Lindbladian Skin Effect in Open Quantum Systems

• Lindbladian Skin Effect is the boundary localization of Liouvillian eigenmodes in operator space, resulting in gap-independent slow relaxation.
• It arises from non-Hermitian dynamics where biorthogonal left and right eigenoperators localize on opposite boundaries, amplifying transient modes exponentially with system size.
• Model studies, from asymmetric dissipative hopping to SSH chains, underscore its topological characteristics and implications for dynamical relaxation in open quantum systems.

The Lindbladian skin effect, also called the Liouvillian skin effect, is the boundary localization of Liouvillian eigenmodes under open boundary conditions in Markovian open quantum systems governed by a Lindblad master equation. It is an operator-space phenomenon: the localized objects are eigenoperators of the Lindbladian superoperator, and their eigenvalues are complex relaxation rates rather than energies. Its central significance is dynamical. In systems with boundary-localized slow modes, the longest relaxation time depends not only on the Liouvillian gap $\Delta$ but also on the localization length $\xi$, so that relaxation can become parametrically slow without Liouvillian gap closing, with $\tau \sim \Delta^{-1}(1+L/\xi)$ (Haga et al., 2020). Subsequent work established exactly solvable many-body realizations, generalized-boundary-condition criteria, topological formulations in terms of point-gap winding, finite-temperature realizations in two-dimensional electron systems, non-Markovian thick-skin regimes, and monitored-trajectory entanglement structures tied to skin accumulation (Mao et al., 2024, Shigedomi et al., 23 May 2025, Kuo et al., 2024, Passarelli et al., 8 Jun 2026).

1. Formal setting in operator space

Open quantum systems in the Markovian regime are described by the Lindblad master equation

$\dot{\rho}(t)=\mathcal{L}(\rho(t)),$

with

$$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$

Right and left eigenoperators are defined by

$$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$

with biorthogonality under the Hilbert–Schmidt inner product. The steady state is the right zero mode, $\mathcal{L}(\rho_{\mathrm{ss}})=0$, and the Liouvillian gap is

$\Delta=|\mathrm{Re}\,\lambda_1|,$

where eigenvalues are ordered by real part (Haga et al., 2020).

Vectorization makes the operator-space structure explicit. Under Choi–Jamiołkowski vectorization, $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$, the Lindbladian becomes a non-Hermitian matrix acting on a doubled Hilbert space. This representation is especially useful in quadratic and integrable problems, where one can identify effective non-Hermitian matrices whose spectra control the Liouvillian rapidities and the spatial profiles of normal master modes (Mao et al., 2024).

The skin effect is defined by exponential concentration of Liouvillian eigenoperators near a boundary under open boundary conditions. In the one-dimensional setting used to expose the mechanism, the slowest right and left modes may localize near opposite edges, with representative envelopes

$$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$

Under periodic boundary conditions the corresponding eigenmodes are extended and the skin effect is absent (Haga et al., 2020).

2. Biorthogonality, boundary localization, and anomalous relaxation

The defining dynamical mechanism of the Lindbladian skin effect is the combination of boundary localization and biorthogonality. Because the slow right and left modes can localize at opposite edges, their overlap is exponentially small in system size,

$\xi$0

For suitably chosen initial states, the expansion coefficient of the slowest mode therefore becomes exponentially large, $\xi$1, even when the Liouvillian gap stays finite. The late-time relaxation criterion is then controlled by $\xi$2 rather than by $\xi$3 alone, which yields

$\xi$4

Equivalently, the asymptotic relaxation velocity satisfies $\xi$5 (Haga et al., 2020).

This result corrects the common gap-only intuition. The statement $\xi$6 is valid when eigenmodes are extended or when left-right overlaps remain $\xi$7, but it fails in the presence of skin-localized slow modes. In this sense, the Lindbladian skin effect decouples transient relaxation scales from the asymptotic decay rate. The same work also noted links to non-normal amplification, pseudospectra, cutoff phenomena in Markov chains, and metastability in open quantum systems (Haga et al., 2020).

A related numerical study of boundary-sensitive $\xi$8-symmetric Lindbladians found the same structural mechanism in single-particle and many-body settings. With feedback, the Liouvillian gap under open boundaries remains finite as $\xi$9, while the relaxation time scales as $\tau \sim \Delta^{-1}(1+L/\xi)$0 and the trace-distance decay shows a cutoff phenomenon: the distance remains approximately constant up to $\tau \sim \Delta^{-1}(1+L/\xi)$1 and then decays rapidly in an $\tau \sim \Delta^{-1}(1+L/\xi)$2 window. In that model the coefficient of the first mode scales as $\tau \sim \Delta^{-1}(1+L/\xi)$3, directly realizing the overlap-based amplification scenario (Feng et al., 2023).

