Partially Entangled Thermal States (PETS)

• PETS are thermal states in the SYK model that interpolate between the thermofield double and pure states via a local operator insertion during Euclidean evolution.
• They enable controlled modulation of entanglement, providing insights into black hole interiors and the dynamics of nearly-AdS₂ gravity.
• Semiclassical analysis using Schwarzian dynamics and JT gravity reveals how operator backreaction adjusts both entanglement entropy and the dual black hole geometry.

Partially Entangled Thermal States (PETS) are a family of states introduced in the SYK model that interpolate between the thermo-field double state and a pure (product) state. They are prepared by a Euclidean path integral describing the evolution over two Euclidean time segments separated by a local scaling operator $\mathcal{O}$. In the holographic description, this class of states consists of two black holes with their interior regions connected via a domain wall, described by the worldline of a massive particle. The construction was introduced in "Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK" (Goel et al., 2018).

1. Definition in the SYK model

The SYK model has $N$ Majorana fermions $\psi_i$ with random all-to-all $q$-body interactions. The construction of PETS uses the SYK low-energy/Schwarzian limit as the boundary theory dual to nearly-$\mathrm{AdS}_2$ gravity (Goel et al., 2018).

A standard thermal pure state in SYK is prepared by evolving a boundary state $|B\rangle$ for Euclidean time $\beta/2$,

$$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$

This is a pure state that looks thermal for a suitable class of observables.

PETS generalize this preparation by inserting a local scaling operator $\mathcal O$:

$$|\Psi\rangle = e^{-\frac12 \beta_L H}\,\mathcal O\, e^{-\frac12 \beta_R H}\,|B\rangle,$$

or equivalently in the energy basis,

$N$0

with

$N$1

The two Euclidean segments have lengths $N$2 and $N$3, and the total inverse temperature of the underlying thermal circle is

$N$4

This family interpolates between two limiting cases. If $N$5, then

$N$6

reduces to the TFD when the partition matches the canonical left-right split and $N$7. If one of the Euclidean segments is taken very large, or if $N$8 is chosen so that the state factorizes, the left and right sectors effectively decouple. In the special boundary-state limit this becomes a product of thermal pure states. PETS are therefore “in between”: the entanglement entropy is smaller than the TFD value but larger than zero.

2. Euclidean preparation and reduced density matrix

The Euclidean preparation is a path integral on a thermal circle cut into two arcs, with $N$9 inserted between them. In the paper’s notation the state is represented schematically as

$\psi_i$0

The Euclidean picture consists of one arc of length $\psi_i$1, the insertion of a local scaling operator $\psi_i$2, and a second arc of length $\psi_i$3 (Goel et al., 2018).

The operator is taken to be a scaling operator of dimension $\psi_i$4, often large:

$\psi_i$5

That regime is important because the operator then backreacts on the bulk geometry in a semiclassical way.

The reduced density matrix for the right system is

$\psi_i$6

and in the energy basis it has the structure

$\psi_i$7

with matrix elements

$\psi_i$8

The reduced state is therefore not simply thermal: it is thermal evolution interrupted by an operator insertion and then re-glued. The stated physical claim is that PETS remain thermal-like for many simple observables, but only partially entangled across the left-right bipartition. For observables that do not probe the fine-grained structure of the insertion, the system still looks approximately thermal.

3. Thermal character and interpolation between two-sided and one-sided geometries

The physical meaning of PETS is organized around partial suppression of left-right entanglement. The operator insertion partially “cuts” the entanglement, so the resulting state is still thermal-looking from each side, but not maximally entangled (Goel et al., 2018).

This interpolation is more specific than a formal deformation of the TFD. PETS were introduced to connect two familiar extremes in holography and SYK: the thermofield double, which is maximally entangled between left and right and is dual to an eternal two-sided black hole, and the thermal pure state, which is a one-sided pure state obtained by projecting one side of the TFD onto a boundary state and is dual to a one-sided black hole geometry.

For SYK, the overlap and correlator structure can be expressed in terms of the operator’s conformal dimension and the Euclidean time separations. This suggests that the entanglement content is controlled jointly by the insertion data and by the asymmetry of the Euclidean preparation. A plausible implication is that PETS furnish a controlled setting in which thermal coarse-grained observables and fine-grained entanglement diagnostics can be varied independently.

4. Schwarzian and Jackiw–Teitelboim description

At low energies, SYK reduces to Schwarzian quantum mechanics, which is dual to Jackiw-Teitelboim gravity in $\psi_i$9. The semiclassical analysis of PETS proceeds through the exact Schwarzian two-point function and its saddle-point geometry (Goel et al., 2018).

A key exact representation for the insertion correlator is

$q$0

In the semiclassical limit, the effective action is

$q$1

with $q$2, $q$3, and $q$4.

The saddle-point equations are

$q$5

and

$q$6

They are also rewritten as

$q$7

$q$8

These equations determine the backreacted geometry, including the opening angles of the Euclidean arcs and the effective temperatures on each side. In the Schwarzian/JT description, the insertion is not a passive probe: it fixes the classical data of the backreacted saddle.

