Emergent Causal Order from Entanglement: Hypothesis and Methodology

This page is the scientific charter for the Schwinger / DMRG / TDVP subsystem. The implementation tracker is in quantum-plan.md; user-facing API and quickstart material is in quantum.md. All three should be read together — this page states what the project is trying to learn; the plan tracks what is currently built; the API page documents how to drive the bits that are built.

Abstract

We investigate whether the partial order induced by Nielsen majorization on reduced-state Schmidt spectra of a real-time-evolved lattice gauge theory agrees with two reference causal structures: the Lieb–Robinson cone of the underlying spin Hamiltonian, and a prescribed causet (causal set) geometry on which the lattice is embedded. The lattice Schwinger model serves as the test system. The framework treats spacetime causal order as a derived quantity of the entanglement structure of a many-body state on a discrete substrate, in the spirit of the entanglement-builds-spacetime program [1, 2], and operationalizes the comparison through tensor-network simulation [3, 4].

The work is exploratory. The hypothesis is falsifiable, but the 1+1-dimensional test bed is the simplest setting in which the question can be posed numerically; agreement between the three orders in this regime would not by itself establish a deeper principle.

1. Hypothesis

Let \(\ket{\psi(t)}\) be the state of a quantum system on a discrete lattice \(\Lambda\) at time \(t\). Fix a family \(\mathcal{F} = \{A\}\) of subsystems (“cuts”) closed under containment. For each \(A \in \mathcal{F}\) let \(\lambda_A(t)\) denote the Schmidt spectrum of \(\ket{\psi(t)}\) across the bipartition \(A \mid \bar A\), sorted nonincreasingly and zero-padded to a common length.

Define the majorization order \(\preceq_{\mathrm{maj}}\) on labels \(\{(A, t)\}\) by

(1)\[(A, s) \preceq_{\mathrm{maj}} (B, t) \;\;\iff\;\; \sum_{i=1}^{k} \lambda_{A,i}^\downarrow(s) \;\le\; \sum_{i=1}^{k} \lambda_{B,i}^\downarrow(t) \quad \forall\, k \ge 1,\]

with the standard normalization \(\sum_i \lambda_{A,i}(t) = 1\) [5, 6].

Let \(\preceq_{\mathrm{LR}}\) denote the partial order induced by the Lieb–Robinson light cone of the lattice Hamiltonian [7, 8], and let \(\preceq_{\mathrm{cs}}\) denote the partial order inherited from a causet \(\mathcal{C}\) on which \(\Lambda\) is embedded [9, 10].

Hypothesis (H). For physical states arising from local-Hamiltonian real-time evolution of a gauge-invariant ground state with a localized charge-anticharge excitation, the three partial orders coincide on the restricted label set obtained by taking \(\mathcal{F}\) to consist of contiguous spatial intervals at sampled times:

(2)\[\preceq_{\mathrm{maj}}\big|_{\mathcal{F}\times T} \;=\; \preceq_{\mathrm{LR}}\big|_{\mathcal{F}\times T} \;=\; \preceq_{\mathrm{cs}}\big|_{\mathcal{F}\times T} \quad \text{up to an entanglement velocity } v_E \le v_{\mathrm{LR}}.\]

Falsification criteria.

1. Strong falsification. If \(\preceq_{\mathrm{maj}}\) contains pairs whose spatial separation exceeds the Lieb–Robinson cone at the corresponding time difference, the hypothesis is wrong: majorization sees a superluminal-in-the-lattice-sense order.

2. Weak falsification. If \(\preceq_{\mathrm{maj}}\) disagrees with \(\preceq_{\mathrm{cs}}\) on a non-vanishing fraction of comparable pairs for non-trivial causet topologies (Phase 5), the prescribed causet geometry does not control the emergent order.

3. Trivial agreement. If \(\preceq_{\mathrm{cs}}\) is forced to coincide with \(\preceq_{\mathrm{LR}}\) by the regularity of the lattice (1+1D total order on time slices), agreement is uninformative; the test must be repeated on causets whose induced order is strictly coarser than the trivial chain.

