Subsystem Operator Rényi Entropy

• Subsystem operator Rényi entropy is a family of measures applied to subsystem-restricted operators or states, quantifying uncertainty, entanglement, and distinguishability.
• It incorporates multiple formulations including operator-algebraic, reduced operator, and basis-dependent approaches to capture various quantum phenomena.
• This framework bridges global phase-space analyses, local operator dynamics, and quantum reference frames, offering actionable insights into entanglement, transport, and thermalization.

Subsystem operator Rényi entropy denotes a family of Rényi-type quantities attached to a subsystem, an operator restricted to a subsystem, or a subsystem state created by an operator. In the literature considered here, the notion appears in several forms: an operator-algebraic entropy of states relative to a chosen reference system on a $C^*$-algebra, a second Rényi entropy of a reduced Heisenberg operator, a basis-dependent Rényi entropy of subsystem measurement probabilities, and relative or Schatten-type Rényi quantities for reduced density matrices after local operator insertions (Mukhamedov et al., 2019, Roy et al., 25 Feb 2026, Tarighi et al., 2022, Zhao et al., 2024).

1. Conceptual scope

A recurrent structure is a bipartition into a subsystem $A$ and its complement, followed by a Rényi functional applied either to a reduced density matrix, to a reduced operator, or to a probability distribution induced by a chosen observable or basis. The basic density-matrix form is

$$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$

while basis-dependent formulations replace $\rho_A$ by subsystem probabilities $p_{I_A}$, and operator-growth formulations replace $\rho_A$ by a normalized reduced operator obtained from $\mathcal O(t)$ (Kim et al., 2023, Tarighi et al., 2022, Roy et al., 25 Feb 2026).

Formulation	Core object	Representative source
Operator-algebraic	$S_\alpha^{\mathcal S}(\varphi)$ for a state $\varphi$ and reference system $S$	(Mukhamedov et al., 2019)
Reduced operator	$A$0 from $A$1	(Roy et al., 25 Feb 2026)
Basis-dependent subsystem entropy	$A$2 from $A$3	(Tarighi et al., 2022)
Observable-based subsystem entropy	$A$4 from subsystem-energy probabilities $A$5	(Najafi et al., 2018)
Relative/distinguishability form	$A$6, $A$7	(Zhao et al., 2024, Zhang et al., 2019)
Global aggregate of all subsystems	$A$8 from the Husimi function	(Zhang et al., 19 Sep 2025)

A common source of confusion is the assumption that subsystem operator Rényi entropy is necessarily basis independent or necessarily a function of a reduced density matrix alone. The basis-dependent subsystem Rényi-Shannon entropy in a configuration basis is explicitly basis dependent, the operator-growth entropy of a Heisenberg operator is state independent, and the operator-algebraic construction depends on a chosen compact convex reference system $A$9 (Tarighi et al., 2022, Roy et al., 25 Feb 2026, Mukhamedov et al., 2019).

2. Operator-algebraic formulation and reference systems

The most general operator-algebraic construction in the sources is the $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$0-algebraic $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$1-mixing Rényi entropy. Let $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$2 be a $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$3-algebra, $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$4 a compact convex set of states, and $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$5. For a countable extremal decomposition

$$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$6

the Rényi entropy relative to the reference system $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$7 is

$$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$8

with value $$S_\alpha(\rho_A)=\frac{1}{1-\alpha}\log\operatorname{Tr}\rho_A^\alpha,$$9 if no countable extremal decomposition exists (Mukhamedov et al., 2019). This extends Ohya’s $\rho_A$0-mixing entropy, is monotonically decreasing in $\rho_A$1, and satisfies

$\rho_A$2

(Mukhamedov et al., 2019).

This framework makes subsystem dependence explicit through the choice of reference system. A spatial subsystem can be modeled by a subalgebra $\rho_A$3 and the restricted state $\rho_A$4, leading to

$\rho_A$5

A dynamical or thermodynamic subsystem can instead be encoded by a distinguished convex subset of states, such as the invariant states $\rho_A$6 or KMS states $\rho_A$7 of a $\rho_A$8-dynamical system $\rho_A$9 (Mukhamedov et al., 2019).

