Complete entropic inequalities for quantum Markov chains

Complete modified log-Sobolev inequality

• Finite-dimensional GNS-symmetric quantum Markov semigroups satisfy a complete modified log-Sobolev inequality with a uniform rate for all environment amplifications.The result resolves the previously open positivity question for the CMLSI constant in finite dimensions.Source: §1 · pp. 3–4

Strong data processing inequality

• Every finite-dimensional GNS-symmetric quantum channel satisfies a complete strong data processing inequality relative to its decoherence-free projection.The contraction constant depends explicitly on the projection index and an L2 contraction condition, and the inequality yields entropy decay and diamond-norm convergence estimates.Source: §1 · p. 4

Approximate tensorization of relative entropy

• Theorem 1.3 establishes a general approximate tensorization inequality for relative entropy over finite-dimensional von Neumann subalgebras without an additive error term.It extends SSA-type control beyond commuting-square settings and remains uniform under all amplifications.Source: §1 · p. 5

Common proof method

• A two-sided relative-entropy estimate through Bogoliubov-Kubo-Mori Fisher information reduces all three inequalities to corresponding spectral-gap conditions.This common method is presented as conceptually simple and potentially useful for other entropic inequalities.Source: §1 · p. 5

Symmetric quantum Markov semigroups

• Theorem 1.4 improves the CMLSI bound for symmetric semigroups by combining approximate tensorization with local double-commutator generators.The bound depends on the Hilbert-space dimension, the maximal noncommutation degree, and the minimum local spectral gap, and is asymptotically better than the general bound for primitive semigroups.Source: §1 · p. 6

Applications and examples

• For SU(2) sub-Laplacians, numerical estimates give a CMLSI constant greater than 0.18 uniformly for all representation dimensions m ≥ 2.The corresponding classical approach is unavailable because the sub-Riemannian setting lacks a curvature lower bound.Source: §1 · p. 6

Local semigroups

• For local subsystem Lindbladians, Theorem 1.6 lower-bounds the global CMLSI constant using local constants, graph degree, an index, and a spectral gap.The construction models 2-local interactions on adjacent graph vertices and applies to connected finite graphs.Source: §1 · pp. 6–7

Applications and examples

• The applications include exponential improvement for random transpositions, improved mixing bounds for continuous-time approximate designs, and CMLSI estimates for quantum Kac generators.For approximate designs, the paper improves a prior bound by a factor n while estimating CMLSI rather than only the spectral gap.Source: §1 · pp. 7–8

2.1. Relative Entropy and Conditional expectation.

• Relative entropy to a von Neumann subalgebra is minimized by the adjoint conditional expectation, via a chain rule over invariant states.The section also characterizes invariant states through the finite-dimensional direct-sum structure of the subalgebra.Source: §2.1 · p. 10

2.2. Subalgebra index and Max-relative entropy.

• The subalgebra index generalizes the Pimsner–Popa index to arbitrary conditional expectations and is equivalent to maximal relative entropy bounds.For invariant state σ, Cτ(M : N) ≤ µmin(σ)^−1C(M : N) and Cτ,cb(M : N) ≤ µmin(σ)^−1Ccb(M : N); these indices are finite exactly when the expectation is faithful.Source: §2.2 · pp. 10–11

2.3. A key lemma.

• Under ρ ≤ cσ, relative entropy admits two-sided estimates in inverse weighted L2 norms, while its upper bound does not require this domination assumption.The comparison relies on the weighted norm induced by the multiplication operator Γρ and yields the basic tool used later for conditional expectations.Source: §2.3 · pp. 12–13

2.4. Detailed balance.

• Detailed balance makes a conditional expectation an L2 projection and makes a quantum channel a weighted-L2 contraction whose multiplicative domain is an isometric ∗-subalgebra.For a full-rank invariant state satisfying σ-DBC, the channel’s restriction to the multiplicative domain is a ∗-isomorphism and L2-isometry, while the resulting spectral-gap condition is independent of the interpolation parameter and invariant state.Source: §2.4 · pp. 15–16
• λ(ρ, s) = λ(σ, s) for each s, with λ independent of s.The argument uses equation (27) and considers r ∈ [0, 1].Source: §2.4 · p. 17
• λ(ρ) ≤ λ(σ, 1) = λ(σ), because Φ∗(id − E_N) is self-adjoint for both ⟨·, ·⟩_σ and ⟨·, ·⟩_σ,s.Its maximal eigenvalue, and thus the associated operator norm, is independent of the Hilbert-space norm choice.Source: §2.4 · p. 17
• ∥(Φ2)∗(id − E_N): L2(σ) → L2(σ)∥ = ∥(Φ2)∗(id − E_N): L2(ρ) → L2(ρ)∥.This equality follows because both σ and ρ are invariant under Φ2.Source: §2.4 · p. 17

