Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Analysis of the conditional mutual information in ballistic and diffusive non-equilibrium steady-states

Malouf, William Tiago Batista, John Goold, Gerardo Adesso, Gabriel Teixeira Landi

Published Jan 1, 2020• 10 citations
Publisher

Abstract

The conditional mutual information (CMI) $\mathcal{I}(A\! : \! C|B)$ quantifies the amount of correlations shared between $A$ and $C$ \emph{given} $B$. It therefore functions as a more general quantifier of bipartite correlations in multipartite scenarios, playing an important role in the theory of quantum Markov chains. In this paper we carry out a detailed study on the behavior of the CMI in non-equilibrium steady-states (NESS) of a quantum chain placed between two baths at different temperatures. These results are used to shed light on the mechanisms behind ballistic and diffusive transport regimes and how they affect correlations between different parts of a chain. We carry our study for the specific case of a 1D bosonic chain subject to local Lindblad dissipators at the boundaries. In addition, the chain is also subject to self-consistent reservoirs at each site, which are used to tune the transport between ballistic and diffusive. As a result, we find that the CMI is independent of the chain size $L$ in the ballistic regime, but decays algebraically with $L$ in the diffusive case. Finally, we also show how this scaling can be used to discuss the notion of local thermalization in non-equilibrium steady-states.

DOI
10.1088/1751-8121/ab93fd

Related papers

Analysis of the conditional mutual information in ballistic and diffusive non-equilibrium steady-states

William T. B. Malouf, John Goold, Gerardo Adesso +1 more

Abstract The conditional mutual information (CMI) I ( A : C | B ) quantifies the amount of correlations shared between A and C given B . It therefore functions as a more general quantifier of bipartite correlations in multipartite scenarios, playing an important role in the theory of quantum Markov chains. In this paper we carry out a detailed study on the behavior of the CMI in non-equilibrium steady-states (NESS) of a quantum chain placed between two baths at different temperatures. These results are used to shed light on the mechanisms behind ballistic and diffusive transport regimes and how they affect correlations between different parts of a chain. We carry our study for the specific case of a 1D bosonic chain subject to local Lindblad dissipators at the boundaries. In addition, the chain is also subject to self-consistent reservoirs at each site, which are used to tune the transport between ballistic and diffusive. As a result, we find that the CMI is independent of the chain size L in the ballistic regime, but decays algebraically with L in the diffusive case. Finally, we also show how this scaling can be used to discuss the notion of local thermalization in non-equilibrium steady-states.

arXiv:1809.09931•Journal of Physics A Mathematical and Theoretical•May 18, 2020•0 citations

Information transport in quantum non-equilibrium steady-states

William T. B. Malouf, John Goold, Gerardo Adesso +1 more

Non-equilibrium steady-states are characterized by the existence of macroscopic heat currents flowing between two or more reservoirs. However, the existence of these currents also implies that information is constantly being transmitted from one reservoir to the other. In this letter we address the transport of information in the context of quantum non-equilibrium steady states. We first show how the ability of the excitations to transfer information can be quantified by the conditional mutual information (CMI), a more general measure of tripartite correlations. We then apply this to an exactly soluble model allowing for both ballistic and diffusive behavior. Our approach allows us to compute the CMI for arbitrary sizes and thus find the scaling rules connecting information transport and diffusivity. Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.

