when acting on qutrits. In this work, definition of the Landau-Streater channel is extended in such a way which retains its equivalence to the Werner-Holevo channel in all dimensions. This channel is then modified to be representable as a model of noise acting on qudits. We then investigate propeties of the resulting noisy channel and determine the conditions under which it cannot be the result of a Markovian evolution. Furthermore, we investigate its different capacities for transmitting classical and quantum information with or without entanglement. In particular, while the pure (or high noise) Landau-Streater or the Werner-Holevo channel is entanglement breaking and hence has zero capacity, by finding a lower bound for the quantum capacity, we show that when the level of noise is lower than a critical value the quantum capacity will be non-zero. Surprizingly this value turns out to be approximately equal to in all dimensions. Finally we show that, in even dimension, this channel has a decomposition in terms of unitary operations. This is in contrast with the three dimensional case where it has been proved that such a decomposition is impossible, even in terms of other quantum maps.
Paper Structure (14 sections, 85 equations, 2 figures)

This paper contains 14 sections, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition of the $SO(d)$ Landau-Streater channels
  3. Spectrum of the channel and its infinitesimal divisibility $\phi_{_x}$
  4. Complementary Channel
  5. The question of extremality and mixed unitary representation
  6. A mixed unitary representation for $SO(4)$ Landau-Streater channel
  7. A mixed unitary representation for $SO(6)$ Landau-Streater channel
  8. Capacities
  9. One shot classical capacity
  10. Entanglement-Assisted Classical Capacity
  11. Bounds for the quantum capacity
  12. The Landau-Streater channel for the group $SU(d)$
  13. Discussion
  14. Acknowledgements

Figures (2)

  • Figure 1: The algorithm for finding the distinct partitions of $d$ points for $d$=even. A point is put at the center and a line connects it with a point on the $d-1$ gon. The other pairs correspond to the lines perpendicular to this line. Here two examples are shown for $d=8$.
  • Figure 2: The shaded region shows the range of the parameter $x$ in which the channel $\Phi_x$ has a positive quantum capacity. Interestingly the cricial value is approximately equal to $0.4$ for all dimensions. Above this critical value, the coherent information is negative and it is not known if the channel $\Phi_{x>x_0}$ has a non-zero quantum capacity.