Purification Phase Transition in Nonunitary Dynamics

• Purification phase transition is a process in monitored quantum systems where repeated measurements drive the evolution from mixed to pure states, characterized by changes in purity, entanglement, and spectral features.
• The transition is diagnosed using metrics such as normalized purity and Rényi entropy, with purification times scaling logarithmically or exponentially depending on system parameters.
• This phenomenon bridges dynamical, spectral, and symmetry mechanisms, offering actionable insights into quantum error correction, scrambling, and the classification of mixed-state phases.

A purification phase transition is a transition in nonunitary quantum dynamics between a regime in which an initially mixed state remains mixed for asymptotically long times and a regime in which arbitrary initial states are driven to purity. In the monitored-dynamics literature this transition is usually formulated for conditional quantum trajectories or monitored channels starting from a maximally mixed state, and it is diagnosed by purity, Rényi entropies, purification time, or equivalent channel-theoretic quantities; in several important settings it coincides with an entanglement transition, while in others it does not. In post-selected non-Hermitian dynamics the same phenomenon can become a sharply defined spectral transition associated with exceptional points and $\mathcal{PT}$- or antiunitary-symmetry breaking rather than only a dynamical crossover (Gullans et al., 2019, Gopalakrishnan et al., 2020, Haldar et al., 2023).

1. Definition and basic diagnostics

In monitored many-body systems the basic object is the conditional state along a measurement record. For a Kraus history $\vec m$, the trajectory state is

$$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$

and purification asks whether this conditional state loses its mixedness under repeated monitored evolution starting from $\rho_0=\mathbb I/2^L$ or an equivalent maximally mixed state (Gullans et al., 2019). In non-Hermitian post-selected dynamics the same question is posed for

$$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$

with normalization imposed when observables are computed (Gopalakrishnan et al., 2020).

A standard diagnostic is the normalized purity

$\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$

or the corresponding second Rényi entropy $S_2=-\log_2\Pi$; in purifying phases $\Pi(t)\to1$, while in mixed phases it does not (Gopalakrishnan et al., 2020). Closely related monitored-circuit works use $\Tr(\rho^2)$ or $S_2=-\ln\Tr(\rho^2)$ as the operational purification observable, often together with a purification timescale defined by when $\vec m$0 vanishes or when purity approaches unity (Lóio et al., 2023).

The phase distinction is dynamical as well as asymptotic. In the purifying phase, local entropy decays at a system-size-independent rate and the total purification time is $\vec m$1 in the random Clifford setting, whereas in the mixed phase the purification time scales as

$\vec m$2

or more generally exponentially in the number of encoded degrees of freedom (Gullans et al., 2019). Other models realize weaker notions of purification: for example, integrable Floquet nonunitary circuits can purify on a timescale proportional to $\vec m$3, and monitored free-fermion circuits can show $\vec m$4 or $\vec m$5 rather than exponential or $\vec m$6 behavior (Zhang et al., 2023, Lóio et al., 2023).

Purification and entanglement transitions are related but not generically identical. In monitored SYK, the distinction is explicit: entanglement can revive after a completely projective measurement if measurements do not occur too often in time, but impurity cannot, because in that setting

$\vec m$7

This yields numerical evidence that purification and entanglement measurement-induced phase transitions are distinct phenomena rather than automatically equivalent reformulations of the same transition (Haldar et al., 2023).

2. Spectral and symmetry mechanisms

In deterministic post-selected dynamics the purification transition can be read directly from the spectrum of the non-Hermitian generator. For continuous measurement with post-selection, the conditional evolution is governed by an effective Hamiltonian $\vec m$8, and the transition occurs when increasing post-selection strength drives the many-body spectrum through an exceptional point. In the weak-post-selection phase all eigenvalues share the same imaginary part, so no eigenvector asymptotically dominates; an initially mixed state remains mixed and pure product states develop volume-law entanglement. In the strong-post-selection phase the imaginary parts split, the slowest-decaying eigenvalue dominates, and every initial state flows to a unique pure weakly entangled steady state (Gopalakrishnan et al., 2020).

