Isotropic quantum spin channels and additivity questions

• Minimum entropy output and minimum entropy gain are defined as two functions obtained by minimizing entropy-related quantities over input states.Source: §1 · p. 1
• Minimum entropy output is important for quantum information problems including entanglement characterization and channel capacities [1].Source: §1 · p. 1
• The additivity question asks whether entangled inputs can improve on product-state strategies for minimizing output entropy.Source: §1 · p. 1
• The paper analyzes bistochastic completely positive maps acting on spin-s systems with rotational invariance.Source: §1 · p. 1
• The paper proves that the function G[Λ] is additive.Source: §1 · p. 1

• Minimum entropy outputs are computed for isotropic channels acting on spin-1/2 and spin-1 systems.Source: §2 · p. 2
• For spin-1/2, rotational invariance and direct evaluation show that every pure-state output has the same spectrum, yielding formula (9).Source: §2 · p. 2
• The spin-1/2 proof represents output entropy through a 3×3 positive matrix Ω of trace one and bounds its operator norm using explicit spin-operator calculations.Source: §2 · p. 2
• The matrix analysis establishes λ = 1 as an eigenvalue of ZZ† and concludes that ∥ZZ†∥ = ∥YY†∥ = 1, proving the stated inequalities.Source: §2 · pp. 2–3
• For spin-1, direct computation gives output entropy log 2, and invariance extends this minimum to all SU(2)-coherent states.Source: §2 · p. 3

• The entropy of two parallel channels is evaluated on the singlet input for spins s = 1/2 and s = 1.Source: §3 · p. 3
• Because the singlet is rotation-invariant, the two-channel output is O(3)-invariant and decomposes across irreducible-representation sectors.Source: §3 · p. 3
• Sector probabilities p_j are obtained from diagonal matrix elements of the output state and the dimensions 2j + 1, then used to compute the entropy.Source: §3 · p. 3
• The singlet-state calculation uses explicit spin-state components and direct evaluation of the relevant probabilities.Source: §3 · p. 3

• The theorem summarizes the properties of the channel function G[Λ] defined by equation (4).Source: §4 · p. 4
• For a finite-dimensional system, G[Λ] is bounded below by zero and vanishes for bistochastic Λ.The first property follows from an invariant state and entropy monotonicity under bistochastic maps.Source: §4 · p. 4
• G[Λ ⊗ Λ′] = G[Λ] + G[Λ′] for arbitrary quantum channels.The proof uses additivity for product states and monotonicity of relative entropy under completely positive maps.Source: §4 · p. 4

5 Con luding remarks

• Minimal entropy output is computed for isotropic channels acting on spin-1/2 and spin-1 systems.Source: §5 · p. 4
• For doubled channels, singlet-state entanglement increases decoherence and produces higher output entropy than optimal product states.The comparison provides some support for the additivity hypothesis.Source: §5 · p. 4
• The paper defines and discusses minimal entropy gain as a new additive function of the channel.Source: §5 · p. 4
• The concluding section acknowledges discussions with Mihaª and Ryszard Horodeki and financial support from Polish and European sources.Source: §5 · p. 4

• [1] addresses the equivalence of additivity questions in quantum information theory.Source: Referen es · p. 4
• [2] is a book on quantum dynamical systems by R. Alicki and M. Fannes.Source: Referen es · p. 4
• [3] cites a Physical Review Letters article by R. Alicki, D. Makowiec, and W. Miklaszewski.Source: Referen es · p. 4
• [4] cites L.I. Schiff’s Quantum Mechanics, while [5] cites G. Lindblad’s work in Communications in Mathematical Physics.Source: Referen es · p. 4