We study the metrological extractability of a relative gravitational-redshift parameter delta encoded in Page–Wootters (PaW) historical states of a clock C and a two-level probe S. Writing the redshifted history state as |Psi(delta)> = exp(-i delta G)|Psi(0)>, with G = dt sum_n n |n><n| (x) H_S, we obtain the exact finite-N quantum Fisher information F_Q = (omega_0 dt)^2 (N-1)(2N-1)/6 tending to (omega_0 T)^2/3 in the continuous limit. Two results follow. First, F_Q vanishes for separable clock-probe states, so that a non-zero joint quantum Fisher information for the relative redshift is equivalent to C–S entanglement entering the correlations; separable-state redshift dynamics, present in earlier PaW treatments, nevertheless survives without providing any joint estimability. Second, a deterministic protocol that reads the clock record n and then performs Ramsey readout on S attains the joint QFI exactly, with no post-selection overhead, yielding a factor ~26 advantage over measuring the reduced probe alone for our toy model. With state-of-the-art optical-lattice clocks (Sr-87, T = 1 s), a 1 m height difference becomes resolvable in tens of shots, placing the PaW resource picture in the same regime as cm-scale clock-array experiments. We contrast this with history-state fidelity, a basis-dependent diagnostic, and discuss decoherence, GHZ scaling and an error budget.
We study the metrological extractability of a relative gravitational-redshift parameter delta encoded in Page–Wootters (PaW) historical states of a clock C and a two-level probe S. Writing the redshifted history state as |Psi(delta)> = exp(-i delta G)|Psi(0)>, with G = dt sum_n n |n><n| (x) H_S, we obtain the exact finite-N quantum Fisher information F_Q = (omega_0 dt)^2 (N-1)(2N-1)/6 tending to (omega_0 T)^2/3 in the continuous limit. Two results follow. First, F_Q vanishes for separable clock-probe states, so that a non-zero joint quantum Fisher information for the relative redshift is equivalent to C–S entanglement entering the correlations; separable-state redshift dynamics, present in earlier PaW treatments, nevertheless survives without providing any joint estimability. Second, a deterministic protocol that reads the clock record n and then performs Ramsey readout on S attains the joint QFI exactly, with no post-selection overhead, yielding a factor ~26 advantage over measuring the reduced probe alone for our toy model. With state-of-the-art optical-lattice clocks (Sr-87, T = 1 s), a 1 m height difference becomes resolvable in tens of shots, placing the PaW resource picture in the same regime as cm-scale clock-array experiments. We contrast this with history-state fidelity, a basis-dependent diagnostic, and discuss decoherence, GHZ scaling and an error budget.
We re-examine whether the internal-clock Page–Wootters (PaW) coupling of an optical lattice clock contributes an additional gravitational-redshift bias when the clock's center of mass (COM) is delocalized in a gravitational potential. We first identify and correct a dimensional error in a frequently used branch propagator: the mass-defect interaction must read H_int=(H_c/mc^2)V_k, so the gravitational and motional branch unitaries are U_g/e=exp(± i (omega_0/2mc^2) V_k t), with a dimensionless coefficient beta=omega_0/(2mc^2) ~ 10^23 J^-1. A propagation factor exp(± i V_k t/(hbar m c^2)) used in earlier estimates carries units of 1/J and is numerically ~2/(hbar omega_0)~10^19 too large, producing spurious order-unity phase wrapping. Using the corrected propagator, exact diagonalization on a Fock basis for ^24Mg and ^87Sr lattice clocks shows that (i) the unconditional reduced-COM mean of V_k equals the bare expectation to numerical accuracy (~10^-42 J), a trivial V_k-with-its-own-exponential identity; (ii) the internal–COM entanglement entropy remains below 10^-3 bit at interrogation times up to 1 s; (iii) the only nonzero PaW quantity is a conditional mean shift whose well-conditioned (high-norm) part scales as beta^2 t^2 Cov(V_k,V_k^2)/(mc^2) ~ 10^-22 (Mg, t=50 ms) and oscillates at the optical frequency omega_0 (period ~2 fs); (iv) this conditional shift averages to ~Delta omega/omega_0 ~10^-6 under any phase-insensitive optical readout and is therefore not a clock observable; (v) finite-temperature thermal moments scale the shift as ~T^3 but remain below 10^-25 at 1 mK. We conclude that internal PaW entanglement introduces no resolvable redshift bias at current or near-future optical clocks; Hu-type COM delocalization decoherence remains the leading quantum-relativistic correction. We provide a full parameter-space exclusion map and a sensitivity roadmap. All numerical results in this paper are illustrative/synthetic, computed with the corrected propagator and realistic Mg/Sr parameters.