5. Bulk dual, domain wall, and interior expansion

The holographic dual of a PETS is a pair of $q$9 regions, or equivalently two black hole exteriors, glued along the worldline of a massive bulk particle created by the operator insertion (Goel et al., 2018). In Euclidean signature, the boundary trajectory is deformed away from a circle into two arcs meeting at a kink. In Lorentzian signature, this becomes a black hole geometry with two horizons and an interior that is enlarged or reduced depending on $\mathrm{AdS}_2$0 and $\mathrm{AdS}_2$1.

The massive operator is represented by a bulk geodesic worldline and acts like a domain wall separating two $\mathrm{AdS}_2$2 patches. Charge conservation in the JT embedding-space formalism gives continuity of the dilaton across the particle worldline, but its derivative jumps by an amount proportional to the mass:

$\mathrm{AdS}_2$3

The geometric regimes are described as follows. For small $\mathrm{AdS}_2$4, the worldline is a mild perturbation of the TFD wormhole. For large $\mathrm{AdS}_2$5, there is strong backreaction, the horizons move apart, and the interior expands. For very large $\mathrm{AdS}_2$6, the geometry approaches a nearly factorized configuration.

The distances between the horizons and between the particle and each horizon obey

$\mathrm{AdS}_2$7

and

$\mathrm{AdS}_2$8

In the large-$\mathrm{AdS}_2$9 regime, the horizon separation grows logarithmically,

$|B\rangle$0

Heavier operator insertions therefore produce a larger interior region. Within the stated framework, the interior size is directly controlled by the same semiclassical parameters that determine the boundary preparation.

6. Entanglement entropy and the meaning of “partial” entanglement

One of the main results is that the entanglement entropy of PETS is controlled by the minimum value of the dilaton in the bulk, exactly as expected from holographic entropy prescriptions (Goel et al., 2018). The modular or Rényi entropy is written as

$|B\rangle$1

For the TFD,

$|B\rangle$2

and in the $|B\rangle$3 limit,

$|B\rangle$4

For PETS, the entropy is not simply thermal. Instead,

$|B\rangle$5

where $|B\rangle$6 and $|B\rangle$7 are the semiclassical momenta associated to the two local minima of the dilaton profile, and $|B\rangle$8 or $|B\rangle$9 depending on which horizon is smaller.

For $\beta/2$0, the small-$\beta/2$1 expansion is

$\beta/2$2

$\beta/2$3

The entropy is then controlled by the smaller one, typically $\beta/2$4 in that regime. In the opposite limit $\beta/2$5,

$\beta/2$6

up to the interchange depending on whether $\beta/2$7 or $\beta/2$8.

The paper explains the term “partially” entangled by comparing PETS to the maximal entropy state with the same coarse-grained energy. If the right side has effective energy

$\beta/2$9

then the thermal state maximizing entropy at that energy has entropy

$$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$0

But the actual PETS entropy is

$$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$1

with $$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$2 in the relevant regime. Hence

$$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$3

This is the operational sense in which PETS are partially entangled: they have less entanglement than the maximally entangled thermal purification compatible with the same coarse-grained energy.

7. One-sided reconstruction and relation to entangled thermal states

A major conceptual point is that one-sided bulk reconstruction can still access interior bulk regions for PETS, unlike in the strict TFD case where a single side only reconstructs the exterior (Goel et al., 2018). The reason given is that the PETS is not maximally mixed on one side; the operator insertion modifies the code subspace and the entanglement wedge. The right side can reconstruct the region inside its entanglement wedge, which includes parts of the black hole interior up to the minimal-dilaton surface. The causal wedge remains smaller and stops at the right horizon, whereas the entanglement wedge can extend behind the right horizon.

The paper connects this to the modular zero-mode construction,

$$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$4

which in the bulk corresponds to an operator localized on the RT surface. Since in $$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$5 the RT surface is a point, the relevant bulk operator sits at the left horizon or minimal dilaton point. The tensor-network picture offered in the paper is that the operator insertion acts like a partially transmitting tensor between the left and right sides, and the horizon with the smallest dilaton acts as the bottleneck.

A recurrent terminological issue is the relation between PETS in SYK and the broader literature on entangled thermal states. The paper "Genuine multipartite nonlocality of entangled thermal states" studies multipartite entangled thermal states built from locally thermal bosonic modes, but it does not introduce PETS as a separate family with an explicit tunable entanglement parameter (McKeown et al., 2010). In that work, the closest analogue of partiality arises from thermal mixture and finite distinguishability, with the control parameter given by $$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$6: when $$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$7 is large compared to $$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$8, the entangled thermal states behave like ideal maximally entangled qubit GHZ, W, or cluster states; when $$|\Psi,\beta\rangle = e^{-\frac{\beta}{2}H}|B\rangle.$$9 is small or $\mathcal O$0 is large, the thermal Gaussians overlap and the entanglement is washed out. This distinction is important. PETS in the SYK/JT setting are defined by Euclidean preparation with a local operator insertion and are interpreted geometrically through a backreacted two-horizon $\mathcal O$1 spacetime, whereas the bosonic entangled thermal states are organized around displaced thermal mixtures, Bell-like inequalities, and multipartite nonlocality.