Quantum Liang Information Flow (QLIF)

• Quantum Liang Information Flow (QLIF) is a directional measure that quantifies causal influence by comparing the von Neumann entropy evolution of a target subsystem under full and frozen dynamics.
• QLIF extends the classical Liang information flow by adopting a freeze/subtract intervention model to isolate the impact of specific subsystems in quantum networks.
• Applications of QLIF include ranking node importance, detecting non-Hermitian directional transport, and differentiating between chaotic and integrable many-body dynamics.

Quantum Liang Information Flow (QLIF) is a quantum generalization of Liang information flow in which the directional causal influence of a subsystem $B$ on a subsystem $A$ is quantified by comparing the entropy evolution of $A$ in the full dynamics with the entropy evolution of $A$ in a counterfactual dynamics where $B$ is frozen, detached, or removed from the evolution mechanism. In the quantum setting, the entropy is the von Neumann entropy, so QLIF is explicitly an entropy-based, intervention-based, and generally asymmetric measure of causal influence rather than a symmetric correlation diagnostic (Yi et al., 2022). Subsequent work has used the same freeze/subtract construction as an operational probe of non-Hermitian directional transport and as a late-time chaos diagnostic in many-body dynamics, while preserving the central interpretation of QLIF as a directional entropy difference under a frozen-source intervention (Yi, 20 Feb 2026, Yi, 16 Mar 2026).

1. Classical origin and quantum generalization

QLIF inherits its conceptual structure from the classical Liang–Kleeman formalism. In the classical setting one decomposes the entropy change of one component into a part due to its own evolution and a remainder attributable to another component. The classical bivariate information-flow formula reviewed in the quantum generalization is

$$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$

with the defining causal idea that subsystem $2$ is excluded or frozen in the counterfactual term. This is tied to Liang’s principle of nil causality: if subsystem $1$ evolves independently of subsystem $2$, then $T_{2\rightarrow 1}=0$ (Yi et al., 2022).

The quantum construction preserves that logic while replacing Shannon entropy by von Neumann entropy. QLIF is introduced to quantify how much one subsystem causally influences another in multipartite quantum dynamics, and more specifically to rank the relative importance of different nodes in a quantum network for influencing a target node. Its operational meaning is the difference between the entropy evolution of the target in the actual quantum network and the entropy evolution of the same target in a modified network where the source node has been removed from the original evolution mechanism (Yi et al., 2022).

This intervention-based structure is the reason QLIF is presented as a causation quantifier rather than a correlation measure. Correlation functions can reveal association, but the defining question in QLIF is counterfactual: what changes in the target’s entropy dynamics when a particular source is disabled? That same counterfactual interpretation is retained in later applications, where the frozen subsystem is implemented directly at the Hamiltonian level and the resulting entropy difference is treated as a directional causal-flow measure (Yi, 20 Feb 2026, Yi, 16 Mar 2026).

2. Formal definition and freezing prescription

The basic rate form of QLIF from $A$0 to $A$1 is

$A$2

where $A$3 is the von Neumann entropy of subsystem $A$4, and $A$5 is the entropy of $A$6 under a modified evolution in which $A$7 is frozen or excluded (Yi et al., 2022). The cumulative version introduced in the original formulation is

$A$8

while later works use the cumulative quantity directly as the primary observable, writing for example

$A$9

or

$A$0

with the second term always computed under a frozen-$A$1 Hamiltonian rather than from a conditional density matrix (Yi, 20 Feb 2026, Yi, 16 Mar 2026).

The frozen evolution is defined by a modified physical dynamics, not by conditioning on an outcome. In the tripartite construction of the original QLIF paper,

$A$2

and the modified unitary is required to factor as

$A$3

In Hamiltonian models this is implemented by deleting the interaction terms involving the frozen node. For a Hamiltonian

$A$4

freezing $A$5 amounts to replacing it by

$A$6

and evolving with $A$7 (Yi et al., 2022).

Later implementations use the same logic in more concrete lattice settings. In the non-Hermitian SSH application, the modified Hamiltonian $A$8 is defined by setting all matrix elements connected to site $A$9 to zero,

$A$0

and QLIF is then computed from the single-particle wavefunction, the single-site reduced density matrix, and the associated site entropy. Because the problem is single-particle, the site entropy reduces to the binary entropy

$A$1

in terms of the occupation probability at site $A$2 (Yi, 20 Feb 2026). In the mixed-field Ising study, the frozen Hamiltonian is constructed by removing all nontrivial terms acting on site $A$3, including both local terms on $A$4 and couplings involving $A$5, and the authors explicitly emphasize that $A$6 because the two objects arise from different dynamical laws (Yi, 16 Mar 2026).

