A completely positive channel N that is covariant for a group G acting on a carrier W = ⊕_s V_s (pairwise inequivalent irreducibles) admits two natural bilinear read-outs of its sector structure: a "realign reading" A_phys[s,t] = Tr[P_s N(P_t)]/d_s, the channel's coefficient matrix on the sector algebra, and a "no-realign reading" A_ext[s,t] = (1/d_s) sum_{a,A} <a|P_s N(|a><A|) P_t|A>, a score. We prove three things. First, covariance forces N(P_t) to be block-scalar, so A_phys is the coefficient matrix c_{st} in N(P_t) = sum_s c_{st} P_s (Schur's lemma). Second, the score is the Gram matrix of the channel's sector-overlap vectors, A_ext[s,t] = (1/d_s) <w_s, w_t> with w_s = (Tr[P_s K_k])_k; hence its diagonal is a manifest sum of squares (nonnegative for any channel) and D·beta is Hermitian positive semidefinite in the sector-trace metric D = diag(d_s), where beta = A_ext. Third, the score equals the sector-return diagonal, A_ext = diag(A_phys), if and only if two independent conditions hold: (I) the overlap vectors of distinct sectors are orthogonal, and (II) within each sector a trace/Frobenius balance sum_k |Tr K_k^{(ss)}|^2 = sum_k ||K_k^{(ss)}||_F^2. Condition (II) has a Schumacher entanglement-fidelity reading: the maximally-mixed entanglement fidelity of the sector-s diagonal sub-channel equals A_phys[s,s]/d_s. We call channels meeting (I)+(II) sector-return. A specific Sp(3) = USp(6)-covariant channel on the 35-dimensional sl(6) adjoint (split 14 + 21) is sector-return exactly, while generic covariant channels — the identity among them — are not; in the correct metric its diagonal sub-channels are scalar-unital and sit exactly at the (II)-balance, and a numerical spectral pass from exact-rational operators resolves them as Hermitian positive semidefinite of Choi rank D^2 = 36 (D = 6 the memory dimension), with dimension-indexed spectra whose leading eigenvalue is the sector-return coefficient. We close with the open problem this sharpens: whether covariant compression through a finite memory (a D-dimensional auxiliary memory, not a minimal Stinespring environment) forces sector-return.
A completely positive channel N that is covariant for a group G acting on a carrier W = ⊕s V_s (pairwise inequivalent irreducibles) admits two natural bilinear read-outs of its sector structure: a "realign reading" A_phys[s,t] = Tr[P_s N(P_t)]/d_s, the channel's coefficient matrix on the sector algebra, and a "no-realign reading" A_ext[s,t] = (1/d_s) sum{a,A} <a|P_s N(|a><A|) P_t|A>, a score. We prove three things. First, covariance forces N(P_t) to be block-scalar, so A_phys is the coefficient matrix c_{st} in N(P_t) = sum_s c_{st} P_s (Schur's lemma). Second, the score is the Gram matrix of the channel's sector-overlap vectors, A_ext[s,t] = (1/d_s) <w_s, w_t> with w_s = (Tr[P_s K_k])_k; hence its diagonal is a manifest sum of squares (nonnegative for any channel) and D·beta is Hermitian positive semidefinite in the sector-trace metric D = diag(d_s), where beta = A_ext. Third, the score equals the sector-return diagonal, A_ext = diag(A_phys), if and only if two independent conditions hold: (I) the overlap vectors of distinct sectors are orthogonal, and (II) within each sector a trace/Frobenius balance sum_k |Tr K_k^{(ss)}|^2 = sum_k ||K_k^{(ss)}||_F^2. Condition (II) has a Schumacher entanglement-fidelity reading: the maximally-mixed entanglement fidelity of the sector-s diagonal sub-channel equals A_phys[s,s]/d_s. We call channels meeting (I)+(II) sector-return. A specific Sp(3) = USp(6)-covariant channel on the 35-dimensional sl(6) adjoint (split 14 + 21) is sector-return exactly, while generic covariant channels — the identity among them — are not; in the correct metric its diagonal sub-channels are scalar-unital and sit exactly at the (II)-balance, and a numerical spectral pass from exact-rational operators resolves them as Hermitian positive semidefinite of Choi rank D^2 = 36 (D = 6 the memory dimension), with dimension-indexed spectra whose leading eigenvalue is the sector-return coefficient. We close with the open problem this sharpens: whether covariant compression through a finite memory (a D-dimensional auxiliary memory, not a minimal Stinespring environment) forces sector-return.
