The Parity-Entanglement Bound (PEB) fixes a minimal relational-information quantum of exactly two bits for a bipartite pure state under a Z2 constraint. We show that this quantum is best understood as a single local-unitary equivalence class of states with fixed mutual information I = 2, and that three features of the framework are faces of this one object: (i) non-locality -- the class is invariant under all local unitaries, so no local operation can change it; (ii) balance without mirror symmetry -- the class contains highly asymmetric representatives whose two sides carry equal entropy yet differ in structure, so global balance needs no symmetric copy; and (iii) sector exchange -- an internal rotation moves weight between the two Z2 parity sectors at conserved total while the state stays in the class. We give the exchange dynamics as a continuity equation, confirm by simulation (including a realistic noise model) that the local/non-local distinction survives realistic decoherence, and address anticipated objections. The significance is structural: this same two-bit class is the quantum from which the framework derives a single time dimension, the invariant speed c, and the Dirac structure. The statement is platform-independent and testable on any entanglement-capable hardware. This work is part of a wider framework built on information conservation (Delta I = 0) and the minimal two-bit quantum.
The Parity-Entanglement Bound (PEB) fixes a minimal relational-information quantum of exactly two bits for a bipartite pure state under a Z2 constraint. We show that this quantum is best understood as a single local-unitary equivalence class of states with fixed mutual information I = 2, and that three features of the framework are faces of this one object: (i) non-locality -- the class is invariant under all local unitaries, so no local operation can change it; (ii) balance without mirror symmetry -- the class contains highly asymmetric representatives whose two sides carry equal entropy yet differ in structure, so global balance needs no symmetric copy; and (iii) sector exchange -- an internal rotation moves weight between the two Z2 parity sectors at conserved total while the state stays in the class. We give the exchange dynamics as a continuity equation, confirm by simulation (including a realistic noise model) that the local/non-local distinction survives realistic decoherence, and address anticipated objections. The significance is structural: this same two-bit class is the quantum from which the framework derives a single time dimension, the invariant speed c, and the Dirac structure. The statement is platform-independent and testable on any entanglement-capable hardware. This work is part of a wider framework built on information conservation (Delta I = 0) and the minimal two-bit quantum.
We prove that any bipartite pure state satisfying a Z2 parity constraint (PC ⊗PS)|Ψ⟩ = +|Ψ⟩ with both sectors populated carries mutual information I(C; S) ≤ 2 bits, with equality if and only if the two Schmidt coefficients are equal (Parity-Entanglement Bound, PEB). Within the Page–Wootters relational-time framework, the PEB implies that the clock–system mutual information is at most 2 bits, achieving exactly 2 bits for the maximum-entropy ground state, independently of system size N and bipartition choice (clock-independence). We further prove that among the uniform ground states of cyclic Zn constraints, Z2 uniquely minimises I(C; S) subject to I(C; S) > 0. N-independence andbclock-independence are verified on IBM Kingston quantum hardware for N = 2, 3, 4, 5 qubits and all clock-qubit positions at N = 4, with all deviations below 0.15σ from the predicted 2 bits. The PEB answers why Moreva et al.’s photon experiment [2] measured I ≈ 2 bits: this is not a feature of their setup but a consequence of the underlying Z2 constraint.
The Parity-Entanglement Bound (PEB) fixes the clock-system mutual information of a ℤ₂ constrained bipartite pure state at exactly two bits. We show that this invariant functions as the zero point of a metric geometry built from information. Six results are given. (i) The deviation Π = 1 - 1/2 I is the unique local-unitary (LU) invariant of the minimal state (pure, ℤ₂, bipartite); relaxing each hypothesis adds a definite number of further axes, which we count numerically (mixedness +1, ℤn+(n-2), tripartite +2; a continuous gauge group opens a continuous axis), (ii) The correlation quantity d = -ln κ is non-negative, symmetric, and vanishes in the coincidence limit; in the gapped regime, at separations above a floor r*(m), it obeys the triangle inequality and is a metric. (iii) The validity window is mapped: the floor r*(m) grows with the gap and the window closes entirely once the correlation length falls below the lattice scale (m ≥ 1.7 in 1D, ≥ 2 in 2D, lattice units); in the opposite, gapless limit the inequality survives with growing slack (0 violations atm m = 0.02) and an asymptotic argument establishes it analytically in the logarithmic regime. Non-Gaussianity per se does not damage the inequality: on a φ⁴ chain the observed violations are reproduced quantitatively by a free field of matched effective gap. (iv) Local noise acts on d as an exactly additive per-region dilation, d → d + δA + δB with δ ≥ 0; pure dephasing leaves d exactly invariant, and local noise can only increase the triangle slack, so the metric is noise-robust by construction. (v) The PEB value equals a combinatorial crossing number of stabiliser generators, hence is independent of system size, clock size, and clock position; it is formally analogous to a topological entanglement entropy. (vi) Following the Dirac-equation construction [6], the two-dimensionality dim ℋC = 2 underlies a distance/time split. The construction is verified by exact and Monte-Carlo simulation (geodesic locking 60/60 across five seeds; zero triangle violations above the floor in more than 1.6 × 10⁴ triples spanning gapped, near-critical, interacting, and noisy regimes) and, as a proof of concept, by a state-preparation study on quantum hardware (IBM FEZ). Formal correspondences to gravitational behaviour geodesic bending in a non-uniform gap and a volume-to-area transition in the crossing count are recorded in Sec. no gravitational derivation is claimed. Statements are labelled [P] proved, [K] known-with-citation, [N] numerical, or [0] open.
