Cardinality-Resolved Interference Accessibility in Multipartite Quantum Systems: A trace-norm framework for measurement-assisted coherence and its use in quantum hardware diagnostics
A single-qubit Ramsey or T2 measurement reports how much interference survives on a qubit, but not why the rest is missing. Phase information lost to an unobserved environment and phase information coherently redistributed into neighbouring qubits produce the same local signature, although only the second can be recovered. We study the interference that a target qubit regains when a controller measures a chosen set of helper systems and applies an outcome-dependent phase correction. For a fixed interference basis and any helper subset T, we prove that the maximum average l1 coherence recoverable this way equals 2‖C_T‖₁, twice the trace norm of the off-diagonal block of the reduced state, and that a projective measurement on T attains it. Imposing a budget on the number of helpers gives a monotone hierarchy I(0) ≤ I(1) ≤ … ≤ I(m) from local coherence to full assistance, from which we derive an access order d_η (the fewest helpers that recover a fraction η of the accessible interference) and an unlocking spectrum π_r. Exact examples show that states with identical local and fully assisted coherence can have opposite access structure, and that coherent leakage and dephasing with the same Ramsey visibility separate at the first helper layer. Exact state-vector calculations on eight qubits show the access order rising from 0.90 for depth-2 local circuits to 3.25 at depth 8 and 4.10 for Haar-random states. Applied retrospectively to published Bell-state tomography from the ibm_brisbane processor, the first layer shows that 91–96% of the accessible interference is invisible in the target marginal. We outline uses in hardware diagnostics: separating redistribution from decoherence, pairwise leakage maps from two-qubit tomography, and sizing mid-circuit readout for feed-forward. Files: the paper (PDF) and CRIA_code.zip, a Python package that regenerates every figure, table and number in the paper and numerically verifies its analytic results (run: python interference_accessibility_reproducibility.py --verify).
A single-qubit Ramsey or T2 measurement reports how much interference survives on a qubit, but not why the rest is missing. Phase information lost to an unobserved environment and phase information coherently redistributed into neighbouring qubits produce the same local signature, although only the second can be recovered. We study the interference that a target qubit regains when a controller measures a chosen set of helper systems and applies an outcome-dependent phase correction. For a fixed interference basis and any helper subset T, we prove that the maximum average l1 coherence recoverable this way equals 2‖C_T‖₁, twice the trace norm of the off-diagonal block of the reduced state, and that a projective measurement on T attains it. Imposing a budget on the number of helpers gives a monotone hierarchy I(0) ≤ I(1) ≤ … ≤ I(m) from local coherence to full assistance, from which we derive an access order d_η (the fewest helpers that recover a fraction η of the accessible interference) and an unlocking spectrum π_r. Exact examples show that states with identical local and fully assisted coherence can have opposite access structure, and that coherent leakage and dephasing with the same Ramsey visibility separate at the first helper layer. Exact state-vector calculations on eight qubits show the access order rising from 0.90 for depth-2 local circuits to 3.25 at depth 8 and 4.10 for Haar-random states. Applied retrospectively to published Bell-state tomography from the ibm_brisbane processor, the first layer shows that 91–96% of the accessible interference is invisible in the target marginal. We outline uses in hardware diagnostics: separating redistribution from decoherence, pairwise leakage maps from two-qubit tomography, and sizing mid-circuit readout for feed-forward. Files: the paper (PDF) and CRIA_code.zip, a Python package that regenerates every figure, table and number in the paper and numerically verifies its analytic results (run: python interference_accessibility_reproducibility.py --verify).
For a qubit undergoing pure dephasing, the trace distance between a natural pair of phase-oppositestates equals the ℓ1 coherence of the qubit. This identity provides a simple way to interpret distinguishabilitybackflow as single-qubit coherence recovery. Using this observation, we motivate aninformation-mirror interpretation of coherence revivals in structured environments: environmentaldegrees of freedom may temporarily store phase information and later make part of it locally visibleagain in the qubit. We define a cycle-resolved reflection coefficient, R = B/L, where L is the coherencelost during a decay segment and B is the coherence recovered during the following revival. Fora recovery cycle referenced to its preceding coherence maximum, ordinary partial recovery satisfies0 ≤ R ≤ 1, with R = 1 corresponding to complete return to the previous maximum. This coefficientdoes not replace standard non-Markovianity measures, but provides a compact diagnostic of howstrongly a structured environment returns previously lost coherence.
