bibkey: rubboli2024magicadditivity
authors: Roberto Rubboli, Ryuji Takagi, Marco Tomamichel
year: 2024
title: "Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond"
doi: 10.22331/q-2024-10-04-1492
url: https://arxiv.org/abs/2307.08258v4
claim: "The α-z Rényi relative entropies of magic D_{α,z}(ρ) = inf over stabilizer states of D_{α,z}(ρ‖σ) are additive for all single-qubit states when |(1−α)/z| = 1. §5 states as open whether they are additive for all two- and three-qubit states in that range, and Table 1 and §10 conjecture that they are."
strata_touched:
  - D5/S3/Quantum/Magic/ThreeQubitStabilizerPolarEscape
  - D5/S3/Quantum/Magic/RenyiBoundaryMagicAdditivityRefutation
license: citation-only
triage: anchor

Rubboli–Takagi–Tomamichel, additivity of magic monotones based on quantum relative entropies

R. Rubboli, R. Takagi and M. Tomamichel, Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond, Quantum 8, 1492 (2024), DOI 10.22331/q-2024-10-04-1492; arXiv:2307.08258v4.

Verified locator

DOI: 10.22331/q-2024-10-04-1492 (Quantum 8, 1492, published 4 October 2024, open access; arXiv v4 carries the accepted text, “Accepted in Quantum 2024-09-19”). Primary version read: https://arxiv.org/abs/2307.08258v4. Source: https://arxiv.org/pdf/2307.08258v4 (text read from the PDF):

• §2, the Clifford group, the stabilizer states, Eq. (2), the DPI region D and Eq. (10);
• §5, the closing paragraph;
• Table 1 and its caption;
• §10.

Source statements

§2: “The n-qubit Clifford group C_n is the group of n-qubit unitaries generated by the single-qubit gates H, S, and the CNOT two-qubit gate. … The pure n-qubit stabilizer states are the ones generated by the Clifford group acting on |0^n⟩ … The set of all (mixed) stabilizer states is formed by all convex combinations (convex hull) of pure stabilizer states, S̄_n := conv{|ψ⟩⟨ψ| : |ψ⟩ ∈ S_n}.”

§2, Eq. (2): “D_{α,z}(ρ‖σ) := 1/(α−1) log Tr[(ρ^{α/2z} σ^{(1−α)/z} ρ^{α/2z})^z] if (α < 1 ∧ ρ ̸⊥ σ) ∨ ρ ≪ σ; +∞ else.”

§2: the data-processing inequality holds for “1. 0 < α < 1 and z ≥ max{α, 1−α}, 2. 1 < α ≤ 2 and α/2 ≤ z ≤ α, and 3. 2 ≤ α < ∞ and α − 1 ≤ z ≤ α”; the line α = 1 is included in the region D. Eq. (10): “D_{α,z}(ρ) := inf_{σ∈S̄} D_{α,z}(ρ‖σ)”.

§5: “It is still an open problem whether the α-z Rényi relative entropies of magic in the range |(1−α)/z| = 1 are additive for all two and three-qubit states. Numerical simulations suggest that additivity holds. We discuss in Section 10 a possible direction to prove this result.”

Table 1 caption: “in Section 10 we conjecture that additivity holds for all states.”