2. Limitations and scope conditions

The following limitations bound the interpretive weight of the result.

• Pure-state, contiguous-cut regime. Nielsen’s theorem is stated for pure bipartite LOCC convertibility [5]. Extensions to mixed states or non-contiguous cut families weaken or break the well-definedness of the \(\preceq_{\mathrm{maj}}\) relation between regions. We restrict to pure states of the full chain and to contiguous intervals throughout.

• Abelian gauge group, 1+1D. The Schwinger model is U(1) with the gauge field eliminable in 1D [11, 12]. Generalization to SU(N) or to higher dimensions is not addressed and is likely to require qualitatively different methods.

• Truncated electric basis. The link Hilbert space is truncated at electric-field cutoff \(\Lambda\); convergence in \(\Lambda\) is a numerical question requiring separate convergence studies [4, 13].

• Trotter and bond-dimension errors. TDVP integration introduces a finite Trotter error and bond-dimension truncation [14]. Both are controllable but must be characterized on each run before attaching meaning to a poset edge.

• No claim of fundamentality. The construction does not derive spacetime from entanglement in any constructive sense; it tests an internal consistency between three independently defined partial orders on a finite system.

3. System and notation

3.1 Lattice Schwinger model

Open boundary conditions, \(N\) staggered sites with lattice spacing \(a\), bare fermion mass \(m\), coupling \(g\), and background field \(L_0\). After a Jordan–Wigner transformation and Gauss-law elimination of the gauge links [4, 11, 12], the Hamiltonian becomes a long-ranged spin Hamiltonian on \(N\) qubits:

(3)\[H \;=\; H_{\mathrm{hop}} + H_m + H_E,\]

with

(4)\[\begin{split}\begin{aligned} H_{\mathrm{hop}} &= \tfrac{1}{4a}\sum_{n=1}^{N-1}\bigl(X_n X_{n+1} + Y_n Y_{n+1}\bigr), \\ H_m &= \tfrac{m}{2}\sum_{n=1}^{N}(-1)^n Z_n, \\ H_E &= \tfrac{g^2 a}{2}\sum_{n=1}^{N-1} L_n^2, \quad L_n = L_0 + \sum_{k=1}^{n}\!\left[\tfrac{1-Z_k}{2} - \tfrac{1-(-1)^k}{2}\right]. \end{aligned}\end{split}\]

Expanding \(L_n^2\) produces a Hamiltonian with \(\mathcal{O}(N^2)\) long-range \(Z_m Z_{m'}\) terms; the resulting MPO bond dimension grows accordingly and is handled with the AutoMPO machinery described below.

This is the same Hamiltonian implemented by tessera.quantum; see quantum.md for the API surface and convention details, and include/quantum/schwinger_model.hpp for the algebraic expansion of \(L_n^2\) into operator and c-number pieces.

3.2 Cut family and Schmidt spectra

The cut family is fixed throughout:

\[\mathcal{F} \;=\; \bigl\{\, [i,j] : 1 \le i \le j \le N \,\bigr\},\]

i.e. all contiguous spatial intervals. For each \(A = [i,j]\) at time \(t\), \(\lambda_A(t)\) is read from the bond singular values of the MPS in left-canonical form at the appropriate cut, with no additional truncation beyond the variational bond dimension.

3.3 Reference causal orders

\(\preceq_{\mathrm{LR}}\) is constructed from the Lieb–Robinson velocity \(v_{\mathrm{LR}}\) extracted from out-of-time-ordered commutator front-propagation in the same simulation. \(\preceq_{\mathrm{cs}}\), when present, is read directly from the causet adjacency provided by the host package; in Phase 4 the chain is regular and the causet order is trivial, so \(\preceq_{\mathrm{cs}}\) becomes informative only in Phase 5.

4. Methodology

4.1 Numerical pipeline

The simulation is structured as a single C++ pipeline using ITensor v3 for all tensor-network operations; Python serves only as a result viewer. The phases below correspond to the implementation plan accompanying this document — see quantum-plan.md for the current build / acceptance status of each phase.