A major caveat concerns detailed balance. For weak-coupling thermal environments satisfying a detailed-balance condition relative to a Gibbs state, right and left eigenmodes cannot localize at opposite boundaries. The relation

$\tau \sim \Delta^{-1}(1+L/\xi)$4

implies, for nondegenerate modes,

$\tau \sim \Delta^{-1}(1+L/\xi)$5

so left and right modes share the same localization and boundary-separation-induced exponentially small overlaps are precluded. This excludes the standard skin-effect mechanism in detailed-balance Liouvillians, although other mechanisms can still generate superexponentially small overlaps (Haga et al., 2020).

3. Canonical models and exactly solvable realizations

Several model families now serve as reference points for the subject.

Model class	Hallmark LSE feature	Source
Asymmetric dissipative hopping of hard-core bosons	$\tau \sim \Delta^{-1}(1+L/\xi)$6 without gap closing	(Haga et al., 2020)
Integrable dissipative XXZ chain with generalized boundary conditions	LSE survives counter-flow boundary hopping and is destroyed by any co-flow boundary hopping in the thermodynamic limit	(Mao et al., 2024)
Dissipative SSH chain	Exact mapping of the damping matrix to a generalized non-Hermitian SSH Hamiltonian	(Yang et al., 2022)

In the prototypical one-particle hopping model, the Hamiltonian is

$\tau \sim \Delta^{-1}(1+L/\xi)$7

and the jump operators are asymmetric stochastic hoppings,

$\tau \sim \Delta^{-1}(1+L/\xi)$8

At $\tau \sim \Delta^{-1}(1+L/\xi)$9, the diagonal sector maps to a non-Hermitian Hatano–Nelson chain. Under periodic boundary conditions, the gap scales as $\dot{\rho}(t)=\mathcal{L}(\rho(t)),$0; under open boundary conditions, the gap approaches the finite limit

$\dot{\rho}(t)=\mathcal{L}(\rho(t)),$1

for $\dot{\rho}(t)=\mathcal{L}(\rho(t)),$2, and the localization length is

$\dot{\rho}(t)=\mathcal{L}(\rho(t)),$3

For $\dot{\rho}(t)=\mathcal{L}(\rho(t)),$4, numerics still show boundary-localized slow modes and $\dot{\rho}(t)=\mathcal{L}(\rho(t)),$5 with $\dot{\rho}(t)=\mathcal{L}(\rho(t)),$6 across a range of parameters (Haga et al., 2020).

An exactly solvable many-body realization was constructed from spin-$\dot{\rho}(t)=\mathcal{L}(\rho(t)),$7 jump operators

$\dot{\rho}(t)=\mathcal{L}(\rho(t)),$8

which, in an invariant diagonal subspace, map to a non-Hermitian XXZ chain. Under open boundaries, the Bethe wavefunction carries an intrinsic factor $\dot{\rho}(t)=\mathcal{L}(\rho(t)),$9 with $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$0, so operator amplitudes are exponentially concentrated near the right edge and $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$1. Under generalized boundary conditions, counter-flow hopping is exponentially suppressed by the imaginary gauge transform and the LSE survives, whereas any finite co-flow boundary hopping restores a PBC-like non-Bloch quantization and destroys exponential boundary accumulation in the thermodynamic limit (Mao et al., 2024).

A third exactly solvable class is the dissipative SSH chain with bond loss and gain. Third quantization reduces the Liouvillian to a damping matrix $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$2 that is unitarily equivalent to a generalized non-Hermitian SSH Hamiltonian, yielding an exact rapidity relation $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$3. Under open boundaries, normal master modes pile up at edges, damping becomes boundary-sensitive, and relaxation times diverge linearly with system size without Liouvillian gap closing. The same framework also shows that the topology of the full Lindbladian problem differs from that of the postselected effective Hamiltonian description (Yang et al., 2022).

4. Boundary conditions, point-gap topology, and symmetry constraints

The Lindbladian skin effect is closely related to the non-Hermitian skin effect, but the relation is not identity. Both phenomena arise from nonreciprocity, boundary sensitivity, and non-Bloch complex momenta; both exhibit sharp distinctions between open and periodic boundaries. The difference is that LSE is formulated in operator space, where right and left eigenoperators, non-normality, and their overlaps directly enter relaxation dynamics (Haga et al., 2020).

