Attempts to understand quantum phase transitions in open systems usually rely on real‑time Lindbladian evolution, which tracks how a quantum state changes as it relaxes toward a steady state. This approach is powerful for studying decoherence, dissipation and long‑time behaviour, but it often fails to reveal the deeper structure of the system including the phase transitions, critical points and hidden quantum order that define its underlying physics.

In this work, the researchers introduce a new framework called imaginary‑time Lindbladian evolution, which allows them to define and classify quantum phases in open systems using the spectrum of an imaginary‑Liouville superoperator. This approach works not only for pure ground states but also for finite‑temperature Gibbs states of stabilizer Hamiltonians, showing its relevance for realistic, mixed‑state conditions.

A key diagnostic in their method is the imaginary‑Liouville gap, the spectral gap between the lowest and next‑lowest decay modes. When this gap closes, the system undergoes a phase transition, a change that is accompanied by diverging correlation lengths and nonanalytic shifts in physical observables. The closing of this gap also coincides with the divergence of the Markov length, a recently proposed indicator of criticality in open quantum systems.

To demonstrate the power of their framework, the researchers map out phase diagrams for systems with

$Z_2^{\sigma }\times Z_2^{\tau }$

symmetry, capturing both spontaneous symmetry breaking and average symmetry‑protected topological phases. Their method reveals universal critical behaviour that real‑time Lindbladian steady states fail to detect, highlighting why imaginary‑time evolution fills a missing piece in the theory of open‑system phases.

Importantly, the authors emphasise that real‑time Lindbladians remain essential for modelling dissipation in practical settings. Their new framework complements this conventional approach, offering a systematic way to study phase transitions in open systems. They also outline how phase diagrams can be constructed using both bottom‑up (state‑based) and top‑down (Hamiltonian‑based) strategies, illustrating the method with a dissipative transverse‑field Ising model.

Overall, this work provides a unified and versatile way to understand quantum phases in open systems, revealing critical behaviour and topological structure that were previously inaccessible. It opens new directions for studying mixed‑state quantum matter and advances the theoretical foundations needed for future quantum technologies.

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Focus on Quantum Entanglement: State of the Art and Open Questions guest edited by Anna Sanpera and Carlo Marconi (2025-2026)

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In the macroscopic world, we see irreversible processes everywhere, heat flowing from hot to cold, gases mixing, systems decaying. Yet at the microscopic level, quantum mechanics is perfectly reversible, with its equations running equally well forwards and backwards in time. How then, does irreversibility emerge from fundamentally reversible dynamics?

A common explanation is coarse-graining, which simplifies a complex system by ignoring microscopic details and focusing only on large-scale behaviour. To make the micro–macro divide precise, however, one must first define what “macroscopic” means. Here it is given a quantitative inferential meaning: a state is macroscopic if it is perfectly inferable from the perspective of a specified measurement and prior. Central to this framework is a coarse-graining map built from the measurement and its optimal Bayesian recovery via the Petz map; macroscopic states are precisely its fixed points, turning macroscopicity into a sharp condition of perfect inferability. This construction is grounded in Bayesian retrodiction, which infers what a system likely was before it was measured, together with an observational deficit that quantifies how much information is lost in forming a macroscopic description.

States that are macroscopically inferable can be characterised in several equivalent ways, all tied to to a new measure of disorder called macroscopic entropy, which captures how irreversible, or “uninferable”, a macroscopic process appears from the observer’s perspective. This perspective is formalised through inferential reference frames, built from the combination of a prior and a measurement, which determine what an observer can and cannot recover about the underlying quantum state.

The researchers also develop a resource theory of microscopicity, treating macroscopic states as free and identifying the operations that cannot generate microscopic detail. This unifies and extends existing resource theories of coherence, athermality, and asymmetry. They further introduce observational discord, a new way to understand quantum correlations when observational power is limited, and provide conditions for when this discord vanishes.

Altogether, this work reframes macroscopic irreversibility as an information-theoretic phenomenon, grounded not in a fundamental dynamical asymmetry but in an inferential asymmetry arising from the observer’s limited perspective. It offers a unified way to understand coarse-graining, entropy, and the emergence of classical behaviour from quantum mechanics. It deepens our understanding of time’s direction and has implications for quantum computing, thermodynamics, and the study of quantum correlations in realistic, constrained settings.

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Focus on Quantum Entanglement: State of the Art and Open Questions guest edited by Anna Sanpera and Carlo Marconi (2025-2026)

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