The minimal illustration is the two-level model

$\vec m$9

for which the nontrivial eigenvalues satisfy

$$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$0

For $$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$1, both modes decay at the same rate and the normalized long-time state remains mixed; for $$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$2, decay rates split and purification occurs onto the slowest-decaying eigenvector; at $$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$3 the matrix becomes non-diagonalizable at an exceptional point. In this sense the purification transition is simultaneously dynamical, spectral, and symmetry-based: it is identified with spontaneous $$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$4-symmetry breaking in the many-body spectrum (Gopalakrishnan et al., 2020).

Floquet nonunitary circuits generalize this mechanism but also show that symmetry breaking alone does not fix the nature of the purifying phase. In the $$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$5D Floquet circuits with antiunitary symmetry $$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$6 satisfying

$$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$7

the symmetry-breaking lines

$$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$8

coincide with purification transitions, yet three distinct outcomes occur. In the Gaussian limit $$\rho_{\vec m}=\frac{K_{\vec m}\rho K_{\vec m}^\dag}{p_{\vec m}},\qquad p_{\vec m}=\operatorname{tr}(K_{\vec m}^\dag K_{\vec m}\rho),$$9, antiunitary symmetry breaking leaves an exponentially large set of co-dominant eigenvalues and there is no purification. In interacting integrable models, the broken phase is weakly purifying with

$\rho_0=\mathbb I/2^L$0

With an antiunitary-symmetric integrability-breaking perturbation, the same broken phase crosses over to strong purification with $\rho_0=\mathbb I/2^L$1 and $\rho_0=\mathbb I/2^L$2 for sufficiently large $\rho_0=\mathbb I/2^L$3 (Zhang et al., 2023).

A plausible implication is that the decisive object is not symmetry breaking alone but the multiplicity structure of the dominant singular or Floquet modes. The data surveyed here support that interpretation: exceptional-point or antiunitary-symmetry breaking creates the possibility of purification, while integrability, Gaussian structure, and eigenvector nonorthogonality determine whether the result is strong purification, weak purification, or no purification at all (Gopalakrishnan et al., 2020, Zhang et al., 2023).

3. Representative models and phase diagrams

The modern literature realizes purification phase transitions in random circuits, chaotic Hamiltonian systems with projective measurements, solvable Brownian models, free-fermion circuits, and spatially disordered hybrid circuits. The table lists representative results that are explicitly reported in the cited works.

Setting	Critical data	Distinctive outcome
1D random Clifford monitored circuit (Gullans et al., 2019)	$\rho_0=\mathbb I/2^L$4, $\rho_0=\mathbb I/2^L$5	Mixed phase with $\rho_0=\mathbb I/2^L$6; pure phase with system-size-independent local purification rate
Large-$\rho_0=\mathbb I/2^L$7 hybrid Brownian circuit (Bentsen et al., 2021)	$\rho_0=\mathbb I/2^L$8, $\rho_0=\mathbb I/2^L$9	Mean-field second-order purification transition from replica-symmetry breaking
Chaotic TFIM with projective measurements (Kuno et al., 2022)	$$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$0, $$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$1	Purity-growth crossover and TMI crossing identify mixed and purified phases
Majorana monitored free fermions (Lóio et al., 2023)	$$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$2, $$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$3	Mixed phase with $$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$4
Dirac monitored free fermions (Lóio et al., 2023)	$$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$5	BKT-like criticality and $$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$6 with $$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$7
Disordered hybrid random Clifford circuit (Anzai et al., 17 Jul 2025)	uniform: $$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$8; modulated: $$\rho(t)=T(t)\rho_0T^\dagger(t),\qquad T(t)=e^{-iH_{\mathrm{eff}}t},$$9	Spatial non-uniformity changes the universality class of the purification transition

These models do not realize a single canonical mixed phase. In the random Clifford circuit the mixed phase carries a nonzero residual entropy density and exponential purification time (Gullans et al., 2019). In the large-$\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$0 Brownian circuit the low-measurement phase is controlled by instantons in a $\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$1D Ising-like effective theory, with

$\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$2

near the transition and a finite-$\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$3 late-time form

$\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$4

once multi-instanton sectors are summed (Bentsen et al., 2021). In the chaotic Hamiltonian model the transition is seen both in the law of purity increase and in the saturation value of the normalized tripartite mutual information, with the purification-phase light cone remaining ballistic while the density of information propagation is reduced on average by projective measurements (Kuno et al., 2022).

