Thermofield Double State

Definition

The thermofield double (TFD) is a pure state on a doubled Hilbert space ${\mathcal{H}}_L\otimes {\mathcal{H}}_R$ whose reduced density matrix on either factor is thermal:

$$|{\mathrm{TFD}}_{\beta }⟩=\frac{1}{\sqrt{Z(\beta )}}\sum _n{e}^{-\beta E_n/2}|n{⟩}_L\otimes |n{⟩}_R,{\mathrm{Tr}}_L|\mathrm{TFD}⟩⟨\mathrm{TFD}|=\frac{e^{-\beta H_R}}{Z(\beta )},$$

with $H|n⟩=E_n|n⟩$ and $Z(\beta )={\sum }_n{e}^{-\beta E_n}$. It is the canonical purification of a Gibbs state, and it goes back to Takahashi and Umezawa’s thermo field dynamics (1975).

In holography, the TFD of two copies of a CFT is dual to the two-sided eternal AdS–Schwarzschild black hole: the two boundary factors are the two asymptotic regions, joined by a (non-traversable) Einstein–Rosen bridge. Geometric connectivity of the bridge is the bulk image of the $L$–$R$ entanglement — the cleanest statement of ER=EPR.

Algebraically the TFD is a cyclic and separating vector for the boundary algebras ${\mathcal{A}}_L,{\mathcal{A}}_R$, and a KMS state at inverse temperature $\beta$ for the time translation that the bulk Killing flow implements. Its modular Hamiltonian is the difference of the two Hamiltonians, $K\propto H_R-H_L$, so modular time translation is ordinary time running oppositely on the two copies. At large $N$ the algebras are type III₁.

Role in Research

The TFD is the single state where almost every thread of the group’s program converges:

• Bell-CHSH on the TFD. The left and right boundary algebras are each other’s commutant, and the modular conjugation $J$ maps one to the other — exactly the structure the group uses to build Bob’s operators from Alice’s with Weyl operators. Course Sem II Wk 8 computes Bell-CHSH between two Rindler wedges in the TFD, a direct extension of Summers–Werner, and it is the obvious arena for bell-chsh-in-holographic-setting.
• Relative entropy and the crossed product. Because the TFD is KMS and type III₁, the Araki–Uhlmann relative entropy is the well-defined information measure; the crossed product of ${\mathcal{A}}_R$ by its modular flow (Chandrasekaran–Penington–Witten) turns the algebra type II${}_{\infty }$ and yields the generalized entropy. The TFD is the reference state of that construction.
• The base geometry for traversability. Deforming the TFD by a double-trace coupling $V=g{\mathcal{O}}_L{\mathcal{O}}_R$ makes the wormhole traversable; this is the starting point of relative-entropy-wormhole-opening and bell-chsh-across-traversable-wormhole.
• A reference, not a catalyst. In the embezzlement protocol the TFD is the state with a unique dual geometry against which the proposed catalyst is contrasted: the catalyst has no geometry of its own, being a broad area/modular-clock wavepacket dressing the TFD wormhole.

Relations

• The flat-space analogue is the Minkowski vacuum restricted to two complementary Rindler wedges: the vacuum is the TFD of the two wedge algebras at the Unruh temperature (Bisognano–Wichmann). This is why free-field two-wedge calculations model the holographic TFD.
• tomita-takesaki-modular-theory — cyclic-separating, with geometric modular flow $K\propto H_R-H_L$; the KMS condition encodes thermality
• type-iii-von-neumann-algebras — the left and right algebras are type III₁ at large $N$
• crossed-product-construction — the gravitational crossed product is built on the TFD modular flow, and its trace gives the generalized entropy
• araki-uhlmann-relative-entropy — computed between excited states and the TFD reference
• large-n-factorization — single-trace operators on each factor factorize at large $N$, which is what makes the algebras type III₁
• er-epr, traversable-wormholes — connectivity from entanglement, and what it takes to send a signal across
• bell-chsh-across-traversable-wormhole, relative-entropy-wormhole-opening — the two TFD wedges as the bipartite arena

Papers

• Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] — the identification with the eternal black hole.
• Takahashi, Umezawa (1975) — the origin of thermo field dynamics, and of the construction itself.
• Chandrasekaran, Penington, Witten (2022), arXiv:2209.10454 — the TFD as a type III₁ KMS state and its crossed product (AQFT course Sem II Block 2).
• 2021-kundu-wormholes-holography — §4.2.1 introduces the TFD as the two-sided eternal black hole.
• 2025-liu-lectures-entanglement-vna — the modular/KMS treatment and the algebraic ER=EPR.

Notes

• A useful slogan for students: the TFD is “entanglement masquerading as geometry”. Everything the group computes in flat space on a pair of complementary wedges has a TFD reading, and conversely every TFD statement has a Rindler analogue that is explicitly calculable with Weyl operators.
• Common confusion: the TFD is a pure state on the doubled space, and its thermality is a property of the reduced state on one factor. The temperature is the modular/KMS temperature of the reference state, which in the holographic dual is the Hawking temperature of the eternal black hole.
• This page absorbed a duplicate, thermofield-double, written a day apart from it in June 2026 for the same object; the aliases of both are kept above.