•Sep 26, 2018•1 citation

Termodinâmica e informação em redes quânticas lineares

William T. B. Malouf

When a quantum system is coupled to several heat baths at different temperatures, it eventually reaches a non-equilibrium steady state (NESS) featuring stationary internal heat currents.From one side, these currents are responsible to cause decorehence and produce entropy in the system.However, their existence also induce correlations between different parts of the system.In this work, we explore this two-folded aspect of NESSs.Using phase-space techniques we calculate the Wigner entropy production on general linear networks of harmonic nodes.Working in the ubiquitous limit of weak internal coupling and weak dissipation, we obtain simple closedform expressions for the entropic contribution of each individual quasi-probability current.Our analysis also shows that, it is exclusively the (reversible) internal dynamics which maintain the stationary (irreversible) entropy production.From the informational point of view, we address how to quantify the amount of information that disconnected parts of a quantum chain share in a non-equilibrium steady-state.As we show, this is more precisely captured by the conditional mutual information (CMI), a more general quantifier of tripartite correlations than the usual mutual information.As an application, we apply our framework to the paradigmatic problem of energy transfer through a chain of oscillators subject to self-consistent internal baths that can be used to tune the transport from ballistic to diffusive.We find that the entropy production scales with different power law behaviors in the ballistic and diffusive regimes, hence allowing us to quantify what is the "entropic cost of diffusivity".We also compute the CMI for arbitrary sizes and thus find the scaling rules connecting information sharing and diffusivity.Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.

•Jun 27, 2019•0 citations

Clustering of Conditional Mutual Information and Quantum Markov Structure at Arbitrary Temperatures

Tomotaka Kuwahara

Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.

arXiv:2407.05835•Physical Review X•Sep 19, 2025•6 citations

Conditional Mutual Information and Information-Theoretic Phases of Decohered Gibbs States

Y. Zhang, Sarang Gopalakrishnan

Conditional mutual information (CMI) is a widely studied multipartite correlation measure with a connection to the Markov property. Intriguingly, CMI can be generated by local dissipation, a phenomenon closely related to teleportation. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for phase transitions, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this Letter, we provide the first rigorous results in this direction. We establish that CMI in high-temperature commuting Gibbs states subject to local dissipation decays exponentially under certain mild restrictions. Conversely, we show that the same setup can sustain long-range CMI at low temperatures. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions.

arXiv:2502.13210•Physical Review Letters•Sep 11, 2025•1 citation

Renormalization group and long-range conditional mutual information in hierarchical models

Yu-Hsueh Chen

A departure of a mixed quantum state from a local Gibbs description is generally invisible to local observables but can be detected by the conditional mutual information (CMI). Here we study the relationship between the renormalization group (RG) and CMI, and in particular, how RG constrains CMI. We first show that the CMI between nonadjacent regions $A$ and $C$, conditioned on the buffer region $B$, is UV-finite whenever the state admits a locally reversible RG with a fixed on-site Hilbert space dimension. We then study two hierarchical models that have long-range CMI and yet admit a simple RG description. The first model has a divergent Markov length at every temperature $0<T<\infty$ but nevertheless flows to an infinite-temperature product state under RG. The second model satisfies the local Markov condition while violating the global one and is stable against weak noise. At the critical noise strength, the two-point CMI decays only polynomially as a function of the system size, while the two-point mutual information vanishes.