This paper extends the derivation of the Lorentz transformation from count-based observables to the quantum level. First, each ingredient of the Bondi derivation is mapped onto the Page-Wootters (PaW) framework, and the k-factor is extracted as a ratio of PaW clock POVM eigenvalues — completing the derivational chain from Ĥ|Ψ⟩ = 0 to t′ = γ(t − vx/c²). Second, the k-factor is promoted to an operator k̂ = (Ĥ_cm + p̂c)/(mc²), revealing a two-sector decomposition of the Lorentz transformation: a time dilation sector (k̂ + k̂⁻¹ = 2E/(mc²), nonlinear in momentum) and a simultaneity sector (k̂ − k̂⁻¹ = 2pc/(mc²), linear in momentum). Third, the Bondi decomposition is applied to the proper time ω₀τ = ω₀Lm/p of a clock traversing baseline L, showing that the standard result separates into a time dilation contribution ω₀(Lm/p + Lp/(mc²)) and a simultaneity contribution −ω₀Lp/(mc²), which sum to the proper time. The two sectors have different behavior under quantum averaging: the simultaneity sector, being linear in p, is protected — its mean receives no corrections from momentum spread σ_p for any symmetric distribution. The time dilation sector, being nonlinear, produces mean corrections at order σ_p²/(m²c²). This asymmetry clarifies the structure of the Pikovski decoherence mechanism: momentum-dependent dephasing of internal clocks originates entirely from the time dilation sector's nonlinearity, while the simultaneity sector contributes a deterministic offset protected by linearity. All corrections vanish in the classical limit σ_p → 0, recovering the companion paper's Bondi derivation exactly.
Gravitational diffusion in optomechanical probes is conventionally modelled as Markovian white noise, but a finite environmental memory can invalidate the interrogation-time optimisation that underlies precision estimates of the gravitational decoherence rate Λg. We treat the diffusion source as an Ornstein–Uhlenbeck colored-noise channel with memory time τc, generalising the single-mode Gaussian covariance of Souza et al. [arXiv:2602.14841] and recovering the Markovian baseline exactly as τc→0, and evaluate the quantum Fisher information (QFI) for Λg under a fixed total experimental budget including preparation overhead. In the underdamped regime of a feedback-cooled femtogram levitated oscillator, we find that the steady-state QFI is independent of τc, whereas the optimal interrogation time at fixed budget shifts by a factor 15–34 and the QFI rate degrades by up to an order of magnitude as τc grows from the Markovian limit to 0.3 s. A two-parameter Fisher analysis further shows that Λg and τc are nearly degenerate during the transient, so that memory acts as a convergence-rate penalty rather than a precision-ceiling penalty. We stress that this work is a protocol-design and quantum-estimation benchmark, not a near-term detection proposal: even at an effective occupation n̄=50 resolving Λg requires of order 10²⁹–10³⁰ repetitions. All numerical results are illustrative/synthetic benchmarks for a 5 K, 100 kHz femtogram oscillator. I. Introduction Gravitationally induced decoherence [1,2] leaves measurable signatures in the Wigner function of a mechanical probe, and quantum metrology asks how precisely the associated diffusion rate Λg can be inferred. Recent work by Souza, de Sá Neto, Russo, Lo Franco and Adesso [3] established the ultimate quantum-Cramér–Rao sensitivity of a single-mode Gaussian probe to Λg within the Diósi/Kafri–Taylor–Milburn master equation, showing that squeezed-vacuum preparation maximises the transient QFI and that the long-time state is independent of the probe. Their analysis, like essentially all optomechanical decoherence metrology, assumes a delta-correlated (Markovian) environment.
