Conditional contraction coefficients and their applications to quantum networks

1.1 Background and motivation

• Conditional contraction coefficients subtract reference-system distinguishability, yielding a nontrivial framework for arbitrary quantum references without requiring equal reference marginals.Naively extending contraction by applying id⊗N can be trivial, while complete contraction coefficients impose equal marginals and may require unbounded reference systems.Source: §1.1 · pp. 3–4

1.2 Summary of contributions

• For relative entropy, conditional contraction equals conditional mutual-information contraction, even when the auxiliary system is quantum, classical, or binary classical.This extends the usual relative-entropy–mutual-information correspondence to conditional mutual information with quantum side information.Source: §1.2 · pp. 4–5
• The framework generalizes contraction-coefficient theory to expansion coefficients, relative contraction coefficients, and divergences beyond relative entropy, including structurally similar f-divergences.For classical references and divergences with the extended direct-sum property, conditional and usual contraction coefficients coincide.Source: §1.2 · pp. 4–5, §1.3 · p. 6
• The paper establishes approximate and several exact tensorization results for relative-entropy SDPI constants, including generalized depolarizing channels, and extends tensorization to conditional mutual information.These results broaden tensorization beyond commonly studied quantities and connect conditional SDPI tensorization with conditional mutual-information contraction.Source: §1.2 · pp. 5–6
• For trace distance, the conditional contraction coefficient is optimized by orthogonal input states, while hockey-stick divergence can be strictly larger conditionally than completely, even classically.The paper also studies relations among divergence- and mutual-information-based contractions and their SDPI analogues.Source: §1.2 · pp. 5–6
• Applications generalize a network-channel contraction bound [26], yield tensor-product bounds and replacing-time results asymptotically equivalent to mixing, and constrain quantum memories and quantum machine learning.The paper organizes these applications across quantum networks, discrete-time quantum Markov chains, and quantum machine learning.Source: §1.2 · p. 6, §1.3 · p. 7

2 Contraction with reference systems

• The section unifies these definitions and properties before specializing to particular divergences in subsequent sections.It introduces generalized SDPI constants, contraction and expansion coefficients, and relative coefficients within one reference-system framework.Source: §2 · p. 7

2.1 Quantum divergences

• Quantum divergences satisfy data processing; direct-sum divergences are jointly convex, providing the structural assumptions used later.Additional properties include non-negativity, faithfulness, additivity, and the direct-sum property.Source: §2.1 · p. 7

2.2 Contraction coefficients

• The framework defines product-reference, complete, and conditional SDPI and contraction coefficients, with conditional quantities subtracting distinguishability already present in the reference system.Ratio optimizations require strictly positive finite denominators; otherwise the coefficient is defined as zero.Source: §2.2 · pp. 8–9

2.3 Properties of contraction coefficients

• Contraction coefficients obey concatenation and tensor-product inequalities, are convex under channel mixing for jointly convex divergences, and tensorization can be strict.The tensor-product statements rely on additivity, while SDPI achievability depends on the precise coefficient definition.Source: §2.3 · pp. 9–10

2.4 Quantum Doeblin coefficient upper bounds on contraction coefficients

• Quantum Doeblin coefficients are semidefinite programs that upper-bound conditional contraction coefficients for jointly convex divergences, improving a prior bound from [16].The bound applies generally to every conditional contraction coefficient based on a jointly convex divergence.Source: §2.4 · p. 11

2.5 Expansion coefficients

• Generalized expansion coefficients extend expansion analysis to reference systems, but they are often zero for the considered divergences and remain generally informative in the trace-distance case.Their usefulness depends strongly on the chosen divergence.Source: §2.5 · pp. 11–12

2.6 Relative contraction and expansion coefficients

• Relative contraction and expansion coefficients extend channel comparisons to quantum reference systems by comparing distinguishability ratios or reference-subtracted ratios for two channels.Relative expansion and contraction coefficients are related, and such quantities have applications to quantum capacities [15].Source: §2.6 · pp. 12–13

