Inaccessible entanglement in symmetry protected topological phases
Bibliographic record
Abstract
Abstract We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group G , which we call G -LOCC, and we define the inaccessible entanglement E inacc as the entanglement that cannot be used for distillation under G -LOCC. We derive a tight bound on E inacc which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>log</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>⩽</mml:mo> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⩽</mml:mo> <mml:mi>log</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , where D ω is the topologically protected edge mode degeneracy of the SPT phase ω with symmetry G . For particular phases such as the Haldane phase, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:msqrt> </mml:math> so the bound becomes an equality. We numerically investigate the distribution of states throughout the bound, and show that typically the region near the upper bound is highly populated, and also determine the nature of those states lying on the upper and lower bounds. We then discuss the relation of E inacc to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".