---
title: Subsystem Operator Rényi Entropy
url: https://www.emergentmind.com/topics/subsystem-operator-renyi-entropy
type: topic
---

# Subsystem Operator Rényi Entropy

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 [1905.03498, 2602.22331, 2203.13124, 2412.10735].

## 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 \(\mathcal O(t)\) [2301.09074, 2203.13124, 2602.22331].

| Formulation | Core object | Representative source |
|---|---|---|
| Operator-algebraic | \(S_\alpha^{\mathcal S}(\varphi)\) for a state \(\varphi\) and reference system \(S\) | [1905.03498] |
| Reduced operator | \(S_{\mathcal O A}^{(2)}(t)\) from \(\rho_{\mathcal O A}(t)\) | [2602.22331] |
| Basis-dependent subsystem entropy | \(Re_\alpha(L)\) from \(p_{I_A}\) | [2203.13124] |
| Observable-based subsystem entropy | \(H_\alpha\) from subsystem-energy probabilities \(p(E_j)\) | [1806.02243] |
| Relative/distinguishability form | \(S_k(\rho_A\Vert\sigma_A)\), \(D_A^{(n)}\) | [2412.10735], [1911.04797] |
| Global aggregate of all subsystems | \(S_W^{(2)}\) from the Husimi function | [2509.16036] |

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 \(S\subseteq\mathcal G\) [2203.13124, 2602.22331, 1905.03498].

## 2. Operator-algebraic formulation and reference systems

The most general operator-algebraic construction in the sources is the \(C^*\)-algebraic \(S\)-mixing Rényi entropy. Let \(\mathcal A\) be a \(C^*\)-algebra, \(S\) a compact convex set of states, and \(\varphi\in S\). For a countable extremal decomposition
\[
\varphi=\sum_k \lambda_k \varphi_k,\qquad \varphi_k\in \mathrm{ex}\,S,
\]
the Rényi entropy relative to the reference system \(S\) is
\[
S_\alpha^{\mathcal S}(\varphi)=
\inf_{\mu\in\mathcal D_\varphi(S)}\frac{1}{1-\alpha}\log\sum_k \lambda_k^\alpha,
\qquad \alpha\in[0,\infty)\setminus\{1\},
\]
with value \(+\infty\) if no countable extremal decomposition exists [1905.03498]. This extends Ohya’s \(\mathcal S\)-mixing entropy, is monotonically decreasing in \(\alpha\), and satisfies
\[
\lim_{\alpha\to1}S_\alpha^{\mathcal S}(\varphi)=S^{\mathcal S}(\varphi)
\]
[1905.03498].

This framework makes subsystem dependence explicit through the choice of reference system. A spatial subsystem can be modeled by a subalgebra \(\mathcal B\subseteq\mathcal A\) and the restricted state \(\varphi|_{\mathcal B}\), leading to
\[
S_\alpha^{\mathcal G(\mathcal B)}(\varphi|_{\mathcal B}).
\]
A dynamical or thermodynamic subsystem can instead be encoded by a distinguished convex subset of states, such as the invariant states \(I(\Theta)\) or KMS states \(K_\beta(\Theta)\) of a \(C^*\)-dynamical system \((\mathcal A,G,\Theta(G))\) [1905.03498].

The same construction recovers familiar entropies in special cases. For a finite probability space with probabilities \(\{p_k\}\), one obtains the classical Rényi entropy
\[
S_\alpha^{\mathcal S}(\varphi)=\frac{1}{1-\alpha}\log\sum_k p_k^\alpha.
\]
For density operators \(\rho\) on \(\mathcal A=\mathcal C(H)+\mathbb CI\), the state \(\varphi_\rho(A)=\operatorname{Tr}(\rho A)\) satisfies
\[
S_\alpha^{\mathcal G}(\varphi_\rho)=S_\alpha(\rho)=\frac{1}{1-\alpha}\log\operatorname{Tr}(\rho^\alpha),\qquad \alpha>1,
\]
and for \(0<\alpha<1\),
\[
S_\alpha^{\mathcal G}(\varphi_\rho)\le S_\alpha(\rho)
\]
[1905.03498].

Reference-system inclusion generates entropy inequalities. For a KMS state \(\varphi\in K_\beta(\Theta)\),
\[
S_\alpha^{K_\beta(\Theta)}(\varphi)\le S_\alpha^{I(\Theta)}(\varphi),\qquad
S_\alpha(\varphi)\ge S_\alpha^{K_\beta(\Theta)}(\varphi),
\]
and under \(G\)-commutativity,
\[
S_\alpha(\varphi)\ge S_\alpha^{I(\Theta)}(\varphi)\ge S_\alpha^{K_\beta(\Theta)}(\varphi).
\]
If the KMS state is unique, then
\[
S_\alpha^{K_\beta(\Theta)}(\varphi)=0
\]
[1905.03498]. 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 \(N\) distinguishable qubits. For a density matrix \(\hat\rho\), the Husimi function is
\[
P_H(\hat\rho,\mathbf n)=\frac{1}{(2\pi)^N}\langle \mathbf n|\hat\rho|\mathbf n\rangle,
\]
and the Wehrl–Rényi entropy of order \(2\) is
\[
S_W^{(2)}(\hat\rho)=-\ln\!\left[\int d\mathbf n\,\big(P_H(\hat\rho,\mathbf n)\big)^2\right]
\]
[2509.16036].

