We study when conjugation by a product Haar unitary on (ℂ^d)^{⊗D} reproduces the scalar mixed-moment behavior of a Haar unitary on the full space as d→∞, with D fixed. On the fully traceless sector—matrices whose one-leg partial traces vanish—we prove sharp global-Haar-scale cancellation: for N=d^D, fixed alternating words have bias O(N^-2) and fixed-L^q fluctuations O(N^-1). If this partial-trace condition is asymptotically stable under centered polynomial evaluation, it yields quantitative scalar asymptotic freeness. For liberation from every bounded deterministic test family, the corresponding partial-trace defect is also necessary at the first nontrivial order and can be reconstructed from D fixed probes, giving a local-unitary second-moment diagnostic related to randomized-measurement methods. For arbitrary bounded deterministic coefficients, we derive a self-contained fixed-order expansion in which finitely many normalized permutation-tuple contractions determine every product-Haar mixed moment up to O(d^-1). Using the exact multi-leg contraction exponent of Collins–Yuan, currently available as a preprint, we obtain a finite necessary-and-sufficient criterion for matching global-Haar mixed moments at every fixed (D,k), and hence an all-orders characterization. In the bipartite k=4 case exactly four tensor-transport contractions survive: the traceless SU(d) exchange interaction activates the two inverse-cycle modes. For Bell-diagonal generators the same modes become explicit ordinary and symplectically twisted autocorrelation energies: centered random sign masks satisfy all four asymptotically almost surely, whereas a classical circulant two-level mask has an exact surviving pairing coordinate 1/3+8/(3d^2). Finally, exact vanishing of the polynomial partial-trace defect is equivalent to local complementarity, yielding a d^2 algebra-dimension bound and extremal connections with distance-two quantum MDS codes and absolutely maximally entangled states.
We study when conjugation by a product Haar unitary on (ℂ^d)^{⊗D} reproduces the scalar mixed-moment behavior of a Haar unitary on the full space as d→∞, with D fixed. On the fully traceless sector—matrices whose one-leg partial traces vanish—we prove sharp global-Haar-scale cancellation: for N=d^D, fixed alternating words have bias O(N^-2) and fixed-L^q fluctuations O(N^-1). If this partial-trace condition is asymptotically stable under centered polynomial evaluation, it yields quantitative scalar asymptotic freeness. For liberation from every bounded deterministic test family, the corresponding partial-trace defect is also necessary at the first nontrivial order and can be reconstructed from D fixed probes, giving a local-unitary second-moment diagnostic related to randomized-measurement methods. For arbitrary bounded deterministic coefficients, we derive a self-contained fixed-order expansion in which finitely many normalized permutation-tuple contractions determine every product-Haar mixed moment up to O(d^-1). Using the exact multi-leg contraction exponent of Collins–Yuan, currently available as a preprint, we obtain a finite necessary-and-sufficient criterion for matching global-Haar mixed moments at every fixed (D,k), and hence an all-orders characterization. In the bipartite k=4 case exactly four tensor-transport contractions survive: the traceless SU(d) exchange interaction activates the two inverse-cycle modes. For Bell-diagonal generators the same modes become explicit ordinary and symplectically twisted autocorrelation energies: centered random sign masks satisfy all four asymptotically almost surely, whereas a classical circulant two-level mask has an exact surviving pairing coordinate 1/3+8/(3d^2). Finally, exact vanishing of the polynomial partial-trace defect is equivalent to local complementarity, yielding a d^2 algebra-dimension bound and extremal connections with distance-two quantum MDS codes and absolutely maximally entangled states.
Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.
Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In 2⊗2 and 2⊗3 systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition
Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on Cd⊗Cd, obtained as the marginal over an environment Cs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level m∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension s=λmd2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to 1 as d grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension d is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.
