Thermo Mutual Information (TMI) Overview

• Thermo Mutual Information (TMI) is defined in the thermo-field double formalism to measure UV-finite correlations between physical and thermo-doubled degrees of freedom.
• It employs holographic and field-theoretic methods to probe inter-boundary entanglement, critical phenomena, and cross-copy correlations in thermal and quantum systems.
• TMI’s versatility is underscored by its applications in many-body dynamics, tripartite mutual information studies, and thermodynamics-inspired dimensionality reduction.

Searching arXiv for recent and foundational papers on “Thermo Mutual Information” and related usages of “TMI”. Thermo mutual information (TMI) is the analogue of ordinary mutual information for a thermal system written in the thermo-field double (TFD) formalism. In that setting it measures correlations between physical, or type-1, degrees of freedom on one copy of the theory and thermo-double, or type-2, degrees of freedom on the second copy, and in holography it probes inter-boundary entanglement in an eternal black hole geometry (Morrison et al., 2012). The acronym is not uniform across the arXiv literature: it also denotes tripartite mutual information in many-body quantum dynamics and “Thermodynamic Manifold Inference” in thermodynamics-inspired dimensionality reduction (Wanisch et al., 2021, Dixit, 2019). In the strict TFD sense, however, TMI is a UV-finite, positive-definite correlation measure defined from reduced density matrices on the doubled Hilbert space, with a direct geometric realization in terms of connected and disconnected extremal surfaces (Morrison et al., 2012).

1. Definition in the thermo-field double formalism

For a system with Hamiltonian $H$, the thermal density matrix is

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$

The TFD construction purifies this mixed state by doubling the Hilbert space,

$H_{\rm TFD}=H_1\otimes H_2,$

with TFD Hamiltonian

$H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$

and state

$$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$

so that $H_{\rm TFD}|\Omega\rangle=0$ (Morrison et al., 2012).

Ordinary mutual information for two disjoint regions $A$ and $B$ on a spatial slice is

$(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$

with

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$

Thermo-mutual information replaces one subsystem by its TFD partner. For regions $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$0 and $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$1 in the original spatial slice, one defines reduced density matrices $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$2, $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$3, and $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$4, and then

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$5

Because $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$6, this can be viewed as mutual information between two independent subalgebras, one from each TFD copy (Morrison et al., 2012).

Several structural properties are emphasized. TMI is non-negative,

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$7

as a consequence of strong subadditivity. It bounds normalized connected correlators between type-1 and type-2 observables. It is UV finite even though each entropy entering the combination is separately divergent, because the divergent terms cancel in the mutual-information combination. The paper also argues that, under the conditions discussed there,

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$8

At $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$9 ($H_{\rm TFD}=H_1\otimes H_2,$0), TMI vanishes because the TFD state becomes a product of ground states. At $H_{\rm TFD}=H_1\otimes H_2,$1 ($H_{\rm TFD}=H_1\otimes H_2,$2), both ordinary MI and TMI go to zero for disjoint regions, although for overlapping regions TMI can diverge (Morrison et al., 2012).

A major technical point is that the reduced density matrices entering TMI admit a clean Schwinger-Keldysh and Euclidean path-integral construction. In Euclidean signature, type-2 insertions are shifted by $H_{\rm TFD}=H_1\otimes H_2,$3 around the thermal circle. This contour formulation ensures that the hybrid density matrix $H_{\rm TFD}=H_1\otimes H_2,$4 is well defined in both Euclidean and Lorentzian signature (Morrison et al., 2012).

2. Field-theoretic and holographic realizations

The original TFD paper develops TMI in three explicit settings: a two-site spin system, a two-dimensional massless Dirac fermion, and a holographic two-dimensional system dual to the eternal BTZ black hole (Morrison et al., 2012).

In the two-site spin-$H_{\rm TFD}=H_1\otimes H_2,$5 chain with

$H_{\rm TFD}=H_1\otimes H_2,$6

the four energy eigenstates consist of one singlet and three triplets with energies

$H_{\rm TFD}=H_1\otimes H_2,$7

The thermal density matrix is

$H_{\rm TFD}=H_1\otimes H_2,$8

Tracing out one site gives

$H_{\rm TFD}=H_1\otimes H_2,$9

The resulting plots show that TMI is positive, lies below ordinary MI, vanishes as $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$0, and also vanishes as $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$1, whereas ordinary MI approaches $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$2 in that limit (Morrison et al., 2012).

For the two-dimensional massless Dirac fermion, the replica trick yields closed-form expressions. For ordinary same-copy MI between intervals $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$3 and $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$4,

$H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$5

whereas the TMI between $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$6 and $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$7 is

$H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$8

In this field-theoretic example, MI bounds TMI from above, both vanish as $H_{\rm TFD}=H_1\otimes 1_2-1_1\otimes H_2,$9, TMI vanishes as $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$0, and TMI decreases with increasing separation (Morrison et al., 2012).