This record gives an exact, machine-checked resource analysis of a specific USp(6) = Sp(3)-covariant quantum compression construction, and the reproducible certificates behind it. The carrier is W = V14 ⊕ V21 (dimension 35), the memory is the sixdimensional irreducible V6, the source is the balanced sector-mixed state with weights (1/28, 1/42), and the figure of merit is the Schumacher entanglement fidelity 𝜔. The construction and its coherent witness are inherited from the companion compression paper (DOI 10.5281/zenodo.21744306), which certifies F*_effEB,V6 ≤ 1/100 < 1/40 < 39587/1382976 = 𝜔(witness): a coherent covariant channel through the six-dimensional memory beats every entanglement-breaking channel through that memory. Four results, each verified exactly over the rationals, answer one coherent question — what preserves the compression advantage in this symmetry-constrained construction: (1) Sector-dephasing benchmark. Removing all coherence between the two output sectors drops the fidelity from 39587/1382976 to the within-sector value 55/3528, which still exceeds the 1/100 entanglement-breaking upper bound; the certified advantage survives, and the coarse output-sector label is blind to the input. (2) Branch entanglement. The two normalized output-sector channels N14 and N21 (output sector restricted, input/reference kept full) are both completely positive, trace-preserving, and have a negative partial transpose; each retains reference entanglement, so neither is entanglement-breaking. (3) Branch replacement. Replacing one branch at a time by an entanglement-breaking channel, at fixed sector probabilities and through the same memory, keeps the V14 branch (M14, 𝜔 = 835/74088 > 1/100) above the bound while the V21-retaining channel (M21, 𝜔 = 263/49392 < 1/100) does not clear it; both complete channels remain NPT. (4) Entanglement-breaking comparator. An explicit admissible covariant entanglementbreaking channel through the same memory strictly outperforms M21, with 𝜔 = 1544075/288004752 > 263/49392 (exact margin 5261/144002376 > 0). This resolves the previously open comparison and brackets the entanglement-breaking optimum as 1544075/288004752 ≤ F*_effEB,V6 ≤ 1/100. The overall lesson is objective-specific: retaining reference entanglement and obtaining a task-specific quantum advantage are distinct here. Every claim is checked by an exact rational verifier (python-flint): Choi positivity and partial-transpose tests use the certificate’s own Gram metric, and the comparator’s score is confirmed by reconstructing the actual channel and re-evaluating the original objective. Scope and caveats: the results have been audited for correctness by an independent reviewer; this record does not claim comprehensive novelty relative to the existing literature on quantum-memory resource theories and operational benchmarks, nor establish journal suitability. Any connection to the author’s broader “Presence Hypothesis” program is
A compact abelian group acts unitarily on a finite-dimensional space. This paper proves that the overlap of two states, one shifted by a group element, is a finite Fourier series whose coefficients are positive semi-definite kernels indexed by the isotypic components of the conjugation representation; the coefficient of the trivial character is the inner product of the twirled states, and the sum of the squared moduli of the other coefficients is the variance of the overlap over the group. For the circle the series is a trigonometric polynomial whose degree is bounded by the total charge, so it is determined exactly by its values at finitely many equally spaced shifts. The same decomposition is applied to a reset channel, a fixed unitary that absorbs encoded inputs one at a time into a hidden register and discards the input between steps. Its readout under a common shift of the inputs is a finite Fourier series whose zero mode is the readout of the jointly twirled input; this need not equal the readout of inputs twirled one at a time, and an explicit Clifford channel separates the two. The degree of the readout is at most the charge bound times the number of steps, and for every charge bound an explicit channel attains it. Every coefficient can be estimated from overlaps measured at finitely many shifts, with a number of runs quadratic in the inverse error and logarithmic in the number of coefficients, and a charge sector of an encoded state can be read at the amplitude level by exact phase estimation of the charge followed by a Hadamard test. Sector kernels of product encodings and Fourier sectors of Clifford channels with Pauli readout are shown to be classically computable in polynomial time. The paper claims no quantum advantage. A final section carries the Fourier argument to the signature of a piecewise-linear path, read as a vector in a Fock space, and decomposes the signature kernel under rotation of the increments into harmonics that are themselves kernels.
