Thermofield Double State

Updated 24 August 2025

Thermofield double state is a purified, entangled state on a doubled Hilbert space that yields thermal density matrices when one copy is traced out.
It employs methods like Bogoliubov transformations and tensor network representations to study entanglement structure, complexity, and thermal correlations in quantum systems.
The state bridges thermal quantum field theory with holographic duality and quantum simulation, offering practical insights into state preparation and gravitational analogs.

The thermofield double (TFD) state is a distinguished entangled pure state defined on a doubled (tensor product) Hilbert space, constructed such that the reduced state on either factor yields a thermal (Gibbs) density matrix. Originally introduced in the context of thermal quantum field theory, the TFD state provides a canonical purification of thermal equilibrium ensembles, with extensive applications across quantum field theory, statistical mechanics, quantum information, tensor network theory, and holographic duality.

1. Formal Definition and Construction

The TFD state for a system with Hamiltonian $\hat{H}$ is defined in the enlarged Hilbert space $\mathcal{H} \otimes \mathcal{H}$ as

$|\mathrm{TFD}(\beta)\rangle = \frac{1}{\sqrt{Z(\beta)}} \sum_n e^{-\frac{\beta}{2} E_n} |n\rangle_L \otimes |n\rangle_R,$

where $\beta = 1/T$ is the inverse temperature, $\{ |n\rangle \}$ are energy eigenstates of $\hat{H}$ with eigenvalues $E_n$ , and $Z(\beta) = \sum_n e^{-\beta E_n}$ is the partition function.

Tracing over the right (or left) factor yields the thermal density operator on the original Hilbert space: $\operatorname{Tr}_R |\mathrm{TFD}(\beta)\rangle \langle \mathrm{TFD}(\beta)| = \frac{e^{-\beta \hat{H}}}{Z(\beta)}.$ This construction provides a purification of the thermal state (Gibbs ensemble), with entanglement between the two copies encoding thermal correlations.

For generic many-body and field-theoretical systems, the TFD is often written as the ground state of a nonlocal Hamiltonian (the "TFD Hamiltonian"), or as the result of acting with an entangling operator—such as a Bogoliubov transformation—in the energy eigenbasis or in terms of ladder operators (Yang, 2017).

2. Complexity, Circuit Geometry, and State Preparation

An essential research direction has investigated the minimal computational resources (circuit complexity) required to prepare the TFD from a simple reference state. Using approaches based on Finsler geometry and Nielsen’s geometric framework, the complexity $\mathcal{C}(|\psi_2\rangle, |\psi_1\rangle)$ between states is defined as the minimal “cost” (geodesic length) of continuous unitary transformations that map $|\psi_1\rangle$ to $|\psi_2\rangle$ , where the space of allowed generators is physically motivated (e.g., ladder operators).

For TFD states in $d$ -dimensional conformal field theory, this leads to

$\mathcal{C}_\text{TFD} \propto T^{d-1},$

with the complexity density finite and matching the scaling predicted by holographic renormalization, provided the circuit is constructed relative to a vacuum state sharing the same ultraviolet (UV) structure (Yang, 2017).

The optimal circuit depends strongly on reference state choice, gate set, and cost function. In free bosonic and fermionic systems, the covariance matrix method allows for explicit computation and reveals that, with appropriate subtraction (complexity of formation), UV divergences cancel and the additional complexity required to prepare the TFD relative to two vacua is proportional to the thermodynamic entropy: $\Delta \mathcal{C} \propto S_\text{th}$ with proportionality constant dependent on model and dimension (Chapman et al., 2018, Jiang et al., 2018).

In experimental and numerical contexts, variational quantum algorithms inspired by the quantum approximate optimization algorithm (QAOA) efficiently prepare TFD states on quantum hardware, alternating between system and entangling Hamiltonians, with circuit depth scaling favorably with system size and temperature (Wu et al., 2018, Zhu et al., 2019, Faílde et al., 2023).

3. Entanglement Structure and Dynamics

The TFD state is maximally entangled at infinite temperature ( $\beta \to 0$ ) and becomes a product state at zero temperature (for Hamiltonians with a unique ground state). Its entanglement dynamics under time evolution with $H_L + H_R$ is closely related to quantum quench phenomena. In integrable models, the time evolution of entanglement entropy is governed by a ballistic quasiparticle picture: $S_\ell(t) = \sum_n \left[ 2t \int_{2|v_n(\lambda)| t < \ell} d\lambda\, |v_n(\lambda)|\, s_n(\lambda) + \ell \int_{2|v_n(\lambda)| t > \ell} d\lambda\, s_n(\lambda) \right],$ with $v_n(\lambda)$ quasiparticle velocities and $s_n(\lambda)$ entropy densities. For the TFD, the entropy density acquires a factor of 2 reflecting the doubling of degrees of freedom (Lagnese et al., 2021).

In free theories, complexity saturates after a time of order $\beta$ , while in strongly coupled (holographic) duals complexity grows linearly up to exponentially long times, reflecting persistent Einstein–Rosen bridge growth in the dual geometry (Chapman et al., 2018, Jiang et al., 2018).

Perturbations such as double-trace operators that couple the two copies can modify entanglement and, in the holographic context, correspond to deformations of bulk wormhole geometries. The entanglement entropy typically equilibrates on “thermal time” scales after such quenches (Dadras, 2019).

