Mixed-State Phase Transitions

• Mixed-state phase transitions are transitions between quantum phases in open quantum systems, marked by non-pure states resulting from decoherence and exhibiting unconventional critical behavior.
• The second Rényi conditional mutual information is employed as a robust metric to quantify the screening of quantum correlations and to identify phase boundaries.
• Numerical methods and tensor network simulations in a doubled Hilbert space provide practical tools for classifying and detecting mixed-state order and critical phenomena.

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 (Kuno et al., 4 May 2025).

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 (Kuno et al., 4 May 2025).