3.1 (Conditional) mutual information contraction

• The section defines mutual-information and conditional-mutual-information contraction coefficients, including classical-variable variants and MIR coefficients for quantum reference systems.The formulations explicitly distinguish unrestricted quantum systems from classical systems and impose shared-reference conditions where specified.Source: §3.1 · p. 14, §3.2 · p. 17
• Theorem 1 equates binary-classical, arbitrary-classical, and quantum-reference formulations of conditional mutual-information contraction for fixed marginal σ_RA.The result extends the relative-entropy/mutual-information contraction correspondence to conditional mutual information with quantum side information.Source: §3.1 · p. 14

3.2 Mutual information with quantum reference systems

• The cited passages identify Theorem 2, Corollary 2, Lemma 5, and Corollary 3 as results concerning quantum channels.Source: §3.2 · pp. 18–19

3.3 Separations between the coefficients

• For qubit depolarizing channels, numerical examples separate some contraction coefficients, but unbounded reference systems leave whether several coefficients differ as an open problem.The numerical search did not find a comparable mutual-information value and therefore supports only a conjecture that the complete contraction coefficient is larger.Source: §3.3 · p. 20

3.4 Partial orders

• Lemma 6 states that for a channel N, the relative-entropy contraction coefficient satisfies D(N) = 1 −sup {ϵ ∈[0, 1] | Eϵ ≥cln N}.Naively building a contraction coefficient from these quantities leads to a trivial result.Source: §3.4 · p. 21
• For erasure channels, the conditional contraction coefficient measures information lost at the output while preserving distinguishability already present in the reference system, and it connects to the Doeblin coefficient and completely less noisy order.The framework also permits binary classical auxiliaries in the conditional-mutual-information definition of completely less noisy.Source: §3.4 · p. 22

3.5 SDPI constants

• The relative-entropy SDPI results yield equalities among established conditional, complete, and mutual-information SDPI constants, while fully quantum CMI and MIR cases require stabilized versions.For relative entropy, stabilization can be reduced to optimization over the auxiliary-system dimension using a maximally mixed state; the classical auxiliary U may be binary, and the MI SDPI constant equals the product-reference SDPI constant.Source: §3.5 · pp. 23–25
• The CMI SDPI constant is a stabilized conditional SDPI constant, and the MIR SDPI constant is a stabilized complete SDPI constant.The need for stabilization arises because fixing σRA fixes the reference-system size and removes flexibility available in the contraction-coefficient setting.Source: §3.5 · pp. 23, 25, 27

3.6 Expansion coefficients and relative contraction coefficients

• Expansion coefficients arise by replacing suprema with infima, but relative-entropy expansion coefficients are generally unhelpful when output dimension does not exceed input dimension unless the channel is unitary.Relative contraction coefficients provide a more informative alternative, and the established identities transfer to relative expansion coefficients.Source: §3.6 · pp. 27–28

3.7 Beyond the relative entropy

• Many results discussed above extend beyond relative entropy to larger classes of divergences whose proofs require two stated properties, including vanishing first derivative.These include several families of f-divergences, including the Petz f-divergence, the Hirche–Tomamichel f-divergence, and the maximal f-divergence.Source: §3.7 · p. 29
• For divergences satisfying (3.147) and (3.148), the contraction framework and many earlier results extend through divergence-specific mutual-information-like quantities.The framework applies to essentially all α-Hellinger divergences, whereas Rényi relative entropies do not satisfy the required class conditions.Source: §3.7 · pp. 29–30
• Theorem 3 establishes equivalence between verifying the relevant contraction condition for binary classical U and for arbitrary finite classical U, with fixed marginal σ_RA.The theorem assumes a divergence satisfying (3.147) and (3.148), channels N and M, a state σ_RA, and η ≥ 0.Source: §3.7 · p. 30
• Corollary 8 gives alternative expressions for the contraction coefficients for every quantum channel when the divergence satisfies (3.147) and (3.148).The stated suprema are restricted to states whose denominator is strictly positive and finite.Source: §3.7 · p. 30
• Analogous extensions apply to expansion coefficients, relative contraction coefficients, Theorem 2 and its consequences for complete contraction coefficients.These extensions follow the same generalization pattern established for the contraction coefficients.Source: §3.7 · p. 31

