We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi--Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack, is asymptotically perfectly distinguishable from the tensor powers of the input: its trace distance from them tends to the maximum value $2$. This maximal separation persists, with a common positive slack, throughout an open neighborhood in the full four-qubit state space. Tracing out one party yields a mixed tripartite example.
The proof connects simultaneous quantum smoothing with polynomial concentration on $\CP^1$. Symmetry reduces the problem to a frame inequality on $\Sym^n(\mathbb C^2)$, where a linear entropy deficit competes with a quadratic cost of polynomial concentration. Bernstein--Markov estimates and a subharmonic barrier control this cost, yielding an exponential obstruction to simultaneous typicality.
These results disprove Dutil's multiparty quantum typicality conjecture, formulated in 2011, and the one-shot simultaneous min-entropy-smoothing conjectures of Drescher--Fawzi and Colomer--Winter. Together with the positive results for smaller systems, they determine the smallest numbers of parties at which simultaneous typicality can fail.
We establish exact thresholds for the failure of simultaneous quantum typicality: three parties for mixed states and four for pure states. For the four-qubit Higuchi--Sudbery state, every sequence satisfying the expected purity bounds on three overlapping marginals, with sufficiently small fixed positive entropy slack, is asymptotically perfectly distinguishable from the tensor powers of the input: its trace distance from them tends to the maximum value 2. This maximal separation persists, with a common positive slack, throughout an open neighborhood in the full four-qubit state space. Tracing out one party yields a mixed tripartite example.
The proof connects simultaneous quantum smoothing with polynomial concentration on \CP1. Symmetry reduces the problem to a frame inequality on \Symn(C2), where a linear entropy deficit competes with a quadratic cost of polynomial concentration. Bernstein--Markov estimates and a subharmonic barrier control this cost, yielding an exponential obstruction to simultaneous typicality.
These results disprove Dutil's multiparty quantum typicality conjecture, formulated in 2011, and the one-shot simultaneous min-entropy-smoothing conjectures of Drescher--Fawzi and Colomer--Winter. Together with the positive results for smaller systems, they determine the smallest numbers of parties at which simultaneous typicality can fail.
We establish two results on the asymptotic equipartition property (AEP) for quantum channels. First, the AEP formulated using smoothing over completely positive trace-preserving maps fails in general. Second, the AEP with smoothing over subchannels (completely positive trace-nonincreasing maps) is equivalent to the strong converse for parallel quantum channel Stein's lemma, to a sharp threshold for the channel hockey-stick divergence, and to an AEP for the channel Lorenz divergence. The counterexample reveals a surprising separation between channel and subchannel smoothing. Although states and classical channels admit a sharp, dimension-independent completion bound, which we establish, the corresponding single-shot gap for quantum channels is unbounded already for qubits. This separation persists asymptotically: we construct qutrit channel pairs with finite max-relative entropy whose CPTP-smoothed rate strictly exceeds the regularized channel relative entropy. The asymptotic gap between channel and subchannel smoothing is unbounded across the family, even when the subchannel approximation error decays exponentially. To establish the equivalences, we develop a uniform filter that converts Lorenz smoothing into a single nearby subchannel. The four conjectured formulations thus share a common asymptotic threshold, whose validity for general channel pairs remains open. Together, these results separate this threshold question from the obstruction imposed by exact trace preservation.
Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem A, the marginal maximal-mixing defect is defined by DA=2∣A∣Tr(ρA2)−1=2∣A∣ρA−2∣A∣IA22, and Qex,εD(n,k) is defined as the maximum number of k-body marginals satisfying DA≤ε in an n-qubit pure state. This counting problem differs from approximate k-uniformity, which requires all k-body marginals to obey a common error bound.
For pure states on 4m qubits, the following local stability inequality is established: ∑i∈TDT∖{i}≥1(∣T∣=2m+1). It follows that, whenever ε<1/(2m+1), the hypergraph of ε-good 2m-subsets is K2m+1(2m)-free. Combined with the known exact eight-qubit construction, this yields the stability plateau Qex,εD(8,4)=56,0≤ε<51. For odd systems of 2k+1 qubits, the exact forbidden hypergraph Hk is used to derive explicit finite-error stability radii. In particular, Qex,εD(9,4)≤120 for 0≤ε<1/17. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.