This project explores a fully timeless formulation of quantum theory motivated by the Page–Wootters mechanism and canonical quantum gravity. The central assumption is that the universe is described by a single global quantum state ∣Ψ⟩ satisfying a Wheeler–DeWitt–type Hamiltonian constraint, with time appearing only as an internal, relational variable defined by “clock” subsystems. I formulate a simple multi–clock framework, introduce a clock neutrality conjecture, and test it in an explicit finite-dimensional toy model. The model consists of two three-tick internal clocks and a qubit system, with joint step relations that encode a static global state over a 3×3 grid of clock readings. Imposing multi-clock consistency selects a constrained family of states in which the two clock evolutions commute on the support of ∣Ψ⟩. Within the decoherent histories framework, I then show that for histories expressible with either clock, the probabilities for relational events agree whenever the two clocks implement the same underlying physical evolution. In this sense the toy model realises a discrete version of clock neutrality: decoherent-histories probabilities are independent of which internal clock is used as the parameter. The construction does not yet yield new empirical predictions, but it isolates a precise structural condition that any timeless, deterministic description of the universe would need to satisfy in order to be compatible with clock-independent low-energy physics.
I. Introduction The problem of time in quantum gravity is the following. General relativity is a diffeomorphisminvariant theory: there is no preferred time variable, and the Hamiltonian constraint HΨ = 0 impliesthat all physical states are ‘frozen’ in the sense that they carry no manifest time dependence [Dirac1964; DeWitt 1967]. Yet quantum mechanics requires time as a background parameter againstwhich evolution is measured. Reconciling these two facts — the timelessness of diffeomorphisminvariant quantum gravity and the temporal structure required for quantum measurement theory—is the problem of time [Isham 1993; Kuchăř 1992]. The problem is not merely philosophical: ithas concrete consequences for the interpretation of quantum cosmological perturbations, includingthose that seed the CMB anisotropies we observe today.One of the most elegant proposed resolutions is the Page–Wootters (PW) mechanism [Page & Wootters 1983]: time is not a background parameter but a relational quantity, encoded in correlationsbetween a ‘clock’ subsystem and the rest of the universe. Physical time is the reading of the clock,and quantum evolution is the conditional state of the system given a particular clock reading. Thisidea has been placed on rigorous mathematical footing by Höhn, Smith & Lock (HSL) [2021], whoproved the equivalence of three apparently distinct quantization procedures — the Dirac constraintquantization, the Page–Wootters mechanism, and the deparametrized Heisenberg picture — undera precise set of conditions. The HSL result reduces the problem of time, for any system satisfyingthose conditions, to the problem of identifying a clock with the right algebraic properties.In Loop Quantum Cosmology (LQC), relational time has been implemented using material degreesof freedom: a minimally-coupled massless scalar field [Ashtekar, Pawlowski & Singh 2006], a GFTcondensate [Gielen & Sindoni 2016], or dust fields [Brown & Kuchăř 1995; Pawłowski 2014]. Allof these are material clocks: dynamical degrees of freedom that must be added to, or identified1within, the matter content of the model. No worked example exists in which the clock arises fromthe algebraic structure of the kinematic Hilbert space itself — as an invariant of the theory’s gaugegroup rather than an additional field. Such an algebraic clock would be conceptually preferable: itrequires no extension of the Hilbert space, its back-reaction on geometry is controlled, and it is notsubject to the multiple-choice