4 On tensorization of SDPI constants

• Tensorization preserves the exponential contraction rate as the number of identically noisy tensor factors grows, motivating its relevance to mixing times.The section notes that this behavior prevents contraction rates from deteriorating with system size when every subsystem experiences the same noise.Source: §4 · p. 31

4.1 Without reference systems

• The no-reference analysis uses a mutual-information correlation coefficient for product channels, whose bounds support the tensorization conclusions but can fail for non-product channels.A replacer channel preparing a maximally entangled state violates the upper bound in the non-product setting.Source: §4.1 · pp. 31–32
• Without reference systems, the relative-entropy SDPI constant admits approximate tensorization, with exact tensorization in several special cases.These include an erasure channel paired with an arbitrary channel, two generalized depolarizing channels, and a quantum-to-classical channel satisfying the stated mutual-information condition.Source: §4.1 · pp. 32–33

4.2 SDPIs with reference systems

• With reference systems, the product-reference SDPI constant and mutual-information SDPI constant tensorize, while complete and conditional relative-entropy constants generally do not.The failure follows from counterexamples in which the reference can be trivial, making the equal-reference-marginal condition ineffective; see [25].Source: §4.2 · pp. 34–35
• Stabilization restores tensorization for conditional SDPI constants and, equivalently, for the CMI SDPI constant, although the stabilized complete SDPI constant still fails in general.The section provides an explicit counterexample for the latter in Section C and summarizes the overall tensorization picture in Fig. 2.Source: §4.2 · pp. 36–37

5 Trace distance and hockey-stick divergences

• For γ ≥1, the hockey-stick contraction coefficient reduces to an optimization over orthogonal pure input states [53].Trace distance is the γ = 1 special case of the hockey-stick divergence.Source: §5 · p. 37
• Reference systems obstruct the general γ reduction: monotonicity of ηγ in γ is open, with preliminary numerical evidence suggesting it may fail with reference systems.The corresponding approach works without reference systems because orthogonal-state optimization is available for every γ ≥1.Source: §5 · pp. 39–40
• With reference systems, Theorem 6 replaces arbitrary state pairs by orthogonal pairs while preserving equal reference marginals when those marginals initially agree.For unrestricted reference marginals, the theorem still supplies orthogonal states for the corresponding comparison.Source: §5 · p. 39
• The resulting trace-distance conditional contraction and expansion optimizations are over mixed orthogonal states, whereas further restriction to pure orthogonal states remains unresolved.The same orthogonal-state reduction applies to relative contraction and relative expansion coefficients, though arbitrary auxiliary channel M limits simplification.Source: §5 · p. 40

5.1 Separating complete and conditional contraction coefficients

• For γ > 1, a classical binary symmetric channel separates complete and conditional hockey-stick contraction coefficients, establishing an affirmative separation result.The channel uses flipping probability p satisfying (1 − p)/p = γ, and its conditional coefficient vanishes even with quantum reference systems.Source: §5.1 · p. 41
• The separation example relies crucially on the hockey-stick divergence being non-faithful; whether faithful divergences admit analogous examples remains open.The conditional lower bound is obtained using a classical reference system and specially chosen binary input distributions.Source: §5.1 · p. 42

6.1 Contraction of mutual information in networks

• Proposition 11 bounds mutual-information and relative-entropy contraction through a quantum network channel using the contraction coefficients of its component channels, with fully quantum reference systems allowed.The quantum and classical formulations differ, although both can be expressed through mutual-information or relative-entropy contraction coefficients.Source: §6.1 · p. 43
• The framework yields tensor-product contraction bounds, including 1 + (1 −p)^2 ≤η_D(D_p^⊗2) ≤1 −2p^2 + p^3 ≤1 −p^2 for the qubit depolarizing channel.The resulting range is illustrated in Figure 3, and the final upper bound is described as the best previously known one.Source: §6.1 · pp. 45–46, Figure 3 · p. 46

