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Multipartite Non-local Magic and SYK Model

Vinay Malvimat, Matthieu Sarkis, Yena Suk, Junggi Yoon

Abstract

We investigate the structure of quantum magic in interacting disordered fermionic systems, quantifying non-stabilizerness via the fermionic stabilizer Rényi entropy (SRE). To resolve the distribution of magic across different scales, we introduce a multipartite non-local magic functional, constructed from an inclusion-exclusion combination of subsystem contributions. This measure serves as a fine-grained diagnostic, isolating genuinely global contributions and revealing nontrivial interactions between local and collective supports of magic. We illustrate the measure on paradigmatic multipartite states and apply these diagnostics to the Sachdev-Ye-Kitaev model and its variants. Crucially, for thermal/typical ensembles, we observe a marked disparity between Thermal Pure Quantum (TPQ) states and the thermal density matrix. This reveals a concealed complexity: the immense computational hardness characterizing the unitary evolution is encoded in the specific microstructure of the black hole microstates, while being washed out in the coarse-grained thermodynamic description. Furthermore, in $\mathcal N=2$ supersymmetric SYK, we show that while fortuitous BPS states exhibit intermediate stabilizer complexity, the multipartite measure unveils a rich, sector-dependent pattern of global correlations, distinguishing them from generic chaotic states.

Multipartite Non-local Magic and SYK Model

Abstract

We investigate the structure of quantum magic in interacting disordered fermionic systems, quantifying non-stabilizerness via the fermionic stabilizer Rényi entropy (SRE). To resolve the distribution of magic across different scales, we introduce a multipartite non-local magic functional, constructed from an inclusion-exclusion combination of subsystem contributions. This measure serves as a fine-grained diagnostic, isolating genuinely global contributions and revealing nontrivial interactions between local and collective supports of magic. We illustrate the measure on paradigmatic multipartite states and apply these diagnostics to the Sachdev-Ye-Kitaev model and its variants. Crucially, for thermal/typical ensembles, we observe a marked disparity between Thermal Pure Quantum (TPQ) states and the thermal density matrix. This reveals a concealed complexity: the immense computational hardness characterizing the unitary evolution is encoded in the specific microstructure of the black hole microstates, while being washed out in the coarse-grained thermodynamic description. Furthermore, in supersymmetric SYK, we show that while fortuitous BPS states exhibit intermediate stabilizer complexity, the multipartite measure unveils a rich, sector-dependent pattern of global correlations, distinguishing them from generic chaotic states.
Paper Structure (27 sections, 6 theorems, 74 equations, 24 figures, 1 table)

This paper contains 27 sections, 6 theorems, 74 equations, 24 figures, 1 table.

Table of Contents

  1. Introduction
  2. Non-Stabilizerness and Stabilizer Renyi Entropy
  3. Non-stabilizerness in finite Majorana systems and the stabilizer Renyi entropy
  4. Clifford Majorana group and Majorana stabilizer states
  5. Majorana spectrum and the stabilizer Rényi entropy
  6. Multipartite Non-Local Magic
  7. Definition and Properties
  8. Examples
  9. GHZ:
  10. W state:
  11. Comments on negativity
  12. The SYK model
  13. Time Evolution: SRE and multipartite non-local SRE
  14. Finite Temperature
  15. Probability Distribution
  16. ...and 12 more sections

Key Result

Proposition 1

$\mathrm M$ satisfies the following properties:

Figures (24)

  • Figure 1: Non-local SRE for a generalized GHZ state
  • Figure 2: Time evolution of the SRE for $N=14$ in the SYK$_4$ model.
  • Figure 3: Stabilizer Rényi entropy as a function of temperature, restricted by the length of Majorana strings. The corresponding values of length of the string are indicated on the right.
  • Figure 4: Stabilizer Rényi entropy as a function of temperature, restricted by the support of Pauli strings on the number of qubits indicated on the right.
  • Figure 5: For $N=14$ ($n=7$ qubits), Multipartite non-local Stabilizer Rényi entropy as a function of inverse temperature. The list on right indicates the number of qubits considered for the computation of multipartite SRE.
  • ...and 19 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2: Tripartite identity
  • proof
  • Proposition 3: States with vanishing local magic
  • proof
  • Proposition 4: Product across a nontrivial partition, general $n$
  • proof
  • Proposition 5: Local free-unitary invariance
  • proof
  • Proposition 6: Additivity over independent $n$-partite states
  • ...and 1 more