The Quantum Wasserstein Distance of Order 1

A. Motivations

• The motivation is to capture locality-sensitive perturbations that unitarily invariant measures miss, since one-qubit operations can change entropy by at most ln 4.The related-work discussion distinguishes this proposal from quantum Wasserstein constructions based on differential structures, couplings, measurements, or basis-vector distances.Source: A. Motivations · p. 1, D. Related works · p. 3

B. The classical Wasserstein distances

• Classical Wasserstein distances arise from couplings minimizing mean pointwise cost and include the W1 distance induced by the Hamming metric.The classical W1 distance is also norm-induced and has applications in ergodic theory, information theory, and GAN training.Source: B. The classical Wasserstein distances · pp. 1–2

C. Our contribution

• The quantum W1 distance is defined through neighboring states that become identical after discarding one qudit, extending classical transport to n-qudit states.Its norm is the largest norm assigning distance at most one to every neighboring pair.Source: C. Our contribution · p. 2
• The distance is invariant under qudit permutations and local unitaries, additive under tensor products, and at most 2k when two states agree after discarding k qudits.Consequently, any operation on k qudits displaces a state by at most 2k in this distance.Source: C. Our contribution · p. 2
• It recovers the Hamming distance on canonical-basis vectors and the classical W1 distance for states diagonal in that basis.The construction therefore retains the expected classical transport behavior on computational-basis probability distributions.Source: C. Our contribution · p. 2
• A quantum Lipschitz constant yields a dual formulation in which the distance is computable through a semidefinite program.The distance equals the maximum expectation-value difference over observables with Lipschitz constant at most one.Source: C. Our contribution · p. 2
• The main entropy result states that states at distance o(n/ln n) can have entropy differences only o(n), making entanglement robust at that scale.The paper also derives quantum Marton and Gaussian concentration inequalities, with eigenvalue concentration controlled by the observable’s Lipschitz constant.Source: C. Our contribution · pp. 2–3
• For tensor powers of one-qudit channels, the proposed distance admits an n-independent contraction bound, while generic shallow circuits are controlled by qudit light-cone size.These results contrast with the trivial large-n behavior of the trace-distance contraction coefficient for tensor-power channels.Source: C. Our contribution · p. 3

II. NOTATION

• The Hilbert space of n qudits is H_n = (C^d)^⊗n, with quantum states represented among positive semidefinite operators of unit trace.The notation also defines marginals over selected qudits and the trace norm through eigenvalue absolute values.Source: §II · p. 3

III. THE QUANTUM W1 DISTANCE

• Two quantum states are neighboring when their marginals agree after discarding one specified qudit.The set of neighboring differences is used to generate the convex body defining the quantum W1 norm.Source: §III · p. 3
• The quantum W1 norm is the unique norm whose unit ball is the convex hull of neighboring-state differences.This construction is valid because the convex body is bounded, closed, centrally symmetric, convex, and has nonempty interior.Source: §III · p. 4
• The induced quantum W1 distance is the norm distance between two quantum states, and equivalent norm formulations support the paper’s calculations.The paper states results primarily for the norm because the corresponding distance statements follow directly.Source: §III · p. 4

A. Relation with the trace distance

• The quantum W1 norm is bounded relative to the trace norm, and the distance between neighboring states is correspondingly constrained.The trace-norm relation supplies comparison bounds for the proposed metric, including the single-qudit case.Source: A. Relation with the trace distance · p. 4

B. Symmetries

• The quantum W1 norm is invariant under qudit permutations and local unitary operations, and it is non-increasing under local quantum channels.Source: B. Symmetries · p. 5

C. Tensorization

• The quantum W1 distance is additive under tensor products, matching the classical counterpart and unlike the trace distance.This additivity is identified as fundamental for distortion measures in rate distortion theory [11, Chapter 5].Source: C. Tensorization · p. 5
• For product states, the tensor-product distance equals the sum of the component distances.Source: C. Tensorization · p. 5

D. Local operations

• For states that coincide after discarding k qudits, their quantum W1 distance is at most 2k.Source: D. Local operations · p. 5
• An operation acting on at most k qudits can displace a state by at most 2k in quantum W1 distance.This follows from continuity under local operations.Source: D. Local operations · pp. 5–6

E. Recovery of the classical W1 distance

• For quantum states diagonal in the canonical basis, quantum W1 recovers the classical W1 distance.Source: E. Recovery of the classical W1 distance · p. 6
• The distance between canonical-basis vectors equals their Hamming distance.Source: E. Recovery of the classical W1 distance · p. 6

• A dual formulation of the quantum Lipschitz constant is provided, extending the classical expectation-difference formulation.Source: §V · pp. 6–7
• The quantum W1 norm is computable through a semidefinite program with the stated dual program.Source: §V · pp. 6–7
• The quantum Lipschitz constant is defined as the dual norm of the quantum W1 norm on observables.Source: §V · p. 6
• For observables diagonal in the canonical basis, the quantum Lipschitz constant recovers the classical Lipschitz constant.Source: §V · p. 6

VI. W1 CONTINUITY OF THE VON NEUMANN ENTROPY

• Operations on k qudits change entropy by at most 2k ln d, reflecting robustness to local perturbations.The proposed quantum W1 distance is described as robust with respect to such perturbations.Source: §VI · p. 7
• The direct classical Shannon-entropy continuity bound cannot hold unchanged for quantum W1.A maximally entangled-state example shows that the classical form requires modification.Source: §VI · pp. 7–8
• The von Neumann entropy satisfies a quantum W1 continuity bound equivalent to the classical bound up to a factor ln n.Source: §VI · p. 8
• For fixed d and large n, entanglement properties are robust under quantum W1 perturbations of size o(n/ ln n).Source: §VI · p. 8