ambiguity [Kuchăř 1992] that afflicts material clock choices when thesystem has multiple degrees of freedom.This paper provides the first such example. The LQC bounce, treated via the Pöschl–Teller (PT)effective potential established in the companion series [Hillard 2026a–d], admits a spectral threshold = √2 that is fixed by the zero of the SU(2) Casimir polynomial W() = ³(²−2)/[C·sinh(/2)],where C = 3.60329… is the Hillard constant [Hillard 2026a, Paper 8]. This zero is a representationtheoretic invariant of the kinematic Hilbert space: it exists prior to any dynamical input, beforeany clock is chosen. We show that instantiates the full HSL relational clock construction. TheClock Decomposition Theorem (Section III) establishes the tensor product factorization H̃_kin H H and constructs the covariant POVM on H. The Temporal Trinity Formula (Section IV)is identified as the HSL clock reading in all three quantization pictures, with explicit clock-changemap T (Appendix A). The result holds exactly: is not a perturbative approximation to a clockbut a genuine algebraic invariant.A relational clock that is an algebraic invariant makes a concrete, falsifiable prediction. The signchange multipole * — the CMB multipole at which the LQC bounce imprint changes sign in thepower spectrum [Hillard 2026a] — is precisely the PW temporal origin in Fourier space: * =/_B. When GFT interactions of strength are included, the PT potential is deformed to U_ =−(2−)sech²(), which remains within the exactly solvable PT family (Section VI). The thresholdshifts to () = √(2−), giving the exact non-perturbative prediction:()/ = √(1 − /2) − 1 −/4 + O(²).* (1)*The sign is negative: GFT interactions push * to lower multipoles. The detection threshold forLiteBIRD is _crit 4/*, placing the prediction within reach of the near-term CMB survey (SectionV). This is the first quantitative prediction from a relational clock interpretation of the LQC bounce.A word on the relationship between this paper and the companion series [Hillard 2026a–d]. ThePWinterpretation of the bounce established here depends on W() only through its functional form:W() ³(²−2). Any function with that polynomial zero structure has a scattering threshold at =√2, and any such function in the Hamiltonian constraint generates the relational clock constructionof Sections III–IV. The validity of the LQG derivation of W in [Hillard 2026a] is therefore not aprerequisite for the results of this paper. The companion series and the present paper are mutuallyreinforcing rather than circularly dependent: [Hillard 2026a] provides the physical motivation andprecise form of W; this paper provides the relational interpretation that explains why has thevalue it has.The paper is organised as follows. Section II reviews the relevant background on LQC, the PTeffective potential, and the HSL formalism. Section III establishes the Clock Decomposition Theorem: spectral factorization (Proposition 1), covariant POVM, and temporal origin. Section IVidentifies the Temporal Trinity Formula in all three HSL pictures (Lemma 1, Proposition 2). Section V derives the physical inner product by group averaging and the full observational prediction(Table 2). Section VI proves that the intertwiner deformation is shape-invariant and promotes theprediction to an exact formula (Proposition 3). Appendix A constructs the clock-change map Texplicitly (Proposition A.1, Corollary A.1).22Main results at a glanceClock Decomposition Theorem (Section III): H̃_kin H H + O(). POVM E(d) =|W()|²/‖W‖² d is covariant. = √2 is the temporal origin.Trinity Formula (Section IV): N_total = N + ln(··k_piv/) is a Dirac observable, clockchange invariant. T acts as identity on (Appendix A).Exact prediction (Sections V–VI): ()/ = √(1−/2) − 1 −/4. LiteBIRD detection threshold:_crit 4/*.First algebraic PW clock in LQC (Section V.4): is an SU(2) Casimir invariant, not amaterial d.o.f. Back-reaction is O(); no Hilbert space extension needed.