quant-pharXiv:2609.02141•Sep 3, 2026•0 citations
The conditional mutual information (CMI) I(A ⁣: ⁣C∣B)\mathcal{I}(A\! : \! C|B)I(A:C∣B) quantifies the amount of correlations shared between AAA and CCC \emph{given} BBB. It therefore functions as a more general quantifier of bipartite correlations in multipartite scenarios, playing an important role in the theory of quantum Markov chains. In this paper we carry out a detailed study on the behavior of the CMI in non-equilibrium steady-states (NESS) of a quantum chain placed between two baths at different temperatures. These results are used to shed light on the mechanisms behind ballistic and diffusive transport regimes and how they affect correlations between different parts of a chain. We carry our study for the specific case of a 1D bosonic chain subject to local Lindblad dissipators at the boundaries. In addition, the chain is also subject to self-consistent reservoirs at each site, which are used to tune the transport between ballistic and diffusive. As a result, we find that the CMI is independent of the chain size LLL in the ballistic regime, but decays algebraically with LLL in the diffusive case. Finally, we also show how this scaling can be used to discuss the notion of local thermalization in non-equilibrium steady-states.
Abstract The conditional mutual information (CMI) I ( A : C | B ) quantifies the amount of correlations shared between A and C given B . It therefore functions as a more general quantifier of bipartite correlations in multipartite scenarios, playing an important role in the theory of quantum Markov chains. In this paper we carry out a detailed study on the behavior of the CMI in non-equilibrium steady-states (NESS) of a quantum chain placed between two baths at different temperatures. These results are used to shed light on the mechanisms behind ballistic and diffusive transport regimes and how they affect correlations between different parts of a chain. We carry our study for the specific case of a 1D bosonic chain subject to local Lindblad dissipators at the boundaries. In addition, the chain is also subject to self-consistent reservoirs at each site, which are used to tune the transport between ballistic and diffusive. As a result, we find that the CMI is independent of the chain size L in the ballistic regime, but decays algebraically with L in the diffusive case. Finally, we also show how this scaling can be used to discuss the notion of local thermalization in non-equilibrium steady-states.
Non-equilibrium steady-states are characterized by the existence of macroscopic heat currents flowing between two or more reservoirs. However, the existence of these currents also implies that information is constantly being transmitted from one reservoir to the other. In this letter we address the transport of information in the context of quantum non-equilibrium steady states. We first show how the ability of the excitations to transfer information can be quantified by the conditional mutual information (CMI), a more general measure of tripartite correlations. We then apply this to an exactly soluble model allowing for both ballistic and diffusive behavior. Our approach allows us to compute the CMI for arbitrary sizes and thus find the scaling rules connecting information transport and diffusivity. Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.
When a quantum system is coupled to several heat baths at different temperatures, it eventually reaches a non-equilibrium steady state (NESS) featuring stationary internal heat currents.From one side, these currents are responsible to cause decorehence and produce entropy in the system.However, their existence also induce correlations between different parts of the system.In this work, we explore this two-folded aspect of NESSs.Using phase-space techniques we calculate the Wigner entropy production on general linear networks of harmonic nodes.Working in the ubiquitous limit of weak internal coupling and weak dissipation, we obtain simple closedform expressions for the entropic contribution of each individual quasi-probability current.Our analysis also shows that, it is exclusively the (reversible) internal dynamics which maintain the stationary (irreversible) entropy production.From the informational point of view, we address how to quantify the amount of information that disconnected parts of a quantum chain share in a non-equilibrium steady-state.As we show, this is more precisely captured by the conditional mutual information (CMI), a more general quantifier of tripartite correlations than the usual mutual information.As an application, we apply our framework to the paradigmatic problem of energy transfer through a chain of oscillators subject to self-consistent internal baths that can be used to tune the transport from ballistic to diffusive.We find that the entropy production scales with different power law behaviors in the ballistic and diffusive regimes, hence allowing us to quantify what is the "entropic cost of diffusivity".We also compute the CMI for arbitrary sizes and thus find the scaling rules connecting information sharing and diffusivity.Finally, we discuss how this new perspective in the characterization of non-equilibrium systems may be applied to understand the issue of local equilibration in non-equilibrium states.
Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.
Conditional mutual information (CMI) is a widely studied multipartite correlation measure with a connection to the Markov property. Intriguingly, CMI can be generated by local dissipation, a phenomenon closely related to teleportation. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for phase transitions, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this Letter, we provide the first rigorous results in this direction. We establish that CMI in high-temperature commuting Gibbs states subject to local dissipation decays exponentially under certain mild restrictions. Conversely, we show that the same setup can sustain long-range CMI at low temperatures. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions.
A departure of a mixed quantum state from a local Gibbs description is generally invisible to local observables but can be detected by the conditional mutual information (CMI). Here we study the relationship between the renormalization group (RG) and CMI, and in particular, how RG constrains CMI. We first show that the CMI between nonadjacent regions AAA and CCC, conditioned on the buffer region BBB, is UV-finite whenever the state admits a locally reversible RG with a fixed on-site Hilbert space dimension. We then study two hierarchical models that have long-range CMI and yet admit a simple RG description. The first model has a divergent Markov length at every temperature 0<T<∞0<T<\infty0<T<∞ but nevertheless flows to an infinite-temperature product state under RG. The second model satisfies the local Markov condition while violating the global one and is stable against weak noise. At the critical noise strength, the two-point CMI decays only polynomially as a function of the system size, while the two-point mutual information vanishes.