Free-fermion circuits show that symmetry can qualitatively change the phase structure. For monitored Majorana circuits with only $\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$5 symmetry, the mixed phase is long-lived but not exponentially stable, with $\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$6 and a finite residual entropy in the window $\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$7. For U(1)-symmetric Dirac circuits, the mixed phase purifies sublinearly at any measurement rate,

$\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$8

and the transition is consistent with BKT criticality rather than conventional power-law scaling (Lóio et al., 2023).

Spatial non-uniformity changes the critical behavior again. In hybrid random Clifford circuits with site-dependent measurement probabilities $\Pi(t)=\frac{\Tr[\rho(t)^2]}{\Tr[\rho(t)]^2},$9, the purification transition survives but its correlation-length exponent changes from the uniform case to a disorder-modified value exceeding two, while spatial modulation of the two-site random Clifford gates can induce a different pure-like phase with residual short-range entanglement (Anzai et al., 17 Jul 2025).

4. Universality, replicas, and limits of the standard mixed-to-pure narrative

One major line of work concerns universal purification dynamics not at the transition itself but deep in the weak-measurement or volume-law phase. In that regime the effective purification time $S_2=-\log_2\Pi$0 or, more generally,

$S_2=-\log_2\Pi$1

sets the scaling variable

$S_2=-\log_2\Pi$2

Using the replica trick, slow purification is mapped to a one-dimensional statistical mechanics of permutation-valued spins or dilute kinks, and two universality classes emerge: Born-rule monitored dynamics, corresponding to the replica limit $S_2=-\log_2\Pi$3, and products of independent random matrices, corresponding to $S_2=-\log_2\Pi$4. They share the same effective adjacency operator in the scaling regime but have different scaling functions because the replica limits differ (Luca et al., 2023).

The generic early-time universal form in that slow-purification regime is logarithmic. For example, the paper reports

$S_2=-\log_2\Pi$5

with $S_2=-\log_2\Pi$6 having mean and variance $S_2=-\log_2\Pi$7 for $S_2=-\log_2\Pi$8; for the second Rényi entropy the Born-rule and independent-random-matrix classes already differ at $S_2=-\log_2\Pi$9 (Luca et al., 2023). A closely related random-matrix treatment of real nonunitary quantum processes shows that the weak phase further splits by symmetry class: for orthogonal dynamics $\Pi(t)\to1$0,

$\Pi(t)\to1$1

while for unitary dynamics $\Pi(t)\to1$2,

$\Pi(t)\to1$3

again in the universal scaling limit inside the weak-measurement phase rather than at the critical point (Gerbino et al., 11 Mar 2026).

These results already imply that “mixed phase” is not a single universal object. A stronger caveat comes from the exactly solvable monitored all-to-all model with weak measurements in random directions. At fixed replica number $\Pi(t)\to1$4, that model shows what looks like an ordinary measurement-driven transition into a purifying or disentangled phase. But in the physical $\Pi(t)\to1$5 limit the action becomes

$\Pi(t)\to1$6

with nonanalytic small-$\Pi(t)\to1$7 behavior $\Pi(t)\to1$8. The consequence is that the putative unbroken saddle at $\Pi(t)\to1$9 is unstable for every finite $\Tr(\rho^2)$0, so the purifying phase disappears and the purification time remains

$\Tr(\rho^2)$1

for all finite measurement strengths (Giachetti et al., 2023).

This directly addresses a common overgeneralization. A purification phase transition is not guaranteed by weak-versus-strong monitoring alone; it can depend qualitatively on the replica prescription, symmetry class, integrability, and whether one studies a Born-rule trajectory ensemble, a deterministic non-Hermitian generator, or a forced or annealed random-matrix process. A plausible implication is that “purification phase transition” names a family of related but not universally identical phenomena rather than a single fixed universality class (Luca et al., 2023, Giachetti et al., 2023, Gerbino et al., 11 Mar 2026).

5. Scrambling, channel capacity, and recoverability

The purification transition has a precise information-theoretic interpretation. For monitored channels with the measurement record retained in an ancilla sector,

$\Tr(\rho^2)$2

so trajectory-averaged entropy equals coherent information for the unraveled channel. In the mixed phase the late-time entropy density stays nonzero on polynomial time scales,

$\Tr(\rho^2)$3

which implies an emergent code space of dimension $\Tr(\rho^2)$4; in the pure phase that capacity density vanishes (Gullans et al., 2019).