VII. QUANTUM MARTON’S TRANSPORTATION

• Theorem 2 establishes a quantum Marton transportation inequality for any state relative to a product reference state σ = σ1 ⊗ ... ⊗ σn.Source: §VII · p. 9
• The quantum Marton inequality improves quantum Pinsker’s inequality in the parameter regime identified by the paper.Source: §VII · p. 9

• The quantum Gaussian concentration inequality bounds the partition function of a quantum Hamiltonian in terms of its quantum Lipschitz constant.Source: §VIII · p. 9
• The proof normalizes H to zero trace and unit quantum Lipschitz norm, then derives the claim using earlier results and a choice of parameters.Source: §VIII · p. 9
• The theorem compares the spectrum’s moment generating function with that of a centered Gaussian having standard deviation √(n ∥H∥L /2).Source: §VIII · p. 9
• Consequently, eigenvalue tails decay at least as fast as Gaussian tails, yielding concentration of the spectrum of H.Source: §VIII · pp. 9–10
• The result applies to local Hamiltonians because their quantum Lipschitz constant is controlled by the maximum norm of terms associated with each qudit.Source: §VIII · p. 10

IX. CONTRACTION COEFFICIENT

• The section motivates quantum W1 contraction by contrasting trace-distance contractivity and its contraction coefficient with the corresponding quantum W1 formulation.Source: §IX · p. 10

A. Tensor power channels

• For any one-qudit channel Φ, the contraction coefficient of Φ⊗n with respect to quantum W1 is at most 1.Source: A. Tensor power channels · p. 10
• Unlike trace distance, quantum W1 retains a nontrivial tensor-power contraction bound as n →∞; for Markov semigroups, this bounds worst-case convergence to equilibrium.Source: A. Tensor power channels · p. 10
• Proposition 11 bounds the tensor-power contraction coefficient using a one-qudit channel, its fixed point, and the replacement channel E.Source: A. Tensor power channels · pp. 10–11
• For amplitude damping, the stated upper bound is trivial when 1/5 ≤ p ≤ 1 because Φp⊗n is already a quantum W1 contraction.Source: A. Tensor power channels · p. 11
• For the quantum depolarizing channel, the quantum W1 contraction coefficient is determined exactly and equals the trace-distance contraction coefficient.Source: A. Tensor power channels · pp. 10–11

• A channel’s expansion factor for quantum W1 is bounded by the size of the light-cone associated with each input qudit.Source: B. Shallow quantum circuits · p. 11
• For shallow local quantum circuits, these light-cone sizes can be bounded readily, providing a practical expansion estimate.Source: B. Shallow quantum circuits · p. 11
• The light-cone is the smallest output-qudit subset whose reduced output vanishes whenever the corresponding input marginal vanishes.Source: B. Shallow quantum circuits · p. 11
• The proof applies the defining light-cone condition together with the earlier quantum W1 bound for operators whose relevant marginal is zero.Source: B. Shallow quantum circuits · p. 12

• Quantum state estimation may require far fewer copies when accuracy is measured as o(n) quantum W1 error rather than o(1) trace-distance error.Source: §X · p. 12
• Quantum W1 distance measures perturbations for quantum machine learning inputs and makes computational-basis encodings continuous with respect to Hamming-distance changes.Source: §X · p. 12
• Because it recovers classical W1 for diagonal states and retains its properties, quantum W1 is proposed as a loss candidate for quantum GANs.Source: §X · p. 12
• Quantum W1 could extend rate-distortion theory beyond the iid regime because it is not limited to tensor-product input states and recovers Hamming distance on basis vectors.Source: §X · p. 12
• The distance generalizes neighboring states beyond product states, suggesting an extension of the connection between quantum differential privacy and measurement gentleness.Source: §X · pp. 12–13
• Its tensor-power contraction bounds motivate studying mixing times of quantum Markov semigroups without requiring a full-rank stationary state.Source: §X · p. 13
• The distance may characterize constant-depth circuit states and extend computational-capability results, while quantum Lipschitz constants may generalize local-Hamiltonian properties.Source: §X · p. 13

APPENDIX A ALTERNATIVE DEFINITION OF NEIGHBORING QUANTUM

• The alternative neighboring relation uses a one-qudit quantum channel connecting two states, and the induced norm is equivalent to the original quantum W1 norm.The relation requires either ρ = Φ(σ) or σ = Φ(ρ), where Φ acts on only one qudit.Source: Appendix A · p. 13

APPENDIX B EFFICIENT ESTIMATION OF THE QUANTUM LIPSCHITZ

• Proposition 15 estimates the quantum Lipschitz constant using channels that replace one qudit with the maximally mixed state, without optimization.The proposition applies to any qudit index i and observable H.Source: Appendix B · pp. 13–14

APPENDIX C LEMMAS

• The supporting lemmas characterize the W1 unit ball through neighboring-state differences, convex decompositions, marginal constraints, and joint convexity.Additional lemmas relate equal reduced states and classical couplings to the construction of these decompositions.Source: Appendix C · pp. 14–15

APPENDIX D PROOF OF THE W1 CONTINUITY OF THE SHANNON

• The proof uses an optimal coupling of two distributions and bounds coordinate disagreements via entropy monotonicity, strong subadditivity, Fano’s inequality, and Jensen’s inequality.The argument applies the concavity of the binary entropy function h2 in its final step.Source: Appendix D · p. 15