The central exact identity is
\[
e^{-S_W^{(2)}(\hat\rho)}
= \frac{1}{(6\pi)^N}\sum_A \operatorname{Tr}(\hat\rho_A^2),
\]
where the sum runs over all \(2^N\) subsystems \(A\subseteq\{1,\dots,N\}\) and \(\hat\rho_A\) is the reduced density matrix on \(A\) [2509.16036]. 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
\[
\operatorname{Tr}(\hat\rho_A^2)=e^{-S_A^{(2)}}.
\]

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, \(S_W^{(2)}\) is subsystem independent in its definition but encodes contributions from all possible subsystems [2509.16036]. The bounds
\[
N\ln(3\pi)\le S_W^{(2)}(\hat\rho)\le N\ln(4\pi)
\]
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-\(N\) limit [2509.16036].

The derivation uses local Haar averaging, SWAP operators, and the identity
\[
\operatorname{Tr}\big[(\hat\rho\otimes\hat\rho)\hat W_A\big]=\operatorname{Tr}(\hat\rho_A^2),
\]
so the same algebraic backbone appears in operator-entanglement and randomized-measurement settings [2509.16036]. 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 \(\mathcal O(t)=e^{i\mathcal H t}\mathcal O e^{-i\mathcal H t}\), a bipartition into \(A\) and \(B\), and normalization \(Z_{\mathcal O}=\operatorname{Tr}[\mathcal O]\), the reduced operator is
\[
\rho_{\mathcal O A}(t)=\frac{\operatorname{Tr}_B[\mathcal O(t)]}{Z_{\mathcal O}},
\]
and the second Rényi subsystem operator entropy is
\[
e^{-S_{\mathcal O A}^{(2)}(t)}=\operatorname{Tr}_A\big[\rho_{\mathcal O A}^2(t)\big]
\]
[2602.22331]. For the local density operator \(\mathcal O=\hat n_l=c_l^\dagger c_l\), the dynamical growth
\[
\Delta S_{\mathcal O A}^{(2)}(t)=S_{\mathcal O A}^{(2)}(t)-S_{\mathcal O A}^{(2)}(0)
\]
is bounded by
\[
\Delta S_{\mathcal O A}^{(2)}(t)\le \ln 2
\]
[2602.22331].

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 [2602.22331].

The same work constructs a unified Schwinger–Keldysh field-theory formalism for \(S_{\mathcal O A}^{(2)}(t)\) 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 [2602.22331]. 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 [2602.22331].

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 \(\rho_A\) and \(\sigma_A\) on an interval \(A\), the relative Rényi entropy used in rational and holographic CFTs is
\[
S_k(\rho_A\Vert\sigma_A)=\frac{1}{k-1}\log\frac{\operatorname{tr}\rho_A^k}{\operatorname{tr}(\rho_A\sigma_A^{k-1})},
\]
and is interpreted there as quantifying how distinguishable two local operator excitations are when restricted to \(A\) [2412.10735]. 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 [2412.10735].

A complementary set of quantities is given by subsystem trace and Schatten distances after local operator quenches. For a nonchiral primary field \(\mathcal O\), the Rényi entropy increase obeys
\[
\Delta S_A^{(n)}(t)=
\begin{cases}
0, & 0<t<\ell,\ t>2\ell,\\
\log d_{\mathcal O}, & \ell<t<2\ell,
\end{cases}
\]
where \(d_{\mathcal O}\) is the quantum dimension [1911.04797]. 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 [1911.04797]. Consequently, the Schatten distances are piecewise constant and, in the orthogonal regimes, the trace distance reaches its maximal value in the source’s normalization [1911.04797].

