---
title: Mixed-State Phase Transitions
url: https://www.emergentmind.com/topics/mixed-state-phase-transition
type: topic
---

# Mixed-State Phase Transitions

Mixed-state phase transitions are transitions between quantum phases that manifest in non-pure (mixed) states, typically arising in open systems subject to decoherence or noise, and reveal critical phenomena with no strict analogue in closed-system, pure-state quantum physics. Recent developments have demonstrated the utility of the second Rényi conditional mutual information (CMI) as a tool for diagnosing phase transitions and classifying non-trivial mixed-state quantum phases, extending the conceptual framework of quantum phase transitions to the regime where a direct interpretation in terms of local order parameters and pure-state entanglement is not generally possible [2505.02125].

## 1. Theoretical Framework for Mixed-State Phase Transitions

Mixed states inherently emerge when quantum systems interact with an environment, leading to decoherence. In such scenarios, the traditional dichotomy of quantum phases classified by pure-state expectation values or entanglement is replaced by a richer structure that is sensitive to the interplay of local recoverability, symmetry properties, and the structure of quantum channels. Non-trivial mixed-state phenomena include average symmetry–protected topological (ASPT) phases, strong-to-weak spontaneous symmetry breaking (SWSSB) unique to open systems, and intrinsic topological phases supported by density matrices that cannot be purified to conventional closed-system order.

To quantify phase transitions in this context, the second Rényi CMI, denoted as $I^{(2)}(A:C|B)$, is employed. For a tripartition of the system into contiguous regions $A$, $B$, and $C$, and for reduced density matrices $\rho_X$, it is defined as:
\[
I^{(2)}(A:C|B) = S^{(2)}_{AB} + S^{(2)}_{BC} - S^{(2)}_B - S^{(2)}_{ABC}
\]
where $S^{(2)}_X = -\log \operatorname{Tr}[\rho_X^2]$ is the second Rényi entropy. This measure captures how quantum correlations between $A$ and $C$ are “screened” by $B$ and, crucially, its spatial decay encodes information about the "gap" of the mixed state.

## 2. Numerical Methodology: Doubled Hilbert Space and CMI Computation

A central computational approach involves mapping the mixed state $\rho$ into a vector $|\rho\rangle\rangle$ within a doubled Hilbert space $\mathcal{H}_u \otimes \mathcal{H}_\ell$ via the Choi–Jamiołkowski isomorphism:
\[
\rho \mapsto |\rho\rangle\rangle = \frac{1}{\sqrt{\mathrm{dim}[\rho]}} \sum_k |k\rangle \otimes \rho |k\rangle
\]
This facilitates the efficient evaluation of the second Rényi entropy for subsystems by acting with maximal depolarization channels (local projectors) on the complementary region, leading to:
\[
S^{(2)}_X = -\log \left( d_{\bar{X}} \cdot \operatorname{Tr}[\rho_X^2] \right)
\]
where $d_{\bar{X}}$ is the Hilbert space dimension of the complement $\bar{X}$. This structure enables the evaluation of $I^{(2)}(A:C|B)$ using powerful matrix product state (MPS) representations and contractible tensor networks, allowing for robust detection of critical signatures in large open systems and quantum circuits.

## 3. Classification and Case Studies: Markov Length and Phase Diagrams

The second Rényi conditional mutual information exhibits characteristic behaviors depending on the underlying phase structure:

- In symmetry-protected topological (SPT) or ASPT mixed states, $I^{(2)}(A:C|B)$ decays exponentially with the width $r$ of region $B$:
  \[
  I^{(2)}(A:C|B)(r) \sim \exp(-r/\xi^{(2)}_M)
  \]
  Here, $\xi^{(2)}_M$ is termed the second Rényi Markov length and serves as an information-theoretic mixed-state correlation length or “gap.”
  
- At a mixed-state phase transition (e.g., between ASPT and trivial phases or across SWSSB boundaries), $\xi^{(2)}_M$ diverges, leading to algebraic decay or saturation in the CMI, signaling criticality.