Abstract We show that a simple telescoping sum trick, together with the triangle inequality and a tensorisation property of expected-contractive coefficients of random channels, allow us to achieve general simultaneous decoupling for multiple users via local actions. Employing both old (Dupuis et al. in Commun Math Phys 328:251–284, 2014) and new methods (Dupuis in IEEE Trans Inf Theory 69:7784–7792, 2023), we obtain bounds on the expected deviation from ideal decoupling either in the one-shot setting in terms of smooth min-entropies, or the finite block length setting in terms of Rényi entropies. These bounds are essentially optimal without the need to address the simultaneous smoothing conjecture, which remains unresolved. This leads to one-shot, finite block length, and asymptotic achievability results for several tasks in quantum Shannon theory, including local randomness extraction of multiple parties, multi-party assisted entanglement concentration, multi-party quantum state merging, and quantum coding for the quantum multiple access channel. Because of the one-shot nature of our protocols, we obtain achievability results without the need for time-sharing, which at the same time leads to easy proofs of the asymptotic coding theorems. We show that our one-shot decoupling bounds furthermore yield achievable rates (so far only conjectured) for all four tasks in compound settings, which are additionally optimal for entanglement of assistance and state merging.
Exact Haar Statistics of Planar k-Purity in Multipartite Quantum Systems
Planar k-uniform states have maximally mixed reductions on every k-site interval of a ring. We introduce planar k-purity---the mean purity of those intervals---as a faithful cost function for this simultaneous constraint. For a Haar-random pure state of n parties with local dimension p, the joint purity of two intervals depends only on their overlap. An exhaustive four-replica calculation therefore yields the complete cyclic covariance kernel and a closed variance for every 1≤k≤⌊n/2⌋. In the balanced qubit case, k=⌊n/2⌋ and N=2n, the variance is asymptotic to 20/(3nN2) for even n and 6/(nN2) for odd n. A site-factorized permutation representation gives all raw moments and permits exact finite-sum evaluation of the skewness. Fixed-seed simulations validate nonbalanced qubit and qutrit cases, while balanced-qubit simulations through n=10 validate the parity formulas and finite-size skewness. Balanced planar and absolute balanced purity have the same Haar mean but different fluctuations; the planar variance is approximately 2.83 times larger at n=10. Thus subsystem incidence, although invisible to every one-cut marginal distribution, controls the collective fluctuations of geometrically related cuts. Together with universal attainability and a linear number of interval constraints at balance, this provides a geometry-aware benchmark for multipartite entanglement beyond AME existence regimes.
At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble {(1−p,ρ1),(p,ρ2)} by c∥H∥h2(p) (with the binary entropy h2). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality ∥[A,logB]∥1≤ch2(p) for positive trace-class operators A≤B with TrA=p and TrB=1 on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be c=1. Here, I prove the latter conjecture using an exact integral representation of the commutator [A,logB] obtained from the operator layer cake theorem, due to Cheng and Liu. The result gives a sharp dimension-independent limit on how rapidly unitary evolution of one component can change the entropy of a binary quantum ensemble. It follows that mixing rates satisfy small incremental mixing with the optimal constant, and entangling rates of bipartite Hamiltonians are bounded by (2logd+1)∥H∥. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.
The strong spin-alignment conjecture fails for three qubits when the subset states are chosen independently. We prove that it holds when they are marginals of one global state: for every three-qubit state, every positive semidefinite qubit operator Q, and every nonnegative set of subset weights, the aligned output majorizes the corresponding marginal mixture. The proof classifies the extreme rays of cones on which a fixed set of aligned eigenvalues is largest. This reduces arbitrary weights to two-term sums and the complete one- and two-body layers. For arbitrary numbers of subsystems, we also characterize when one unital quantum channel simultaneously realizes all marginal replacements. This criterion proves alignment for fully PPT states. A separate argument proves alignment for locally rotated stabilizer states in every prime local dimension, at maximally mixed background. These results determine the minimum output Rényi entropies of the associated three-qubit replacement channels. Under either of the two input restrictions above, they also determine the many-use classical capacity and the exact strong-converse exponent for correlated replacement channels of arbitrary size.