Gapped area laws in two dimensions and polynomial PEPS

Source and verification

This article explains family 265 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Entanglement on a square lattice

A pure many-body state may require exponentially many coefficients even when its Hamiltonian is local. This family relates a spectral gap to two forms of structure: an entropy bound proportional to boundary size, and a polynomial-bond tensor-network approximation on the original square grid. The second manuscript uses additional information and compression estimates beyond the area inequality itself. (OpenAI, 2026) (OpenAI, 2026b)

The prerequisites are finite-dimensional Hilbert spaces, tensor products, spectral decompositions, and elementary probability. We introduce the quantum-information quantities before discussing the proof.

Let $\Lambda \subset {\mathbb{Z}}^2$ be finite, with edges between nearest neighbors that both belong to $\Lambda$. At each site place ${\mathbb{C}}^q$, giving

$${\mathcal{H}}_{\Lambda }=\underset{x\in \Lambda }{⨂}{\mathbb{C}}^q.$$

For a unit vector $\Omega$, its density matrix is $P_{\Omega }=|\Omega ⟩⟨\Omega |$. The reduced density matrix on $A\subseteq \Lambda$ is ${\rho }_A={\mathrm{Tr}}_{\Lambda \setminus A}{P}_{\Omega }$. Its entanglement entropy is

$$S(A)=-\mathrm{Tr}({\rho }_A\log {\rho }_A).$$

Logarithms are natural and $0\log 0=0$. The Schmidt decomposition across $A$ and its complement writes $\Omega ={\sum }_j\sqrt{p_j}{u}_j\otimes v_j$ with orthonormal vectors and ${\sum }_j{p}_j=1$. The nonzero eigenvalues of both reduced matrices are the $p_j$, so complementary regions have equal entropy. The elementary bound is $S(A)\le |A|\log q$.

For disjoint regions, mutual information and conditional mutual information are

$$\begin{array}{c}I(X:Y)\\ =S(X)+S(Y)-S(XY),\\ I(X:Z\mid Y)\\ =S(XY)+S(YZ)\\ -S(Y)-S(XYZ).\end{array}$$

Juxtaposition denotes union of disjoint systems. Strong subadditivity makes conditional mutual information nonnegative. Discarding a system cannot increase ordinary mutual information, but conditioning is a different operation: a bound on $I(X:Z)$ does not automatically bound $I(X:Z\mid Y)$.

🏷️ The global gap and the area theorem

Let $d_{\Lambda }$ be distance in the induced graph, with distance infinity between different components. Consider

$$\begin{array}{c}H=\sum _{\begin{array}{c}\varnothing \ne X\subseteq \Lambda \\ {\mathrm{diam}}_{d_{\Lambda }}X\le R\end{array}}{h}_X,\\ h_X=h_X^{\ast },\Vert h_X\Vert \le J.\end{array}$$

There is one designated term per support, possibly zero. For fixed range $R$, a graph ball has uniformly bounded size, so this convention bounds the number and total norm of interactions meeting each site. It rules out hiding arbitrarily many independent norm-$J$ terms on the same support.

The unique ground vector $\Omega$, up to phase, has energy $E_0$. A global spectral gap $\Delta >0$ means

$$H-E_{0}I\ge \Delta (I-P_{\Omega }).$$

In particular, a normalized vector $\psi$ with excitation energy at most $\delta$ obeys $1-|⟨\Omega ,\psi ⟩{|}^2\le \delta /\Delta$. This is how low energy forces closeness to the ground-state direction. No gap is assumed for Hamiltonians restricted to subregions.

Let ${\partial }_{\Lambda }A$ be the nearest-neighbor edges with one endpoint in $A$ and the other in $\Lambda \setminus A$.

Area law attributed to the first manuscript

For fixed $q,R,J,\Delta$, there is a finite constant $C$ such that $S(A)\le C|{\partial }_{\Lambda }A|$ for every subset of every finite induced square-lattice domain satisfying the stated unique-ground-state and global-gap assumptions. The same constant works for all domain sizes, shapes, holes, and disconnected components. (OpenAI, 2026a)

The induced-graph metric matters: a short path through a missing part of the ambient square lattice cannot justify an interaction. If the edge boundary is empty, $A$ is a union of graph components. The Hamiltonian splits across components, and uniqueness makes its ground vector a tensor product, so $S(A)=0$. The cases $q=1$, empty domains, and range-zero on-site Hamiltonians are also immediate.