3. Directionality, asymmetry, and causal interpretation

QLIF is directional because the frozen-source construction is not symmetric under interchange of source and target. In general,

$A$7

and this asymmetry is one of its defining features (Yi, 20 Feb 2026). In the original quantum-network formulation, the causal interpretation is justified by a quantum nil-causality theorem: if the full evolution factors as

$A$8

then the evolution of $A$9 is independent of $B$0, and therefore

$B$1

This is the direct quantum analogue of Liang’s nil-causality principle (Yi et al., 2022).

The sign of QLIF is also physically meaningful. A positive value means that the presence or dynamics of the source increases the entropy of the target; a negative value means that the source decreases the target entropy. The later NHSE study stresses that the cumulative QLIF itself can be positive or negative at intermediate times, with negative values arising from interference and wavepacket-redistribution effects rather than from a failure of the formalism (Yi, 20 Feb 2026). The original network examples make the same point in a different language: negative cumulative flow means that the presence of a given node reduces the target’s uncertainty (Yi et al., 2022).

In bipartite systems, QLIF has a particularly simple form. Freezing $B$2 forces the frozen evolution to factor as a product of local unitaries, so the frozen entropy of $B$3 is constant and

$B$4

For closed pure bipartite states, $B$5, so the two directions coincide; for mixed states the symmetry is generally lost (Yi et al., 2022). The CNOT example given in the original paper makes that asymmetry explicit: with a mixed initial control-target state, the cumulative flows satisfy $B$6 and $B$7 bit (Yi et al., 2022).

This directional, intervention-based structure distinguishes QLIF from several neighboring notions of quantum information flow. The non-Hermitian SSH work contrasts it with symmetric two-point correlators such as $B$8 and with mutual-information-like quantities, both of which are described as unable or ill-suited to capture directional bias (Yi, 20 Feb 2026). The mixed-field Ising comparison similarly positions QLIF against the out-of-time-order correlator: OTOC probes operator spreading, whereas QLIF asks how much the local state of $B$9 would have differed had $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$0’s dynamics been removed (Yi, 16 Mar 2026). Broader process-theoretic approaches such as the signaling power of quantum channels are also directional, but they quantify causal influence operationally at the level of channels, superchannels, or process matrices rather than through the freeze/subtract entropy construction that defines QLIF (Santos et al., 2024).

4. Representative physical applications

The first applications of QLIF were small quantum networks and a common-bath open-system model. In three-qubit XY-coupled spin systems, QLIF ranks node importance by comparing how strongly different senders influence a target. With $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$1 and $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$2, the cumulative flows satisfy $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$3, matching the stronger coupling. The same models show that joint influence is generally nonadditive: $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$4 and in the explicit examples the joint flow exceeds the sum of the individual flows. In a five-qubit star geometry, adding a strong $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$5-$$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$6 coupling suppresses $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$7 and $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$8, enhances $$T_{2\rightarrow 1} = \frac{dS_{1(\mathrm{classical})}}{dt} - \frac{dS_{1\not{2}(\mathrm{classical})}}{dt},$$9 and $2$0, and can even drive $2$1, indicating that the strongly coupled branch reduces the uncertainty of the target. In the common-bath example of two noninteracting qubits embedded in a zero-temperature bosonic reservoir, the instantaneous rates and the cumulative flows rank the two directions differently: the rate $2$2 from the weakly coupled qubit to the strongly coupled qubit has the higher peak, but the cumulative flow $2$3 eventually dominates as equilibrium is approached (Yi et al., 2022).

A later application studies QLIF as a directional probe of the non-Hermitian skin effect in a non-reciprocal SSH chain with intracell hoppings $2$4. There the central observable is the asymmetry

$2$5

which vanishes in the reciprocal limit and satisfies the sign rule

$2$6

The paper reports a “scissors effect”: in the Hermitian case the two directional QLIF curves overlap, whereas for $2$7 they split after an initial overlap. For small $2$8, the asymmetry is approximately linear in $2$9, but over the full parameter range it is non-monotonic, reaching its largest magnitude at moderate non-reciprocity $1$0–$1$1 or, equivalently, at moderate skin length

$1$2

with $1$3. The same study extracts onset-time velocities from the threshold condition

$1$4

and finds the ordering

$1$5

which is interpreted as NHSE-induced blocking of information flow against the skin direction. Three temporal regimes are identified: light-cone-bounded spreading, a $1$6-dependent stabilization or quasi-stationary regime, and coherent oscillations (Yi, 20 Feb 2026). A notable methodological point is that all relevant sites are chosen on the same sublattice; mixed-sublattice choices produce asymmetry even at $1$7 because of the bipartite structure rather than non-Hermiticity (Yi, 20 Feb 2026).