Pedagogical Guide, Foundations Audit, Uniqueness & Substitutability Analysis, Literature Comparison, and Reading Guidance : A Pedagogical Guide to Understanding the OT–GKSL Architecture This paper develops a comparative interpretation of two recent Oxford research directions through the certified source–readout architecture of the Einstein-locked OT–GKSL framework: the experimental extraction of classical ticks from a quantum clock studied by Wadhia, Meier, Ares and collaborators, and the post-selected repulsive gravitational signal proposed by Saldanha, Marletto and Vedral. The comparison is structural rather than mechanistic. OT–GKSL starts from open quantum-state dynamics rather than from a primitive spacetime geometry. Its native objects are density states, the complete GKSL generator, and, in the detailed-balance dissipative sector, a positive quantum optimal-transport geometry. Classical events, clocks, Lorentzian geometry and gravitational observables are reconstructed only through stable physical records, event individuation, calibration and certification. The relevant hierarchy is ρ → ω(ρ) = M(ρ) → [ρ]evt → Xᵃ → s → eᵃ → g_ro, with the native OT geometry and the Lorentzian readout metric kept strictly distinct. The native process carries a directed semigroup ordering, but not a primitive Lorentzian causal structure. For temporal readout, the framework distinguishes native entropic ordering, a clock-like record, and calibrated classical time: t_ent ≠ X⁰ ≠ t_ro. This separation provides a natural operational language for the quantum-clock experiments, where microscopic clockwork dissipation and the energetic/informational burden of extracting a classical record are experimentally distinguishable. The associated OT–GKSL reporting ledger is C_tick = C_nat + C_ext + C_cert, where C_nat denotes the native clockwork contribution, C_ext the extraction and amplification burden, and C_cert the additional cost of maintaining a stable, calibrated and auditable record. This is a reporting ledger, not a universal microscopic thermodynamic identity. The corpus explicitly identifies the experimentally realized distinction clockwork cost ≠ classical-record extraction cost. For the gravitational proposal, each branch of the Saldanha–Marletto–Vedral protocol remains individually attractive, δ_A > 0, δ_B > 0, while post-selection and coherent amplitude interference can produce an effective conditional shift δ_eff = (β δ_B − α δ_A)/(β − α), which may be negative. The paper emphasizes the distinction between the unconditional joint dynamics and the conditioned measurement record. In particular, for complementary post-selection outcomes f and f̄, P_f ⟨p⟩_f + P_f̄ ⟨p⟩_f̄ = α² δ_A + β² δ_B, for a probe initially centered at zero momentum in the ideal branch-kick model. Thus a negative conditional momentum shift is not by itself equivalent to a negative gravitational coupling, a repulsive force on every branch, or a complete negative momentum balance for the total source–probe–apparatus system. Within OT–GKSL, the post-selection instrument is treated as an imported quantum-measurement structure rather than as an independently derived prediction of the framework. Measurement is represented as a directed certified readout between physical subsystems, A ← B, where the record-bearing subsystem must satisfy stability, accessibility, rank and bridge conditions. Classicalization is required only for the declared record degrees of freedom and does not imply complete dephasing of all quantum degrees of freedom. The resulting mirror is therefore: Ares/Wadhia:native clockwork → detector interaction → classical record → calibrated temporal estimate, Saldanha–Marletto–Vedral:coherent source–probe dynamics → measurement/post-selection → conditional record → gravitationally interpreted probe observable. The common lesson is not that the two systems share a physical mechanism, nor that either experiment confirms OT–GKSL. Rather, both highlight the importance of distinguishing native quantum dynamics from the physical process by which a classical statement becomes accessible. OT–GKSL extends this distinction to spacetime itself: classical Lorentzian geometry is treated as a post-constructed, certified readout rather than as the primitive background of the native dynamics. The paper explicitly identifies the remaining open bridge problem for the gravitational protocol: embedding the SMV preparation into a native OT–GKSL source state, deriving the corresponding gravitational branch interaction from the certified source/readout construction, and embedding the post-selection and probe records into the framework. A complete source–probe–apparatus energy–momentum closure is outside the scope of the present comparative study. 