Proof strategy

• The proof uses entropy production, weighted derivation estimates, spectral-gap control, and completely bounded index constants.The argument reduces the lower bounds to key estimates and replaces the ordinary index by its completely bounded version for CMLSI.Source: §3 · pp. 18–20

Definition and scope

• CMLSI requires relative-entropy decay uniformly after tensoring the semigroup with arbitrary finite-dimensional reference systems.This complete formulation is equivalent to uniform control over all amplifications and is stable under tensorization.Source: §3 · pp. 18–19

Main theorem

• Every GNS-symmetric finite-dimensional quantum Markov semigroup satisfies the complete modified logarithmic Sobolev inequality under the stated extension and fixed-point assumptions.The theorem supplies bounds for both the ordinary and completely bounded modified logarithmic Sobolev constants.Source: §3 · p. 20

Scope

• The framework covers derivation triples and symmetric quantum Markov semigroups on finite von Neumann algebras when the relevant completely bounded index is finite.For the latter class, trace symmetry ensures a derivation representation of the Lindbladian.Source: §3 · p. 21

Consequences

• The bounds are the first generic nontrivial lower bounds for non-primitive quantum Markov semigroups and do not depend on environment size.The paper notes that sharper bounds can follow from approximate tensorization, while the asymptotically optimal general scaling remains open.Source: §3 · p. 21

Main theorem

• A GNS-symmetric quantum channel admits an explicit complete strong data processing inequality whenever its weighted L2 contraction coefficient is below one.The coefficient depends on a domination constant and the L2 contraction parameter, with a value strictly below one when the latter is below one.Source: §4 · p. 22

Proof strategy

• The argument converts χ2 or weighted L2 contraction into relative-entropy contraction by interpolating between states with the same decoherence-free part.The proof uses second-derivative comparisons along state interpolations and bounds the relevant domination parameter by the minimum eigenvalue of the reference state.Source: §4 · pp. 22–23

Tensorization

• The complete SDPI constant is stable under tensorization, unlike the ordinary formulation's general behavior.A chain-rule argument bounds the product-channel constant by the larger of the component constants.Source: §4 · pp. 23–24

Spectral-gap bounds

• For GNS-symmetric channels, both SDPI and complete SDPI constants are bounded by explicit functions of the spectral gap and ordinary or completely bounded index.The same estimate applies to SDPI after replacing the completely bounded index by its ordinary counterpart.Source: §4 · p. 24

Scope and comparison

• The results provide explicit constants for general non-ergodic GNS-symmetric channels independently of environment size.For primitive unital channels, prior bounds can be generically better, but the present result covers the broader non-ergodic setting and preserves tensorization.Source: §4 · pp. 24–25

Approximate tensorization

• Theorem 5.1 establishes that L2-clustering, with constant λ < 1, implies complete approximate tensorization, with c bounded between 1/(1−λ^2) and 2Cτ(M:N)/(1−λ)^2.The bounds also tensorize completely, replacing Cτ(M:N) by Cτ,cb(M:N) for matrix-amplified states.Source: §5 · p. 26
• Theorem 5.3 improves the constants when λ < 1/√2, using Cmax and Cmax,cb, and the resulting bound converges to 1 as λ → 0.Thus the refinement recovers the optimal commuting-square limit, unlike the preceding general theorem.Source: §5 · p. 27

Channel-based bounds

• Theorem 5.4 derives complete approximate tensorization from a GNS-symmetric quantum channel Φ whose adjoint is sandwiched in completely positive order around an EN-invariant conditional expectation.Corollary 5.5 extends this to intersections of finitely many subalgebras and provides bounds independent of the index Cτ,cb(M:N), though not exact for commuting expectations.Source: §5 · pp. 29–31

Entropic uncertainty relations

• Approximate tensorization tightens entropic uncertainty relations for two projective measurements with quantum memory, and the resulting inequality is tight for ρAM = 1/dA ⊗ρM.The measurement maps are pinchings onto different orthonormal bases, with the overlap matrix controlling the clustering constant.Source: §5 · pp. 32–33
• The same framework extends uncertainty relations with quantum memory to three measurements by combining the corresponding overlap matrices and applying the multi-subalgebra tensorization bounds.The construction uses three orthonormal bases and a bound on the associated completely bounded constant.Source: §5 · pp. 33–34

• The stated assertion follows from Theorem 5.4 and the chain rule, paralleling the proof of Corollary 5.5.Source: §5 · p. 34
• Remark 5.10 states that the constant λ1 in Corollary 5.9 can be written explicitly.Source: §5 · p. 34
• Corollary 5.9 remains valid when ΦW is defined using the products of EX, EY, and EZ in any permutation of XYZ.Source: §5 · p. 34

