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) 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.
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 parity constraint to 2 bits, and its symmetry extension gives I(C;S) = 2 log₂ n for ℤₙ and a representation-dependent reduction I = 2 log₂|G| − (2/|G|) Σ_ρ d_ρ² log₂ d_ρ for a finite group G. We carry this program to a continuous, non-abelian gauge group, SU(2). The finite-group sum over irreps is replaced by an integral over the group with the Haar measure, and the invariant clock–system state is fixed, uniquely, by maximum entropy at fixed quadratic Casimir — the SU(2) heat kernel, with block weight P_j ∝ d_j² e^(−β j(j+1)). We prove that the relational information I(C;S) = 2S(ρ_C) is finite for every β > 0, that the non-abelian censorship ΔI_NC = 2 Σ_j P_j log₂ d_j is manifestly non-negative, and that both reduce to the finite-group results in the appropriate limits, recovering S₃ ≈ 3.84 bits. The construction is realized on IBM Kingston: for the j ≤ 1 truncation the measured relational information is I = 5.22 bits against a predicted 5.15 (a 1.3% deviation), and an X-basis coherence witness W = 0.72 excludes a classical mixture. The result places the discrete symmetry–information correspondence inside a continuous gauge group whose representations organize both spin and the electroweak interaction.
The parity–entanglement bound (PEB) fixes the relational information of a Zₙ-constrained uniform state at I(A:B) = 2 log₂ n bits for every bipartition and every system size. The topological entanglement entropy of a Zₙ gauge theory is γ = log n, likewise independent of the size and shape of the region. We show that these are one constant: I_PEB = 2γ, exactly, for every n, and that the agreement is not numerology but the same superselection structure read twice — the cut crosses n charge sectors, and both quantities are the logarithm of that count. Extending to a finite group G, the programme's decomposition gives I = 2 log₂|G| − ΔI_NC with ΔI_NC = 2 Σ_ρ P_ρ log₂ d_ρ, so the non-abelian censorship of the programme is exactly the shortfall of the bound below 2γ, and it vanishes precisely on the abelian groups. Both statements are verified exactly, by singular values for n = 2–5 at every cut and size and by representation theory for nine groups [N]. Both sides are then measured on IBM Heron processors, with nothing fitted [N]. Abelian: nine independent cuts across three sizes give a weighted mean of 1.999 bits with a spread of 0.004 bits, encoded Z₃ and Z₄ registers give 3.199 and 3.978 bits against 3.170 and 4.000, and the reading is unchanged at 27 qubits and depth 107. Non-abelian: on the S₃-constrained state the two informations are read from one tomography and differ by 1.329 ± 0.016 bits against the predicted ΔI_NC = 1.3333, a 0.3σ agreement and 83σ from zero, while the same analysis on an abelian control returns −0.008 ± 0.004; the censored value agrees with the one prepared by a different route on a third processor in earlier work. The practical content is that the programme's two-bit constant is a quantity other fields have already realised, without reference to this programme, as topological order on three platforms and as a dense-coding capacity on a fourth [K]; what is new is that one constant, read three ways across seven platforms, agrees throughout, and that the censorship term acquires a standard identity as the edge-mode contribution. Statements are labelled [P] proved, [K] known, [N] numerical or measured.
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) fixes the clock–system relational information of a ℤ₂ constraint to 2 bits; its symmetry extensions give 2log₂n for ℤₙ, a representation-dependent reduction for finite non-abelian groups, and a heat-kernel form for the continuous group SU(2). We take the next step in the gauge-group ladder to SU(3), the gauge group of the strong interaction. The invariant clock–system state is the SU(3) heat kernel, block weight P_(p,q) ∝ d²_(p,q) e^(−βC₂(p,q)), which we confirm by aggressive Monte Carlo to be the maximum-entropy state at fixed quadratic Casimir — the same variational principle that fixes the SU(2) state, with no additional constraint. The relational information is finite for all β>0 and the non-abelian censorship ΔI_NC = 2ΣP log₂d ≥ 0 follows the single formula that runs unbroken from ℤ₂ through the finite groups to SU(2) and SU(3); abelian groups (ℤₙ, U(1)) give exactly zero. A structural feature distinguishes SU(3) from SU(2): SU(3) possesses complex (conjugate-pair) representations, absent in SU(2) where every irrep is self-conjugate. These complex representations — the quark/antiquark-type pairs — carry the majority of the censorship, a fraction that grows toward unity as β→0. The construction is realized in a Kingston-noise simulation (j-truncated to dimensions 1, 3, 3̄, 8; 8 qubits, depth 3–4): relational information I = 7.84 against a predicted 7.79, with an X-basis coherence witness W = 0.86 excluding a classical mixture. Across the series SU(N), SU(2) is the unique member with no complex representations; SU(3) is the first with an asymmetric censorship.
In the Page–Wootters relational-time framework a ℤ₂ parity constraint (P_C ⊗ P_S)|Ψ⟩ = +|Ψ⟩ ties a clock C to a system S. We show that the constraint controls one specific term of the entanglement entropy — the sector (charge) entropy — which obeys H({p_e, p_o}) ≤ 1 bit with equality iff the two parity sectors are equally weighted, for every system size N, every clock size k, and every state in the constrained subspace. The familiar value I(C;S) = 2 bits is recovered exactly when the complementary configurational term vanishes, i.e. on the rank-2 family, to which the uniform ℤ₂ ground state belongs; there I = 2 bits independently of N, of k and of which qubits form the clock. We further prove an exact complementarity V² + P² = 1 between the clock–system fringe visibility and the sector predictability. On IBM quantum hardware we measure the quantity the theory actually predicts: the weight of the reduced clock state inside the rank-2 subspace. For an 8-dimensional (k = 3) clock register at N = 5 we find 0.920, against 0.250 expected if the register carried the naive 2k = 6 bits; the two surviving eigenvalues are 0.445 and 0.475. A depolarisation parameter calibrated from an independent stabilizer fidelity witness returns I = 1.998, 1.999, 1.997 bits for k = 1, 2, 3. A separable control state is included: a single-basis clock-marginal estimator returns 1.9991 bits on it, while the tomographic estimator used here returns 0.835 bits and V² + P² = 0.0005, so the protocol separates the constrained state from a classical mixture.