- In specific scenarios such as the TFIM under competition of $ZZ$ and $X$ decoherence, the CMI transitions from exponential decay (finite Markov gap) in the paramagnetic regime to a constant (saturating at $\ln 2$) in the SWSSB phase, indicative of a breakdown in local recovery and a qualitative change in order.

These behaviors provide a classification axis for mixed-state phases: finite $\xi^{(2)}_M$ indicates recoverable phases (from an information-theoretic perspective), while a diverging or infinite $\xi^{(2)}_M$ marks the onset of phases where local Petz recoverability fails.

| Phase Type              | Behavior of $I^{(2)}(A:C|B)$       | Markov Length $\xi^{(2)}_M$         |
|------------------------|-------------------------------------|-------------------------------------|
| ASPT, trivial, paramagnetic | Exponential decay                    | Finite                              |
| SWSSB                   | Saturates at constant (e.g., $\ln 2$) | Diverges (infinite)                 |
| At criticality          | Power-law or weak decay                | Diverges                            |

## 4. Illustrative Models: Cluster Hamiltonians and TFIM with Decoherence

- **Cluster model under odd-site Z decoherence:** Starting from a $\mathbb{Z}_2\times\mathbb{Z}_2$ SPT phase, local dephasing channels render the pure ground state into a mixed state. Numerical analysis tracks $I^{(2)}(A:C|B)$ as a function of some driving parameter (e.g., transverse field $h_x$), detecting sharp peaks where $\xi^{(2)}_M$ diverges, effectively locating the transition between ASPT and trivial mixed-state phases.

- **TFIM with $ZZ$ and $X$ decoherence:** By tuning the probabilities $p_{zz}$ and $p_x$ of decoherence channels, the system transitions between a paramagnetic mixed phase (finite Markov length, exponential decay of CMI) and the SWSSB phase (CMI saturation). The latter displays robust, size-independent CMI, revealing the universal non-decaying signature of strong-to-weak symmetry breaking mixed-state order.

These archetypal calculations—using the doubled Hilbert space CMI formalism and MPS simulations—underscore the broad applicability and diagnostic power of the second Rényi CMI and Markov length.

## 5. Implications for Quantum Information and Open Quantum Systems

The Rényi CMI and Markov length serve as operationally meaningful diagnostics for mixed-state phases. A short Markov length implies efficient recovery of global properties from local information (i.e., robust quantum memory against decoherence, via the Petz map). Divergence of the Markov length marks a qualitative change in the structure of correlations and signals a breakdown of such recoverability. This provides a precise information-theoretic translation of the phase boundary for mixed-state phases, extending beyond pure-state entanglement-based characterizations.

The techniques outlined also provide an avenue for the systematic classification of mixed-state strongly correlated systems and for detecting novel critical phenomena in measurement-induced phase transitions, open-system SPT orders, and mixed-state topological order. 

## 6. Future Directions and Open Questions

Advancing this framework raises several questions and opportunities:

- Refinement of the operational meaning of the Markov length, particularly its relationship to recoverability and the construction of optimal recovery maps in practical settings.
- Extension to higher Rényi indices and multipartite CMI to probe richer correlation structures in mixed states.
- Application to measurement-induced criticality and non-equilibrium phase transitions in hybrid quantum circuits.
- Exploration of connections between mixed-state CMI-based gaps and conventional physical correlation lengths, as well as their role in determining the stability and coherence times of quantum memories subject to general non-unitary channels.
- Experimental validation in programmable quantum simulators, where open-system dynamics and mixed-state properties are accessible.

## 7. Summary

The second Rényi conditional mutual information and its associated Markov length provide a rigorous, efficiently computable probe for nontrivial mixed-state quantum phases and their transitions. By employing the doubled Hilbert space formalism and tensor network methods, complex open-system phenomena—such as the emergence of ASPT, SWSSB, and other intrinsic mixed-state orders—can be characterized and classified with precision. The divergence of the Markov length at criticality offers a clear operational signature of mixed-state phase transitions, anchoring a new methodology for understanding and engineering robust quantum order in the presence of decoherence [2505.02125].

Source: https://www.emergentmind.com/topics/mixed-state-phase-transition