The same construction recovers familiar entropies in special cases. For a finite probability space with probabilities $p_{I_A}$0, one obtains the classical Rényi entropy

$p_{I_A}$1

For density operators $p_{I_A}$2 on $p_{I_A}$3, the state $p_{I_A}$4 satisfies

$p_{I_A}$5

and for $p_{I_A}$6,

$p_{I_A}$7

(Mukhamedov et al., 2019).

Reference-system inclusion generates entropy inequalities. For a KMS state $p_{I_A}$8,

$p_{I_A}$9

and under $\rho_A$0-commutativity,

$\rho_A$1

If the KMS state is unique, then

$\rho_A$2

(Mukhamedov et al., 2019). This shows that subsystem operator Rényi entropy can quantify uncertainty relative not only to a spatial restriction but also to an admissible decomposition structure.

3. Global phase-space formulations and subsystem-independent aggregation

A different construction appears in the Wehrl–Rényi-2 entropy of the Husimi function for $\rho_A$3 distinguishable qubits. For a density matrix $\rho_A$4, the Husimi function is

$\rho_A$5

and the Wehrl–Rényi entropy of order $\rho_A$6 is

$\rho_A$7

(Zhang et al., 19 Sep 2025).

The central exact identity is

$\rho_A$8

where the sum runs over all $\rho_A$9 subsystems $\mathcal O(t)$0 and $\mathcal O(t)$1 is the reduced density matrix on $\mathcal O(t)$2 (Zhang et al., 19 Sep 2025). The same quantity can therefore be read as a subsystem-independent scalar built from the entire system, or as a weighted aggregate of all subsystem Rényi-2 data via

$\mathcal O(t)$3

This exact relation is notable because it replaces the choice of a privileged subsystem by a sum over all subsystem purities. In the terminology of the source, $\mathcal O(t)$4 is subsystem independent in its definition but encodes contributions from all possible subsystems (Zhang et al., 19 Sep 2025). The bounds

$\mathcal O(t)$5

follow from purity bounds, and explicit examples show that Haar-random states approach the upper bound, while GHZ and W states approach the lower bound in the large-$\mathcal O(t)$6 limit (Zhang et al., 19 Sep 2025).

The derivation uses local Haar averaging, SWAP operators, and the identity

$\mathcal O(t)$7

so the same algebraic backbone appears in operator-entanglement and randomized-measurement settings (Zhang et al., 19 Sep 2025). This suggests a bridge between global phase-space Rényi entropies and aggregated subsystem operator Rényi data.

4. Reduced operators, operator growth, and Schwinger–Keldysh formalisms

In non-interacting fermionic systems, subsystem operator Rényi entropy is defined directly from a time-evolved operator rather than from an operator-state mapping. For a Hermitian operator $\mathcal O(t)$8, a bipartition into $\mathcal O(t)$9 and $S_\alpha^{\mathcal S}(\varphi)$0, and normalization $S_\alpha^{\mathcal S}(\varphi)$1, the reduced operator is

$S_\alpha^{\mathcal S}(\varphi)$2

and the second Rényi subsystem operator entropy is

$S_\alpha^{\mathcal S}(\varphi)$3

(Roy et al., 25 Feb 2026). For the local density operator $S_\alpha^{\mathcal S}(\varphi)$4, the dynamical growth

$S_\alpha^{\mathcal S}(\varphi)$5

is bounded by

$S_\alpha^{\mathcal S}(\varphi)$6

(Roy et al., 25 Feb 2026).

This formulation is explicitly state independent. The entropy depends on the Heisenberg operator, the spatial bipartition, and the Hamiltonian, rather than on an initial many-body state. The source emphasizes that it encodes both spatial and temporal information and therefore directly connects to transport for a local operator related to a conserved quantity (Roy et al., 25 Feb 2026).