Phase	Object	Operation	Reference
1	Schwinger MPO	AutoMPO assembly with \(U(1)\) charge conservation	[4]
2	Ground state \(\ket{\Omega}\)	Two-site DMRG sweeps to convergence	[3, 15]
3	Initial quenched state	Local creation of \(q\bar q\) pair at separation \(d\) with Gauss-compatible electric string	[16, 17]
4	Time evolution	Two-site TDVP with adaptive bond dimension; one-site TDVP after saturation	[14]
5	Per-step observables	\(\langle L_n \rangle_t\), \(\langle Z_n \rangle_t\), contiguous-cut Schmidt spectra	[16]
6	Majorization poset	Pairwise test on padded sorted spectra; transitive reduction to Hasse form	[5, 6, 18]
7	Lieb–Robinson order	\(v_{\mathrm{LR}}\) from OTOC front; cone-induced partial order on labels	[7, 8]
8	Causet order (optional)	Inherited from tessera adjacency on a non-regular causet embedding	[9, 10]
9	Comparison	Kendall \(\tau\), discordant-pair fraction, Hasse graph edit distance

4.2 Initial state construction

The \(q\bar q\) creation operator is

(5)\[\ket{q\bar q;\, i_0,d} \;=\; \mathcal{N}^{-1}\, \sigma^+_{i_0}\, \Bigl[\textstyle\prod_{n=i_0}^{i_0 + d - 1} U_n\Bigr]\, \sigma^-_{i_0 + d}\, \ket{\Omega},\]

where \(U_n\) is the gauge link in the original lattice formulation and \(\mathcal{N}\) is the norm; in the Gauss-eliminated representation this reduces to the appropriate shift in the cumulative electric-field variables [16]. Skipping the string yields a Gauss-violating state and is a known failure mode.

4.3 Acceptance and validation

Two external benchmarks gate the analysis:

1. Spectral benchmark. Ground-state energy per site for \(N=20\), \(m/g \in \{0,\, 0.125,\, 0.25\}\) must agree with [4] to within \(10^{-3}\).

2. String benchmark. In the heavy-quark regime \(m/g \gg 1\) at \(d=4\), \(\langle L_n \rangle_t\) should exhibit a flat plateau of value \(\pm 1\) between \(i_0\) and \(i_0 + d\) at intermediate times, consistent with [16]. Energy conservation must hold to relative tolerance \(10^{-3}\) over the full evolution.

Failure of either benchmark invalidates the upstream construction and must be addressed before any conclusion is drawn from the poset comparison.

4.4 Comparison statistics

For each pair of labels \(((A,s),(B,t))\), the three orders return a value in \(\{\prec,\, \succ,\, =,\, \mathrm{incomparable}\}\). We report:

• Kendall \(\tau\) between the linear extensions of each pair of orders (per time slice and pooled);

• Discordant fraction — pairs comparable in one order and not the other, or oppositely ordered;

• Hasse graph edit distance between the transitive reductions, normalized by edge count.

Statistical uncertainty is estimated by bootstrap over Trotter step sequences and over initial \(i_0\) within a translation-invariant subregion.

4.5 Threats to validity

• Spurious agreement from regularity. On a translationally invariant lattice the three orders may coincide for reasons unrelated to the hypothesis. Phase 5 with non-trivial causet topologies is required to distinguish.

• Truncation-induced order violations. Aggressive bond-dimension truncation can perturb Schmidt spectra enough to flip a borderline majorization comparison. Comparisons within \(\epsilon = 10^{-6}\) of the decision boundary are flagged and excluded.

• Quantum-number sectoring. All three orders are reported globally; a refinement that majorizes within fixed \(U(1)\) charge sectors may be more physically meaningful and is deferred.

5. Deliverables and reproducibility

Each run produces (i) the converged ground-state energy and bond-dimension profile, (ii) per-step electric-field and charge profiles, (iii) the full time series of contiguous-cut Schmidt spectra, (iv) the three Hasse diagrams, and (v) the comparison statistics with bootstrap intervals. Random seeds, Trotter schedule, bond-dimension caps, and cutoff \(\Lambda\) are recorded with each output for reproducibility.

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