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Convolutional strong subadditivity conjecture

Let ρ\rho, \σ\σ, and τ\tau be three nn-qudit quantum states. For the convolution operations below, define

ρ⊠τ⊠σ:=(ρ⊠s,tτ)⊠l,mσ,\rho\boxtimes\tau\boxtimes\sigma:=(\rho\boxtimes_{s,t}\tau)\boxtimes_{l,m}\sigma, ρ⊠σ:=ρ⊠s,tσ,σ⊠τ:=σ⊠s,tτ,\rho\boxtimes\sigma:=\rho\boxtimes_{s,t}\sigma,\qquad \sigma\boxtimes\tau:=\sigma\boxtimes_{s,t}\tau,

with s2+t2≡1mod  ss^2+t^2\equiv 1\mod s, s≡tmod  ds\equiv t\mod d, m≡lsm\equiv ls, and l2+m2≡1mod  dl^2+m^2\equiv 1\mod d.

Convolutional strong subadditivity. One has

S(ρ⊠τ⊠σ)+S(σ)⩽S(ρ⊠σ)+S(σ⊠τ).S(\rho\boxtimes\tau\boxtimes\sigma)+S(\sigma)\leqslant S(\rho\boxtimes\sigma)+S(\sigma\boxtimes\tau).

The paper introduces this as a balanced version of the preceding, non-balanced entropy inequality. Its status is not resolved in the supplied text.

Sources & referencesView supporting material

Primary source

Kaifeng Bu, Weichen Gu and Arthur Jaffe, “Quantum Ruzsa Divergence to Quantify Magic”, arXiv:2401.14385 (2026).

quantum-convolutionquantum-entropyquantum-information

Progress summary

Refreshed
Open

The conjecture remains open for general quantum states, with proofs known only for two restricted families.

The conjecture is the balanced entropy inequality introduced in “Entropic Quantum Central Limit Theorem and Quantum Inverse Sumset Theorem.” It is proposed for arbitrary quantum states and would imply the triangle inequality for quantum Ruzsa divergence.

Known results

  • All three states stabilizer states: proved as Proposition 37.
  • All three states diagonal in the computational basis: proved as Proposition 37.
  • No general proof or counterexample is reported.

Current status (as of August 2026): the conjecture is established only for stabilizer and computational-basis diagonal states; its validity for arbitrary states remains open.

Sources

Solutions 0

No solutions have been posted yet.