Multipartite Thermal Pure Quantum States

• The MTPQ state is a pure state framework extending TPQ to multiple quantum systems, encoding both thermal equilibrium and genuine multipartite entanglement.
• It utilizes a half-Boltzmann filter applied to a Haar-random seed state, ensuring nonfactorizing thermal correlations that align with holographic dual expectations.
• The approach finds practical applications in quantum many-body physics and holographic models, with validations in multi-copy Sachdev-Ye-Kitaev systems.

The Multipartite Thermal Pure Quantum (MTPQ) state is a construction that generalizes the concept of the thermal pure quantum (TPQ) state from single-system or bipartite quantum statistical mechanics to the multipartite regime. MTPQ provides a framework for encoding thermal equilibrium properties and multipartite entanglement at finite temperature within a single pure state distributed over multiple parties, with direct applications to quantum information, many-body physics, and holographic duality. Its construction, properties, and diagnostic role with respect to genuine multipartite entanglement measures such as Latent Entropy (L-entropy) have been recently established, with detailed application to models including the multi-copy Sachdev-Ye-Kitaev (SYK) system (Basak et al., 2024, Basak et al., 31 Jan 2026).

1. Formal Definition and Construction

The MTPQ state extends the TPQ ansatz to $n$ parties. Given $n$ quantum systems (parties), each with a Hilbert space $\mathcal H^{(k)}$ and local Hamiltonian $H^{(k)}$ (typically sharing an identical spectrum), one samples a Haar-random global pure state $|\psi\rangle$ in $\bigotimes_{k=1}^n \mathcal H^{(k)}$. The MTPQ state with inverse temperature parameter $\alpha$ is constructed by applying the "half-Boltzmann" filter independently to each subsystem:

$$|\Psi_\alpha\rangle = \left(\prod_{k=1}^n e^{- \frac{\alpha}{2} H^{(k)}}\right) |\psi\rangle.$$

This construction produces a normalized pure state whose reduced density matrices capture thermal statistics for individual parties and whose multipartite entanglement is diagnostic of genuine quantum correlations at nonzero temperature.

To ensure compatibility with correlations characteristic of holographic duals (e.g., multi-boundary wormholes), a state-dependent assignment for the energy basis is essential. For each party, one performs a Schmidt decomposition of $|\psi\rangle$ across the bipartition $k$ vs.\ the rest, then maps the local Schmidt basis into the energy eigenbasis with properly aligned local unitaries (see (Basak et al., 2024), Sec.~1). The local Hamiltonian $H^{(k)}$ acts on the party $k$ by conjugation with this unitary, ensuring that the spectral structure relevant for thermalization is respected within each subsystem.

2. Factorization Problem and Its Resolution

A naive, state-independent product of single-party thermal factors acting on random states leads to reduced density matrices that always factorize, i.e., for any two parties $A$ and $B$,

$$\rho_{AB} = \rho_{\text{th}}^{(A)}(\beta) \otimes \rho_{\text{th}}^{(B)}(\beta),$$

where $$\rho_{\text{th}}^{(A)}(\beta) \propto e^{-\beta H^{(A)}}$$. This absence of cross-party entanglement contradicts the expectations from holography, where the wormhole dual should induce nontrivial multipartite correlations (“factorization problem”).

The MTPQ construction resolves this by tying the energy basis assignment to the Schmidt decomposition of the initial random state. The local Boltzmann filters thus act on correlated local degrees of freedom in a manner explicitly dependent on the global seed state $|\psi\rangle$. In this scenario, correlation functions across parties, such as

$$\langle\Psi_\alpha | \mathcal O^{(1)} \mathcal O^{(2)} \cdots | \Psi_\alpha\rangle,$$

no longer factorize upon random averaging, enabling the pure state to reproduce connected expectations matching the thermofield double or multi-boundary wormhole correlations (Basak et al., 2024). For example, in the bipartite case, this approach exactly reproduces thermal double correlators.

3. Entanglement Structure and L-Entropy

Genuine multipartite entanglement in MTPQ states is quantitatively analyzed using Latent Entropy (L-entropy), an entanglement monotone constructed from the reflected entropy of pairs and extended via the geometric mean to $n$ parties. The L-entropy between parties $A_i$ and $A_j$ is

$$\ell_{A_iA_j} = \min\{ 2 S(A_i), 2 S(A_j) \} - S_R(A_i : A_j),$$

where $S_R$ denotes the reflected entropy of the two-party reduced state. For the MTPQ construction, the two-party reductions are thermal and factorized, leading to

$$\ell_{A_iA_j}^{(\text{MTPQ})} = 2\, \min\{ S_{\text{th}}(\beta_i), S_{\text{th}}(\beta_j) \},$$

with $S_{\text{th}}(\beta)$ denoting the thermal entropy at inverse temperature $\beta$.