4. Holography, Black Holes, and Gravitational Duals

In AdS/CFT, the TFD state is dual to an eternal black hole geometry with two asymptotically AdS boundaries, each corresponding to one Hilbert space copy. The bulk dual of time evolution in the TFD corresponds to dynamical evolution of the black hole interior, with the Einstein–Rosen bridge (wormhole) length increasing monotonically with time.

Proposals for the dual of quantum circuit complexity in the boundary theory include the "complexity = volume" (CV) and "complexity = action" (CA) conjectures, which posit that boundary circuit (or path) complexity corresponds to maximal volume slices or on-shell action evaluated in Wheeler–DeWitt patches of the bulk spacetime. Field-theoretic measures of complexity can be UV-finite and reproduce the correct temperature and volume scaling if constructed relative to the appropriate reference state (Yang, 2017, Chapman et al., 2018, Chapman et al., 2019, Doroudiani et al., 2019).

Notably, the equivalence between fidelity susceptibility and circuit complexity for TFD states provides a field-theoretic explanation for their common gravitational dual: both have been shown to match maximal bulk volumes in AdS (Yang, 2017).

RG flows of the TFD state under relevant deformations correspond, in the bulk, to scalar field profiles sourcing geometry inside the black hole, with the IR fixed point residing behind the event horizon. Correlation functions and entanglement between the two copies directly probe the bulk geometry interior (Das et al., 2021).

5. Tensor Network Representations and Quantum Simulation

The TFD state is naturally encoded in tensor network representations, particularly projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansätze (MERA). In the context of Rokhsar–Kivelson-type wavefunctions and models exhibiting quantum criticality, the TFD construction allows mapping quantum phase transitions in $d$ dimensions to thermal transitions in $d+1$ dimensions, extending the classical-quantum correspondence.

The $N$ th Rényi entropy of a TFD-based PEPS maps to a partition function of a $d+1$ -dimensional classical statistical model (e.g., a 3D $\mathbb{Z}_2$ gauge-Higgs model for the toric code with field deformations), allowing quantum universality classes to be studied through classical simulations (Xu et al., 2020). Entanglement renormalization circuits applied to TFDs provide explicit renormalization group (RG) flows for thermal states and demonstrate the loss of long-range order (e.g., topological order) at high temperatures (Lin et al., 2021).

Variational quantum algorithms, Hamiltonian forging, and entanglement forging ansätze—where circuits of width $N$ construct TFD states efficiently—enable near-term and NISQ device applications (Faílde et al., 2023).

6. Extensions: Symmetries, Charges, and Geometric Contexts

TFD states accommodate additional symmetries and external fields. For systems with $U(1)$ (or other) charges, one constructs charged TFD states (cTFD) with chemical potential $\mu$ , leading the state to factorize into two sectors with shifted effective temperatures and times: $\beta_k^{(\pm)} = \beta (1\pm \mu/\omega_k), \quad t_k^{(\pm)} = t (1 \pm \mu/\omega_k)$ (Chapman et al., 2019, Doroudiani et al., 2019). The complexity of such states, when compared to entropies or to their holographic black hole duals (charged Reissner–Nordström spacetimes), reveals nontrivial structure (e.g., nonanalytic scaling, matching divergence structure under suitable cost functions).

The TFD formalism has been extended to background-independent settings such as group field theory (GFT), where equilibrium states are defined via modular theory, and TFD states become entangled squeezed vacua, interpreted as condensate states with temperature parameter $\beta$ (Guo, 2019).

Euclidean path integral derivations connect the TFD state to Rindler wedges, causal diamonds, and the Unruh effect, elucidating the universality of TFD construction for systems with horizons or bounded regions (Valdivia-Mera, 2020, Chakraborty et al., 2023). The periodicity of the Euclidean time coordinate encodes the temperature and leads to the TFD structure through conformal symmetry and analytic continuation.

7. Limitations, Singularities, and Universality

Analysis of TFDs in black hole backgrounds demonstrates that noncanonical temperatures introduce anomalous singularities in field propagators at the horizon, leading to infrared-dominated divergences that affect the regularized stress-energy tensor and imply strong backreaction unless the temperature matches the Hawking value. This constrains the physical realization of TFDs in curved and thermal spacetimes (Anempodistov, 2020).

Projective measurements, double-trace deformations, and external perturbations can induce phase transitions in the entanglement wedge in holographic duals, correspond to teleportation protocols, and realize phase transitions in the bulk geometry as the TFD is manipulated (Antonini et al., 2022).

The universality of the TFD construction arises from its role in purifying thermal states, its geometric realization via Euclidean path integrals and modular theory, and its interpretation as encoding horizon thermality in a wide variety of geometric and dynamical contexts, including Rindler wedges, black holes, and causal diamonds (Chakraborty et al., 2023, Valdivia-Mera, 2020).

In summary, the thermofield double state is a foundational object providing a canonical purification of equilibrium (thermal) states, bridging concepts in quantum information, statistical mechanics, quantum simulation, and geometric duality. Its entanglement structure, complexity, and dynamical evolution encode deep connections between thermal field theory, tensor network methods, and the geometric structure of spacetime—in particular within the framework of holography and quantum gravity.