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 [2011.11006]. In the CFT regime, the leading correction to \(\mathcal F^{(n)}_{A,\mathcal X}\) is
\[
\mathcal F^{(n)}_{A,\mathcal X}
=1-\frac{\pi^2(n^2-1)\Delta_{\mathcal X}}{3n}\,x^2+o(x^2),
\qquad x=\frac{\ell}{L},
\]
so the leading short-interval behavior depends only on the scaling dimension \(\Delta_{\mathcal X}\) [2011.11006]. In the extremely gapped limit of the harmonic chain, by contrast, the leading behavior depends only on the total number of excited quasiparticles \(R\), both for Rényi entropies and for subsystem Schatten distances [2011.11006]. 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 \(|g\rangle=\sum_I a_I|I\rangle\) written in a local product basis, the subsystem probabilities are
\[
p_{I_A}=\sum_{I_{\bar A}} P_{I_A I_{\bar A}},\qquad P_I=|a_I|^2,
\]
and the subsystem operator Rényi entropy is
\[
Re_\alpha(L)=\frac{1}{1-\alpha}\ln\sum_{I_A} p_{I_A}^\alpha
\]
[2203.13124]. At a critical point, this obeys
\[
Re_\alpha(L)=a_\alpha L + x_\alpha \ln L + \mathcal O(1),
\]
with universal logarithmic coefficient \(x_\alpha\) [2203.13124]. For critical quadratic fermions with \(U(1)\) symmetry,
\[
x_\alpha=
\begin{cases}
\frac{c}{8}, & \alpha\le 4,\\[0.5ex]
\frac{\alpha}{\alpha-1}\frac{c}{8}, & \alpha>4,
\end{cases}
\]
where \(c\) is the central charge. Without \(U(1)\) symmetry,
\[
x_\alpha=
\begin{cases}
\frac{b(\alpha)}{8}, & \alpha\le 1,\\[0.5ex]
\frac{\alpha}{\alpha-1}\frac{c}{8}, & \alpha>1.
\end{cases}
\]
These formulas exhibit non-analytic changes at \(\alpha=4\) or \(\alpha=1\), respectively [2203.13124].

An observable-based formulation is the Rényi entropy of subsystem energy. For a truncated subsystem Hamiltonian \(H_l\) with eigenstates \(|e_j\rangle\), subsystem-energy probabilities are
\[
p_j=\langle e_j|\rho_l|e_j\rangle,
\]
and
\[
H_\alpha=\frac{1}{1-\alpha}\ln\sum_j p_j^\alpha
\]
[1806.02243]. This quantity obeys an area law in gapped phases and, at criticality,
\[
H_\alpha(l)=\epsilon(\alpha)\ln l+\beta_\alpha
\]
with universal coefficient \(\epsilon(\alpha)\) that scales with the central charge in the Ising and XX universality classes [1806.02243]. 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 [1806.02243]. 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 [1806.02243].

Taken together, these formulations show that subsystem operator Rényi entropy need not refer to the spectrum of \(\rho_A\) 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 \(\mathcal H_A\otimes\mathcal H_B\) with dimensions \(m\) and \(n\), the average subsystem Rényi entropy
\[
S_\alpha(m,n)=\mathbb E[S_\alpha(\rho_A)]
\]
admits an exact solution for \(m=\alpha=2\) and an analytic approximation for general \(\alpha\) [2301.09074]. In the large-\(n\) limit,
\[
\widetilde S_\alpha(m,n)\sim \ln m-\frac{\alpha}{2n}(m-m^{-1}),
\]
which matches the asymptotic Page behavior at \(\alpha\to1\) [2301.09074]. 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 [2301.09074].

Thermal subsystem Rényi entropy in all-to-all systems was analyzed for SYK-like models. For a subsystem \(A\) of \(M\) modes inside \(N\) total modes, the second Rényi entropy
\[
\mathcal S_A^{(2)}=\frac{1}{1-2}\log\operatorname{Tr}_A(\rho_A^2)
\]
is computed from a replicated large-\(N\) path integral with subsystem-dependent bilocal fields [2003.09766]. For Majorana SYK\(_q\) with \(q\ge4\), the small-subsystem limit is maximally mixed,
\[
\mathcal S_A^{(n)}(\lambda\to0)=\frac{M\log2}{2},\qquad \lambda=\frac{M}{N},
\]
and for \(M\le N/2\) the results are well approximated by a thermal entropy with an effective temperature determined by energy matching [2003.09766].

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
\[
S_A^{(2)}=\int_0^1 d\lambda\, \langle \mathcal S_{\rm kick}\rangle_{Z_A^{(2)}(\lambda)},
\]
and is computed for extended subsystems in one and two dimensions [2302.10940]. 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 [2302.10940]. 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 [2302.10940].

Observer dependence appears explicitly in quantum reference frames. There, one considers relational subsystems \(X\subseteq SR_{\bar i}\), dephases the reduced relational state \(\rho_X^{(R_i)}\) in a group-label basis, and defines the diagonal Rényi entropy
\[
S_\alpha(\Delta\rho_X^{(R_i)}).
\]
For ideal frames, these are frame-independent diagonal Rényi invariants:
\[
S_\alpha(\Delta\rho_X^{(R_i)})=S_\alpha(\Delta\rho_Y^{(R_j)})
\]
for the corresponding subsystem \(Y\) in frame \(R_j\) [2603.23598]. The same source proves a coherence–entanglement tradeoff,
\[
S_\alpha(\Delta\rho_X^{(R_i)})=S_\alpha(\rho_{\overline X}^{(R_i)})+C_\alpha(\rho_X^{(R_i)}),
\]
and for non-ideal frames derives a bound on observer-dependent entropy differences in terms of the effective relational Hilbert-space dimension [2603.23598]. 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.

Source: https://www.emergentmind.com/topics/subsystem-operator-renyi-entropy