Abstract We develop a complementarity-based description of open two-qubit dynamics built aroundthe budget Bk(t) = P2k(t)+V 2k(t)+C2(t) and the associated deficit Dk(t) = 1−Bk(t). The mainoutcome is not a new universal witness of non-Markovianity, but a more useful split betweenstructure observables and flow observables. Using explicit prototype amplitude-damping anddephasing channels, together with asymmetric couplings and mixed initial states, we showthat deficit-based distances such as ∆D = maxt|Dk(t) − SL(t)| mainly encode state- andchannel-dependent geometry, whereas positive-variation functionals such as NC and N−Drespond more directly to memory-driven recovery. In addition, for an exactly solvable Bell-statedephasing sector we prove that the revival windows of C and −DA coincide and obtain closedexpressions for BA and DA. The paper reframes complementarity dynamics as a two-levelproblem: geometry of resource partition on one side and temporal backflow on the other. Theframework is tied to a same-dataset BLP-like benchmark and to two analytical propositionsthat anchor the relation between concurrence revival and deficit flow in solvable sectors and ongeneral revival intervals
Abstract We develop a complementarity-based description of open two-qubit dynamics built aroundthe budget Bk(t) = P2k(t)+V 2k(t)+C2(t) and the associated deficit Dk(t) = 1−Bk(t). The mainoutcome is not a new universal witness of non-Markovianity, but a more useful split betweenstructure observables and flow observables. Using explicit prototype amplitude-damping anddephasing channels, together with asymmetric couplings and mixed initial states, we showthat deficit-based distances such as ∆D = maxt|Dk(t) − SL(t)| mainly encode state- andchannel-dependent geometry, whereas positive-variation functionals such as NC and N−Drespond more directly to memory-driven recovery. In addition, for an exactly solvable Bell-statedephasing sector we prove that the revival windows of C and −DA coincide and obtain closedexpressions for BA and DA. The paper reframes complementarity dynamics as a two-levelproblem: geometry of resource partition on one side and temporal backflow on the other. Theframework is tied to a same-dataset BLP-like benchmark and to two analytical propositionsthat anchor the relation between concurrence revival and deficit flow in solvable sectors and ongeneral revival intervals
We study complementarity dynamics in open two-qubit systems through the local budgetBA = P2A+ V 2A + C2 and its deficit DA = 1 − BA. The purpose is not to define a newnon-Markovianity measure, but to distinguish two diagnostic roles. Structure observablesdescribe how predictability, visibility, concurrence, and deficit are arranged along a statespacetrajectory, whereas flow observables record temporal recovery. Amplitude-damping anddephasing prototypes, finite parameter scans, BLP-like comparisons, and two analyticallysolvable Bell-sector results show that ΔD is mainly structural, while NC and N−D respond torevival windows. The framework therefore clarifies what complementarity budgets certify inopen two-qubit dynamics.
We present a conservative reformulation of the complementarity analysis containedin the underlying manuscript materials. The claim is restricted to the part that isdirectly supported by the available derivations: an exact reduced-state balance for localpredictability, local visibility, and entanglement in an exactly solvable mini-reservoir modelfor non-Markovian two-qubit dynamics. For each local qubit, the reduced budget is definedas Bk(t) = P2k (t) + V 2k (t) + C2(t) and is completed by a remainder Mk(t) = 1 − Bk(t).Within the Hilbert–Schmidt bookkeeping adopted from the source material, this remainderis written as the nonlocal contribution beyond concurrence, Mk(t) = 2Cnlhs(t)− C2(t). Theresult is model bound and should not be read as a universal conservation law for arbitrarycombinations of coherence, discord, and entanglement measures. Instead, it isolates a precisecomplementarity identity inside a controlled Markov-embedded open-system setting.
Complementarity relations constrain the distribution of coherence, predictability, and openness in quantum systems. Here we show that, in open quantum systems, these local constraints acquire a geometric interpretation through quasistatic transport. For a driven dissipative qubit, the complementarity variables define cylindrical coordinates on the Bloch sphere, while openness appears geometrically as a radial deficit associated with reduction from a larger Hilbert space. Quasistatic driving induces a work connection on the resulting steady-state manifold whose curvature determines the cyclic response. Hamiltonian-aligned dissipation produces an exact work connection and vanishing cyclic work, whereas fixed pointer-basis dissipation generates non-integrable transport, finite curvature, and holonomic response. The resulting curvature admits a phase-resolved representation on the triality manifold and develops perturbatively with pointer--Hamiltonian mismatch. In the weak-mismatch limit, the curvature is governed by a competition between coherence-preserving and pure-dephasing channels, producing symmetry-related positive- and negative-curvature sectors. These results establish a direct connection between complementarity, dissipation, and geometric thermodynamic response, and show that cyclic quasistatic work provides an operational probe of nonequilibrium quantum geometry.
