Approximate tensorization of the relative entropy for noncommuting conditional expectations

Abstract

In this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions

“…In fact, it was shown in [6] that the above conditional expectation coincides with that E D A for the Davies semigroup previously introduced:…”

“…(iii)⇒(iv) classically follows from standard arguments of finite speed of propagation of local perturbations [84,41] (see also [53,54] in the quantum setting). However, the implication (iv)⇒(i) is still unknown to hold under a good notion of decay of correlations and has been the subject of recent focus [6,5,20]. The goal of this paper is precisely to fill in this missing gap.…”

Learning quantum many-body systems from a few copies

“…and refer to the corresponding non-commutative weighted L 2 space with inner product X, Y σ := tr σ 1 2 X † σ 1 2 Y as L 2 (σ). The Petz recovery map corresponding to a subregion A ⊂ V with respect to σ(µ) is defined as R A,σ(µ) (X) = tr A (σ(µ)) −1/2 tr A (σ(µ) 1/2 Xσ(µ) 1/2 ) tr A (σ(µ)) −1/2 , We will also need the notion of a conditional expectation E A with respect to the state σ(µ) into the region A ⊂ Λ (see [64,65] for more details). For instance, one can choose E A := lim n→∞ R n A,σ(µ) , where R A,σ(µ) is the Petz recovery map of σ(µ).…”

“…In other words, the map E A is a completely positive, unital map that projects onto the algebra N A of fixed points of R A,σ(µ) . This algebra is known to be expressed as the commutant [65] N A := {σ(µ) it B(H A )σ(µ) −it ; t ∈ R} .…”

Weak quasi-factorization for the Belavkin-Staszewski relative entropy

“…The previous inequality was extended in [BCR20] to the more general context of finite-dimensional von Neumann algebras: Let M ⊂ N 1 , N 2 ⊂ N be von Neumann subalgebras of the algebra of linear operators acting on a finite-dimensional Hilbert space H and let E M , E 1 , E 2 be corresponding conditional expectations onto M, N 1 , N 2 , respectively. Then, a weak approximate tensorization for the relative entropy with parameters c ≥ 1, d ≥ 0 is satisfied (and denoted by AT(c,d)) if, for any ρ ∈ D(H), it holds that (4) D(ρ||E M * (ρ)) ≤ c (D(ρ||E 1 * (ρ)) + D(ρ||E 2 * (ρ))) + d , where the maps E M * , E 1 * , E 2 * are the Hilbert-Schmidt duals of E M , E 1 , E 2 , respectively (see also [Lar19], where a strong version of approximate tensorization, with d = 0, is considered).…”