1. Foundations of the Architecture: Foundations |GKSL/Lindblad ; Carlen–Maas ; Jacobson ; Sakharov ; Donoghue ; Lovelock) Establishes the core Einstein-locked OT/GKSL architecture for certified geometric readout and coherence-dependent gravitational sourcing. Optimal-Transport Gravity Trilemma | Identifies the certified operational boundary of geometric readout by proving the fundamental trade-off between temporal resolution, coframe stability, and bridge fidelity. 2. Emergence and Recovery of Classical Physics: Exact Reduced OT/GKSL Equations | Mori–Zwanzig/projection operators ; effective field theory ; Carlen–Maas ; Wilsonian reduction / Demonstrates the controlled recovery of classical Newtonian and gravitational sectors as exact non-linear reductions of the native OT/GKSL state dynamics. Certified Einstein Non-Linear Readout | Lovelock ; Bianchi identities ; Donoghue EFT ; Jacobson thermodynamic gravity// Develops the full non-linear Einstein-locked readout closure for the metric sector. Non-Linear Dynamics and Readout | Dynamical systems, center manifold/effective reduction ; quantum Markov semigroups ; non-linear open-system reductions // Explores the exact reduced non-linear evolution on collective state manifolds. The Seeley–DeWitt Bridge | Seeley–DeWitt heat-kernel ; Vassilevich // Formalizes the operational connection between native state dynamics and the effective classical readout. The SDW Bridge: Composite Brout–Englert–Higgs Dynamics, Spectral Separation, and the Emergent Graviton | Formalizes the emergence of the Brout-Englert-Higgs composite scalar and the spin-2 graviton via the Seeley-DeWitt expansion, strictly preserving the Einstein-Lock. Bridge between QCD and OT/GKSL Readout | Wilson lattice gauge theory ; Gross–Wilczek–Politzer asymptotic freedom ; Kogut–Susskind Hamiltonian lattice gauge theory // Connects the Optimal Transport / GKSL framework to Quantum Chromodynamics, exploring the constitutive bridge and effective low-energy dynamics. 3. The Certified Boundary and Structural Limits: Certified Spacetime Readout on Finite Support: A Unified Temporal and Geometric Boundary | Decoherence / Quantum Darwinism ; quantum reference frames ; finite information bounds ; Jacobson // Unifies the temporal and geometric branches of classical readout into a single certified spacetime problem. Introduces the unified spacetime readout burden and derives the central unified certified-budget inequality, proving that temporal precision, geometric coframe nondegeneracy, and bridge compatibility draw from the same finite entropic and informational resources and cannot be made simultaneously ideal. Certified Causality, Locality, Nonlocality, and Relativity in the Einstein-Locked OT/GKSL Framework | Algebraic QFT/locality ; operational quantum theory ; quantum reference frames ; relativistic causality tests // Determines the exact status of causality, locality, nonlocality, and the principle of relativity within the Einstein-locked OT/GKSL architecture. Shows that causal-local spacetime semantics is a certified readout property rather than a primitive native axiom; proves a patchwise gluing theorem for certified local causal structure; and derives a unified finite-budget inequality showing that temporal precision, geometric certification, bridge admissibility, and overlap compatibility all compete for a single residual causal-local headroom on finite effective support. Entropic Tick Cost and Certified Temporal Readout in the Einstein-Locked OT/GKSL Framework | Demonstrates that classical ticks are finite-resource readout objects extracted from native entropic ordering, rather than primitive background parameters. Decomposes the entropic tick cost into native, extraction, and certification branches, and derives a theorem-level certified temporal budget inequality connecting temporal resolution, finite effective support, and certification margins. Entropic Tick Cost & Spectral Budget | Page–Wootters time ; thermal time hypothesis ; quantum clocks ; Salecker–Wigner bounds // Establishes a theorem-strength certified boundary for classical spacetime by proving a fundamental trade-off between entropic tick resolution, coframe stability, and finite informational budget. Toy Certified Pipeline from Optimal Transport QCD | Provides a protocol-level implementation and scaling model for certified bridge margins. Certified Spectral Boundary from Heat-Kernel Budgets and Entropic Transport in the Einstein-Locked OT/GKSL Framework | Heat-kernel spectral budgets; entropic OT/GKSL transport; certified spectral boundary; Einstein-locked readout. Develops a spectral-geometric control layer for the OT/GKSL framework, where the native heat trace bounds finite spectral resources, the cutoff gap defines a certification margin, and entropic transport controls the drift of readout-support budgets without inducing a state-dependent Einstein–Hilbert kinetic term. Correlation Separation in the Einstein-Locked OT/GKSL Framework | Establishes a theorem-level distinction between native, readout, and causal-local correlations, and reframes t