Maldacena-Qi Coupling & Traversable Wormholes

• Maldacena-Qi coupling is defined as a left–right interaction in dual quantum systems (e.g., SYK copies) that energetically favors a thermofield double ground state.
• It employs a bilinear term (or Hermitian hopping) and a shared Schwarzian low-energy sector to analytically bridge microscopic correlations with macroscopic wormhole traversability.
• The coupling produces distinct transport signatures and a first-order phase transition between wormhole-like and black-hole-like regimes, linking quantum entanglement with geometric connectivity.

Searching arXiv for Maldacena-Qi coupling and closely related wormhole/SYK papers. arxiv_search(query="Maldacena Qi coupling SYK eternal traversable wormhole", max_results=10, sort_by="relevance") Searching arXiv for the original Maldacena-Qi paper and subsequent analyses. arxiv_search(query="(Maldacena et al., 2018) Eternal traversable wormhole Maldacena Qi", max_results=10, sort_by="relevance") The Maldacena–Qi coupling is a left–right interaction introduced for two identical quantum systems, most prominently two copies of the Sachdev–Ye–Kitaev model, with the purpose of driving the coupled ground state toward a thermofield double state of the uncoupled theory and, in the nearly-$AdS_2$/Jackiw–Teitelboim description, supporting an eternal traversable wormhole. In its canonical SYK form, the coupling is the bilinear term

$H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$

added to matched left and right Hamiltonians, while closely related complex-fermion realizations use a Hermitian hopping term between corresponding modes. Across the literature, the coupling is used to relate microscopic left–right pairing, gapped low-energy dynamics, traversability, and the distinction between connected wormhole and disconnected black-hole phases (Maldacena et al., 2018, Alet et al., 2020, Zhou et al., 2020).

1. Canonical definition and boundary realizations

In the original coupled-SYK construction, two copies of SYK with matched microscopic couplings are coupled by

$$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$

This is the canonical Maldacena–Qi interaction in the Majorana formulation. A closely related version considered in tunneling spectroscopy uses complex fermions and writes the coupled Hamiltonian as

$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$

so that the left–right term is an explicit Hermitian hopping operator between corresponding fermionic modes (Maldacena et al., 2018, Zhou et al., 2020).

The broader two-copy perspective emphasizes the same structural feature: one begins with identical left and right Hamiltonians and adds a special interaction that energetically favors left–right pairing. For the SYK example this is written as

$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$

with identical disorder realization on the two sides. The same design principle is then extended to spins, fermions, and bosons by engineering the interaction so that the paired left–right state is annihilated by the coupling and therefore becomes the ground state when the coupling dominates (Alet et al., 2020).

On the gravity side, the corresponding interaction is a double-trace coupling between the two asymptotic boundaries,

$$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$

The boundary and bulk descriptions share the same low-energy sector, so the Maldacena–Qi coupling is simultaneously a microscopic left–right interaction in SYK and a boundary-to-boundary coupling in nearly-$AdS_2$ gravity (Maldacena et al., 2018).

2. Thermofield-double structure and effective low-energy theory

A central property of the coupling is that it drives the coupled ground state toward a thermofield double state of the uncoupled Hamiltonian. In the coupled SYK model, the ground state becomes the maximally entangled infinite-temperature thermofield double as $\mu\to\infty$, while for finite $\mu$ it remains very close to a thermofield double at an effective inverse temperature $\beta(\mu)$, chosen to maximize

$H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$0

The numerical study reported for $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$1 and $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$2 pushes to $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$3 Majoranas and $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$4, respectively, and finds that the overlap is exactly $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$5 in the limits $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$6 and $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$7, dips at intermediate coupling, and nevertheless remains extremely high: at the largest sizes, the deviation is about $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$8 for $H_{\rm int}=i\mu \sum_j \psi_L^j\psi_R^j ,$9 and about $$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$0 for $$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$1 (Alet et al., 2020).

In the large-$$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$2 limit, the coupled SYK ground state is described exactly by a thermofield double at a $$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$3-dependent temperature, and the overlap saturates to $$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$4. The quoted analytic relation is

$$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$5

with

$$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$6

The numerical $$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$7 agrees well with this formula once $$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$8 is not too small, and for $$H_{\rm total}=H_{\rm L,SYK}+H_{\rm R,SYK}+H_{\rm int}, \qquad H_{\rm int}= i\mu \sum_j \psi_L^j \psi_R^j .$$9 a $$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$0 extrapolation aligns well with the large-$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$1 prediction; the main disagreement at small $$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$2 comes from finite-size effects and enhanced sample-to-sample fluctuations (Alet et al., 2020).