For compiled nested-measurement circuits, or reflection towers, the companion paper established that every discrepancy between the coherent Born law tr(M rho) and the record-dephased collapse self-model tr(M Delta_J rho) is governed by reduced difference operators derived from X_J = rho − Delta_J rho. This paper extends that exact but rigid subsystem theory into an operational theory of accessibility. First, we prove the channel-interface theorem: for any quantum channel Lambda modeling access, the two laws agree for every effect available after Lambda if and only if Lambda(X_J) = 0; the maximal bias is delta_J(Lambda) = (1/2)||Lambda(X_J)||1, attained by a Helstrom measurement. By data processing, this quantity is an information monotone. Second, we introduce operator-valued Mobius elements Z_J and prove a localization theorem showing that Z_J is supported exactly on branch pairs whose records differ on every element of J. This yields a corrected hierarchy of classicality orders: state-level Mobius support is always the simplicial complex generated by the branch-difference sets, and therefore has order one whenever two branches coexist, whereas the Frauchiger–Renner obstruction has order two at the question level, equivalently at the interface level of Wigner’s measurement channel, whose Mobius weights are exactly |eta_M(J)|. Third, we prove that pairwise visibility equals the Uhlmann fidelity of the inaccessible-side branch states, v_rr′(S) = F(sigma{\bar S,r}, sigma_{\bar S,r′}), thereby placing screening, independent-noise product laws—which become exact equalities—and correlated record environments, including all-or-nothing secret-sharing records, within a single formula. Fourth, we construct an explicit compiled tower exhibiting exact operator-level cancellation at a budget-complete interface with unit pairwise visibilities, proving that the family of agreeing interfaces is a down-set but is not closed under unions. Fifth, we prove robustness: diamond-norm gate errors epsilon_n change every discrepancy by at most the sum over n of epsilon_n, converting exact screening into quantitative approximate screening. Finally, we develop cost-constrained discrepancies, including an exact exponential implementability gap of 2^(−c) for product measurements on a depth-c read-chain; the accessibility hypergraph, in which budgets correspond to covering and screening to hitting; the parameter geometry of the Frauchiger–Renner family, where the order-two stratum has codimension two and full classicality is impossible on the open simplex; and two-sided asymptotic detection exponents controlled solely by the single-copy discrepancy.
This technical report packages a set of hardware component-witness experiments run on IBM superconducting quantum processors. The experiments test whether a hidden auxiliary phase can affect a later system-qubit measurement, whether that phase can be recovered from basis-resolved readout, and whether two auxiliary registers can produce a nonadditive response rather than a simple sum of their individual effects. The package includes the report draft, figures, raw hardware result JSONs, raw-count appendix, and scripts for regenerating the figures and count tables. The central result chain is: echo/phase response on real hardware, basis-resolved phase readout without density-matrix tomography, and a two-register Y-basis nonadditivity test where the first run remains a frozen FAIL and an independent higher-shot replication reaches STRONG_PASS. The longer-term motivation is to develop small, testable quantum hardware primitives for systems that can preserve, read out, and combine latent histories or unchosen alternatives, instead of reducing every intermediate state to a single classical output immediately. In that sense, these experiments are intended as early building blocks for future quantum-assisted sequence or language-model architectures. This work does not claim a completed quantum language model, natural-language processing on a quantum computer, consciousness, selfhood, classical impossibility, or quantum advantage. It reports only hardware component witnesses on specific IBM backends, edges, and calibration windows, with the possibility of richer classical explanations left open.
