Approximate invertible map as a recovery quantum channel

• Related to Reversing quantum channels?

import Quantum_channel

/-- Invertible quantum channel -/
def invertible [quantum_channel Λ] :=
∀ [quantum_state ρ], Λ⁻¹(Λ(ρ)) = ρ

-- "non-invertibility as a witness of non-Markovianity."

• Let quantum_channel Λ{t,0} with t ≥ 0 model system evolution from time 0 to time t.

  • the inverse map Λ⁻¹{t,0} is in general not a quantum_channel.

import Depolarizing_channel

/-- completely depolarizing channel is non-invertible -/
lemma not_invertible_completely_depolarizing :
  ...
sorry  -- "it sends all input states to the maximally mixed state."

/-- reversible quantum_channel -/
def reversible [quantum_channel Λ] :=
∃ [quantum_channel Λ*],
∀ [quantum_state ρ], Λ∗(Λ(ρ)) = ρ

lemma quantum_channel_reversible_iff_unitary [quantum_channel Λ] :
  reversible Λ ↔ unitary Λ := sorry

/--
"Maps describing noisy dynamics
e.g. the non-fully Depolarizing channel
are invertible but not reversible."
-/
lemma invertible_of_non_completely_depolarizing :=
  ¬ completely_depolarizing Λ → invertible Λ := sorry

lemma not_reversible_of_non_completely_depolarizing :=
  ¬ completely_depolarizing Λ → ¬ reversible Λ := sorry

• Attentuated non-Markovian effects;

• “Use Fidelity of quantum state to measure how well the Petz recovery map recovers a random quantum state.”

  • Mean fidelity as a measure of reversibility for quantum channels.

  • We can find the best Petz recovery strategy: The strategy that optimizes the mean fidelity.”

/--
"If the channel is invertible then 
the intermediate map is given by Λ{t,s} = Λ{t,0} Λ⁻¹{s,0}."
-/
def intermediate_map [invertible Λ] (s t : ℝ) :=
Λ{t,0} Λ⁻¹{s,0}

• Idea: “substitute the non-physical inverse map by a physical recovery map.”

• “The intermediate map Λ{t,s} is memoryless: depends only on times t and s; Does not retain any information about previous times.”

• “Relax the definition: Λ doesn’t have to be CPTP.

/--
"Another possible way to relax the definition of reversible maps 
is by requiring it to hold only for a subset of all quantum_states.

Given a channel Λ,
we say it is recoverable
if there exists another channel Λ' such that
∀ ρ ∈ S, Λ'(Λ(ρ)) = ρ
where S ⊆ D(H)."
-/
def recoverable_states_of_channel [quantum_channel Λ] (S : set quantum_state) :=
∃ [quantum_channel Λ'],
∀ ρ ∈ S, Λ'(Λ(ρ)) = ρ

-- recoverable set for the Depolarizing channel?

• “The most well-known class of recovery maps are the so-called Petz recovery maps”

• “Given a noisy process described by a non-CP invertible map Λ, what is the best choice of reference state σ that makes ΛσP to act as close as possible to a reverse map Λ∗?”

• “An input state ρ i goes through a noisy channel Λ. The output state: ρo = Λ(ρ i).

  • Let ΛσP be a recovery channel such that ρr = ΛσP (ρo).

  • How close are ρr and ρ i?

• Search numerically for the best recovery channel:

  • “optimize FΛ over a uniform distribution of mixed input states.

  • generate random 1-qubit mixed states,

    • generate 2-qubits pure states chosen randomly using the Haar measure;

    • trace out the second qubit.

/-- let dμρ be a uniform measure over the input states -/
def average_fidelity [quantum_channel Λ] [quantum_state σ] :=
∫ dμρ F(ρ, ΛσP(Λ(ρ)))

/-- optimal reference state σ∗ -/
def optimal_ref_state [quantum_state σ∗] :=
average_fidelity Λ σ∗ = max {average_fidelity Λ σ | quantum_state σ}

• “Given that we are taking a uniform measure over mixed states, and that the channels are unital, it is expected that the best Petz recovery map is obtained by taking the reference state as the maximally mixed one.”

  • confirm numerically (start a repl):

  • “It is reasonable to suppose that the optimal Petz recovery channel is obtained with a reference state that is located on the z-axis.”

    • “Although the Amplitude damping channel is not unital, it maps a uniform distribution of states onto a distribution which is invariant under rotations around the z-axis of the Bloch sphere.

  • use this parametrization to numerically obtain the best reference state for different values of the noise strength p.”

  • let the reference state be σ = (1-q)|0⟩⟨0| + q|1⟩⟨1|.

    • for q = 1/2 this is the maximally mixed state;

    • parametrization suggested by the azimuthal symmetry of the output distribution of states for both noisy channels.

  • “evaluate the average fidelity as a function of q for both channels.