The $n$-partite L-entropy is then the appropriate mean over all pairs:

$$\ell_{A_1 \cdots A_n} = \left( \prod_{1 \leq i < j \leq n} \ell_{A_iA_j} \right)^{2 / n(n-1)}.$$

This structure realizes a finite-temperature generalization of 2-uniform entanglement, where all two-party marginals are maximally (thermally) mixed. At high temperature (small $\alpha$), $S_{\text{th}}(\alpha) \to \ln D$ (maximal entropy), and the L-entropy saturates its maximal value; as temperature decreases (large $\alpha$), it exhibits a Page-curve-like fall-off (Basak et al., 31 Jan 2026).

4. Application to the Multi-Copy Sachdev-Ye-Kitaev (SYK) Model

A leading testbed for the MTPQ construction is the coupled multi-copy SYK model,

$$H^{(k)} = \sum_{i<j<k<\ell} J_{ijkl} \chi^i_{(k)} \chi^j_{(k)} \chi^k_{(k)} \chi^\ell_{(k)},$$

where each party hosts a set of Majorana fermions with random quartic interactions. The MTPQ construction yields a multipartite pure state in the tensor product of the Fock spaces for each copy.

Numerical diagonalization for $n=3$–5 parties shows:

• Each party’s entanglement entropy $S(A_k)$ tracks the thermal SYK entropy at its own effective temperature, obtained by minimizing the relative entropy $S(\rho_{A_k} \Vert \rho_{\text{th}}(\beta_k))$.
• The multipartite L-entropy $\ell_{1 \cdots n}(\alpha)$ displays the expected crossover: for $n=5$ parties, it saturates the maximal $2\log d$ value over a broad $\alpha$ range, indicating robust finite-temperature 2-uniformity.
• Mutual information $I(A_i:A_j)$ reflects this structure, with a two-step transition as a function of $\alpha$ consistent with the formation and transition of bulk geometric structures in the dual AdS wormhole scenario (Basak et al., 2024).

5. Thermodynamic and Holographic Properties

The MTPQ state reproduces thermal correlators for local observables under random averaging, as in the conventional TPQ construction:

$$\frac{\langle\Psi(\beta)|\,\mathcal O\,|\Psi(\beta)\rangle}{\langle\Psi|\Psi\rangle} = \frac{\mathrm{Tr}[ \mathcal O\, e^{-\beta H} ]}{Z(\beta)},$$

ensuring the correct thermal expectation for one-point functions. For multipartite observables, the state-dependent construction prevents naive factorization and allows for genuine thermal correlations matching holographic expectations.

In the context of AdS/CFT, MTPQ states model microstates of $n$-boundary wormholes at finite temperature. The L-entropy Page-like curve is interpreted as the entanglement signature of the competition between homology saddles or island transitions in the holographic bulk, reflecting emergent geometric phenomena (Basak et al., 2024).

6. Assumptions, Limitations, and Generalizations

The construction of MTPQ states presumes:

• The existence of a single global Haar-random reference state in the full composite Hilbert space of dimension $D^n$.
• Concentration-of-measure phenomena (Haar typicality), which become exact as $D\to\infty$.
• Absence of direct cross-party couplings in the Hamiltonians; the construction is exact only if filters $e^{-\alpha H^{(i)}/2}$ act on strictly distinct subsystems.

Numerical validations are typically carried out at large $N$ in the SYK model, and $1/N$ corrections are neglected. A plausible implication is that for intermediate system sizes, corrections to factorization and entanglement measures may become non-negligible.

7. Significance and Future Directions

MTPQ provides a pure-state formalism for multipartite quantum thermodynamics and finite-temperature entanglement, bridging concepts from quantum typicality, entanglement monotones, and holographic emergence. It delivers a flexible, tunable parameterization (via $\alpha$) that interpolates between maximally entangled (2-uniform) and unentangled states, enables diagnostic study through measures such as L-entropy, and faithfully reproduces key features expected from holographic duals of multipartite entangled systems. Prospective research directions include extensions to systems with cross-party interactions, exploration in large-scale quantum simulation platforms, and refined analysis of entanglement phase transitions in higher-dimensional and non-integrable models (Basak et al., 2024, Basak et al., 31 Jan 2026).