For comparison, a product state has zero entropy across every cut. A product of Bell pairs on disjoint nearest-neighbor edges has entropy $\log 2$ times the number of those pairs crossing the cut. This elementary example exhibits the boundary scaling without suggesting that general ground states factor into pairs.

🏷️ Positive constraints from a possibly frustrated Hamiltonian

The original terms need not share a common minimizing vector. The first manuscript nevertheless constructs positive operators $k_i$ that annihilate $\Omega$ separately and retain the global gap when summed. Their supports have rapidly decaying tails; they are not finite-range terms of a newly assumed frustration-free Hamiltonian. (OpenAI, 2026a)

Here is the spectral mechanism. Choose a real even smooth frequency cutoff $\chi$ with $\chi (0)=1$ and support in $(-\Delta /2,\Delta /2)$, and a time kernel $f$ with $\chi (\omega )=\int f(t)e^{it\omega }dt$. For each original term define

$$\begin{array}{c}{M}_i=\int f(t)e^{itH}{h}_i{e}^{-itH}dt\\ -⟨\Omega ,h_i\Omega ⟩I.\end{array}$$

The cutoff removes every ground-to-excited matrix element; subtracting the scalar removes the ground component. Thus $M_i\Omega =0$. Since the sum of the time-evolved terms is $H$ and $\int f=1$, one also has ${\sum }_i{M}_i=H-E_{0}I$.

Choose a uniform $c_{\ast }$ bounding all $\Vert M_i\Vert$ and set $k_i=|M_i|/c_{\ast }$. The spectral theorem gives $|M_i|\ge M_i$ and identical kernels, hence

$$\begin{array}{c}0\le k_i\le I,k_i\Omega =0,\\ \sum _i{k}_i\ge c_{\ast }^{-1}(H-E_{0}I).\end{array}$$

Locality comes from propagation estimates: the influence of a time-evolved local operator is exponentially small outside a distance proportional to elapsed time. Fix an integer $k\ge 1$ and put $\alpha =k/(k+1)$. The source chooses the cutoff with quantitative derivative bounds giving $|f(t)|\le C{e}^{-c|t{|}^{\alpha }}$; smoothness alone would not give that particular rate. Combine this decay with propagation and then control the absolute-value operation by the operator square-root estimate. The resulting tail has the form

$$\Vert k_i-{\mathbb{E}}_{N_r(a_i)}{k}_i\Vert \le C{e}^{-c{r}^{\alpha }}.$$

Here $a_i$ is an interaction anchor, $N_r(a_i)$ is a graph ball, and ${\mathbb{E}}_{N_r(a_i)}$ traces out the exterior and tensors back its normalized identity. All constants are independent of total volume. These positive constraints allow local contraction products to suppress excitation while keeping the desired vector fixed. (OpenAI, 2026a)

🏷️ Collar information and entropy cancellation

A collar is a buffer around a region. The analytic part first obtains a subvolume entropy exponent for suitable boxes, then improves it through a finite randomized scan across collars. The scan moves partitions on many copies of the state. A common interpolation controls both the information gain and the excitation energy of the same transported vectors; two bounds on their logarithmic norms force an improved information estimate. These are substantial manuscript inputs. (OpenAI, 2026a)

The schedule has deterministic fill moves and random moves that transfer the remaining part of a nearby interaction support together. Random offsets make an interaction rarely split, while random support selection prevents it from remaining split too long. A support move never shifts the predetermined fill schedule. The entire finite system and schedule are fixed before the number of copies tends to infinity.

