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Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map
David Sutter, Marco Tomamichel, Aram W. Harrow
TL;DR
Relative-entropy data processing is tight precisely when a recovery map reverses the channel on both states. This paper constructs a pinching-based recovery map from rotated Petz maps. Its strengthened recoverability bound is strictly stronger than the fidelity bound except under perfect recovery.
Problem
The data-processing inequality does not by itself quantify how well a channel's effects can be reversed, motivating bounds that relate its entropy loss to recoverability.
Source: §1 · p. 1Method
The paper constructs a recovery map using pinching and rotated Petz recovery maps, with a convex-sum weight that may depend on ρ.
Source: §2 · p. 3, §2.1 · p. 4, §3 · p. 6Results
The measured-relative-entropy bound is strictly stronger than the fidelity bound except under perfect recoverability, and the gap can be of order log d.
Source: §3 · p. 6Takeaways & Limitations
The theorem reproduces a conditional-mutual-information lower bound and additionally shows that the recovery map sends ρB to ρBC.
Source: §3 · p. 6The theorem's recovery map may depend on ρ, so it is not universal in the sense established for the conditional-mutual-information bound.
Source: §3 · p. 6
Abstract
from arXiv · showhide
The quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy. It is known that this inequality is only tight when there is a "recovery map" that exactly reverses the effects of the quantum channel on both states. In this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version. The recovery map is a convex combination of rotated Petz recovery maps and perfectly reverses the quantum channel on the second state. As a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved in [Fawzi and Renner, Commun. Math. Phys., 2015]. Our proof only relies on elementary properties of pinching maps and the operator logarithm.
1 Introduction
Relative entropy obeys data processing, with equality exactly when a recovery map reverses the channel on both states. The introduction presents a strengthened recoverability result proved using pinching maps.
1 Introduction
- Equality in data processing holds if and only if a recovery map simultaneously recovers both input states from their channel outputs.On the support of the recovered second state, the map can be taken to have the Petz recovery-map form.Source: §1 · p. 1
- The measured relative entropy is optimized over projective measurements, and restricting to projective measurements does not change the supremum over POVMs.The measurement projectors can be taken to be rank one.Source: §1 · p. 2, §1.1 · p. 2
- Prior results established rotated-Petz recovery bounds, including a general relative-entropy statement whose recovery map can depend on ρ [11] [34] [37].The map in that statement perfectly reconstructs σ from N(σ).Source: §1 · p. 2
1.1 Main result.
- The main result strengthens an earlier recoverability inequality and directly implies it, using a recovery map constructed from pinching maps.The proof is conceptually different from previous proofs and relies on properties of pinching maps.Source: §1.1 · pp. 2–3
1.2 Notation.
- The paper assumes finite-dimensional Hilbert spaces and defines Sσ(A) as states whose support is contained in the support of σ.It also uses S(A) for density operators and P(A) for non-negative operators.Source: §1.2 · p. 3
2 Variations on the Petz Recovery Map
The section introduces pinching and rotated Petz recovery maps, then connects them by expressing the pinching recovery map as a convex combination of tensor products of rotated Petz maps.
2 Variations on the Petz Recovery Map
- The section introduces pinching and rotated Petz recovery maps as two variants of the Petz map that play a key role in the work.The paper also establishes a close connection between these two maps.Source: §2 · p. 3
2.1 Pinching recovery map.
- The pinching map is built from the spectral projectors of a self-adjoint operator, removes off-diagonal blocks, and has properties used in the recovery construction.The section highlights commutation and trace-preservation properties, along with a pinching operator inequality.Source: §2.1 · pp. 3–4
2.2 Rotated Petz recovery map.
- The rotated Petz construction uses unitaries defined from the eigenspace projectors of σ and N(σ), which commute with those operators.The paper defines a convex hull of rotated Petz recovery maps.Source: §2.2 · p. 4
2.3 Connection between pinching and rotated Petz recovery map.
- The pinching recovery map can be written as a convex combination of tensor products of rotated Petz recovery maps.Integrating over the rotations yields the pinchings used in the construction.Source: §2.3 · p. 5
3 Main Result and Proof
The section bounds relative-entropy loss using recovery from the processed state, with the main lower bound expressed through a limit involving the pinching recovery map. It also gives a conditional-mutual-information specialization and an upper bound based on max-relative entropy.
3 Main Result and Proof
- The relative-entropy loss is lower-bounded using a limit of relative entropies between ρ⊗n and states recovered from N(ρ)⊗n by the pinching recovery map.Theorem 3.3 uses a recovery map that is a convex combination of rotated Petz maps and perfectly recovers σ from N(σ).Source: §3 · pp. 5–6
3.1 Proof of Proposition 3.1
- Proposition 3.1 uses the pinching recovery map to prove monotonicity through properties of pinching maps and the operator logarithm.The proof invokes operator Jensen inequality and the logarithm’s monotonicity and concavity.Source: §3 · pp. 5–6, §3.1 · p. 7
3 Main Result and Proof
- The measured-relative-entropy bound is strictly stronger than the fidelity bound except when recovery is perfect; their gap can be of order log d.In the commutative case, the bound coincides with the best known classical result.Source: §3 · p. 6
- For conditional mutual information, Theorem 3.3 reproduces the bound in (2), and the recovery map can also map ρB to ρBC.This follows by choosing ρ = ρABC, σ = idA ⊗ρBC, and N(·) = trC(·).Source: §3 · p. 6
3.3 Proof of Proposition 3.8
- Proposition 3.8 gives an upper bound on the relative-entropy difference in terms of max-relative entropy and recovery maps.The stated comparison uses a minimization over recovery maps, which can be bounded using any such map.Source: §3 · pp. 6–7, §3.3 · p. 9
3.2 Proof of Theorem 3.3
- Theorem 3.3’s proof combines a variational characterization of measured relative entropy with the convex hull of rotated Petz recovery maps.The proof uses a minimax step and Lemma 3.11.Source: §3.2 · p. 8
4 Discussion
A universal recovery map for the conditional-mutual-information case was established in [34]. Theorem 3.3 and techniques from [34] imply a universal version of the theorem [20].
A Alternative proof of Theorem 3.3
The alternative proof formulates the recovery maps as a convex hull of rotated Petz maps and uses probability measures over their rotations. Compactness and lower semicontinuity ensure the minima are attained, while the proof establishes the two inequalities of Theorem 3.3 through separate arguments.
A Alternative proof of Theorem 3.3
- The proof takes the convex hull of rotated Petz recovery maps and considers probability measures over the rotation parameters.It invokes Theorem 1 of [31] as part of the proof setup.Source: A Alternative proof of Theorem 3.3 · p. 9
- The minima over probability measures are attained because the rotation domain and its probability-measure space are compact, and the objective is lower semicontinuous.The argument uses continuity of expectation values and lower semicontinuity of relative entropy.Source: A Alternative proof of Theorem 3.3 · p. 9
- Iterating the argument from Equation (54) and combining it with Proposition 3.1 and Lemma 2.1 proves the first inequality of Theorem 3.3.The second inequality follows from Rényi-divergence monotonicity in the order parameter and an optimal measurement that does not increase fidelity.Source: A Alternative proof of Theorem 3.3 · pp. 9–10