This operator-space formulation admits topological characterizations. In a two-dimensional electron system with Rashba spin-orbit coupling, in-plane magnetic field, and substrate-mediated dissipation, the vectorized quadratic Liouvillian can be brought to a triangular form whose point-gap topology is fully captured by a non-Hermitian damping matrix $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$4. For fixed $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$5, the integer invariant is

$$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$6

A nonzero winding gives a $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$7 Lindbladian skin effect; when transposed time-reversal symmetry is present, a $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$8 invariant at $$\mathcal{L}(\rho):=-i[H,\rho]+\sum_\alpha \left(L_\alpha \rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$9 yields a $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$0 skin effect in which Kramers pairs localize on opposite edges (Shigedomi et al., 23 May 2025).

That same work identified symmetry constraints that force $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$1 for all reference points and therefore forbid the $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$2 skin effect. Breaking the relevant constraints by simultaneously taking $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$3 and $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$4 opens point gaps at generic $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$5 and produces nonzero winding. With $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$6 and $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$7, the system instead supports the $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$8 skin effect at $$\mathcal{L}(\rho_j^{\mathrm R})=\lambda_j \rho_j^{\mathrm R},\qquad \mathcal{L}^\dagger(\rho_j^{\mathrm L})=\lambda_j^* \rho_j^{\mathrm L},$$9 (Shigedomi et al., 23 May 2025).

A complementary one-dimensional classification compares the full Lindbladian winding to the postselected non-Hermitian one. For quadratic fermionic systems with loss and gain, the matrix controlling normal-master-mode profiles is

$\mathcal{L}(\rho_{\mathrm{ss}})=0$0

and the Lindbladian point-gap winding is

$\mathcal{L}(\rho_{\mathrm{ss}})=0$1

In the absence of gain, $\mathcal{L}(\rho_{\mathrm{ss}})=0$2; in the absence of loss, $\mathcal{L}(\rho_{\mathrm{ss}})=0$3. When both gain and loss are present, $\mathcal{L}(\rho_{\mathrm{ss}})=0$4 remains quantized and can flip sign at a point-gap-closing transition that reverses the Lindbladian skin localization and is invisible under postselection (Chaduteau et al., 9 Jul 2025).

These topological results also sharpen a common misconception. The postselected no-jump Hamiltonian can reproduce the skin effect only in restricted limits. In general, the mixed-state topology of the full Lindbladian contains transitions and localization reversals that the postselected description misses (Chaduteau et al., 9 Jul 2025).

5. Dynamical manifestations, alternative mechanisms, and exceptions

The most direct signatures of the Lindbladian skin effect are dynamical. In the two-dimensional electron model, a quench in the in-plane Zeeman field generates edge charge accumulation under open boundaries. Reversing the sign of $\mathcal{L}(\rho_{\mathrm{ss}})=0$5 reverses the side of accumulation. The temperature dependence is explicit: the skin effects become pronounced when $\mathcal{L}(\rho_{\mathrm{ss}})=0$6, and they are suppressed when thermal broadening homogenizes the dissipative rates and closes the point gap. In the reported numerics, pronounced $\mathcal{L}(\rho_{\mathrm{ss}})=0$7 skin appears at $\mathcal{L}(\rho_{\mathrm{ss}})=0$8 and is suppressed by $\mathcal{L}(\rho_{\mathrm{ss}})=0$9, while the $\Delta=|\mathrm{Re}\,\lambda_1|,$0 case is visible at $\Delta=|\mathrm{Re}\,\lambda_1|,$1 and suppressed at $\Delta=|\mathrm{Re}\,\lambda_1|,$2 (Shigedomi et al., 23 May 2025).

That same model also realizes a critical variant. The average localization length grows linearly with system size, $\Delta=|\mathrm{Re}\,\lambda_1|,$3, so the usual estimate

$\Delta=|\mathrm{Re}\,\lambda_1|,$4

saturates to $\Delta=|\mathrm{Re}\,\lambda_1|,$5 at large $\Delta=|\mathrm{Re}\,\lambda_1|,$6. The relaxation time therefore becomes independent of system size even though the system exhibits a Liouvillian skin effect. This was attributed to a Liouvillian analogue of the critical non-Hermitian skin effect (Shigedomi et al., 23 May 2025).

Not every boundary-asymmetric relaxation pattern corresponds to standard boundary-localized Liouvillian eigenoperators. In quadratic Lindbladian SSH systems with identical coupling to the environment in each cell, the steady state is spatially uniform and individual normal modes on top of it do not show boundary asymmetry. The observed chiral damping instead arises from interference between biorthogonal Liouvillian normal modes,

$\Delta=|\mathrm{Re}\,\lambda_1|,$7

so the skin effect is dynamical rather than a stationary localization of single modes (Zhou et al., 2021).