6.1. Symmetric Lindbladians.

• Theorem 6.1 bounds the modified log-Sobolev constant of a symmetric quantum Markov semigroup using component spectral gaps, approximate tensorization, and complete amplification estimates.The resulting estimates remain valid uniformly for all environment dimensions.Source: §6.1 · pp. 35–36
• When each Lindblad operator fails to commute with at most m others, Corollary 6.2 uses the detectability lemma to provide an alternative bound.Its conditional-expectation product estimate depends on m and the spectral gap of the averaged generator.Source: §6.1 · p. 37
• Theorem 6.1 is preferable for highly noncommuting generators, whereas Corollary 6.2 can be tighter when most generators commute.The section also notes that the general constants need not improve on those of Theorem 3.3.Source: §6.1 · p. 37

6.2. Groups and transference.

• Transference embeds a symmetric quantum Markov semigroup into a classical Markov semigroup tensored with an identity map, with the invariant algebra given by the representation commutant.Proposition 6.5 transfers relevant functional-inequality estimates from the classical semigroup.Source: §6.2 · pp. 37–38

6.3. Semigroups transferred from the sub-Laplacian on SU(2).

• The sub-Laplacian itself is known to be hypoelliptic, but whether it satisfies CMLSI remains open.The transferred finite-dimensional generators nevertheless have a uniform spectral gap of 1 for all m.Source: §6.3 · pp. 39, 41
• For larger SU(2) representations, transference from the torus yields a dimension-free numerical CMLSI bound that improves on the Schur-multiplier bound for m>4.The transferred generators have integer spectra, enabling comparison with torus estimates.Source: §6.3 · p. 41

7. Local semigroups

• Graph-local semigroups combine Theorem 3.3’s local CMLSI constants with sharpened approximate tensorization to obtain asymptotically tight bounds.The approach targets symmetric semigroups whose Lindblad operators act on graph edges and is independent of the total environment size.Source: §7 · p. 42
• The quantized graph-Laplacian construction transfers classical CMLSI estimates, with α_CMLSI(∆G)/(1 + 5π^2α_CMLSI(∆G)) ≤ α_CMLSI(∆q_G) ≤ α_CMLSI(∆G).This is one of two locality-preserving quantum constructions; the other is the subsystem Lindbladian acting on graph-edge pairs of an n-qudit system.Source: §7 · pp. 42–43
• Theorem 7.1 gives exponentially tighter CMLSI bounds for subsystem Lindbladians by exploiting locality, spectral gaps, approximate tensorization, and the detectability lemma.For uniformly gapped expander families, the resulting CMLSI constant has constant-order asymptotics, recovering the classical expander scaling.Source: §7 · p. 43

7.1. Random permutations.

• Nearest-neighbor quantum random transpositions transfer classical permutation-group estimates, yielding bounds independent of the local dimension d and exponentially better in n than generic graph-based estimates.The model realizes adjacent register swaps on graph edges and is identified with the natural action of Sn on the n-qudit space.Source: §7.1 · pp. 45–46

7.2. Approximate unitary designs.

• For approximate unitary designs, the section introduces entropic design notions and relates relative-entropy contraction to stronger design criteria through Lemma 7.8.It then constructs local subsystem Lindbladians from universal symmetric measures on SU(d^2), extending local random circuits to continuous time.Source: §7.2 · pp. 47–49
• The local CMLSI constant and spectral gap for the local Brownian generator are independent of k, while Bourgain–Gamburd’s theorem gives a uniform gate-set spectral gap.The gate-set result requires symmetry and algebraic entries; the Brownian dynamics arise by transferring heat flow on U(d^2) through tensor-power representations.Source: §7.2 · pp. 50–51
• Theorem 7.9 assumes that the graph G has a linear subgraph whose vertices can be listed consecutively with edges joining successive vertices.It also specifies ε(k) := ∥Φ(k) µ − D(k) Haar : L2 → L2∥ < 1 and a symmetric universal gate set whose elements have algebraic entries.Source: §7.2 · pp. 51–52
• The resulting continuous-time evolutions become ε-approximate unitary k-designs, with the local-Brownian example converging in diamond norm under the derived time bound.The estimates use the CMLSI and spectral-gap bounds from Theorem 7.9 and extend to arbitrary system sizes and tensor multiplicities as stated in the supplied passages.Source: §7.2 · p. 53
• The design bounds are worst-case nearest-neighbour estimates, and greater graph connectivity is expected to improve the gap while leaving the CMLSI constant essentially unchanged on complete graphs.The text identifies complete-graph behavior as the motivation for the following subsection, not as an established theorem here.Source: §7.2 · p. 53

