We study a four-qubit product-EPR holographic code whose reconstructing region contains matter and one leg of a geometry bond. Seven deformations compare local, bipartite, bond-stabilizing, and bond-moving operators. Exact spectra show that only matter-controlled bond motion produces a leading quadratic logical-state dependence of the boundary entropy. We select one state-independent physical recovery by optimizing coherent information of the channel Choi state. A separate semidefinite program maximizes entanglement fidelity over all channels. At fixed coupling the physical recovery attains that optimum to numerical precision, while recovered-entropy subtraction leaves a nonzero variational proto-area response. Total stabilizer Renyi magic is also nonzero for deformations with no boundary response, so it does not characterize the effect. Projecting the exact stabilizer-Renyi quadratic form away from pairwise Pauli tangents defines a leading-order boundary witness. Its global convex minimum is positive for the bond-moving deformation and zero for the six comparisons. For the reference-matter-geometry resource partition, however, pairwise tangents span the full projective tangent space at the product-EPR base, so the witness vanishes for every Hermitian deformation. A two-bound extension retains this degeneracy but exhibits a split-geometry structural residual and a logical-state-dependent cut switch. These results establish a finite-code witness, not a universal magic law or continuum gravitational dynamics.
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The paper asks which deformations of a minimal holographic-code model make a boundary entropy depend on the logical state while preserving recoverable matter information, and whether a partition-adapted magic diagnostic distinguishes this effect. The model consists of a logical matter qubit and a Bell geometry bond, with a four-qubit product-EPR Choi state. Seven unitary deformation families are compared using exact reduced spectra.
Only the tested matter-controlled bond-moving deformation produces logical-state-dependent boundary entropy, with peak-to-peak response $\Delta_q S(A)=\epsilon^2/2+O(\epsilon^4)$; the six comparisons have zero response. A single recovery per channel is selected by numerically optimizing Choi-state coherent information within a unitary-plus-discard ansatz. At $\epsilon=\pi/8$, it achieves $I_c=0.921821$ bits and entanglement fidelity $0.990393$, matching an independent all-CPTP fidelity SDP within numerical tolerance. Subtracting recovered-matter entropy leaves a proto-area response of $0.084924$ nats.
Total stabilizer Rényi magic does not distinguish responding from nonresponding deformations. Instead, projecting its exact quadratic expansion away from pairwise boundary Pauli tangents gives a globally minimized witness of $1/(4\ln2)$ for the bond-moving generator and zero for the comparisons. However, under the reference–matter–geometry resource partition, pairwise tangents span the entire projective tangent space, making this witness zero for every Hermitian deformation. A two-bond extension exhibits a structural residual and a logical-state-dependent candidate-cut switch, but no nonzero resource witness. The construction supplies neither a finite-coupling magic monotone, a universal magic–geometry relation, nor continuum gravitational dynamics.
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We study a four-qubit product-EPR holographic code whose reconstructing region contains matter and one leg of a geometry bond. Seven deformations compare local, bipartite, bond-stabilizing, and bond-moving operators. Exact spectra show that only matter-controlled bond motion produces a leading quadratic logical-state dependence of the boundary entropy. We select one state-independent physical recovery by optimizing coherent information of the channel Choi state. A separate semidefinite program maximizes entanglement fidelity over all channels. At fixed coupling the physical recovery attains that optimum to numerical precision, while recovered-entropy subtraction leaves a nonzero variational proto-area response. Total stabilizer Renyi magic is also nonzero for deformations with no boundary response, so it does not characterize the effect. Projecting the exact stabilizer-Renyi quadratic form away from pairwise Pauli tangents defines a leading-order boundary witness. Its global convex minimum is positive for the bond-moving deformation and zero for the six comparisons. For the reference-matter-geometry resource partition, however, pairwise tangents span the full projective tangent space at the product-EPR base, so the witness vanishes for every Hermitian deformation. A two-bound extension retains this degeneracy but exhibits a split-geometry structural residual and a logical-state-dependent cut switch. These results establish a finite-code witness, not a universal magic law or continuum gravitational dynamics.
The paper asks which deformations of a minimal holographic-code model make a boundary entropy depend on the logical state while preserving recoverable matter information, and whether a partition-adapted magic diagnostic distinguishes this effect. The model consists of a logical matter qubit and a Bell geometry bond, with a four-qubit product-EPR Choi state. Seven unitary deformation families are compared using exact reduced spectra.
