Comparing two noisy quantum reference frames as statistical experiments depends on the dimension of the ancillary memory available to the decision procedure. For finite-dimensional channels A and B with invertible A, we show that exact simulation of all measurements assisted by an r-dimensional ancilla is equivalent to r-positivity of the unique factor Gamma = BA^{-1}. The hierarchy can be realized by physical channel pairs: every unital, trace-preserving map that is k-positive but not (k+1)-positive embeds as the factor between the channels D_a and Gamma composed with D_a on an exact interval determined by the smallest Choi eigenvalue. For depolarizing source and target channels D_a and D_b, including negative and singular source parameters, the phase boundary is $\mathcal{D}_a \succeq_r \mathcal{D}_b \Longleftrightarrow -1/(dr-1) \leq b/a \leq 1$ for $a\neq 0$. We derive closed formulas for the restricted level-r deficiency and for the distance to every physical post-processing, $δ_{\mathrm{phys}}(\mathcal{D}_b\mid\mathcal{D}_a)=(1-1/d^2)\operatorname{dist}(b,I_a)$, where $I_a=\operatorname{conv}\{a,-a/(d^2-1)\}$. The largest physical conversion cost hidden from all tests through level k is $(d-k)/[d(d^2-1)]$. An untouched m-level spectator changes the first detecting external level from k+1 to $\lfloor k/m\rfloor+1$. A transpose--depolarizing construction shows that the separation is not confined to depolarizing factors. The results quantify the distinction between ancilla-restricted statistical simulation and implementation by a single quantum channel.
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12 pages, 3 figures; includes ancillary verification code and a deterministic regression certificate
Comparing two noisy quantum reference frames as statistical experiments depends on the dimension of the ancillary memory available to the decision procedure. For finite-dimensional channels A and B with invertible A, we show that exact simulation of all measurements assisted by an r-dimensional ancilla is equivalent to r-positivity of the unique factor Gamma = BA^{-1}. The hierarchy can be realized by physical channel pairs: every unital, trace-preserving map that is k-positive but not (k+1)-positive embeds as the factor between the channels D_a and Gamma composed with D_a on an exact interval determined by the smallest Choi eigenvalue. For depolarizing source and target channels D_a and D_b, including negative and singular source parameters, the phase boundary is Da⪰rDb⟺−1/(dr−1)≤b/a≤1 for a=0. We derive closed formulas for the restricted level-r deficiency and for the distance to every physical post-processing, δphys(Db∣Da)=(1−1/d2)dist(b,Ia), where Ia=conv{a,−a/(d2−1)}. The largest physical conversion cost hidden from all tests through level k is (d−k)/[d(d2−1)]. An untouched m-level spectator changes the first detecting external level from k+1 to ⌊k/m⌋+1. A transpose--depolarizing construction shows that the separation is not confined to depolarizing factors. The results quantify the distinction between ancilla-restricted statistical simulation and implementation by a single quantum channel.
A finite quantum reference token with group-valued misalignment induces a random-unitary channel, but optimal degradation is generally an optimization over all quantum post-processings. For a unitary representation U of a finite group G, we prove that the following conditions are equivalent: U contains every irreducible type; one ancilla-assisted input has an orthonormal G-orbit; signed group measures embed isometrically into channels in diamond norm; and, for every pair of noise laws p,q, infΛ∈CPTP21∥ΦqU−Λ∘ΦpU∥⋄=minr∈P(G)21∥q−r∗p∥1. Thus representation completeness is the exact carrier condition for universal reduction of quantum post-processing to classical convolution. We determine the minimum ancilla dimensions for an orthogonal orbit and for an invariant calibration seed. For an incomplete carrier, with visible Plancherel dimension S(U), we derive the exact conditional-expectation distance 21∥id−ΦuU∥⋄=1−1/S(U) and an explicit quantum--classical deficiency gap. For irreducible carriers the deficiency is obtained in closed form; the faithful two-dimensional representation of S3 yields an exact ten-percent reduction relative to classical convolution. We also characterize law identifiability through the conjugation representation, provide finite linear programs and decision witnesses, and establish both a finite-dimensional obstruction and stable visible-band reconstruction for infinite compact groups. Deterministic ancillary code reproduces the finite-group examples and numerical regression checks.
