Reversing quantum channels

Take two Quantum states ρ and Οƒ. Let each pass through a Quantum channel 𝒩. Measure the decrease in Quantum relative entropy between the two states after the channel acts. Is the decrease relatively small? Then it is possible to perform a recovery channel: To perfectly recover one state and approximately recover the other. β€œThis can be interpreted as quantifying how well one can reverse the action of a Quantum channel.”

import Quantum_state
import Quantum_channel

variables
(ℋ₁ β„‹β‚‚ : Type) 
[complex_Hilbert_space ℋ₁]
[complex_Hilbert_space β„‹β‚‚]
(ρ Οƒ: ℋ₁ β†’β‚—[∁] ℋ₁)
[quantum_state ρ]
[pos_semidef Οƒ]
[supp ρ βŠ† supp Οƒ]
(𝒩 : (ℋ₁ β†’β‚—[∁] ℋ₁) β†’β‚—[∁] (β„‹β‚‚ β†’β‚—[∁] β„‹β‚‚))
[quantum_channel 𝒩]

theorem exists_recovery_channel :
βˆƒ (β„› : (β„‹β‚‚ β†’β‚—[∁] β„‹β‚‚) β†’β‚—[∁] (ℋ₁ β†’β‚—[∁] ℋ₁)) [quantum_channel β„›], 
D(ρβˆ₯Οƒ) - D(𝒩(ρ)βˆ₯𝒩(Οƒ)) β‰₯ - log F(ρ,β„› 𝒩 ρ)
∧
β„› 𝒩 Οƒ = Οƒ :=
begin
  sorry,
end

def Petz_recovery_map (Οƒ) (𝒩) :=
Ξ» Q : β„‹β‚‚ β†’β‚—[∁] β„‹β‚‚, 
Οƒ^{1/2} * 𝒩†([𝒩 Οƒ]^{-1/2} * Q * [𝒩 Οƒ]^{-1/2}) * Οƒ^{1/2}

notation `𝒫` := Petz_recovery_map
  • The Petz recovery map is linear, completely positive and trace non-increasing;

β€œWe define a rotated or β€œswiveled” Petz map, which plays an important role in the construction of a recovery channel:

def partial_isometric_map (Οƒ) (t : ℝ) :=
Ξ» M : ℋ₁ β†’β‚—[∁] ℋ₁, 
Οƒ^{i*t} * M * Οƒ^{-i*t}

notation `𝒰` := partial_isometric_map

def rotated_Petz_recovery_map (Οƒ) (𝒩) (t : ℝ) :=
(𝒰 Οƒ -t) β—¦ (𝒫 Οƒ 𝒩) β—¦ (𝒰 (𝒩 Οƒ) t)

notations `β„›` := rotated_Petz_recovery_map

theorem perfect_recovery :
βˆƒ t,
(β„› Οƒ 𝒩 t) (𝒩 Οƒ) = Οƒ :=
begin
  sorry,
end

  • the reference state Οƒ is always recovered perfectly with the constructed Petz map;

  • Is Reversing quantum channels the same as Quantum error correction?

  • Approximate invertible map as a recovery quantum channel;

  • β€œFor an ensemble of commuting quantum_state, {p i, ρ i}, the optimal recovery channel is obtained with the reference state being ρ = βˆ‘ i, pi ρi.”

  • Paper by Fzi and Renner show approximate version of … rotated_Petz_map.

References