---
myst:
html_meta:
description: |
Hypothesis and methodology for testing whether the causal order
defined by majorization on Schmidt spectra of a real-time-evolved
lattice Schwinger-model state agrees with the Lieb-Robinson light
cone and with a prescribed causet structure.
---
# Emergent Causal Order from Entanglement: Hypothesis and Methodology
```{contents}
:local:
:depth: 2
```
This page is the scientific charter for the [Schwinger / DMRG / TDVP
subsystem](quantum.md). The implementation tracker is in
[quantum-plan.md](quantum-plan.md); user-facing API and quickstart
material is in [quantum.md](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 {cite}`VanRaamsdonk2010Building, VanRaamsdonk2018BCbit`,
and operationalizes the comparison through tensor-network simulation
{cite}`Schollwock2011MPS, Banuls2013Schwinger`.
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
```{math}
:label: maj-def
(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 $\sum_i \lambda_{A,i}(t) = 1$
{cite}`Nielsen1999LOCC, MarshallOlkinArnold2011`.
Let $\preceq_{\mathrm{LR}}$ denote the partial order induced by the
Lieb–Robinson light cone of the lattice Hamiltonian
{cite}`LiebRobinson1972, HastingsKoma2006`, and let $\preceq_{\mathrm{cs}}$
denote the partial order inherited from a causet $\mathcal{C}$ on which
$\Lambda$ is embedded {cite}`Ambjorn2004CDT, Regge1961`.
**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:
```{math}
:label: hypothesis
\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 {cite}`Nielsen1999LOCC`. 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 {cite}`KogutSusskind1975, BanksKogutSusskind1976`.
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
{cite}`Banuls2013Schwinger, ZoharCiracReznik2016`.
- *Trotter and bond-dimension errors.* TDVP integration introduces a finite
Trotter error and bond-dimension truncation
{cite}`Haegeman2016TDVP`. 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
{cite}`KogutSusskind1975, BanksKogutSusskind1976, Banuls2013Schwinger`,
the Hamiltonian becomes a long-ranged spin Hamiltonian on $N$ qubits:
```{math}
:label: ham
H \;=\; H_{\mathrm{hop}} + H_m + H_E,
```
with
```{math}
:label: ham-parts
\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}
```
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](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:
```{math}
\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](quantum-plan.md) for the current
build / acceptance status of each phase.
```{list-table}
:header-rows: 1
:widths: 18 22 35 25
* - Phase
- Object
- Operation
- Reference
* - 1
- Schwinger MPO
- AutoMPO assembly with $U(1)$ charge conservation
- {cite}`Banuls2013Schwinger`
* - 2
- Ground state $\ket{\Omega}$
- Two-site DMRG sweeps to convergence
- {cite}`White1992DMRG, Schollwock2011MPS`
* - 3
- Initial quenched state
- Local creation of $q\bar q$ pair at separation $d$ with Gauss-compatible electric string
- {cite}`Buyens2014StringBreaking, Pichler2016RealTime`
* - 4
- Time evolution
- Two-site TDVP with adaptive bond dimension; one-site TDVP after saturation
- {cite}`Haegeman2016TDVP`
* - 5
- Per-step observables
- $\langle L_n \rangle_t$, $\langle Z_n \rangle_t$, contiguous-cut Schmidt spectra
- {cite}`Buyens2014StringBreaking`
* - 6
- Majorization poset
- Pairwise test on padded sorted spectra; transitive reduction to Hasse form
- {cite}`Nielsen1999LOCC, MarshallOlkinArnold2011, Bhatia1997MatrixAnalysis`
* - 7
- Lieb–Robinson order
- $v_{\mathrm{LR}}$ from OTOC front; cone-induced partial order on labels
- {cite}`LiebRobinson1972, HastingsKoma2006`
* - 8
- Causet order (optional)
- Inherited from tessera adjacency on a non-regular causet embedding
- {cite}`Ambjorn2004CDT, Regge1961`
* - 9
- Comparison
- Kendall $\tau$, discordant-pair fraction, Hasse graph edit distance
-
```
### 4.2 Initial state construction
The $q\bar q$ creation operator is
```{math}
:label: qqbar
\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
{cite}`Buyens2014StringBreaking`. 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
{cite}`Banuls2013Schwinger` 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 {cite}`Buyens2014StringBreaking`. 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.
## References
```{bibliography}
:style: unsrt
```