bibkey: caofitzsimmonsmannpereiralaflamme2023cptphptp
authors: Ningping Cao; Maxwell Fitzsimmons; Zachary Mann; Rajesh Pereira; Raymond Laflamme
year: 2023
title: "Quantum Maps Between CPTP and HPTP"
doi: null
url: https://arxiv.org/abs/2308.01894v1
claim: "A Hermiticity-preserving trace-preserving map is semi-positive (SP) if some invertible density has an invertible density image, and semi-nonnegative (SN) if some density has a density image (Definitions 1–2); SN is equivalent to Ψ ∘ Φ = Ξ for CPTP Φ, Ξ (Theorem 2). The authors conjecture that non-SP SNTP maps still signal no information backflow in higher dimensions (Outlook)."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/SemiNonnegativeBackflowRefutation
license: citation-only
triage: anchor

Cao, Fitzsimmons, Mann, Pereira and Laflamme, quantum maps between CPTP and HPTP

N. Cao, M. Fitzsimmons, Z. Mann, R. Pereira and R. Laflamme, Quantum Maps Between CPTP and HPTP, arXiv:2308.01894 [quant-ph] (v1 3 August 2023, the only version). The arXiv record lists no DOI or journal reference.

Verified locator

Primary version read: https://arxiv.org/abs/2308.01894v1; the text was read from the PDF.

• page 3: Definition 1 (SP and SPR);
• page 4: Definition 2 (SN) and Theorem 2;
• page 5: Example 2;
• pages 6–7: the Outlook, with the conjecture;
• the supplementary material on Example 2.

Source statements

Definition 1: “Let H and K be finite-dimensional Hilbert spaces. An HP map Ψ : B(H) → B(K) is said to be semi-positive (SP) if there exists an invertible density matrix ρ such that Ψ(ρ) is also an invertible density matrix. The map Ψ is said to be semi-positive after reduction (SPR) if Ψ as a map into B(K_Ψ) is SP, where K_Ψ = Span(⋃_{a∈B(H)} Range(Ψ(a))).”

Definition 2: “A HPTP map Ψ : B(H) → B(K) is said to be semi-nonnegative (SN) if there exists a density matrix ρ such that Ψ(ρ) is also a density matrix.”

Theorem 2: “Let H and K be finite dimensional Hilbert spaces and let Ψ be an HPTP map from B(H) to B(K). Then TFAE 1. Ψ is semi-nonnegative. 2. There exist CPTP maps Φ : B(H) → B(H) and Ξ : B(H) → B(K) such that Ψ ∘ Φ = Ξ.”

Outlook: “The single qubit non-SP SNTP maps only send one pure state to a pure state … For the physical scenarios that these maps can characterize, information backflow would not happen, and these maps are not uniquely defined. … Non-pure, not full-rank states comprise flat facets on the surface of D. A non-SP SNTP map can map one of those facets (or part of it) to valid quantum states. We conjecture that non-SP SNTP maps still signal no information backflow and are not uniquely defined even for higher dimensional H.”

The source cites for information backflow [11] H.-P. Breuer, E.-M. Laine, J. Piilo and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016), and [20] S. Chakraborty and D. Chruściński, Phys. Rev. A 99, 042105 (2019).

Reading used by the repository

Information backflow is the increase of the trace distance D(ρ, σ) = ‖ρ − σ‖₁/2 between two states under the intermediate map Ψ of a context Ψ ∘ Φ = Ξ (Theorem 2), with Φ and Ξ given by finite Kraus families. The repository addresses the first conjunct of the conjecture, “signal no information backflow”; a counterexample to it refutes the conjunction.