An opposite caution is supplied by dissipative systems with a collective Hermitian jump operator. There, the effective non-Hermitian Hamiltonian can display the ordinary non-Hermitian skin effect, but the full Lindblad evolution washes out boundary-condensed Liouvillian eigenmodes. The trajectory-averaged populations spread symmetrically, the steady state becomes maximally mixed, and only hidden signatures remain in short-time coherence decay and long-time entropy selection (Longhi, 2020).

A third exception occurs in postselected dynamics with partial loss of quantum jumps. The resulting nonlinear master equation supports a postselected skin effect with a steady-state density fitted by a scale-invariant tanh profile, but this is not a linear Liouvillian spectral skin effect. The asymmetry is generated by the effective Hatano–Nelson Hamiltonian embedded in the nonlinear conditioned evolution, whereas the unconditioned Lindblad steady state of the same model is uniform (Liu et al., 2024).

6. Higher-dimensional, non-Markovian, and many-body extensions

The mechanism underlying the standard slowdown formula is not restricted to one dimension. For rectangular two-dimensional systems, if right and left modes localize on opposite edges, the replacement $\Delta=|\mathrm{Re}\,\lambda_1|,$8 the perpendicular linear dimension preserves the form of the result. For interacting hard-core particles at $\Delta=|\mathrm{Re}\,\lambda_1|,$9, the many-body generalization reduces to ASEP under open boundaries, where the spectral gap stays nonzero in the thermodynamic limit and the exponentially small left-right overlaps persist, supporting the same scaling logic. The genuinely interacting $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$0 many-body case remains open (Haga et al., 2020).

Non-Markovian environments qualitatively modify the skin effect. Using hierarchical equations of motion for a bosonic bath with Drude–Lorentz spectral density,

$$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$1

one finds a thick skin effect: the dominant skin mode broadens and shifts into the bulk, with an HEOM localization length $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$2 larger than its Markovian counterpart. The relaxation time still scales linearly with system size,

$$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$3

but strong coupling moves $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$4 closer to zero and increases $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$5, producing slower relaxation. Cross-site skin-mode coherence appears only when the system-bath coupling contains counter-rotating terms, and both skin-mode and steady-state coherence remain robust against additional local dephasing (Kuo et al., 2024).

Skin steady states can also delocalize. In a two-leg Lindbladian ladder with opposite skin directions on the two legs, increasing the interchain coupling $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$6 gradually cancels the opposing accumulations. In the single-particle sector this changes the Liouvillian-gap scaling from $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$7 to $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$8; in the many-body setting it changes the steady-state entanglement scaling from an area law to a logarithmic law. Because the corresponding effective non-Hermitian Hamiltonian exhibits critical NHSE, the work predicted a critical Liouvillian skin effect: even arbitrarily small $$|\rho\rangle=\sum_{ij}\rho_{ij}|i\rangle\otimes|j\rangle$$9 induces dramatic changes in the thermodynamic limit (Feng et al., 2024).

A related monitored-fermion study with directed particle-conserving dissipation and power-law coherent hopping showed that the geometry of many-body skin states controls trajectory entanglement. Short-range hopping is consistent with complete skin accumulation and area-law entanglement, whereas sufficiently long-range hopping produces a finite bulk tail and algebraic sub-volume-law entanglement. The distinction is encoded by the thermodynamic imbalance, with complete LSE corresponding to $$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$0 and incomplete LSE to $$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$1 (Passarelli et al., 8 Jun 2026).

Experimental relevance has been emphasized across several platforms. Proposed or discussed implementations include ultracold atoms in optical lattices with laser-assisted asymmetric dissipative hopping, photonic lattices, driven-dissipative bosonic circuits, superconducting quantum simulators, semiconductor two-dimensional electron gases on substrates, and monitored fermionic chains. Diagnostics include extracting $$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$2 from late-time decay of local observables, $$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$3 from steady-state spatial profiles, $$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$4 from relaxation curves, edge-resolved charge accumulation after quenches, and trajectory-resolved entanglement measures in monitored settings (Haga et al., 2020, Shigedomi et al., 23 May 2025).

In aggregate, the Lindbladian skin effect has developed from a boundary-spectral anomaly into a broad dynamical framework. Its defining content is the boundary localization of relaxation modes in operator space, but its consequences now include gap-independent slow relaxation, generalized-boundary-condition fragility and robustness, point-gap topology with $$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$5 and $$|\langle x|\rho_1^{\mathrm R}|y\rangle|\sim e^{-(2L-x-y)/\xi},\qquad |\langle x|\rho_1^{\mathrm L}|y\rangle|\sim e^{-(x+y)/\xi}.$$6 indices, critical and thick-skin regimes, and many-body entanglement structures that depend on how dissipation, coherent transport, and monitoring are combined.