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

Yifan Zhang, Sarang Gopalakrishnan

Abstract

Classical and quantum Markov networks -- including Gibbs states of commuting local Hamiltonians -- are characterized by the vanishing of conditional mutual information (CMI) between spatially separated subsystems. Adding local dissipation to a Markov network turns it into a \emph{hidden Markov network}, in which CMI is not guaranteed to vanish even at long distances. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for teleportation, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this work we provide the first rigorous results in this direction. We establish that CMI in high-temperature Gibbs states subject to local dissipation decays exponentially, (i) for classical Hamiltonians subject to arbitrary local transition matrices, and (ii) for commuting local Hamiltonians subject to unital channels that obey certain mild restrictions. Conversely, we show that low-temperature hidden Markov networks can sustain long-range CMI. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions. We also show several applications in quantum information and many-body physics.

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

Abstract

Classical and quantum Markov networks -- including Gibbs states of commuting local Hamiltonians -- are characterized by the vanishing of conditional mutual information (CMI) between spatially separated subsystems. Adding local dissipation to a Markov network turns it into a \emph{hidden Markov network}, in which CMI is not guaranteed to vanish even at long distances. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for teleportation, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this work we provide the first rigorous results in this direction. We establish that CMI in high-temperature Gibbs states subject to local dissipation decays exponentially, (i) for classical Hamiltonians subject to arbitrary local transition matrices, and (ii) for commuting local Hamiltonians subject to unital channels that obey certain mild restrictions. Conversely, we show that low-temperature hidden Markov networks can sustain long-range CMI. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions. We also show several applications in quantum information and many-body physics.

Paper Structure

This paper contains 18 sections, 32 theorems, 135 equations, 5 figures.

Table of Contents

  1. Generalized Cluster Expansion Under Local Unital, Commutation-Preserving Channels
  2. Preliminaries
  3. Generalized Cluster Expansion
  4. Connected Clusters
  5. Bounding Cluster Derivative Under Unital Channels
  6. Proof of Theorem \ref{['thm:decay_cmi_informal']} (formally Theorem \ref{['thm:decay_cmi']})
  7. Finite Markov Length in Classical Gibbs Distribution at High Temperature
  8. Converting to Pinned Hamiltonian
  9. Generalized Cluster Expansions Under Pinning
  10. Connected Clusters
  11. Bounding Cluster Derivative Under Pinning
  12. Proof of Theorem \ref{['thm:classical_decay_cmi']}
  13. Long-Range CMI at Low Temperature Through Fault-Tolerant MBQC
  14. Preliminaries
  15. Proof of Theorem \ref{['thm:long_range_cmi_informal']} (formally Theorem \ref{['thm:long_range_cmi']})
  16. ...and 3 more sections

Key Result

Theorem 1

(Informal) Consider the Gibbs state $\rho \propto e^{- \beta H}$ of some local commuting Hamiltonian $H$. Let $\mathcal{E} = \otimes_i \mathcal{E}_i$, where $\mathcal{E}_i$ is a unital channel acting on site $i$ We demand that $\{\mathcal{E}_i\}$ is commutation-preserving. Consider any three subsyst Where $d_{AC}$ is the distance between $A$ and $C$, and $\xi=O(\beta/\beta_c)$ is called the Markov

Figures (5)

  • Figure 1: (a) An example of partitioning a Markov network exhibiting conditional independence. $\textcolor{Green}{Y}$ separates $\textcolor{Cerulean}{X}$ and $\textcolor{Goldenrod}{Z}$ and we have $I(\textcolor{Cerulean}{X}:\textcolor{Goldenrod}{Z}|\textcolor{Green}{Y})=0$. (b) An example of hidden Markov chain. $WXYZ$ forms a Markov chain and each site is subject to a transition matrix. (c) An example of quantum hidden Markov chain. $\rho_{ABCD}$ forms a quantum Markov chain and each site is subject to a quantum channel. Note that we use the double leg to denote the bra and ket, so fixing the index $mn$ corresponds to choosing $\ket{m}\bra{n}$. (d) An example of classical hidden Markov network. Grey dashed lines separate sites and we partition the system into $\textcolor{Cerulean}{X}$, $\textcolor{Green}{Y}$, and $\textcolor{Goldenrod}{Z}$. The white boxes represent a channel that takes two bits as input and outputs their parity. (e) An example of quantum hidden Markov network. Grey dashed lines separate sites and we partition the system into $\textcolor{Cerulean}{A}$, $\textcolor{Green}{B}$, and $\textcolor{Goldenrod}{C}$. The circuit rotates the computation basis to the Bell basis and then measures it.
  • Figure 2: (a) Geometry of the MBQC cluster state we consider. The bulk region $\textcolor{Green}{B}$ is measured, generating entangled code states between $\textcolor{Cerulean}{A}$ and $\textcolor{Goldenrod}{C}$ which sit at the boundary. We show a three-dimensional cluster state as an example, but in general the dimensionality is arbitrary as long as a "time-like" direction exist, along which quantum information propagates. (b) The unit cell of the three-dimensional cluster state that encodes a Toric code in MBQC. The graph that defines the cluster state has vertices on the edge and the face of the unit cell.
  • Figure 3: (a) An example of a cluster with one element supported on qubit 1, 2, one element supported on qubit 2, 3, 4, 5, and two elements supported on qubit 4, 6. (b) An example of a disconnected cluster. (c) An example of a cluster connecting $\textcolor{Cerulean}{A}$ and $\textcolor{Goldenrod}{C}$ but is disconnected. (d) An example of a connected cluster but does not connect $\textcolor{Cerulean}{A}$ and $\textcolor{Goldenrod}{C}$.
  • Figure 4: (a) The factor graph of a Gibbs states with four bits in a line and nearest-neighbor interactions. We partition the four bits into $\textcolor{Cerulean}{X}$, $\textcolor{Green}{Y}$, and $\textcolor{Goldenrod}{Z}$. (b) The marginal distribution on $\textcolor{Cerulean}{X}\textcolor{Goldenrod}{Z}$. (c) The Gibbs distribution is subject to two transition matrices $T_2$ and $T_3$ acting on the two bits in $\textcolor{Green}{Y}$. (d) The distribution on $\textcolor{Cerulean}{X}\textcolor{Goldenrod}{Z}$ conditioned on post-selecting on $y_2y_3$ in $\textcolor{Green}{Y}$. (e) Identifying the post-selected transition matrix as a new term in the factor graph. (f) The post-selected marginal distribution on $\textcolor{Cerulean}{X}\textcolor{Goldenrod}{Z}$ becomes the marginal distribution of a new Gibbs distribution, after applying the identification in (e).
  • Figure 5: (a) A Quasi-local channel that connects two states. (b) A schematics of mixed-state phase diagram

Theorems & Definitions (52)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Theorem A.1
  • Lemma A.1
  • Proposition A.1
  • proof
  • Proposition A.2
  • ...and 42 more