Only the tested matter-controlled bond-moving deformation produces logical-state-dependent boundary entropy, with peak-to-peak response ΔqS(A)=ϵ2/2+O(ϵ4); the six comparisons have zero response. A single recovery per channel is selected by numerically optimizing Choi-state coherent information within a unitary-plus-discard ansatz. At ϵ=π/8, it achieves Ic=0.921821 bits and entanglement fidelity 0.990393, matching an independent all-CPTP fidelity SDP within numerical tolerance. Subtracting recovered-matter entropy leaves a proto-area response of 0.084924 nats.
Total stabilizer Rényi magic does not distinguish responding from nonresponding deformations. Instead, projecting its exact quadratic expansion away from pairwise boundary Pauli tangents gives a globally minimized witness of 1/(4ln2) for the bond-moving generator and zero for the comparisons. However, under the reference–matter–geometry resource partition, pairwise tangents span the entire projective tangent space, making this witness zero for every Hermitian deformation. A two-bond extension exhibits a structural residual and a logical-state-dependent candidate-cut switch, but no nonzero resource witness. The construction supplies neither a finite-coupling magic monotone, a universal magic–geometry relation, nor continuum gravitational dynamics.
A holographic code assigns a state-dependent entropy remainder to every boundary region, but such remainders need not be the extremal lengths of any one bulk metric. We study this common-source question for the proto-area of a calibrated recovery, the entropy of a boundary region minus that of the matter recovered from it. At zeroth order the reference values must match the lengths of one reference geometry in a fixed renormalization scheme, and this matching also determines the entropy-to-length factor. On a graph with a fixed minimum-cut chamber the data derivatives must lie in the image of the cut-incidence matrix, whose left null vectors are witnesses. On the hyperbolic disk the first variation of renormalized length is the rank-two geodesic X-ray transform, with closed range in an adapted data space, and the geodesic displacement, fixed by a forced Jacobi equation, enters at second order. The resulting normal Hessian gives a gauge-invariant criterion, necessary and sufficient, for a regular pair of response coefficients to come from one metric two-jet. For the constant-curvature family generated by a boundary reparametrization the Hessian has a nonzero rank-four mode, so the image of the length map is curved. For a fixed recovery the proto-area Hessian is minus the Bogoliubov-Kubo-Mori information lost in recovery, which the Petz map decomposes orthogonally at a product reference, and which, on a finite-dimensional space of logical directions with positive minimum retained fraction, is at most a sharp multiple of the retained information. Geometric realization requires this loss to reproduce the normal length Hessian on every interval. Two codes show that exact regional area operators can fail to sew into one graph geometry, and that a matched reference with vanishing first response can fail at second order. All criteria refer to one recovery branch and one extremal-surface branch.
Quantum error-correcting codes provide a powerful framework for emergent spacetime, yet existing subsystem erasure-correcting holographic code models describe only quantum fields on a fixed background: in such codes, the entropic area term is state independent and cannot capture gravitational backreaction. We argue that this limitation is intrinsic to exact subsystem complementary recovery and that incorporating backreaction naturally leads to approximate quantum error correction. We introduce a Ryu-Takayanagi-like entropy decomposition for approximate subsystem erasure-correcting codes, defining bulk matter entropy via optimal recovery and a complementary proto-area entropy as the difference between boundary entropy and recoverable bulk entropy. For a broad class of skewed quantum codes obtained by small nonlocal perturbations of exact codes, the proto-area increases monotonically with bulk entropy, closely aligning with the changes in extremal surface areas in gravitational systems. We identify the origin of this response as a form of tripartite non-local magic in the Choi state of the encoding map, which vanishes in stabilizer codes and controls the leading matter-geometry coupling in approximate subsystem erasure-correcting codes.