A different line of work compares QLIF with OTOC as a chaos diagnostic in the mixed-field Ising chain

$1$8

In that setting the cumulative QLIF

$1$9

shows early-time power-law growth

$2$0

with nearly identical exponents and wavefront propagation in integrable and chaotic regimes. The front follows

$2$1

for the parameter choice $2$2, $2$3, so early-time QLIF is attributed to local Hamiltonian structure rather than chaos. The raw QLIF magnitude depends strongly on the initial state, spanning four orders of magnitude across product-state, eigenstate, and quench protocols. The main positive result appears at late times in the integrated quantity

$2$4

For chaotic dynamics, $2$5 grows linearly or at least monotonically, reflecting irreversible thermalization; for integrable dynamics it saturates or oscillates, reflecting reversible quasiparticle dynamics. On that basis the paper presents time-integrated QLIF as a late-time chaos diagnostic complementary to OTOC rather than a replacement for it (Yi, 16 Mar 2026).

5. Relation to neighboring quantum information-flow frameworks

QLIF belongs to a broader family of quantum information-flow concepts, but its defining feature remains the frozen-source entropy comparison. This makes it distinct from open-system backflow measures based on distinguishability, from operational signaling measures on channels, and from graph-based correlation heuristics in quantum circuits.

One neighboring framework is the signaling power of quantum channels and higher-order maps. There the basic quantity is

$2$6

with the maximization taken over channels in the opposite causal direction. That measure is faithful to non-signaling, operationally interpretable through dense coding, and extendable to superchannels, memory channels, and indefinite causal order. It is therefore a rigorous directional information-flow measure, but it is not entropic and not Liang-style: its primary objects are channels and processes rather than subsystem entropy dynamics under freezing interventions (Santos et al., 2024).

Another neighboring line is the BLP-type notion of quantum information flow as the rate of change of distinguishability,

$2$7

which can be rewritten in terms of the rate of conditional min-entropy. That framework interprets positive flow as information backflow from environment to system and is well suited to non-Markovianity and subsystem-environment exchange, but it does not define pairwise directional flows $2$8 and $2$9 via node freezing. It is therefore conceptually adjacent to QLIF, yet formally different (Roy et al., 2018).

A further contrast appears in local-information and information-lattice approaches to quench dynamics, where information is decomposed into local scale-resolved densities and currents. Those constructions provide a hydrodynamic picture of information transport and can isolate interface and topological information channels, but they are based on inclusion–exclusion decompositions of subsystem information rather than on freezing a source and subtracting the resulting target entropy. This suggests a possible complementarity between QLIF’s intervention-based directionality and information-lattice locality, but the two frameworks are not identical (Bauer et al., 1 May 2025).

6. Limitations, caveats, and open directions

QLIF requires both full knowledge of the dynamics and a well-defined intervention. The original formulation explicitly identifies these as core requirements, and it does not provide a general estimator that would infer QLIF directly from observational data without knowing the dynamics a priori or performing the frozen-node intervention (Yi et al., 2022). The construction is most straightforward when the Hamiltonian can be cleanly decomposed into local terms and interaction terms so that “freezing” has an unambiguous implementation.

Later applications reveal additional technical caveats. In the non-Hermitian SSH study, the standard calculations use a single-particle initial state, open boundaries, exact diagonalization, and a site-based freezing prescription. The paper notes finite-size effects, sensitivity to sublattice geometry, and the absence of a deeper treatment of alternative non-Hermitian dynamical formalisms such as postselected trajectories, unravelings, or fully normalized density-matrix evolution under gain/loss. Its conclusions are therefore tied to an effective single-particle non-Hermitian framework (Yi, 20 Feb 2026). In the chaos-diagnostic comparison, the numerics are limited by system size and bond dimension, with late-time chaotic dynamics potentially affected by truncation, and the main chaotic benchmark is concentrated around a particular parameter choice $T_{2\rightarrow 1}=0$0, so broader universality remains open (Yi, 16 Mar 2026).

Open directions recur across the literature. The original quantum-network paper leaves open whether QLIF can be estimated without a direct intervention, analogous to classical statistical Liang estimators (Yi et al., 2022). The NHSE study highlights interacting many-body non-Hermitian generalizations, Fock-space skin effects, higher-order skin effects, and experimental verification in photonic and topolectrical platforms (Yi, 20 Feb 2026). The mixed-field Ising study points toward broader parameter scans and a more systematic understanding of late-time integrated QLIF beyond the specific free-fermion and MPS-accessible regimes examined there (Yi, 16 Mar 2026).

Taken together, these developments define QLIF as a directional entropy-based intervention measure built from a frozen-source counterfactual. Its central formal identity,

$T_{2\rightarrow 1}=0$1

makes it naturally asymmetric, causation-oriented, and sensitive to phenomena that are invisible to symmetric correlators. The existing literature shows that this basic construction can rank node importance in small quantum networks, resolve directional transport in non-reciprocal lattices, and separate late-time thermalization from integrable recurrence in many-body dynamics, while also making clear that QLIF remains methodologically demanding and still open-ended as a general many-body framework (Yi et al., 2022, Yi, 20 Feb 2026, Yi, 16 Mar 2026).