We develop a complete first-order theory of observer-relative interference for nested, Wigner’s-friend–type measurement scenarios, modeled as compiled reflection towers: quantum circuits in which every internal measurement is implemented by a transversal record copy and every internal classical control is implemented by a record-controlled unitary. For any set J of records, the discrepancy between the coherent Born law, tr(Mρ), and the record-dephased self-model, tr(MΔ_Jρ), is determined by the single Hermitian traceless operator X_J = ρ − Δ_Jρ. Our central result, the exact interface theorem, states that an interface—either a subsystem S or a unital *-subalgebra A—is indifferent between the two laws if and only if the reduced difference operator tr_{\bar S}(X_J), respectively E_A(X_J), vanishes; moreover, the maximal bias over all effects equals the trace distance and is attained by a Helstrom measurement. Expanding the reduced difference in terms of branch cross-operators Γ^S_{rr′} identifies four distinct mechanisms: combinatorial screening by inaccessible record copies, residual which-path suppression, question-specific cancellation, and multipartite synergy. The combinatorial “copy budget” of earlier treatments is recovered as the zeroth-order approximation. We also prove a multiplicative suppression law for noisy records, replacing the hard combinatorial accessibility threshold with an exponential classicalization depth; a Möbius localization theorem, which assigns every certainty paradox a well-defined classicality order and shows that the Frauchiger–Renner obstruction is irreducibly second-order, with Möbius weight (1/2)√(p00 p11) = 1/6; a non-robustness theorem showing that single-record empirical adequacy in the Frauchiger–Renner state is a codimension-one coincidence; and an algebraic agreement theorem stating that, on the record-invariant algebra, coherent execution and fully collapsed sequential execution agree exactly at every depth.
A quantum record can have locally accessible populations while its distinguishing phase remains correlated with a distant source. We study this access boundary in a specified finite recording apparatus with a returning, vacuum/one-excitation source. Source-position weights give finite-time cutoff bounds without removing any physical hopping or return coupling. For an initially far-tagged record and its incoherent mixture, every reader in a declared position-preserving local-control class obeys a phase-visibility envelope and a necessary access-time inequality. A separate finite local probe shows why that state-discrimination limit is not an unrestricted limit on distinguishing two dynamical laws: fresh local coherence can reveal a stipulated added dephaser without recovering the old record phase. We construct a bounded, degree-three reader accessing only source positions 1–32. With the initial tag fixed at position 65, exact finite-source arrival certificates and an analytical tail comparison yield recovery uniformly over every finite total source length at least 65. Accessible control faults are charged against the arrived signed-phase norm, while source-moving drift is charged against the full norm. At the fixed deadline 25.02/J, the matched phase-dependent difference between coherent and archive-gated dynamics exceeds 0.005, including the stated simultaneous control faults and four additional absolute response allowances of 0.008 each. Complementary results cover a vacuum-ready source and a stronger endpoint-prepared finite example. The conclusions are conditional on the supplied model, preparations, common realized controls and calibration bounds. They are not a switchable-fundamental-law experiment, a nuisance-independent classifier, an optimal-speed theorem or an end-to-end formal verification. The supporting Lean companion contains 59 declarations in eight modules with accepted returned Mac compilation, type and axiom evidence; physical analytical interfaces remain explicit hypotheses or outside that subset. The deposit files include the main manuscript, technical supplement and full research/reproduction package.
When a quantum system interacts with its environment it may incur in decoherence. Quantum erasure makes it possible to restore coherence in a system by gaining information about its environment, but measuring the whole of it may be prohibitive: Realistically, one might be forced to address only an accessible subspace and neglect the rest. In such a case, under what conditions will quantum erasure still be effective? In this work we compute analytically the largest recoverable coherence of a random qubit plus environment state and we show that it approaches 100% with overwhelmingly high probability as long as the dimension of the accessible subspace of the environment is larger than D, where D is the dimension of the whole environment. Additionally, we find a sharp transition between a linear behavior and a power-law behavior as soon as the dimension of the inaccessible environment exceeds the dimension of the accessible one. Our results imply that the typical states of a qubit plus environment system admit a measurement spanning only about D degrees of freedom, any outcome of which projects the qubit on a maximally coherent state. This suggests, for instance, that in the dynamics of open quantum systems, if the interactions are known, it would in principle be possible to gain sufficient information and restore coherence in a qubit by dealing with a fraction of the physical resources.