The same work constructs a unified Schwinger–Keldysh field-theory formalism for $S_\alpha^{\mathcal S}(\varphi)$7 and for state Rényi and von Neumann entanglement entropies. For non-interacting systems, the resulting correlation-matrix formulas have the same structure, but the operator entropy is written in terms of infinite-temperature Keldysh Green’s functions, whereas state entanglement entropies are written in terms of vacuum Green’s functions (Roy et al., 25 Feb 2026). This produces explicit formulas for Aubry–André and Anderson models and shows that subsystem operator Rényi entropy can capture ballistic, diffusive, anomalous diffusive, and localized behavior through finite-size scaling of saturation times and through temporal growth profiles (Roy et al., 25 Feb 2026).

A related distinction is important: operator-growth entropies of this type are not relative entropies and are not defined by choosing a basis of subsystem measurement outcomes. They are reduced-operator Rényi entropies in real time.

5. Local operator quenches, relative Rényi entropy, and subsystem distances

In two-dimensional CFT, local operator quenches lead to subsystem Rényi quantities that measure either entanglement or distinguishability of operator-generated states. For two reduced density matrices $S_\alpha^{\mathcal S}(\varphi)$8 and $S_\alpha^{\mathcal S}(\varphi)$9 on an interval $\varphi$0, the relative Rényi entropy used in rational and holographic CFTs is

$\varphi$1

and is interpreted there as quantifying how distinguishable two local operator excitations are when restricted to $\varphi$2 (Zhao et al., 2024). In rational CFTs it is zero before the light cone reaches the entangling point and becomes a constant at late times for several operator families, with the late-time value controlled by finite-dimensional matrices of two-point coefficients. In holographic CFTs, the collision relative entropy reconstructs the entanglement wedge and induces a metric proportional to the Bures metric on the corresponding bulk region (Zhao et al., 2024).

A complementary set of quantities is given by subsystem trace and Schatten distances after local operator quenches. For a nonchiral primary field $\varphi$3, the Rényi entropy increase obeys

$\varphi$4

where $\varphi$5 is the quantum dimension (Zhang et al., 2019). The same analysis shows that the reduced density matrix of an interval hosting a quasiparticle is orthogonal to the reduced density matrix of the interval without quasiparticles, and that reduced density matrices hosting quasiparticles at different positions are also orthogonal to each other (Zhang et al., 2019). Consequently, the Schatten distances are piecewise constant and, in the orthogonal regimes, the trace distance reaches its maximal value in the source’s normalization (Zhang et al., 2019).

For the free non-compact boson and its harmonic-chain discretization, exact excited-state Rényi entropies and subsystem Schatten distances were obtained for several low-lying multi-particle states, together with short-interval expansions for general excited states (Zhang et al., 2020). In the CFT regime, the leading correction to $\varphi$6 is

$\varphi$7

so the leading short-interval behavior depends only on the scaling dimension $\varphi$8 (Zhang et al., 2020). In the extremely gapped limit of the harmonic chain, by contrast, the leading behavior depends only on the total number of excited quasiparticles $\varphi$9, both for Rényi entropies and for subsystem Schatten distances (Zhang et al., 2020). This contrast isolates two regimes: conformal-energy control in the gapless theory and purely combinatorial quasiparticle counting in the deeply gapped theory.

6. Basis-dependent and observable-based many-body formulations

A distinct usage of subsystem operator Rényi entropy appears in basis-dependent Rényi-Shannon entropies. For a ground state $S$0 written in a local product basis, the subsystem probabilities are

$S$1

and the subsystem operator Rényi entropy is

$S$2

(Tarighi et al., 2022). At a critical point, this obeys

$S$3

with universal logarithmic coefficient $S$4 (Tarighi et al., 2022). For critical quadratic fermions with $S$5 symmetry,

$S$6

where $S$7 is the central charge. Without $S$8 symmetry,

$S$9

These formulas exhibit non-analytic changes at $A$00 or $A$01, respectively (Tarighi et al., 2022).