The spectral weight with which a many-body eigenstate contributes to a given channel is not fixed by the smooth one-point ETH envelope, and its higher-order correlations---the connected cumulants of three and more channel-resolved spectral functions---have lacked a systematic organization. We construct one. In random free fermions, where every channel overlap is a squared Slater minor of a Haar-orthogonal one-body eigenvector matrix, the r-resolvent moment admits an exact replica-sector decomposition: a sum over the double-coset sectors of the orthogonal Weingarten algebra, in which the permutation sectors are the replica contraction classes and each sector value is determined by the overlap pattern of the channels and of the eigenstates. The sector coefficients follow from an exact occupancy-matrix assembly---a signed convolution over the slot permutations, a decomposition into connected blocks, and a finite-state chain transfer matrix whose local transition rules are independent of r and k---so the sector theorem holds at arbitrary finite multiplicity, realized exactly at r=3 (k=2,3,4), r=4 and r=5 (k=2). The structure carries the physical hierarchy: the fully coincident sector is the projection that yields the Selberg moment ratios qr, the levels are nested by exact marginalization, and the sectors are the multi-energy spectral cumulants behind the resolvent cumulants, with the irreducible self-energy vertices of the Feshbach ladder as the connected kernel. The multi-resolvent correlations of random free fermions are therefore organized, level by level, by one replica-sector algebra---an exact microscopic benchmark for the higher-order fluctuation sector of eigenstate thermalization, with its microscopic sector geometry made explicit.
We study the canonical USp(4)-covariant quantum compression problem in which the fifteen-dimensional carrier W = V5 ⊕ V10 is encoded through a four-dimensional covariant memory M = V4 and decoded, and we ask for the optimal entanglement fidelity F*(p) of the composite channel on a trace-one, sector-weighted reference state ρ_p. This problem is the operational face of a structural fact: the same pair of sectors 5 ⊕ 10 that makes USp(4) on C^4 state-homogeneous but not observable-uniform is exactly the modal content a minimal covariant memory is asked to carry. Our contribution is not a resolution of the underlying capacity question but a demonstration that this problem admits an exact reduction and a machine-checkable global bound: the reduction and the final lower- and upper-bound certificates are exact, while only the intermediate relaxation-solving step is numerical. We give an exact representation-theoretic reduction — Wedderburn splitting of the covariant commutant over Q(i), an induced-involution positivity bridge, and an exact trace-preservation reduction to fifteen free rational coordinates — turning F*(p) into a bilinear polynomial optimization over a product of two explicit semialgebraic bodies, each the affine preimage of the product cone (R+ × SOC4 × SOC4). Exploiting correlative sparsity we build an order-two sum-of-squares relaxation, solve it numerically for discovery, reconstruct the resulting SOS upper bounds in exact rational arithmetic, and separately certify exact rational feasible lower bounds from explicit witnesses. At p = 1/2 and p = 7/8: 0.0749 ≤ F*(1/2) ≤ 0.21, and 0.1546 ≤ F*(7/8) ≤ 0.643 (exact rationals in the paper). The upper bounds are Positivstellensatz certificates whose polynomial identity and positive semidefiniteness are verified with no optimizer in the loop by a standalone verifier we provide. Throughout we keep three visibly separate registers — exact theorems, numerical diagnostics, and open or conditional interpretations — and we are explicit that the exact optimum, the exactness of the relaxation, and the width of the certified interval remain open. Package: this record includes the manuscript, a single 30-page combined archival PDF (cover + manuscript + auditable reduction-supplement listing) inside the package, and a verification package with two independent machine-checkable obligations (the exact reduction chain and the certified two-sided enclosure). Manuscript licensed CC BY 4.0; code licensed MIT.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
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.