The Parity–Entanglement Bound (PEB) fixes a ceiling: any ℤ₂-constrained bipartite pure state with both sectors populated carries I(C;S) ≤ 2 bits. Whether the 2-bit value is also a floor — a genuine quantum of information — has remained implicit in the program's language. Here we settle it, by proof and by hardware, in both directions at once. There is no unconditional floor: I takes every value in (0,2] on the constrained family, and we measure states down to I = 0.394 bits. There are, however, exactly two conditional floors, each activated by a different operational requirement. Floor I (locality): every local k=1 autoencoder of the balanced state has reconstruction fidelity exactly 1/4 — a flat floor, proved in three lines and measured flat at 0.250 ± 0.009 across five distinct local encoders on ibm_fez, while a single cut-crossing gate restores the device ceiling (0.973 vs. the k=2 control's 0.975: global compression is free). Floor II (clock function): the rank-2 rigidity bound I ≥ 2 − 2log₂(1+ε), with ε the tick overlap, holds at every point of a measured Schmidt sweep (8/8, forbidden region empty), with the two independently measured curves I = 2h(b) and ε = |2b−1| tracking theory to ≤ 0.03 bits and ≤ 0.007 respectively. The two floors meet the PEB ceiling at a single point: ε = 0 ⇔ b = 1/2 ⇔ I = 2. The 2-bit quantum is therefore not the floor of the constraint but the pinch of the unconditional ceiling with the conditional floors — information below 2 bits is always paid for, exactly, in tick overlap, and the shared bits can be globally compressed for free but locally not at all. Statements are labelled [P] proved, [K] known, [N] measured.
The Parity–Entanglement Bound (PEB) established that a Z2 parity constraint in the Page–Wootters framework forces the clock–system mutual information to I(C; S) = 2 bits. We extendthat result. Placing the PEB in the stabilizer formalism and examining the Schmidt structuredirectly, we show that the 2-bit value rests on a rank-2 (uniform) condition, not on the constraintalone, and that it is invariant under in-plane extension and under Z2 concatenation across scales.We prove a Zn bound, I(C; S) ≤ 2 log2 n, saturated by the uniform ground state independentlyof the system size, the clock size, and the clock position; and we compute the non-abelian value,finding for S3 a relational information of ≈ 3.84 bits, strictly below the abelian value 2 log2 6. Werecover the Z2 topological entanglement entropy γ = log2 2 from the minimal toric code. The resultsform a discrete-symmetry analogue—distinct from Noether’s theorem—in which a symmetry fixesa definite relational information. This work extends, and does not supersede, the original Z2 PEB.
The Parity-Entanglement Bound (PEB) and its Zn / non-abelian extensions fix the clock-system mutual information I(C;S) of a symmetry-constrained pure state to an absolute, calculable value: 2.000 bits for a uniform Z2 constraint, 2 log2(n) for Zn, and representation-dependent values (e.g., 3.84 bits for S3). These invariants are independent of qubit number, clock size, clock position, and physical platform. Here we convert these mathematical results into a platform-agnostic, rank-agnostic benchmarking protocol for quantum processors. A single experimental run provides three-axis error diagnosis - passive decoherence, active gate noise, and logical survivability - all referenced to the absolute invariant, without any external baseline or classical simulation. The same protocol also enables lightweight entanglement certification in quantum networks (single-qubit, two-basis measurement). The method is demonstrated on IBM Kingston hardware for N = 2-5 qubits; all measured values lie within 0.15 sigma of the target, and a Schmidt-balance sweep follows the theoretical curve with R^2 = 0.998. The PEB thus supplies a universal ruler for calibration, error diagnosis, and cross-platform comparison - a tool that assists quantum computation without being a fault-tolerance code.
The Parity–Entanglement Bound (PEB) program takes the two-bit parity stabilizer—the membrane—as its primitive: prior work derives relational time, an emergent metric, and a geometric field law once the membrane is assumed. Here we ask where the membrane itself comes from, and answer with a selection principle. We define a resource-bounded survival dynamics on random local check structures whose only currency is relational information retained per unit resource under noise, with every rule fixed a priori and no tuned weight. Across independent random histories the dynamics converges, 12/12, to a single structure: a crossing Z-type generator (the membrane) and a crossing X-type logical (the exchange), mutually commuting, of minimal weight—exactly the stabilizer pair {ZᵢZ_N, XᵢX_N} of the [N,N−1,2] parity code, with e_{C|S}=2 and I=2 bits. The membrane is thus not an assumption but an attractor: the most information-efficient survivor of a competition among structures. The convergence is stable under larger populations and longer runs, and identical under depolarizing, dephasing, and amplitude-damping retention—a property of the structure, not of the noise model. The commutativity that lets the exchange move sector weight without annihilating the membrane—"exchange without annihilation" in the vacuum-asymmetry analysis—emerges from selection rather than being imposed. We further map the framework's own viability: on the emergent-geometry side the metric axioms hold only on an interior window of the information axis, Π∈(0,Π_c), with a parameter-free coupling that diverges at both walls—a measured sensitivity diagram for the structure's constants. Statements are labelled [P] proved, [K] known, [N] numerical.