In the holographic example, the field theory lives on the two disconnected asymptotic boundaries of the eternal BTZ black hole. The Ryu-Takayanagi prescription computes $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$1 from bulk geodesics, and TMI is realized by the competition between disconnected geodesics on the two boundaries and connected geodesics stretching through the Einstein-Rosen bridge. This gives a direct geometric meaning to cross-boundary entanglement. The same analysis also notes that, at leading large $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$2, holographic TMI can vanish in regimes where subleading field-theory correlations still exist, because the RT prescription captures only the $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$3 contribution (Morrison et al., 2012).

3. Holographic TMI in eternal black holes and shock-wave geometries

Recent holographic work develops TMI as a probe of inter-boundary correlations in more elaborate black-hole backgrounds. In both the five-dimensional Einstein-Maxwell-dilaton 1RC black hole and the charged AdS black hole deformed by a homogeneous string cloud, the setup is a TFD state dual to a two-sided eternal geometry, and the definition remains

$$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$4

with $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$5 and $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$6 taken as identical strips on opposite boundaries (Pant et al., 2023, Kaushal et al., 14 Jul 2025).

The 1RC black-hole analysis uses the competition between a disconnected surface $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$7 and a connected wormhole surface $$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$8. For positive TMI one chooses the connected wormhole surface, and the TMI becomes zero when

$$|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum_n e^{-\beta E_n/2}\,|E_n\rangle_1\otimes |E_n\rangle_2,$$9

The strip width $H_{\rm TFD}|\Omega\rangle=0$0 is related to the turning point $H_{\rm TFD}|\Omega\rangle=0$1 by

$H_{\rm TFD}|\Omega\rangle=0$2

In that model, TMI increases with the width $H_{\rm TFD}|\Omega\rangle=0$3, increases with the criticality parameter $H_{\rm TFD}|\Omega\rangle=0$4, vanishes below a critical width $H_{\rm TFD}|\Omega\rangle=0$5, and $H_{\rm TFD}|\Omega\rangle=0$6 decreases as $H_{\rm TFD}|\Omega\rangle=0$7 increases. The TMI remains finite in the critical limit $H_{\rm TFD}|\Omega\rangle=0$8 (Pant et al., 2023).

The shock-wave analysis deforms the Kruskal coordinate as

$H_{\rm TFD}|\Omega\rangle=0$9

and defines a regularized entropy contribution

$A$0

so that

$A$1

The numerical conclusion is that HTMI decreases as $A$2 increases, disappears beyond a critical $A$3, and that $A$4 increases with $A$5. The paper interprets this as a slowing of TMI disruption near the critical point: increasing $A$6 slows the disruption of TMI under shock perturbations (Pant et al., 2023).

The string-cloud background exhibits a related but not identical pattern. The metric is written in Poincaré coordinates as

$A$7

with blackening factor

$A$8

where $A$9 is the string-cloud/backreaction parameter and $B$0 is the charge parameter. TMI is again nonzero only when the connected wormhole surface dominates. The analysis finds a critical-width transition $B$1 for $B$2, suppression of TMI with increasing charge, a more intricate dependence on backreaction, and a sharp transition under shock-wave perturbations: inter-boundary entanglement is entirely disrupted beyond a critical shock strength, and that threshold decreases with increasing charge (Kaushal et al., 14 Jul 2025).

Holographic setup	Static TMI behavior	Shock-wave behavior
1RC black hole near critical point	TMI increases with $B$3 and $B$4; $B$5 for $B$6; $B$7 decreases as $B$8 increases	HTMI decreases with $B$9; $(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$0 increases with $(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$1; disruption slows as $(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$2 increases
Charged AdS black hole with string cloud	Critical-width transition; increasing charge suppresses TMI; backreaction tends to strengthen correlations	TMI vanishes beyond a critical shock strength; the threshold decreases with increasing charge

4. Mutual information rewritten in thermodynamic language

A distinct thermodynamic line of work does not start from the TFD doubled Hilbert space, but instead rewrites mutual information itself in the language of equilibrium thermodynamics. In “Shannon Meets Carnot: Mutual Information Via Thermodynamics” the Gaussian channel is represented as an equivalent thermal system, with signal-to-noise ratio mapped to inverse temperature,

$(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$3

and the posterior $(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$4 written as a Boltzmann distribution (0806.3133).

For the Gaussian channel

$(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$5

the effective energy is identified as

$(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$6

The presence of the $(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$7 term makes the Hamiltonian temperature dependent, so the standard relation $(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$8 is replaced by the generalized second law

$(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{A\cup B}),$9

The paper interprets this as a separation between heat that changes occupation probabilities and heat that shifts the energy levels themselves (0806.3133).

