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.
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.
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.
This preprint studies quantum Markov dynamics and quantum channels that preserve the Lee–Yang property. For systems of two or more qubits, preservation for all Lee–Yang density matrices forces tensor-product dynamics, with local generators characterized by explicit semidefinite conditions. This ordinary preservation property is equivalent to preservation in the presence of arbitrary inert reference systems. For systems containing higher spins, preservation of all pure Lee–Yang inputs excludes intersite interactions and forces isotropic diffusion with axial drift at higher-spin sites. The paper also develops classifications of static maps with stable full Choi tensors, including singular one-qubit maps and independently covariant maps between finite spins, and establishes quantitative rigidity results for diagonal Hamiltonians. The proofs combine positivity of quantum maps, polynomial root-collision arguments, and coherent tensor contraction. The manuscript includes a disclosure of the AI tools used in its development.
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.
We derive the Born rule as the unique outcome statistics induced by a canonical symmetry projection beneath nonselective measurement and unobservable symmetry structure. Concretely, given a compact symmetry group acting by conjugation, we identify the quantum channel that removes symmetry-orbit dependence while preserving exactly the invariant operator algebra. In a standard von Neumann algebraic setting, we prove that Haar averaging yields a normal, unital, completely positive idempotent map whose range is the fixed-point algebra, and hence a conditional expectation onto the invariant subalgebra. Moreover, any normal channel that is invariant under the symmetry action and fixes the invariant algebra pointwise must coincide with this Haar-induced projection. Specializing to projective measurement, Haar averaging over the associated phase-rotation symmetry yields the Lüders (pinching) reduction as the unique normal update map satisfying the same invariance and fixpoint requirements. The induced outcome weights are characterized as the unique probabilities specified by the restriction of the quantum state to the commutative outcome algebra, and they are preserved by the canonical reduction. As a further structural consequence of this symmetry projection, the von Neumann entropy is monotone under the induced reduction. The channel-based derivation applies in all finite dimensions, including qubits, and addresses a structural question not resolved by Gleason-type and positive operator-valued measure (POVM)-based classification results: it identifies the unique normal symmetry-forced reduction channel associated to a fixed measurement context, rather than characterizing admissible global probability functionals. License note: Distributed under CC BY-NC-ND 4.0.
This paper proves a finite-dimensional carrier-descent theorem for canonical projective quantum measurement channels. Under explicit universal-carrier, Ward-admissibility, and finite projective-sector hypotheses, each nonzero sector carries a compact U(1) phase carrier, and a finite family of orthogonal sectors determines a central sectorwise phase torus. Haar averaging over this compact carrier gives the conditional expectation E_P(X) = ∑ⱼ Πⱼ X Πⱼ, whose trace-dual state map is the Lüders nonselective measurement channel Δ_P(ρ) = ∑ⱼ Πⱼ ρ Πⱼ. The fixed observable algebra is the block algebra B_fix = {X : [X, Πⱼ] = 0 for all j}, while the classical outcome algebra is its center Z_out = span{Π₁, …, Πₘ}. The Born weights pⱼ(ρ) = Tr(Πⱼρ) are the unique coefficients representing the restricted state on Z_out, and selective measurement is conditioning on the corresponding boundary record. Except in the trivial one-sector case, the Lüders projection is idempotent, noninvertible, and nonmultiplicative on the full matrix algebra; hence the associated boundary-channel datum is a boundary-channel projection rather than full reversible Heisenberg–Schrödinger collapse, i.e. full-algebra unitary equivalence. The paper proves that canonical finite projective measurement is the endpoint of a carrier-to-phase-to-Haar boundary-channel descent: phase selection determines Haar projection, Haar projection determines the Lüders channel, central restriction determines the Born weights, and the collapse criterion certifies the obstruction to full reversible Heisenberg–Schrödinger collapse in every nontrivial projective context. License note: Distributed under CC BY-NC-ND 4.0.
Quantum private broadcasting (\textsf{QPB}) asks whether a quantum message can be encrypted once, broadcast to several recipients, and decrypted locally while the joint ciphertext reveals no plaintext information. Broadbent, González-Guillén, and Schuknecht showed that the full tensor-power space Hd⊗t is generally unsuitable for \textsf{QPB}, identified Symt(Hd) as a natural plaintext space, introduced symmetric unitary t-designs, and asked which other subspaces support private broadcast. Using Schur-Weyl duality, we classify all subspaces S⊆Hd⊗t on which the collective Haar twirl T(t) is a replacement channel. Precisely, every Schur-Weyl block weight must be independent of the encoded state, and in each active block Vλ⊗Wλ, the reduced state on the multiplicity space Wλ must be fixed. This yields explicit two-block and general multiblock constructions and realizes the classification as the Schur-Weyl form of the Knill-Laflamme conditions for a complementary channel of T(t). We also introduce a canonical reduction under deletion of a recipient and characterize privacy heredity, design heredity, and \textsf{QPB}-heredity for exact restricted designs. The symmetric tower is fully \textsf{QPB}-hereditary, whereas the antisymmetric tower is fully privacy-hereditary and is design-hereditary from level s to s−1 exactly when s≤(d+1)/2. For canonical carriers in arbitrary Schur-Weyl blocks, hereditary reduction is governed by multiplicity-free branching along the Young graph and inclusion of the associated Peter-Weyl coefficient spaces.