6.2 Limits on quantum memories for preserving resources even with post-processing

• The memory results bound a time after which no resource can be distilled from noisy memories, regardless of the initial shared state, by combining one-shot distillation bounds with mutual-information or conditional-mutual-information contraction.The governing coefficient depends on the task and allowed post-processing operations, as summarized in Table 1.Source: §6.2 · p. 47, Table 1 · p. 48

6.2.1 Limits on entanglement preservation with non-entangling map post-processing

• After a sufficient depth, non-entangling post-processing cannot distill a state with singlet fraction greater than 1 −ε for any input state, although the bound excludes the interval endpoint ε = 1/2.The no-go result uses mutual-information contraction, while the ε = 1/2 limitation follows from the existence of a product state with singlet fraction 1/2.Source: §6.2.1 · p. 49

6.2.2 Limits on entanglement preservation with LOCC post-processing

• For LOCC post-processing, squashed entanglement replaces the looser mutual-information bound, yielding a no-go theorem for distilling sufficiently entangled states after a sufficient memory depth.The theorem applies to every initial state and uses the conditional-mutual-information contraction coefficient; its stated restriction is ε ∈(0, 0.07).Source: §6.2.2 · pp. 50–51

6.2.3 Limits on preserving resources for secret key with LOCC post-processing

• The section motivates separate secret-key preservation limits because key distillation can be easier than entanglement distillation, and introduces squashed entanglement as the relevant upper-bound tool for LOPC.The one-shot use of squashed entanglement is described as an approximate upper bound on key distillation.Source: §6.2.3 · p. 51
• Conditional contraction bounds show that secret keys of fidelity > 1−ε cannot be distilled from sufficiently old quantum memories, including under LOCC and one-way LOCC post-processing.The LOCC result applies to inhomogeneous and homogeneous Markov chains, while the one-way result gives a corresponding time threshold and restricts ε to a stated parameter range.Source: §6.2.3 · pp. 51–53

6.3 Limits on quantum memories for quantum secret sharing schemes

• For (t,m)-quantum secret sharing, the mutual-information contraction coefficient of the noise on t memories controls when no encoding remains ε-recoverable.The argument bounds recoverability through singlet fraction and symmetric extendibility, and Theorem 11 applies to arbitrary encodings once the noisy state is sufficiently close to a two-symmetrically extendible state.Source: §6.3 · pp. 55, 56, 58
• The product-channel formulation captures error-correcting behavior but is difficult to compute, so the α+ Doeblin coefficient supplies bounds based only on individual memory channels.This reflects that secret-sharing encodings can simultaneously function as error-correcting codes against network noise; Corollary 15 provides the resulting individual-channel relaxation.Source: §6.3 · p. 55

6.4 Quantum channel capacities

• The capacity discussion relates contraction coefficients and channel partial orders to additivity questions, including informational degradability, regularized less noisy channels, and completely less noisy channels.Prior work [15] constructed a less noisy but non-degradable channel, while Proposition 13 considers a corresponding candidate construction for completely less noisy channels from degradable channels.Source: §6.4 · pp. 58–59
• Complete less noisiness implies that a channel is informationally degradable and regularized less noisy.Source: §6.4 · p. 59
• The proof follows essentially the argument of [15, Proposition 6.1].Source: §6.4 · p. 59
• An example in [73] is completely less noisy but not degradable, and the construction can guide searches for further examples.Source: §6.4 · p. 59

7.1.1 Replacing, mixing, and their equivalence

• Replacing differs from mixing by incorporating a reference system and omitting an explicit fixed point, yet both describe convergence to a replacer channel.The equivalence is established using finite-dimensional equivalence of channel norms, including the diamond norm.Source: §7.1.1 · pp. 60–61