The effective low-energy theory of the coupled system is formulated in terms of the shared Schwarzian sector. Both the nearly-$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$3 gravity system and the coupled SYK model reduce to

$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$4

A symmetric classical solution takes

$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$5

and the $$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$6 constraint gives

$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$7

This fixes the wormhole scale and the gap,

$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$8

while the ground-state energy is

$$\hat H_{\text{MQ}} = \sum_{i<j;k<l}J_{ij,kl}\Big( \hat c^\dagger_{L,i}\hat c^\dagger_{L,j}\hat c_{L,k}\hat c_{L,l} + \hat c^\dagger_{R,i}\hat c^\dagger_{R,j}\hat c_{R,k}\hat c_{R,l} \Big) + \mu\sum_i\Big( \hat c^\dagger_{L,i}\hat c_{R,i} + \hat c^\dagger_{R,i}\hat c_{L,i} \Big),$$9

In this form, the Maldacena–Qi coupling appears as an interaction-induced potential in the universal low-energy sector (Maldacena et al., 2018).

3. Traversability and the nearly-$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$0 wormhole

The gravitational role of the Maldacena–Qi coupling is to generate the negative null energy required for an eternal traversable wormhole. In the nearly-$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$1 analysis, traversability requires violating the averaged null energy condition. The relevant JT gravity equation is

$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$2

which integrates to

$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$3

For a wormhole whose dilaton grows toward both boundaries, the left-hand side is negative, so the integrated null energy must also be negative. The boundary coupling $$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$4 produces this effect: correlated quantum fields generate a negative expectation value of the null energy in the bulk, and the resulting stress tensor deforms the nearly-$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$5 geometry into one with two boundaries that remain causally connected (Maldacena et al., 2018).

In this description, the coupling is not merely a bookkeeping device for correlations. It is the mechanism that stabilizes a connected throat and makes the wormhole traversable and eternal. The same paper describes the interaction as a direct boundary-to-boundary tunneling term that lowers the energy when the two systems are strongly correlated, generates a gap, and drives the ground state close to a thermofield double. The time for a signal to cross is set by the emergent scale $$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$6, and left–right commutators vanish before the signal has had time to traverse the wormhole and become nonzero afterward. That operational criterion identifies traversability in the coupled system (Maldacena et al., 2018).

A useful contrast is provided by the broader ER=EPR literature. “Electric fields and quantum wormholes” studies entangled charged matter in two disconnected boxes and defines a non-perturbative wormhole susceptibility

$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$7

showing that entanglement plus Gauss-law charge correlations can mimic the electric response of a wormhole. However, that work explicitly does not study the Maldacena–Qi coupled-SYK model or its specific coupling mechanism: there is no explicit double-trace coupling, no traversable-wormhole protocol of the Gao–Jafferis–Wall or Maldacena–Qi type, and no SYK-like interaction between two Hamiltonians (Engelhardt et al., 2015). The distinction matters because the Maldacena–Qi construction is a specific inter-copy coupling, not merely any entangled two-sided state.

4. Phase structure, gap formation, and thermodynamics

At low temperature, the coupled model is in a connected wormhole-like phase. In gravity this is a global-$$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$8-like geometry with two boundaries connected by a throat; in SYK it is a gapped ground state $$\hat{H}_{\rm coupled} = \hat{H}_{\rm L}\otimes 1 + 1\otimes \hat{H}_{\rm R} + \hat{H}_{\rm int}, \qquad \hat{H}_{\rm int} = i\mu \sum_{j=1}^{N/2}\hat{\chi}_{\rm L}^j \hat{\chi}_{\rm R}^j,$$9 that is very close to a thermofield double at an effective inverse temperature $$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$0. At sufficiently high temperature, the system instead behaves like two disconnected black holes, one on each side, with free energy approximately that of two decoupled SYK systems and entropy

$$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$1

plus thermal corrections (Maldacena et al., 2018).

The canonical ensemble exhibits a first-order Hawking–Page-like transition between these regimes. At low temperature,

$$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$2

whereas at high temperature

$$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$3

Equating the two contributions gives the estimate

$$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$4

In the microcanonical ensemble, by contrast, the two phases are continuously connected, and the entropy $$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$5 is continuous; the large-$$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$6 analysis shows a smooth interpolation between the wormhole-like and black-hole-like regimes (Maldacena et al., 2018).

The finite-temperature behavior described in transport language is consistent with this phase structure. In the tunneling-spectroscopy analysis of the complex-fermion realization, when $$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$7 the system is in the wormhole phase and the two SYK copies remain strongly correlated; when $$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$8, there is a first-order transition into a black-hole phase where the two sides become effectively disconnected, corresponding to two separate Rindler-like geometries (Zhou et al., 2020).

