Correction to: Ruminations on matrix convexity and the strong subadditivity of quantum entropy (Letters in Mathematical Physics, (2023), 113, 1, (18), 10.1007/s11005-023-01638-2)

We correct here a mistake which is present in Eq. (5.7) of our paper [1]. The error has only a minor local effect on the streamlined derivation of SSA presented there. However, the correction is of independent interest, as it leads to a decomposition of the quantum mutual information of a composite system into a sum of two terms, one of a distinctly quantum nature and another expressing a purely classical correlation effect. There is an error in Eq. (5.7) of [1], to which our attention was called independently by Seth Lloyd and Eric Carlen. It does not invalidate the main points made there. However, Eqs. (5.5)–(5.7) and their enclosing paragraph should be replaced by the text bracketed below by <<<.. >>>. <<< Given a state operator ρ₁₂ on H₁₂=H₁⊗H₂, let ρ^₁₂ be the operator which is diagonal in the tensor product’s orthonormal basis consisting of functions |k,l⟩₁₂=|k⟩₁⊗|l⟩₂, where |k⟩₁ and |l⟩₂ are normalized eigenvectors of ρ₁ and ρ₂, with the matrix elements (Figure presented.) The mapping ρ₁₂↦ρ^₁₂ can be accomplished through a probability average over transformations of ρ₁₂ under unitary mappings. For that, let νρ₁₂(dU) be the Haar probability measure over the group of unitaries which in the above basis acts as multiplication operators of independently varying phases (Figure presented.) In these terms, ρ^₁₂ is presentable as the “pinched state”: (Figure presented.) The mapping ρ₁₂↦ρ^₁₂ is easily seen to have the following properties: Unitary operators of the form (5.6’) commute with ρ₁⊗ρ₂. The states ρ^₁₂ and ρ₁₂ coincide on the commuting algebra of functions F(X₁,X₂) of the operators X₁=∑_kk·|k⟩⟩k|⊗1 and X₂=∑_ll·1⊗|l⟩⟩l|. Being diagonal in the above basis, the state ρ^₁₂ is presentable as a classical probability distribution of a pair of random variables of values corresponding to l and k, with Pr{(X₁,X₂)=(k,l)}=(ρ12)_kl,kl. The restrictions of the two states on each of the two subsystems coincide, that is: (Figure presented.) >>> Unitary operators of the form (5.6’) commute with ρ₁⊗ρ₂. The states ρ^₁₂ and ρ₁₂ coincide on the commuting algebra of functions F(X₁,X₂) of the operators X₁=∑_kk·|k⟩⟩k|⊗1 and X₂=∑_ll·1⊗|l⟩⟩l|. Being diagonal in the above basis, the state ρ^₁₂ is presentable as a classical probability distribution of a pair of random variables of values corresponding to l and k, with Pr{(X₁,X₂)=(k,l)}=(ρ12)_kl,kl. The restrictions of the two states on each of the two subsystems coincide, that is: (Figure presented.) Adjustments: The original equation (5.7) was referred to in the published paper in two places, where the following adjustments should be made: i) In the justification for Eq. (5.11) (the subadditivity of entropy): the original (5.7) should be replaced by the modified (5.7’), resulting in (Figure presented.) Here, the first relation follows from (5.7’) by the Jensen inequality, the second holds by the classical subadditivity of entropy, and the equality is by (5.8’). ii) Within the proof of Theorem 5.3 (the strong subadditivity of entropy): the reference to (5.7) there was only optional, and should be dropped. The proof proceeds unchanged with the choice ρ~₃=(dimH3)^-11 for Eq. (5.21). In contrast to the strong subadditivity (SSA), the proof of simpler subadditivity (SA) has not been in need of simplification. Nevertheless, the modified Eq. (5.11’) presented above leads to a decomposition of potential interest of the quantum mutual information between the two subsystems, which is defined as (Formula presented.) The inequalities in Eq. (5.11’) allow to split the above difference into a sum of two positive terms, of which the first is of quantum nature and the second is purely classical: (Formula presented.) The first summand is the increase in the quantum state’s entropy which results from the measurement of the pair of commuting observables (X₁,X₂) (adapted to ρ). The second summand is the increase in the classical entropy when a joint distribution of a pair of classical variables is replaced by the (uncorrelated) product of its marginal probability distributions. We should, however, hasten to add that decompositions into classical versus quantum parts of correlations are not unique. In particular, the one explained above differs from the two that were introduced in [2, 3], guided by considerations of information transmission and state teleportation.