For the controlled geometric templates, the scan yields $I(X:\Lambda \setminus S_0)\le C{n}^{1-\epsilon }$, where $S_0$ consists of the target and a sublinear collar. An amplification argument then obtains, for any fixed $M$, inverse-power information $C{n}^{-M}$ between the target and any physical set beyond a slightly larger collar. It uses typical spectral subspaces, the positive constraints, and a localized comparison that leaves the remote system untouched. The bound applies jointly to a remote union, not merely to each remote piece separately. (OpenAI, 2026a)

The final conversion into an area bound is short enough to prove completely. Suppose

$$\begin{array}{c}A=D\dot{\cup }{A}_0\dot{\cup }{A}_1,\\ A_f={\dot{\bigcup }}_{i\in {\mathcal{I}}_f}{X}_i,\end{array}$$

and each family has an order. Put $B=\Lambda \setminus A$. Assume

$$I\left(X_i:B\cup \bigcup _{j<i,j\in {\mathcal{I}}_f}{X}_j\right)\le {\epsilon }_i.$$

The entropy chain rule gives, with $E_f={\sum }_{i\in {\mathcal{I}}_f}{\epsilon }_i$,

$$\begin{array}{c}S(B{A}_f)\\ \ge S(A)+\sum _{i\in {\mathcal{I}}_f}S(X_i)-E_f.\end{array}$$

Indeed, each conditional entropy added in the chain equals $S(X_i)$ minus its mutual information with the preceding systems, and purity gives $S(B)=S(A)$. Conversely, purity and subadditivity give

$$\begin{array}{c}S(B{A}_f)=S(D{A}_{1-f})\\ \le S(D)+\sum _{i\in {\mathcal{I}}_{1-f}}S(X_i).\end{array}$$

Add the two inequalities for $f=0,1$. Every individual tile entropy cancels, leaving

$$S(A)\le S(D)+\frac{1}{2}\sum _i{\epsilon }_i.$$

The planar construction supplies these two ordered families with $|D|\le C|{\partial }_{\Lambda }A|$ and summable errors ${\sum }_i{\epsilon }_i\le C|{\partial }_{\Lambda }A|$. Each tile’s collar avoids both the exterior and the earlier tiles of its own family. Sparse strips and recursive repairs near point contacts establish this ordering for arbitrary cuts. Since $S(D)\le |D|\log q$, the area theorem follows. The difficult geometry is what makes the elementary cancellation applicable. (OpenAI, 2026a)

🏷️ Polynomial PEPS approximation

A projected entangled-pair state, or PEPS, places a virtual pair ${\sum }_{a=1}^{D_e}|a⟩\otimes |a⟩$ on each lattice edge. At each site an arbitrary linear map takes its incident virtual spaces into the physical space ${\mathbb{C}}^q$. Contracting all paired indices produces a vector. The largest $D_e$ is the bond dimension; the maps need not be isometries and the vector need not initially be normalized.

PEPS consequence attributed to the companion

Fix $q\ge 2$ and $J,\Delta >0$. For every open $L\times L$ square grid, $L\ge 2$, with on-site and nearest-neighbor terms of norm at most $J$, a unique ground vector and global gap at least $\Delta$, there is a nonzero PEPS $\Phi$ on that same grid with bond dimension at most $C{L}^c$ and

$$\underset{\theta }{\min }\Vert \frac{\Phi }{\Vert \Phi \Vert }-e^{i\theta }\Omega \Vert \le L^{-1}.$$

Constants depend only on $q,J,\Delta$. This is an existence statement for global vector error, without an efficient tensor-finding or contraction algorithm. (OpenAI, 2026b)

No translation invariance, commutativity, frustration freeness, subregion gap, or gapped path from a product state is assumed. The PEPS theorem’s domain and interaction class are narrower than those of the area theorem.

Why does entropy alone not complete the approximation? If Schmidt probabilities are decreasing, their tail beyond rank $D$ obeys only the elementary estimate

$$\sum _{j>D}{p}_j\le \frac{S(A)}{\log (D+1)}.$$

For $j>D$, $p_j\le 1/(D+1)$, so those terms contribute at least $\log (D+1)$ times their total mass to entropy. Even an entropy of order $\log L$ does not make this bound small to an arbitrary inverse power of $L$ at polynomial rank. Moreover, separate truncations at different cuts need not be compatible.

The companion supplies two stronger inputs. First, mutual information becomes inverse-polynomially small for regions separated by buffers that taper linearly toward finitely many junctions, with minimum width a sufficiently large multiple of $\log L$. Its proof establishes the required conditional-information estimates rather than simply conditioning an ordinary information bound.