An observable-based formulation is the Rényi entropy of subsystem energy. For a truncated subsystem Hamiltonian $A$02 with eigenstates $A$03, subsystem-energy probabilities are

$A$04

and

$A$05

(Najafi et al., 2018). This quantity obeys an area law in gapped phases and, at criticality,

$A$06

with universal coefficient $A$07 that scales with the central charge in the Ising and XX universality classes (Najafi et al., 2018). The same work shows that the largest subsystem-energy probabilities closely mimic the largest Schmidt coefficients and that truncated Shannon and truncated von Neumann entropies are almost indistinguishable (Najafi et al., 2018). Because the relevant observable is the truncated Hamiltonian itself, the source emphasizes that this entropy is associated with a natural observable and can be connected to a Loschmidt-echo protocol (Najafi et al., 2018).

Taken together, these formulations show that subsystem operator Rényi entropy need not refer to the spectrum of $A$08 alone. It may instead quantify the Rényi complexity of subsystem measurement outcomes in a chosen basis or of a distinguished subsystem observable.

7. Typicality, thermal ensembles, and observer dependence

Typicality results provide useful baselines. For a Haar-random pure state on $A$09 with dimensions $A$10 and $A$11, the average subsystem Rényi entropy

$A$12

admits an exact solution for $A$13 and an analytic approximation for general $A$14 (Kim et al., 2023). In the large-$A$15 limit,

$A$16

which matches the asymptotic Page behavior at $A$17 (Kim et al., 2023). The source explicitly notes that these results can be repurposed for subsystem operator Rényi entropies whenever an operator is purified or vectorized into a bipartite pure state (Kim et al., 2023).

Thermal subsystem Rényi entropy in all-to-all systems was analyzed for SYK-like models. For a subsystem $A$18 of $A$19 modes inside $A$20 total modes, the second Rényi entropy

$A$21

is computed from a replicated large-$A$22 path integral with subsystem-dependent bilocal fields (Zhang et al., 2020). For Majorana SYK$A$23 with $A$24, the small-subsystem limit is maximally mixed,

$A$25

and for $A$26 the results are well approximated by a thermal entropy with an effective temperature determined by energy matching (Zhang et al., 2020).

For the Hubbard model, a different replica path integral introduces the Rényi entanglement through a local kick term between replicas. Within inhomogeneous DMFT, the second Rényi entropy is extracted as

$A$27

and is computed for extended subsystems in one and two dimensions (Bera et al., 2023). In the correlated metallic phase, the subsystem-size scaling is described by the crossover formula interpolating between the volume-law thermal Rényi entropy and the universal boundary-law Rényi entanglement entropy with logarithmic violation (Bera et al., 2023). The same framework yields a Rényi mutual information that shows hysteresis across the first-order Mott transition and non-monotonic temperature dependence near the critical endpoint (Bera et al., 2023).

Observer dependence appears explicitly in quantum reference frames. There, one considers relational subsystems $A$28, dephases the reduced relational state $A$29 in a group-label basis, and defines the diagonal Rényi entropy

$A$30

For ideal frames, these are frame-independent diagonal Rényi invariants: $A$31 for the corresponding subsystem $A$32 in frame $A$33 (Hamette, 24 Mar 2026). The same source proves a coherence–entanglement tradeoff,

$A$34

and for non-ideal frames derives a bound on observer-dependent entropy differences in terms of the effective relational Hilbert-space dimension (Hamette, 24 Mar 2026). This makes explicit that subsystem operator Rényi entropy can be basis dependent and observer dependent even when it is frame-invariant in a diagonalized form.

Across these formulations, the common core is a Rényi functional applied after restriction: to a subalgebra, to a reduced state, to a reduced operator, to a subsystem probability distribution, or to a dephased relational state. What changes from framework to framework is the meaning of the restriction, the role of the operator, and the physical interpretation of the resulting entropy—mixing, entanglement, distinguishability, transport, thermalization, or observer dependence.