7.1 On hierarchies of operational times

• Mixing, replacing, and related asymptotic notions such as decoupling are equivalent for time-homogeneous quantum Markov chains because the channel powers converge to a replacer channel.The paper also characterizes mixing by eventual nontriviality of the quantum Doeblin coefficient.Source: §7.1 · p. 60, The equivalence · p. 61

The equivalence

• The equivalence proof shows that convergence of channel powers to the fixed-point replacer in diamond norm yields the replacing condition for arbitrary reference states.The resulting trace-distance excess over the reference marginal vanishes asymptotically.Source: The equivalence · p. 62

7.1.2 Finite-time bounds

• The D-replacement time measures how many channel applications are needed for every input state to approach the unique fixed point within a divergence threshold, and it is bounded using the conditional SDPI constant.For relative entropy, replacement and decoupling times coincide, while the replacement time is controlled by the product SDPI constant and tensorizes under channel powers.Source: §7.1.2 · pp. 62–64

7.1.3 Doeblin coefficient is eventually non-trivial for mixing channels

• A quantum channel is mixing if and only if some finite iterate has a strictly positive quantum Doeblin coefficient.The result addresses the coefficient’s possible non-faithfulness at a single application by showing eventual positivity for sufficiently contractive iterates.Source: §7.1.3 · p. 65

7.2 Correlation decay across a 1D open chain

• For one-dimensional open quantum chains, the paper proves exponential end-to-end correlation decay without requiring maximally entangled input states, provided the channel sequence has Choi operators bounded below by a strictly positive minimum eigenvalue.A full-rank Choi operator ensures the required positive contraction condition for each channel.Source: §7.2 · pp. 66–67
• Correlations across a 1D open chain decay exponentially fast in the depth.Source: §7.2 · p. 67
• Compared with [23], the result requires neither an entangled initial state nor channel unitality, but it does require full-rank Choi states.Because α+ is a semidefinite program, an exponential-decay bound can in principle be computed.Source: §7.2 · p. 67

8.1 Stability of quantum learning algorithms

• For a fixed quantum learning channel, contraction coefficients quantify how the noisy model and mutual-information terms from data encoding determine algorithmic stability.For layered channels N = P_d ◦ U_d ◦ ··· ◦ U_1 ◦ P_1, the stability factor includes ∏_{i=1}^d(1 − α_+(P_i)) multiplying I(ST2; T1), and the associated generalization error scales as √γ′.Source: §8.1 · p. 69

8.2 Information decay of layered quantum circuits

• When noisy circuit encoders create correlations between accessible and inaccessible systems, conditional contraction coefficients extend prior information-decay analyses and guide circuit-depth and encoder choices.When the relevant contraction is small, the earlier conclusions include exponential sample requirements for error mitigation and noisy barren plateaus; the framework also provides insight into minimizing information decay.Source: §8.2 · p. 70

8.3 Information scrambling in quantum circuits with prior correlations

• Scrambling must be analyzed through reduced channels for restricted observers, because unitary evolution preserves globally accessible information even when it becomes locally inaccessible.The reduced-channel setting studies information about C accessible to collections of B_i given prior correlations R, using conditional contraction coefficients.Source: §8.3 · pp. 70–71
• The resulting coefficients distinguish information loss from scrambling: information survives globally but is hidden from small regions when global contraction is near one and regional contraction is small.If both relevant quantities are small, the circuit destroys information; if the global quantity is near one while regional quantities are near one, information survives globally but remains hidden from all small regions.Source: §8.3 · p. 71

• The study shows that quantum reference systems create a rich contraction structure that clearly departs from classical contraction coefficients in network and multi-user settings.Source: §9 · p. 71

SDPI tensorization

• Fully quantum mutual-information and conditional-mutual-information SDPI constants tensorize, unlike commonly considered classical-quantum SDPI constants in general; related constructions appear in [18].This indicates that tensorization depends crucially on how quantum side information is incorporated.Source: §9 · p. 71

Applications

• The applications support a compositional theory in which local contraction properties control information propagation and lifetime in larger quantum networks.A central open question is when such local-to-global principles hold with quantum side information, especially for SDPI constants.Source: §9 · p. 72