  • Optimal reference state for the dephasing and depolarizing channels.

  • For the dephasing channel, as all the states of the form of σ are preserved, any value of q is also an optimal choice,”

  • “Using the maximally mixed state as reference state, and the unitality of such maps, we obtain Λ I/2 P (ρ) = Λ†(ρ).”

  • optimal Petz recovery map for these noise models is again to apply the noisy map: No recovery is actually obtained.

  • The identity channel as a “recovery map” is more advantageous than taking the optimal Petz when considering the set of all single qubit states as input.

  • Compare recovery performance of the optimal Petz map for a given p with two other channels: The identity channel and the completely_depolarizing_channel.

  • “From Fig. 3, the latter is clearly the worst strategy, despite the fact that the maximally mixed state is indeed the average state for the input distribution of states.”

  • Petz does better than identity if there is a rotation next to the dephasing, depolarizing maps.

  • Petz works better than identity for Amp damp with small q.

def self_dual [quantum_channel Λ] :=
Λ† = Λ

import Dephasing_channel

lemma self_dual_depolarizing_channel :
  self_dual depolarizing_channel := sorry

lemma self_dual_dephasing_channel :
  self_dual dephasing_channel := sorry

• Fig. 4:

  • “For weak amplitude damping, the optimal_ref_state is |0⟩⟨0|.

  • As the noise strength increases, the optimal_ref_state moves towards the maximally mixed state.”

• Fig. 5:

  • “Compare the optimal Petz map for a given p with the previous strategies: the identity map as the recovery map (always returning ρr = ΛAD (ρ)), and returning ρr = 1/2 independently of the initial state.”

• Fig. 3 and Fig. 5:

  • “the average fidelity for the dephasing map is always higher.

  • So the dephasing channel can be better recovered.”

• “employ Petz recovery maps to obtain approximations for Λ{t,s}.”

  • “Λapprox{t,0} = Φ{t,s,0} Λ{s,0}.”

  • “As Φ{t,s,0} may depend on the time interval [0, s], the map Λapprox{t,0} is not necessarily CP-divisible (Markovian).

• “take the Trace distance between two states as a quantifier of their distinguishability.

  • For a NM dynamics it can be shown that ∀ t' ≥ t, ∥Λ{t,0}(ρ₁) - Λ{t,0}(ρ₂)∥₁ ≤ ∥Λ{t',0}(ρ₁) - Λ{t',0}(ρ₂)∥₁.

  • This should be contrasted with the Markovian case, where ∀ t' ≥ t, ∥Λ{t,0}(ρ₁) - Λ{t,0}(ρ₂)∥₁ ≥ ∥Λ{t',0}(ρ₁) - Λ{t',0}(ρ₂)∥₁.”

• “The Petz recovery map in the context of non-Markovian evolution.”

• See also Recoverability of Gaussian states;

• Petz map cannot in principle be the identity map?

  • Petz map is a sepific instance of quantum_channel; May be you cannot arrive at the identity channel this way?

• In non-Markovian dynamics, replace non-physical inverse map with physical Petz map; Approximate dynamics;

  • Markovian dynamics to simulate non-Markovian dynamics;

import Markovian

def approximate_map := sorry

notation `Λapprox` := approximate_map

/--
The approximate map is not CP divisible.
-/
lemma non_Markovian_approximate_map :
¬ CP_divisible Λapprox{t,0} :=
begin
  -- Note that Λapprox{t,0} = Λ{t,s} Λ{s,0} ΛσP{0,s} Λ{s,0}
  -- so the non-CP map Λ{t,s} is still contained in Λapprox{t,0};
  sorry
end

-- "The approximate map is completely positive by construction";

• How well does Λapprox{t,0} approximate Λ{t,0}?

• compare the non-Markovianity of both channels.”

• Derive dephasing models from Lindblad generators;

  • Choice of decay rates lead to certain probabilities;

import Trace_distance

/--
For Markovian dynamics, 
trace distance monotonically decreases with evolution time;
-/
lemma trace_dist_decreases_monotonically_with_evolution_time :
∀ [quantum_state ρ σ], 
∥Λ(ρ) - Λ(σ)∥₁ ≤  ∥ρ - σ∥₁ := sorry

-- Data processing inequality?

• RHP measure for non-Markovianity;

  • Related to Choi distance: distance between Choi matrices of the original and approximate maps;

    • Plot the Choi distance against evolution time for the chosen noise models;

  • A measure of positivity;

• Information backflow: Distinguishability difference between …

  • For the two chosen noise models the approxmate and original dynamics sometimes have the same Choi distance but different state distinguishability:

    • Possible implication: Information backflow is not a good way to tell if an evolution is non-Markovian;

BLP measure for non-unital maps;

• What about qudit channels?

References

• Lautenbacher2021;