Non-Markovian optical channels can redistribute interferometric visibility,which-path information, and entanglement in ways that are not captured byconcurrence dynamics alone. Motivated by cavity-QED, waveguide-feedback,and related photonic reservoir architectures, we study an exactly solvabletwo-qubit mini-reservoir model and interpret its reduced dynamics as an effectivelocal photonic feedback channel. For the reduced two-qubit state, predictabilityPk, visibility Vk, and concurrence C obey the mixed-state relationP2k (t)+V 2k (t)+C2(t) = Bk(t) ≤ 1, where Bk(t) is the locally accessible complementaritybudget. Within the Hilbert–Schmidt formulation of completecomplementarity, we show that the corresponding deficit is closed exactly byMk(t) ≡ 1 − Bk(t) = 2Cnlhs(t) − C2(t). The remainder therefore identifiesthe part of the nonlocal optical coherence sector that is not already encodedin concurrence. Exact trajectories for Bell, nonmaximally entangled parity,superposition-of-Bell, and Werner initial states reveal memory-inducedcollapse-and-revival structure in the accessible budget, in the visibility sectorwhen local coherence is present, and in the remainder itself. Comparisonswith one-sided discord and with the purity deficit show that Mk(t)is a distinct reduced-state diagnostic rather than a relabeling of entanglement,discord, or mixedness. Because all quantities are reconstructed fromthe same reduced two-qubit state, the framework is naturally compatible with interferometric and tomographic readout in memory-bearing quantumopticalplatforms.
The author proposes two postulates for the foundation of time and spacetime in open quantum systems. Postulate P1 (Time): the proper time of an open quantum system is generated by the integral of its boundary structural-persistence functional Φ on the variationally selected autonomous boundary. Postulate P2 (Spacetime): spacetime is the synchronisation pattern of a network of open-system boundary Φ-clocks coupled through shared baths. From P1 the author derives (i) a thermodynamic clock τ_∂ = ∫ Φ ds, whose sign and magnitude classify structure-building, structure-maintaining, and structure-dissolving regimes; and (ii) two channels of local arrow-inversion windows (monitored feedback Φ_fb > 1; non-Markovian backflow Φ_NM > 1), bounded in duration by the local entropy budget and consistent with the global second law (Sagawa-Ueda for feedback; memory-induced negative σ_irr for backflow). From P2 the author derives (iii) a synchronisation functional Σ_ij between two Φ-clocks coupled through a shared bath, with the Connes-Rovelli thermal time hypothesis recovered in a triple limit (joint KMS state + Σ → 1 + Spohn-modular reduction); and (iv) an induced graph distance d_ij = -log|<X_i X_j>|/N that satisfies the triangle inequality on a metric-compatibility regime operationally characterised by joint CP-divisibility and absence of BLP non-Markovianity witnesses on the relevant triangles. The two witness conditions are jointly necessary for the triangle inequality on this regime; their sufficiency is conjectured under qualifications stated in the manuscript and listed as an open problem. The author provides a complete epistemic-status audit of every quantitative claim and an explicit comparison with the principal existing programmes on time and emergent spacetime: Connes-Rovelli thermal time hypothesis, Page-Wootters conditional time, Van Raamsdonk / AdS-CFT entanglement-based emergence, causal set theory (Sorkin), Verlinde / Padmanabhan thermal gravity, and Wolfram hypergraph rewriting. The audit of falsifiable predictions, the experimental discriminators (3-qubit feedforward, directional asymmetry, triangle inequality), and the worked-case reanalysis of Micadei et al. (2019) are the subject of the companion Preprint II.
We consider a single qubit undergoing pure dephasing with fixed populations and show that, fora natural pair of phase-opposite states, the trace distance coincides exactly with the ℓ1-norm ofcoherence. This allows the Breuer–Laine–Piilo (BLP) non-Markovianity indicator to be expresseddirectly in terms of single-qubit coherence dynamics within this setting, avoiding the usual optimizationover initial states. We also derive a simple relation between the entropy rate and coherence,showing that coherence revivals necessarily correspond to intervals of decreasing von Neumann entropy.Together, these results provide a transparent way to connect coherence, distinguishability,and information flow in dephasing qubits.