A common self-consistent channel diagnostic uses a reference contrast matrix R and a processed contrast matrix P. Under a fixed d-dimensional carrier, a common preparation/readout interface, and an invertible reference, one obtains W = Tr(P R^{-1}) = Tr T, where T is the transfer block on the d^2 − 1 traceless operator coordinates. Every entanglement-breaking d-dimensional channel obeys W ≤ d − 1. The inequality is useful only if the dimension and interface assumptions are part of the experimental contract. We construct an exact countermodel when they are not. For every d ≥ 2, set m = d^2 − 1 and let n_1, …, n_{m+1} be the vertices of a regular simplex in R^m. An affine encoding of a local m-dimensional signal ball into the m + 1 = d^2 classical simplex, paired with binary linear readouts, gives an invertible isotropic contrast matrix R = β I_m. The stochastic memory transition K_q = q I_{m+1} + (1 − q) 11^T/(m+1), 0 ≤ q ≤ 1, acts as r ↦ q r on the encoded local chart. Hence for every depth n the observed contrasts satisfy P_n = q^n R, all semigroup identities P_{n+k} = P_n R^{-1} P_k, and the apparent transfer spectrum is exactly that of the d-dimensional depolarizing channel. Yet the physical realization is a measure-and-prepare channel on a d^2-state classical memory and is therefore entanglement breaking for every q. A naive d-dimensional trace witness returns W = (d^2 − 1)q, which exceeds the legitimate entanglement-breaking threshold d − 1 whenever q > 1/(d + 1). At q = 1 the apparent violation factor is d + 1 although the physical memory is classical. Under affine mixture-preserving encodings, d^2 classical states are also minimal for faithfully representing an open (d^2 − 1)-dimensional interface chart. The result does not contradict valid dimension witnesses or trusted-input quantum-memory tests. It identifies a precise failure mode of transferring a dimension-conditioned trace certificate outside its interface contract: repetition consistency, spectral consistency, and arbitrarily deep Markov consistency do not by themselves recover the missing dimension assumption.
Quantum reference frame (QRF) transformations are usually presented as unitary or gauge-fixing changes of coordinates. That picture is exact for ideal frames but fails for finite clocks, gyroscopes, POVM-defined tokens, observers correlated with the targets they describe, and relativistic frames with quantum boost and clock degrees of freedom; the obstruction is loss of distinguishability. We model a non-ideal frame by a normal covariant perspective channel. A deterministic transformation A → B then exists exactly when the kernel of A's channel is contained in the kernel of B's channel, and is unique on the preparable image. Ancilla-stable transformations are complete-order quotient morphisms, while a laboratory implementation on a chosen output space is a separate Choi-Arveson extension problem. The common invariant content of a family of frames is a Heisenberg operator system, carrying an algebraic product only on its product-compatible core. For finite frames in the regular representation, concrete covariant degradability A → B is decided by a group-convolution linear program with explicit dual witnesses ; for Abelian groups this becomes a Fourier mode-support resource lattice. Our main result is that for every finite group this convolution linear program already equals the full diamond-norm deficiency: coherent, multiplicity-exploiting post-processings never outperform random translations, so the resource theory of frame degradation is entirely classical. The same quotient principle governs process tensors, protected relational sectors, entangled observers, relativistic covariance — the boosted-spin perspective being an explicit U(1) dephasing channel — and observer networks. Reproducible numerical certificates accompany every analytic claim.
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state A, published as a 1,472-byte witness, with certified Ic>0 at the exact rational p=16239/250000=0.064956 (per-Pauli convention; total error rate 3p=0.194868) -- beyond both the best value printed in the Agarwal et al. paper (0.064657) and the strongest state in the same authors' public repository (≈0.064911), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in p on [0,1/4], so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of A and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than 1/25000 -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.
How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.
A Lorentz boost acts on the canonical spin of a massive particle through a momentum-dependent Wigner rotation. We show that, for one fixed observer boost, reducing over an uncertain momentum can strictly contract every pairwise spin-state trace distance without producing a channel that is degradable from the less contracted one. For spin 1/2, we first characterize the exact inversion-symmetric channel cone generated by a fixed Wigner angle and transverse momentum directions. Inside this cone lies the Pauli family Mα=diag(1−α,1−α,1−2α), 0≤α<1/2. For 0<α<β<1/2, all trace distances between distinct spin states are strictly smaller after Mβ than after Mα, yet the unique linear post-processing factor has a negative normalized Choi eigenvalue. We solve the optimization over all physical converters exactly: 21infΛ∈CPTP∥Φβ−Λ∘Φα∥⋄=2−3αα(β−α), whereas the reverse deficiency is β−α. Thus the identity dominates the family, while all positive-noise members are pairwise incomparable under CPTP post-processing. The ideal construction is realized as the narrow-packet limit of pure, normalizable five-component momentum states, and explicit perturbation and finite-shot tomography bounds certify an open set of examples. Separately, every nonidentity member fails embedding in a time-homogeneous Pauli-diagonal Lindblad semigroup. Hence ordering all unassisted spin distinguishabilities does not determine the quantum statistical post-processing order.