The large-$\Tr(\rho^2)$5 Brownian hybrid circuit makes the same idea concrete in subsystem language. In the mixed phase there is a threshold fraction $\Tr(\rho^2)$6 such that only subsystems larger than $\Tr(\rho^2)$7 retain information about the reference on polynomial times. Near criticality,

$\Tr(\rho^2)$8

and the averaged Rényi-2 mutual information between the erased complement and the reference vanishes,

$\Tr(\rho^2)$9

consistent with a dynamically generated quantum error-correcting encoding (Bentsen et al., 2021).

Scrambling diagnostics sharpen this picture. In the chaotic measured Hamiltonian, the long-time normalized TMI remains negative on both sides of the transition but its saturation value is much more weakly negative in the purified phase, while the spacetime light-cone pattern remains essentially ballistic. Measurements therefore reduce the amount of scrambling without visibly deforming the propagation front (Kuno et al., 2022). This matches the mixed-phase interpretation in which purification is prevented not by absence of transport but by nonlocal encoding.

Noisy monitored circuits extend the same logic from trajectory purity to channel recoverability. For the noisy brick-wall circuit with measurements, conventional noise, and ancilla-assisted “quantum-enhanced” operations, the coherent information

$S_2=-\ln\Tr(\rho^2)$0

undergoes a transition from a recoverable phase with $S_2=-\ln\Tr(\rho^2)$1 to an irrecoverable phase with $S_2=-\ln\Tr(\rho^2)$2. For the reported Clifford simulations at $S_2=-\ln\Tr(\rho^2)$3, the critical values include

$S_2=-\ln\Tr(\rho^2)$4

at $S_2=-\ln\Tr(\rho^2)$5, and

$S_2=-\ln\Tr(\rho^2)$6

for depolarizing noise at $S_2=-\ln\Tr(\rho^2)$7 (Qian et al., 2024). This is a channel-capacity generalization of purification physics: the order parameter is no longer simply whether the state becomes pure, but whether quantum information remains transmittable at all.

6. Broader purification perspectives and adjacent usages

A broader strand of work uses purification not as the object that dynamically grows under monitoring, but as an organizing principle for mixed-state phases themselves. In the purification-based classification of one-dimensional open systems with $S_2=-\ln\Tr(\rho^2)$8 symmetry, a mixed state is represented as

$S_2=-\ln\Tr(\rho^2)$9

and the enlarged pure system has eight purified fixed-point Hamiltonians labeled by

$\vec m$00

Interpolations between these vertices form a cube of mixed-state phases, with edge transitions corresponding to single topological indices and two distinct edge criticalities reported as $\vec m$01 for trivial$\vec m$02SWSSB and $\vec m$03 for the conventional SPT-type transition (Guo et al., 25 Feb 2026).

A related purification perspective maps spontaneous strong-to-weak symmetry breaking in mixed states to SPT order in the purified state. In the two-dimensional global $\vec m$04 construction, the mixed-state Rényi-2 correlator is mapped to a classical Ising correlator and the decoherence threshold is

$\vec m$05

This is a sharp transition in mixed-state order diagnosed through purification, but it is not the standard monitored-circuit purification-time transition (Sala et al., 2024).

An even more distant but conceptually adjacent usage appears in conformal field theory. There, a pure state $\vec m$06 is constructed from a mixed $\vec m$07 by subtracting “undetectable regions” rather than adding ancillas, and the resulting entropy is

$\vec m$08

This quantity equals the entanglement wedge cross-section in the stated setting and undergoes a transition at

$\vec m$09

which is a purification-related phase transition in holographic mixed-state entanglement rather than in monitored dynamics (Jiang et al., 2024).

Taken together, these works suggest two complementary meanings of purification phase transition. The dominant one in quantum dynamics concerns the competition between scrambling and nonunitarity, producing mixed, weakly purifying, strongly purifying, or non-purifying phases. A broader usage treats purification as a structural embedding of mixed states into enlarged pure states, allowing mixed-state topology, symmetry breaking, and holographic entanglement transitions to be reformulated in terms of purified parent states. A plausible implication is that the term has become a bridge concept linking nonunitary dynamics, spectral theory, quantum error correction, open-system phase structure, and mixed-state entanglement geometry (Guo et al., 25 Feb 2026, Sala et al., 2024, Jiang et al., 2024).