Open problems

• Can the hockey-stick contraction optimization be restricted to orthogonal states, thereby making its contraction coefficient monotone in γ and enabling applications similar to?Source: §9 · p. 72
• It remains open whether a faithful divergence can satisfy ηc D(N) < η∆ D(N), particularly for relative entropy, because unbounded reference systems hinder confirmation.Source: §9 · p. 72
• Future work should characterize when relative-entropy SDPI constants tensorize or fail, and which divergence and side-information structures determine that behavior.Source: §9 · p. 72
• Non-linear strong-data-processing inequalities with reference systems, including those studied in [86], remain unexplored.Source: §9 · p. 72

• ChatGPT (GPT-5.6 Sol) contributed to conceiving Lemma 12 and Proposition 26, polishing figures, correcting typos, and detecting an error in an earlier draft.Source: Statement on AI usage · p. 72

A Contraction coefficients with classical reference systems

• Classical-reference complete and conditional contraction coefficients equal the standard contraction coefficient, so complete or conditional coefficients are unnecessary in this setting.This conclusion follows for the defined classical–quantum optimizations and establishes equivalence with the standard coefficient.Source: A Contraction coefficients with… · pp. 78–79
• The appendix defines generalized complete and conditional contraction coefficients for a quantum channel and generalized divergence over classical–quantum states.The constructions explicitly restrict the reference system to be classical.Source: A Contraction coefficients with… · p. 78
• The equality proof assumes the divergence has the extended direct-sum property and uses classical–quantum state decompositions together with equality of the reference divergences.The lemma applies the direct-sum structure to tuples of quantum states and a positive function g.Source: A Contraction coefficients with… · pp. 78–79
• The required property holds for Umegaki and Belavkin–Staszewski quantum relative entropies and many quantum α-Hellinger divergences, but not trace distance or hockey stick divergences.For relative entropies, g(p(r), q(r)) = p(r); for quantum α-Hellinger divergences, g(p(r), q(r)) = p(r)^αq(r)^(1−α).Source: A Contraction coefficients with… · p. 79

• The argument chooses λ ∈ [0, 1] so that the interpolated state ρ(λ) is full rank.Source: B Limit property of relative entropy… · p. 79
• Equation (B.9) expresses the resulting second-order quantity through Tr[(ln ρ(λ) − ln σ)(σ − ρ)].Source: B Limit property of relative entropy… · p. 80
• For general divergences, the section shows that one manifestation of the stated condition implies the other.Source: B Limit property of relative entropy… · p. 80
• Starting from full-rank σ, a sufficiently small ε ensures the perturbed operator is a quantum state, while µ(0) = 0 and µ′(0) = 1/ε enable a chain-rule argument.Source: B Limit property of relative entropy… · pp. 80–81
• The final equality in the derivation follows from the assumed vanishing first derivative property.Source: B Limit property of relative entropy… · p. 81

Failure of tensorization

• The stabilized complete relative-entropy SDPI constant does not tensorize in general.This follows from a channel and state supplied by Proposition 26 and the resulting two-copy violation.Source: C Failure of tensorization for the… · p. 81

One-copy analysis

• For one copy, dephasing the input preserves the numerator and cannot increase the denominator, reducing the optimization to dephased states.The argument then exploits the pure reference states and equal-reference-marginal constraint to characterize admissible blocks.Source: C Failure of tensorization for the… · p. 82
• The one-copy value 1/2 is attained by an explicit state construction.The construction also preserves the reference marginal, enabling comparison with the two-copy case.Source: C Failure of tensorization for the… · p. 83

Two-copy construction

• A two-copy perturbation of the uniform classical label distribution preserves positivity and the reference marginal while changing selected probabilities by relative amounts +ε, +ε, and −2ε.Because the conditional quantum states are identical for each label pair, the relative-entropy calculation reduces to classical relative entropy and yields the tensorization violation.Source: C Failure of tensorization for the… · pp. 83–84