Second, near a patch of logarithmic radius, an accurate projection is a sum of mutually orthogonal cylindrical subspaces. Each summand fixes the inside to one of finitely many vectors and leaves the outside unrestricted. The inside boundary may vary among nested squares, while the total number of inside vectors is polynomial in $L$. The source does not assert an equally accurate truncation at one prescribed boundary or low tensor complexity of each inside vector. (OpenAI, 2026b)

🏷️ Distribution, compression, and return to the original grid

The proof organizes physical registers among abstract parties attached to dyadic squares. A party may perform arbitrary private contractions. Initially one party holds the whole vector $\Omega$, with auxiliary copies privately initialized elsewhere. This is legitimate for a representation theorem; it is not a state-preparation algorithm.

Small-information buffers permit approximately splitting purifications and changing which party owns a region. Auxiliary copies handle places where a third party would otherwise obstruct the move, and the adaptive patch projections handle junctions. Each party participates in only boundedly many operations involving other parties over its entire lifetime. Every such operation has a polynomial-size expansion using private contractions and normalized bipartite resources, whose Schmidt ranks can still be large. (OpenAI, 2026b)

There are at most $C{L}^3$ changes. With error $O(L^{-20})$ per change, contractivity makes the vector errors add to $O(L^{-17})$. The retained reference vector has norm close to one, so the procedure does not rely on renormalizing an exponentially small successful outcome.

A separate density-compression theorem replaces the bipartite resources by polynomial sums of product operators in the final density calculation. For a chosen set of replaced resources, only gates involving their endpoint parties are expanded. Bounded lifetime participation controls the coefficient cost. Gaussian second-moment estimates then bound the trace-norm error independently of private dimensions and the original resource Schmidt ranks. The result is an operator tensor network $\sigma$ with polynomial links and

$$\Vert \sigma -P_{\Omega }{\Vert }_1\le C{L}^{-5}.$$

The sampled operator need not be positive or Hermitian. Nevertheless, one product-basis column gives a vector approximation. This useful linear-algebra step has an elementary proof. Set $E=\sigma -P_{\Omega }$, $\Vert E{\Vert }_1\le \eta <1$, and $a_z=⟨\Omega |z⟩$. Then

$$\begin{array}{c}\sum _z\Vert E|z⟩{\Vert }^2\le {\eta }^2,\\ \sum _z|a_z{|}^2=1.\end{array}$$

Some $z$ with $a_z\ne 0$ satisfies $\Vert E|z⟩\Vert \le \eta |a_z|$. Therefore $w=\sigma |z⟩/a_z$ has $\Vert w-\Omega \Vert \le \eta$ and is nonzero. Normalization costs at most another $\eta$. Fixing this product-basis column in the operator network preserves every virtual dimension, so it yields the required ket network without a lower bound on $|a_z|$. (OpenAI, 2026b)

Finally, links between nearby parties at equal or adjacent dyadic scales are routed horizontally and then vertically through their square anchors. Nonunit scale-$2^j$ anchors lie on rows or columns with coordinate $2^{j-1}(2s+1)$, which identifies the scale uniquely. Each grid edge therefore carries only a constant number of routed links. Padding to a power-of-two square and reflecting back increases congestion by at most a constant factor and fixes the original physical sites.

If each link has dimension at most $C_1{L}^r$ and at most $\chi$ links traverse any grid edge, combining their indices gives bond dimension at most $(C_1{L}^r{)}^{\chi }$. Constant congestion is what preserves polynomial growth. For sufficiently large $L$, the normalized error $C{L}^{-5}$ is below $L^{-1}$; finitely many smaller sizes are absorbed into the constant using exact finite representations. (OpenAI, 2026b)

🏷️ Key steps and proof outlines

Two scanner probabilities control different errors

The scanner needs both a low probability of creating a split interaction and a definite probability of repairing one. These are different estimates. At scale $n$, the source uses an interaction radius $r_0$, a lookahead depth $D$, and an offset window $m$, with

$$r_0\ll D\ll m.$$

The nominal front follows a fixed list of site slots. Moving an entire interaction support early does not change that list: its later fill slot is simply consumed without moving anything.