A broader generalization appears in coupled gauged matrix models and vector models. There the left–right coupling can still create a thermofield-double-like ground state, but the entanglement is tied to whether the color degrees of freedom are confined. Below the deconfinement temperature, the system remains in a low-energy confined phase and the ground state is thermofield-double-like; as temperature or energy is raised, a subset of color degrees of freedom deconfines and the entanglement between the two copies is reduced correspondingly. This leads to the argument that deconfinement is associated with loss of entanglement and, in the dual gravity picture, with the disappearance of the wormhole (Alet et al., 2020).

5. Spectral, transport, and bulk signatures

The coupling produces distinctive dynamical signatures in tunneling spectroscopy. In the transport setup, each SYK side is attached to a lead, a voltage $$S_{\rm int}= g\sum_{i=1}^N \int du\, O_L^i(u)\,O_R^i(u).$$9 is applied to the left lead, and the current into the right lead is measured. The current operator is

$AdS_2$0

and the resulting expectation value is

$AdS_2$1

The tunneling probability is

$AdS_2$2

so left–right transmission is directly controlled by the retarded MQ Green’s functions (Zhou et al., 2020).

In the wormhole phase, the low-energy theory is governed by the nearly-$AdS_2$3/JT limit, and the retarded Green’s functions have poles at

$AdS_2$4

These poles are identified as an $AdS_2$5-protected tower. The conformal tunneling probability exhibits sharp peaks near the discrete wormhole resonances for small $AdS_2$6, and the paper emphasizes that the resulting oscillatory conductance is an unambiguous transport signature of the eternal traversable wormhole and of the underlying $AdS_2$7 symmetry. As $AdS_2$8 increases, the peaks broaden, and at large $AdS_2$9 the dominant low-energy peaks shift to

$\mu\to\infty$0

which is interpreted as a crossover to behavior controlled by an operator of scaling dimension $\mu\to\infty$1 (Zhou et al., 2020).

In the black-hole phase, the same coupling is much less effective in producing left–right transport. For $\mu\to\infty$2 and small $\mu\to\infty$3,

$\mu\to\infty$4

and the tunneling probability is suppressed and scales as $\mu\to\infty$5. The conductance then has no oscillatory $\mu\to\infty$6-type pattern and instead shows a single peak near $\mu\to\infty$7 (Zhou et al., 2020).

The same work also gives a bulk derivation. In the wormhole geometry, a bulk Dirac fermion propagates on global $\mu\to\infty$8 with

$\mu\to\infty$9

and the resulting bulk transmission probability matches the boundary answer after the stated identifications. In the black-hole phase, the bulk action is a sum over two Rindler wedges, and the two boundaries are coupled by a boundary term

$\mu$0

with $\mu$1, which is the gravity-dual version of the Maldacena–Qi coupling (Zhou et al., 2020).

6. Generalizations, confinement, and conceptual boundaries

Although the phrase “Maldacena–Qi coupling” usually refers to the SYK bilinear $\mu$2, the underlying strategy is more general. The two-copy framework of “Entanglement and Confinement in Coupled Quantum Systems” shows that the detailed nature of the uncoupled system matters much less than the fact that the interaction energetically rewards left–right pairing. That is why the same qualitative behavior appears for spin chains, free fermions, and harmonic oscillators (Alet et al., 2020).

For spin chains, the interaction is built from

$\mu$3

which makes the infinite-temperature thermofield double the zero-energy state in the strong-coupling limit. For coupled harmonic oscillators, a quadratic left–right deformation with parameters $\mu$4 yields a ground state that is exactly a thermofield double for a special relation between those parameters. These examples clarify that the Maldacena–Qi term is the canonical instance of a broader design principle rather than an isolated algebraic curiosity (Alet et al., 2020).

The most conceptually developed extension concerns gauged matrix and vector models, where gauge invariance induces confinement/deconfinement structure. In the microcanonical picture of the gauged Gaussian matrix model, partial deconfinement is described by an $\mu$5 subgroup deconfining while the rest remains confined, leading to the entanglement estimate

$\mu$6

or, since $\mu$7,

$\mu$8

For vector models, the remaining entanglement scales like

$\mu$9

or

$\beta(\mu)$0

These formulas support the interpretation that as deconfinement progresses, the quantum entanglement responsible for connecting the two sides is gradually lost, corresponding in the dual gravity picture to a wormhole that narrows and eventually disappears (Alet et al., 2020).

A recurring misconception is to identify every two-sided entangled system with the Maldacena–Qi mechanism itself. The comparison with “Electric fields and quantum wormholes” makes the conceptual boundary explicit: that paper studies entangled charged matter with no geometric connection and defines a wormhole susceptibility that diagnoses electric-flux response, but it does not contain the direct inter-copy coupling characteristic of the Maldacena–Qi model (Engelhardt et al., 2015). A plausible implication is that the term “Maldacena–Qi coupling” should be reserved for the specific left–right interaction that both favors a thermofield-double-like ground state and enters the low-energy gravitational subsector as the interaction supporting an eternal traversable wormhole.