Within this mapping, conditional entropy becomes an integral over internal-energy derivatives,

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$0

and mutual information takes the thermodynamic form

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$1

With the generalized second law included,

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$2

This framework also yields a thermodynamic proof of the Guo-Shamai-Verdú theorem,

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$3

and recovers, for Gaussian input,

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$4

This thermodynamic reformulation is conceptually adjacent to thermo-mutual information, but it addresses mutual information via temperature-dependent Hamiltonians rather than cross-copy correlations in a TFD geometry (0806.3133).

5. Finite-temperature mutual information as a diagnostic of criticality

Another closely related line of work studies mutual information directly in thermal many-body states. In the Lipkin-Meshkov-Glick model,

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$5

mutual information is used because the model is fully connected and has no natural notion of locality. For a bipartition $S(\rho)=-\mathrm{Tr}(\rho\log \rho).$6,

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$7

with $S(\rho)=-\mathrm{Tr}(\rho\log \rho).$8. The phase boundary is

$S(\rho)=-\mathrm{Tr}(\rho\log \rho).$9

The principal finding is that $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$00 is finite everywhere except on the phase transition line, where it diverges logarithmically with system size. For thermal critical points $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$01,

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$02

while at the zero-temperature quantum critical point $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$03,

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$04

(Wilms et al., 2011).

For interacting quantum systems at finite-temperature critical points, a separate analysis studies Rényi mutual information

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$05

with

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$06

Using

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$07

the paper shows that for $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$08 critical behavior appears at two temperatures,

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$09

because the replicated region $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$10 has effective temperature $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$11. In the XXZ model with Ising anisotropy, the area-law coefficient has a $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$12 singularity, the corner term has a logarithmic divergence with Cardy-Peschel coefficient

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$13

and a constant term associated with broken symmetries jumps between

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$14

across $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$15 and $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$16 (Singh et al., 2011).

These results do not define thermo-mutual information in the TFD sense, but they establish a broader thermodynamic role for mutual information: at finite temperature it captures total correlations, survives beyond purely entanglement-based diagnostics, and can detect thermal criticality through universal singular structures (Wilms et al., 2011, Singh et al., 2011).

6. Terminological ambiguity: tripartite mutual information and Thermodynamic Manifold Inference

A recurrent source of confusion is that “TMI” frequently denotes tripartite mutual information rather than thermo-mutual information. For three disjoint subsystems $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$17,

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$18

or equivalently

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$19

In long-range interacting $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$20 chains, the sign of TMI distinguishes whether information is predominantly shared globally or remains accessible through local pieces: positive TMI means information about $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$21 is more accessible from $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$22 and $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$23 separately, while negative TMI means information about $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$24 is only recoverable by joint measurements on $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$25, signaling delocalized quantum information (Wanisch et al., 2021).

The sign structure is highly model dependent. In permutation-symmetric many-qubit states, typical entanglement scales only as $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$26 because a $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$27-qubit reduced density matrix lives in a $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$28 symmetric subspace, with

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$29

As a consequence, random permutation-symmetric states are marginally entangled and their TMI is typically positive, even though the kicked top can show exponential OTOC growth and equilibration to random symmetric-state values (Seshadri et al., 2018). In non-Markovian open spin chains, negative TMI is not always a suitable indicator of quantum scrambling because TMI is built from von Neumann entropy and counts both quantum and classical correlations; the paper introduces tripartite logarithmic negativity,

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$30

as a more quantum-specific scrambling witness (Han et al., 2022). In integrable quenches, negative tripartite mutual information at intermediate times is tied to entangled quasiparticle multiplets whose entanglement content is not directly related to the Generalized Gibbs Ensemble, and the paper explicitly describes TMI as an “ideal lens” for observing the weakening of the relationship between entanglement and thermodynamics (Caceffo et al., 2023).

The acronym also appears in a completely different sense in “TMI: Thermodynamic Manifold Inference,” where TMI is a manifold-learning and dimensionality-reduction method rather than a correlation measure. There the model approximates positive data by a Gibbs-Boltzmann form

$$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$31

learns sample-specific intensive variables $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$32 and state-space extensive variables $$\rho=e^{-\beta H}, \qquad Z=\mathrm{Tr}\,e^{-\beta H}.$$33, minimizes a KL-divergence objective, and equips the reduced intensive-variable space with a thermodynamic/Fisher-Rao Riemannian metric supporting geodesics and volume elements (Dixit, 2019).

This terminological plurality suggests a practical distinction. In TFD and holographic finite-temperature work, TMI usually means thermo-mutual information. In scrambling and many-body dynamics, it usually means tripartite mutual information. In thermodynamics-inspired machine learning, it can mean Thermodynamic Manifold Inference. The underlying commonality is the use of entropy, relative entropy, and Gibbs-type structures, but the mathematical objects and physical interpretations are not interchangeable (Morrison et al., 2012, Wanisch et al., 2021, Dixit, 2019).