We develop an exact finite-dimensional theory of the replica mechanics underlying measurement-induced entanglement transitions in monitored quantum circuits, separating three structures that are often conflated: linear outcome-summed replica channels, normalized conditional trajectories, and asymptotic large deviations. For fixed-basis projective measurements we determine the minimal invariant unital algebra of the one-wire feature space and prove that independently Haar-averaged two-site gates on a perfect matching close the n-replica permutation space exactly for every replica number, with an explicit Weingarten bond channel. The resulting one-period transfer operators have real, positive, semisimple nonzero spectra that compress to bond-label space; at n=2 the model is exactly the anisotropic triangular-lattice Ising model of Bao, Choi and Altman, with a closed-form critical point and a closed free-fermion spectrum. Carrying the same computation through n=3,4,5 locates the annealed critical points pc(2)=0.2338 <pc(3)=0.305(3)<pc(4)≈0.383<pc(5)≈0.47 with the Potts-class exponents at n=3,4, and supports, at the three-size level, the first-order prediction of the q=n-Potts identification at n=5: an exact three-size test (L=4,6,8) on a symmetry-reduced orbit basis of dimension 3,865 over the 2.1×108 bond-label space shows the two-phase gap closing faster than the continuous baseline at every size, with the scaled gap falling below the continuous envelope at the matched sizes and dual effective closing rates ordered consistently with the continuous, marginal and first-order behaviours at matched sizes---a directional trend, with the registered conditional that completes it: if the closing is genuinely exponential, a fourth size (L=10) will show the effective closing rate stabilizing at a positive interface tension, and if the asymptotic is a slowly decaying power law, that rate will continue to decay---and at local dimension d=3 the same three-size test keeps the gap level below the d=2 values at every size. We further prove a no-freezing theorem for finite-reachable record processes, establish the worst-case complexity of conditional-trajectory decision, derive the exact replica interpolation identity with its measurable self-averaging criterion, and give the first numerical quenched record scaled cumulant generating functions for Haar circuits, which exhibit genuine record multifractality; the Clifford record-count disorder statistics are closed exactly through a verified conjugation 3-design.dummy
We determine which representation sectors control the slowest convergence of random Pauli rotations and several local random quantum circuits. For d=2n, the continuous Pauli walk has exact all-representation gaps ΔSU(d)=8(d−1)d−8(n≥4),ΔPU(d)=8(d2−1)d(d−3)(n≥3). The special-unitary gap is attained in ⋀8Cd and is invisible to balanced moments, while the projective gap is already attained in a balanced fourth-moment representation. We obtain exact discrete and orthogonal Pauli analogues, exact finite-Clifford chain/path gaps, and dimension-independent all-representation Haar brickwork lower bounds; on three qubits every weighted Haar-SU(4) edge walk reduces to one universal principal-angle constant. For real complete-graph models, the entire ordinary fourth-tensor gap reduces to a quadratic-size occupation operator, is simple for every n≥4, and has an expansion beginning Δn(4)=9n10−105n288+O(n−3), with competing branches separated only beyond all algebraic orders. For the two finite real Clifford complete-graph walks, this fourth-tensor mode is the exact all-representation group gap for every n≥144; in the same range it simultaneously governs every ordinary even tensor degree k≥4. Thus different representation scopes can have genuinely different slow modes.
The entanglement structure of a bipartite pure state is characterized by its entanglement spectrum, or equivalently, by the family of Rényi-k entanglement entropies with integer k. Taking the difference of two such entropies defines a quantity known as an entanglement spread, that is monotonous under local operations, and that vanishes if and only if the entanglement spectrum of the state is flat. We introduce a multipartite generalization of this notion which, for a fixed number D≥3 of parties, results in a collection of maps indexed by pairs of D-edge-colored graphs (obeying some condition). Each member of this collection takes the form of a difference of two multipartite Rényi entanglement entropies, each associated to a local unitary polynomial invariant (known in this context as a trace-invariant), and is shown to be monotonous under local operations. We then argue that a previously introduced family of so-called hypergraph-tensor (HT) states can be understood as a multipartite counterpart to the family of flat bipartite states. Indeed, we first prove that, when D=3, a tripartite pure state is HT if and only if it minimizes a particular (infinite) family of entanglement spreads. Second, we embed the set of HT states into the much larger family of spectral hypergraph-tensor (SHT) states, defined by a new Ansatz we introduce. We then prove that HT states can be uniquely characterized as the minimizers of some fixed entanglement spread among SHT states, for arbitrary D≥3. We also investigate SHT states in their own right: in particular, we characterize their orbits under local unitary transformations. Finally, along the way, the asymptotic large-N expectation values of a number of entanglement spreads are computed in the Haar-random state of local dimension N.