Fix one interaction. Because the initial offset is uniform among $m$ choices, only $O(D)$ offsets put its support in the region where a split can occur. Thus its splitting probability is $O(D/m)$. On a good history, the list of eligible interaction anchors contains at most $CnD$ slots and includes every currently split term. Choosing a side and then an anchor therefore repairs each such term with probability at least $c/(nD)$.

The remaining issue is a chain of charges that runs far ahead of the nominal front. The source bounds these exceptional histories by any prescribed inverse power of $n$, after choosing the scales. Away from them, the two estimates above control how much interaction energy can accumulate while the metric is transported. Crucially, these are probabilities on the original finite history tree, before transporting the quantum state. Reweighting them by an evolving state without proof would invalidate the argument. The common interpolation across this tree is what lets the energy comparison and entropy comparison refer to the same vectors (OpenAI, 2026a).

Replacing an entangled measurement by prepared resources

The PEPS compression theorem allows a gate to consume a normalized entangled bra $⟨\eta |$. Its sampling argument instead needs prepared pair vectors. An elementary averaging device bridges the two descriptions without a factor depending on the Schmidt rank of $\eta$.

Append $m-1$ copies of $\eta$ to an input pair register and average the $m$ cyclic permutations of all pair registers. Call the resulting contraction $T_m$. Each permutation is a product of local permutations at the two parties, and

$$T_m\eta ={\eta }^{\otimes m}.$$

For $v\perp \eta$, the $m$ vectors obtained by inserting $v$ in different positions are orthogonal. Their average consequently has norm $\Vert v\Vert /\sqrt{m}$. Therefore

$$\Vert T_m-|\eta {⟩}^{\otimes m}⟨\eta |\Vert \le m^{-1/2}.$$

This operator-norm estimate also holds with arbitrary spectator registers.

For a gate expanded as ${\sum }_{\xi }{c}_{\xi }{M}_{\xi }$, with at most $r$ such effects per monomial, telescoping gives total replacement error at most

$$\frac{r}{\sqrt{m}}\sum _{\xi }|c_{\xi }|.$$

Every branch must have the same garbage output: unused occurrences receive their full fixed stack of $\eta$‘s. Otherwise the branchwise approximations would not combine into an approximation of the original coherent sum. Expanding the cyclic averages increases the number of terms by at most $m^r$, but does not increase their absolute coefficient sum (OpenAI, 2026b).

Why compression costs a polynomial per error location

After this replacement, the density calculation samples prepared pair operators. Expand its error according to the set of positions at which a centered sampling correction is used. A term with $s$ corrections touches at most $2s$ endpoint parties. If each party participates in at most $b$ nonprivate gates during its lifetime, only $2bs$ gates need to be expanded; all other gates remain contractions.

This is the operational reason for the lifetime participation condition. The coefficient cost is a fixed polynomial per corrected position, while the source’s second-moment estimate contributes $k^{-s/2}$ when each source is sampled $k$ times. Choosing polynomial $k$ sufficiently large lets the sum over subsets of error locations converge to the desired inverse-polynomial bound. Expanding every gate in advance would lose this control. The resulting operator network can then enter the column-extraction and constant-congestion routing arguments proved above (OpenAI, 2026b).

🏷️ Connections and proof scope

• Coulomb ionization also concerns the full interacting ground state. Its continuum fermionic model and its energy/radius conclusions differ from the finite-dimensional lattice gap studied here.
• Binary information contraction studies classical mutual information and data processing. Here reduced density matrices replace probability distributions, and strong subadditivity supports the entropy cancellation.

These notes verify the displayed elementary deductions and audit selected statements and proof interfaces. The replica interpolation, scanner comparisons, amplification, recursive geometry, and density-compression estimates are attributed to the manuscripts; their complete proofs are not independently verified here.

🏷️ Manuscripts and verification scope

• A two-dimensional area law from a global spectral gap — pinned manuscript and companion source.
• Polynomial PEPS approximation of gapped square-grid ground states — pinned manuscript and companion source.

No family scope page was present at lean/docs/265.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A two-dimensional area law from a global spectral gap.

🐻  OpenAI 2026b. Polynomial PEPS approximation of gapped square-grid ground states.