We formulate the proposed connection between quantum magic and matter-responsive geometry at a finite, already-prepared quantum carrier interface. The complete output is an isometry V from C to R tensor B, and residual response means that its complete complementary channel is not constant on the logical code. Exact recovery from R is equivalent to the absence of that response.The proposed unconditional implication from non-Clifford resources to residual response is false. We prove explicit counterexamples: a Clifford copying circuit has a nonconstant complement, whereas non-Clifford logical evolution—and even a fixed syndrome containing bipartite non-local magic—can coexist with a constant complement and exact logical recovery. We therefore separate magic, residual response, and physical geometric backreaction.For a specified controlled-rotation family, we construct a non-Clifford responding program and obtain the exact formulas: delta leak of theta equals sine theta divided by two, and epsilon rec R of theta equals one minus cosine theta, divided by two, for theta between zero and pi over two, inclusive. A general first-order residual-observable witness gives a quantitative sufficient condition for response under a declared deformation. A Pauli-compression criterion separately characterizes constant complements for Clifford encoders.Destructive Born–Lüders readout consumes measured quantum names and returns a classical interface. Readout-and-drop of an unobserved residual is exactly partial trace and does not further change the radiation channel. The four-Majorana recovery obstruction, finite-channel certificates, access-preserving endpoint continuity, and finite-proxy recurrence are retained with their proper scope. No metric, field-to-memory receiver, admitted analytic quantum coefficient, or gravitational backreaction theorem is constructed.
Pati (2026) proved analytically that any generalized Hilbert-space inner product with positive operator A≠I permits superluminal signaling, uniquely fixing the standard inner product as the only one consistent with relativistic causality. We provide a numerical and analytic companion to this result. Using adversarial variational search over an EPR-steering protocol, we confirm zero signal at A=I to machine precision, find nonzero signal at every tested deviation down to 10⁻⁶ with no safe region at any scale, determine the universal leading-order coefficient (exactly 1.000000, confirmed to 8 decimal places), show the same structure persists from qubits to qutrits, and find that the optimal metric deviation and optimal probe observable are exactly orthogonal on the Bloch sphere. Extending the protocol to a logical Bell pair between two [[5,1,3]] holographic code blocks, we prove and numerically confirm an exact immunity theorem: the signal vanishes identically when the inner-product deformation is confined to k physical qubits for k=1,2, and becomes genuinely nonzero exactly at k=3 — precisely the code's erasure-correction threshold. We show this generalizes, without further computation, to every quantum error-correcting code, via the general erasure-correction theorem of Knill-Laflamme and Bennett-DiVincenzo-Smolin-Wootters — resting on the same mathematical foundation used to derive the Ryu-Takayanagi holographic entanglement-entropy formula from quantum error correction. Simulation/computation-only results; no hardware component.
When absolute maximal entanglement is unavailable in a state family, different objectives can select different distributions of residual entanglement. We determine exact trade-offs for qutrit stabilizer states and binary--ternary product states. For eight qutrits, evaluation of the complete published classification gives three nondominated balanced-cut entanglement profiles, characterized by the probabilities of obtaining three or four maximally entangled qutrit pairs. We relate the squared entropy-deficit objective to code weight enumerators and use a uniformity-forcing bound to prove that its eleven-qutrit minimum is six. The unique minimizing Clifford class is an already classified code, whose entropy profile also attains a sharp bound on average reference--erasure mutual information among four-uniform eleven-qutrit stabilizer states. In local dimension six, explicit sector alignments attain 56 maximally mixed four-party reductions, the maximum for arbitrary binary--ternary product states. Within the stabilizer product family, the exact cost--count Pareto boundary consists of (44,48) and (48,56). These results identify when aggregate optimization determines operational entanglement properties and when it does not. Published classifications supply coverage; analytical bounds, explicit witnesses, and independent arithmetic checks establish the stated consequences.
We study the holographic map in AdS/CFT, as modeled by a quantum error correcting code with exact complementary recovery. We show that the map is determined by local conditional expectations acting on the operator algebras of the boundary/physical Hilbert space. Several existing results in the literature follow easily from this perspective. The Black Hole area law, and more generally the Ryu-Takayanagi area operator, arises from a central sum of entropies on the relative commutant. These entropies are determined in a state independent way by the conditional expectation. The conditional expectation can also be found via a minimization procedure, similar to the minimization involved in the RT formula. For a local net of algebras associated to connected boundary regions, we show the complementary recovery condition is equivalent to the existence of a standard net of inclusions -- an abstraction of the mathematical structure governing QFT superselection sectors given by Longo and Rehren. For a code consisting of algebras associated to two disjoint regions of the boundary theory we impose an extra condition, dubbed dual-additivity, that gives rise to phase transitions between different entanglement wedges. Dual-additive codes naturally give rise to a new split code subspace, and an entropy bound controls which subspace and associated algebra is reconstructable. We also discuss known shortcomings of exact complementary recovery as a model of holography. For example, these codes are not able to accommodate holographic violations of additive for overlapping regions. We comment on how approximate codes can fix these issues.