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.
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.
We provide a complete set of game-theoretic conditions equivalent to the existence of a transformation from one quantum channel into another one, by means of classically correlated preprocessing and postprocessing maps only.Such conditions naturally induce tests to certify that a quantum memory is capable of storing quantum information, as opposed to memories that can be simulated by measurement and state preparation (corresponding to entanglement-breaking channels).These results are formulated as a resource theory of genuine quantum memories (correlated in time), mirroring the resource theory of entanglement in quantum states (correlated spatially).As the set of conditions is complete, the corresponding tests are faithful, in the sense that any non-entanglement-breaking channel can be certified.Moreover, they only require the assumption of trusted inputs, known to be unavoidable for quantum channel verification.As such, the tests we propose are intrinsically different from the usual process tomography, for which the probes of both the input and the output of the channel must be trusted.An explicit construction is provided and shown to be experimentally realizable, even in the presence of arbitrarily strong losses in the memory or detectors.
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.
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
High-dimensional entanglement offers significant advantages over its low-dimensional counterpart in various information-processing tasks. However, to harness these advantages, it is crucial that the quantum channels used to store or transmit the subsystems of an entangled system not only preserve entanglement but also maintain its dimensionality above a certain threshold. The maximum entanglement dimension that a channel can preserve is referred to as its effective dimensionality, since the channel cannot be used to transmit information of dimension greater than that in a single use. In this work, we present a method to certify whether a quantum channel can preserve entanglement dimension above a given threshold. Unlike existing approaches, our method is faithful, i.e., it can be applied to any channel, and avoids common assumptions such as preparation of entangled states, auxiliary side channels, or perfect measurement devices. Moreover, the method can be extended to faithfully certify other classes of non-resource-breaking channels, such as non-nonpositive-partial-transpose-breaking channels (non-NPT-breaking channels). Finally, we discuss possible experimental realizations of our certification scheme through explicit examples.
We characterize two-step quantum processes with a prescribed first channel whose unperturbed final evolution exactly recovers the input. For a system of dimension d and a first channel of Kraus rank k > 1, the compatible family is affinely bijective to the set of channels from dimension d(k−1) to dimension d. For qubit dephasing with residual coherence r, an intermediate X intervention not only reconstructs the parametrizing channel but also attains the optimal operational distance, equal to (1−r²) times the diamond distance between the parametrizing channels. We determine the classical subfamily, its minimum register cardinality, and the maximal temporal negativity (1−r²)/2. For imperfect recovery, we prove O(√δ) proximity to the exact family while preserving the first channel, with dimension-independent constants. An explicit two-qubit example shows that the exponent 1/2 cannot be uniformly improved. Finally, we derive a Pauli-cycle witness whose exact classically attainable region is characterized by a quadratic inequality involving the cyclic coherence and two return probabilities. The witness requires neither population preservation nor perfect recovery. A linear version uses twelve experimental settings and admits explicit finite-sample significance bounds with calibrated control errors. Analytical proofs are accompanied by reproducible code. No experimental data or departures from standard quantum mechanics are claimed.
When does a readable record survive the actual physical process, and when does memory change that survival? Present readability, temporal label retention and process memory answer separate questions. Their analysis requires a physical carrier, an identity-relevant record quotient and its actual continuation. We develop their common reduction architecture and derive an explicit memory correction to temporal record certificates. On a classical record quotient, the channel defect equals the largest class re-identification error. Its additive composition bound holds for the actual composed quotient process; it need not hold for individually calibrated steps whose preparations erase the correlations reached in a real run. A two-CNOT construction makes every freshly calibrated step preserve the record perfectly while the actual two-step continuation erases it. The missing term is a process-composition gap, which we define and bound by discrepancies between actual joint preparations and the preparations used by a reduced model. The same construction identifies a memory sector that must be retained to obtain a faithful record continuation. We also prove that decoherence and broadcasting do not fix temporal record identity; that CP-divisibility depends on the carrier and preparation; and that memory can be irrelevant to a protected identity register. A finite dephasing model has no trace-distance backflow on its declared interval yet changes record retention under a correlation-severing intervention. These results separate memory magnitude, causal record relevance and Constitution change. The derivations use unchanged finite-dimensional quantum dynamics and retain explicit scope conditions for every identity claim.