7.3. Quantum Kac model.

• The quantum Kac model assigns each complete-graph collision an energy-preserving, time-reversible, swap-symmetric unitary distribution, producing a subsystem Lindbladian for pairwise interactions.These three conditions respectively encode elasticity, self-adjointness, and symmetry between the colliding particles.Source: §7.3 · pp. 54–55
• For Haar collisions, the local channel replaces each edge state by the maximally mixed state, and the global semigroup converges to the maximally mixed state d^-n 1.In this case local partial traces commute, enabling the relative-entropy estimate stated after Theorem 5.3.Source: §7.3 · p. 55
• The complete graph can be decomposed into at least floor(n/4) connected spanning subgraphs of maximal degree 3 with mutually disjoint edge sets, supporting analysis of general collision specifications.This decomposition is introduced immediately before the proof of the general quantum Kac theorem.Source: §7.3 · p. 56
• For prime p, the edge sets E_l form mutually disjoint cyclic subgraphs, and for general n they are enlarged to connect all vertices.Bertrand’s postulate supplies a prime ⌊n/2⌋ < p < n, enabling the construction for arbitrary n.Source: §7.3 · p. 56
• Universal collision specifications with decay of correlations satisfy a CMLSI bound with a constant λ0 independent of the system size.The proof combines a uniform spectral-gap lower bound on connected linear subgraphs with a decomposition of the complete graph into commuting bounded-degree subgraphs.Source: §7.3 · p. 57
• Complete-graph k-design Lindbladians satisfy the theorem’s assumptions, yielding a CMLSI constant that remains Ω(1) as n grows.Universality is preserved under U ↦ U^⊗k, while decay of correlations follows from [18, Lemma 18].Source: §7.3 · pp. 57–58
• The complete-graph heat semigroup has spectral gap Ω(n), improving over the result of [92].This bound follows by applying the same method with the reverse detectability lemma [4].Source: §7.3 · p. 58

Established results

• The paper’s established CMLSI result applies to finite-dimensional GNS-symmetric quantum Markov semigroups and controls all amplifications, unlike the cited L2-log-Sobolev bound.The amplification control is stated as a distinction from L2-log-Sobolev or hypercontractivity estimates.Source: §8 · p. 58

CMLSI

• The paper leaves open whether the CMLSI lower bound can be improved asymptotically beyond the stated dimension dependence.The proposed target is α_CMLSI(L) ≥ λ(L)o(d^2)^−1.Source: §8 · p. 58

SDPI

• A further open problem asks whether SDPI constants for non-primitive channels admit lower bounds in terms of modified log-Sobolev constants.A positive answer would potentially strengthen known comparisons and yield CSDPI estimates from spectral gap and index bounds.Source: §8 · pp. 58–59
• The discussion also asks whether a uniform α′(Φ)<1 controls relative entropy for all state pairs sharing the same decoherence-free part.This would be stronger than the paper’s α_S​​DPI notion and is linked to global quantum χ2 contraction.Source: §8 · p. 59

Detectability estimate

• If each projection E_i commutes with all but at most g others, Theorem A.1 provides a detectability bound involving the spectral gap λ(L).The proof orders the projections and controls the accumulated error through the at-most-g noncommuting neighbors.Source: Appendix A · pp. 59–60

Proof strategy

• The proof moves id−E_i through the ordered product until it reaches E_i, where the term vanishes, while bounding each step by the noncommuting-neighbor count.Triangle and Jensen inequalities control the resulting sum.Source: Appendix A · p. 60

Ordering

• The appendix notes that the ordering of projections in the adjoint product may be arbitrary.This observation is attributed to the cited prior result.Source: Appendix A · p. 60

Module-Choi approach

• The first strategy treats a finite-dimensional von Neumann algebra with a trace and conditional expectation, deriving order relations for bimodule maps from norm estimates.Its key setting is a subalgebra N⊂M with trace-preserving expectation E_N.Source: Appendix B · pp. 60, 62

Comparison

• Lemma B.1 and Lemma B.2 supply complementary order-estimate mechanisms based respectively on module Choi operators and standard Choi matrices.The appendix closes by comparing these approaches in the trivial-subalgebra and compact-group representation settings.Source: Appendix B · pp. 62–64

Module-Choi approach

• For symmetric channels close to E_N in L2 norm, iterating the channel yields bounds controlled by the cb-index and powers of the contraction parameter.The relation (Φ−E_N)^k=Φ^k−E_N is used to transfer the estimate to iterates.Source: Appendix B · p. 63

Standard Choi approach

• The second strategy specializes to M=B(H) and uses the standard Choi-Jamiołkowski matrix, whose positivity characterizes complete positivity and whose normalization characterizes channels.This gives a direct route to completely positive order comparisons with conditional expectations.Source: Appendix B · p. 63

Examples

• For compact-group representations, the fixed-point algebra is the commutant, and the appendix describes its decomposition into matrix-algebra blocks and associated conditional expectation.For the finite-group left regular representation, the commutant is isomorphic to the group algebra.Source: Appendix B · p. 64