We establish two concentration principles for almost i.i.d. quantum sources under the weakly almost i.i.d. condition, the broadest condition in the established hierarchy of almost i.i.d. sources. Our results therefore apply also to Wasserstein- and Mazzola-Sutter-Renner almost i.i.d. sources. Weakly almost i.i.d. sources are sequences of multipartite states whose fixed-size marginals converge, on average, to tensor powers of a reference state, while allowing arbitrary global correlations and entanglement. We prove a noncommutative weak law of large numbers for empirical observables, and a universal entropy-concentration principle showing asymptotic concentration on subspaces of exponential dimension governed by the von Neumann entropy of the reference state. These concentration principles provide a unified and conceptually transparent approach to several information-theoretic applications beyond the i.i.d. setting, including direct proofs of universal compression within classes of weakly almost i.i.d. sources sharing a common reference state and asymmetric quantum hypothesis-testing bounds. They also yield concentration of empirical local-measurement frequencies and of empirical averages of commuting single-site observables, with applications to single-site generalized Gibbs ensembles, together with bounds on smooth and spectral entropies.
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27 pages;Revised version with corrections and clarifications. Main results unchanged
We establish two concentration principles for almost i.i.d. quantum sources under the weakly almost i.i.d. condition, the broadest condition in the established hierarchy of almost i.i.d. sources. Our results therefore apply also to Wasserstein- and Mazzola-Sutter-Renner almost i.i.d. sources. Weakly almost i.i.d. sources are sequences of multipartite states whose fixed-size marginals converge, on average, to tensor powers of a reference state, while allowing arbitrary global correlations and entanglement. We prove a noncommutative weak law of large numbers for empirical observables, and a universal entropy-concentration principle showing asymptotic concentration on subspaces of exponential dimension governed by the von Neumann entropy of the reference state. These concentration principles provide a unified and conceptually transparent approach to several information-theoretic applications beyond the i.i.d. setting, including direct proofs of universal compression within classes of weakly almost i.i.d. sources sharing a common reference state and asymmetric quantum hypothesis-testing bounds. They also yield concentration of empirical local-measurement frequencies and of empirical averages of commuting single-site observables, with applications to single-site generalized Gibbs ensembles, together with bounds on smooth and spectral entropies.
We study the robustness of asymptotic entanglement manipulation for Mazzola--Sutter--Renner (MSR) almost-iid sources, which allow a sublinear number of deviations from tensor-power structure. For pure sources along ∣φ⟩AB, every concentration rate below S(φA) is achievable by a single Schur--Weyl protocol depending only on the rate and local Hilbert spaces. For general sources along ρAB, every distillation rate below ED(ρAB) is achievable by a single LOCC protocol sequence independent of the particular source and defect size. Its error vanishes exponentially and uniformly for each prescribed defect bound rn=o(n), yielding an optimal universal distillation rate equal to ED(ρAB). For dilution, the entanglement cost of every MSR almost-iid target sequence along ρAB is at most EF∞(ρAB), the regularized entanglement of formation of the reference state. Consequently, every MSR almost-iid sequence along a pure reference state ∣φ⟩AB is asymptotically reversible, with distillable entanglement and entanglement cost equal to S(φA), even when its individual states are mixed. The proofs establish structural and entropic properties of MSR almost-iid sources that may be useful in other information-theoretic settings.
Independent and identically distributed (i.i.d.) states are ubiquitous in quantum information theory. However, in a practical setting, the i.i.d. assumption is too stringent, and possibly not realistic. A physically more compelling class of 'almost i.i.d.' sources was recently proposed by [Mazzola/Sutter/Renner, arXiv:2603.15792]. In this paper, we introduce two alternative definitions of almost i.i.d. states, based on the normalised quantum Wasserstein distance and on the idea of looking at the average k-body marginal. We explore some basic properties of these notions and prove a strict hierarchical relation among them, with Mazzola et al.'s notion being the strictest, the one based on k-body marginals the loosest, and the one based on the quantum Wasserstein distance in between. Strict separation is established by means of explicit examples.
The generalised quantum Stein's lemma is a key result in quantum hypothesis testing, and connects this fundamental primitive of quantum information processing with quantum resource manipulation, a task that is central for technological applications. Prior works have proved this statement in the idealised setting of independent and identically distributed (i.i.d.) sequences of quantum states, and recent extensions consider also sources that are 'close' to i.i.d., according to the strict notion put forth by Mazzola, Sutter, and Renner. For several applications, however, one would need to consider yet more general sources. We establish a version of the generalised quantum Stein's lemma that is conceptually much simpler and general, as it applies to any source that is asymptotically close to an i.i.d. state with respect to the normalised quantum Wasserstein distance of order 1. As an immediate consequence, we solve the Stein exponent of a scenario where the null hypothesis is arbitrarily varying, expressing it in terms of i.i.d. Stein exponents corresponding to arbitrary states in the convex hull of the null hypothesis base set.
The independent and identically distributed (iid) paradigm for asymptotic communication through quantum channels describes n uses of a channel Φ by Φ⊗n. Since exact tensor-power behaviour is an idealization, we ask whether the asymptotic capacities of Φ remain achievable for channel sequences that are almost-iid along Φ. We consider sequences of Choi-almost-iid channels whose normalized Choi states are Mazzola-Sutter-Renner (MSR) almost-iid along the normalized Choi state of Φ, with defect parameters rn=o(n). In Renner's exponential de Finetti theorem for quantum states, almost-iid states appear in the approximating mixture and motivate the MSR almost-iid class. Choi-almost-iid channels play the corresponding role in an exponential de Finetti theorem for quantum channels established in a companion paper. In this paper, we prove that the universal unassisted classical and quantum capacities of the class of Choi-almost-iid channels along Φ equal those of the channel Φ. Here universality is within this class: for each rate below the relevant capacity, a code sequence depending on Φ and the rate, but not on the particular channel sequence, achieves vanishing error for every sequence in the class.
We study the von Neumann entanglement entropy generated by a Haar distributed passive interferometer acting on n equally squeezed input modes with fixed nonzero squeezing strength s. Previous work established proportional weak typicality for integer R'enyi orders α≥2 and stated a sublinear von Neumann result, while the proportional von Neumann case remained open. For a subsystem of kn modes satisfying kn/n→r∈(0,1), we prove that, for every ε>0 and all sufficiently large n, P(ES1,nS1,n−1≥ε)≤2exp[−log2(en)cs,rε2n2]. The proof represents the entropy as a singular value statistic of a principal block of UUT, where U denotes the unitary interferometer. It regularizes the logarithmic singularity at the endpoint corresponding to a pure Gaussian mode and applies concentration on the unitary group. The result establishes proportional von Neumann weak typicality and further implies almost sure convergence of S1,n/ES1,n to 1, a typical volume law, and the variance bound Var(S1,n)=Os(log2n). An accompanying Lean 4 development verifies the proof chain.
Standard finite quantum de Finetti theorems approximate the k-system marginals of permutation-invariant states of n-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/n error bound in max-relative entropy improves on the previously best known k2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗n, where dA=dimA, we prove that the k-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of r defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2 and decays exponentially in n for a suitable choice of parameters r and k.