Averaged Accessible Information (AAI)

• AAI is defined as the Haar-averaged Rényi-2 accessible information, relying solely on the purity of the local reduced density matrix.
• It employs randomized measurements, such as classical shadows with Pauli protocols, to efficiently estimate information recovery in many-body systems.
• AAI serves as a scalable diagnostic for quantum information scrambling, distinguishing between chaotic, ballistic, and localized dynamics.

Averaged Accessible Information (AAI) is a randomized-measurement information diagnostic introduced for quantum many-body dynamics to quantify how much information about an initially local perturbation remains recoverable from a local subsystem when the measurement basis is chosen blindly and randomly (Chen et al., 13 May 2026). In the formulation of “Probing Quantum Information Scrambling via Local Randomized Measurements” (Chen et al., 13 May 2026), the quantity is denoted $\chi_2$ and is defined as the Haar-averaged Rényi-2 accessible information. For a local reduced density matrix $\rho_A$, it reduces to the closed form

$$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$

so that the diagnostic depends only on subsystem purity rather than on an explicitly optimized measurement protocol (Chen et al., 13 May 2026).

1. Definition and formal construction

The starting point for AAI is the standard Holevo quantity for an ensemble

$\mathcal{E}=\{p_i,\rho_i\},$

written as

$$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$

In the AAI framework, this quantity represents the optimal accessible information, namely the maximum information one could extract with the best possible measurement. The motivation for AAI is that Holevo information requires an optimal, state-dependent measurement, which is not practical in many-body experiments (Chen et al., 13 May 2026).

AAI replaces that optimization by averaging over randomized measurements. For a projective measurement in a basis $\mathcal{M}=\{|\alpha_j\rangle\}$, and for an ensemble of pure states

$$\mathcal{E}_{\mathrm{pure}}=\{p_i,|\psi_i\rangle\},$$

the classical outcome probabilities are

$$P(j|i)=|\langle \alpha_j|\psi_i\rangle|^2, \qquad P(j)=\sum_i p_i P(j|i)=\langle \alpha_j|\rho|\alpha_j\rangle.$$

The basis-dependent information functional is the Rényi-2 classical mutual information

$$I_2(\mathcal{E}:\mathcal{M}) = H_2(P(j))-\sum_i p_i H_2(P(j|i)),$$

with

$H_2(q)=-\log\!\left(\sum_j q_j^2\right).$

AAI is then defined by averaging this quantity over Haar-random measurement bases in the “annealed” form

$\rho_A$0

The use of Haar-random bases makes the result universal and basis-independent, and the Rényi-2 structure makes it directly compatible with randomized-measurement protocols (Chen et al., 13 May 2026).

2. Haar-random formula and purity reduction

The defining averages of the construction are

$\rho_A$1

and, for any pure state $\rho_A$2,

$\rho_A$3

Substituting these identities yields the central analytical result

$\rho_A$4

The paper also gives the equivalent spectral expression

$\rho_A$5

where $\rho_A$6 are the eigenvalues of $\rho_A$7 (Chen et al., 13 May 2026).

Because $\rho_A$8 is a function only of $\rho_A$9, AAI becomes a purity-based information measure. The paper states that $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$0 is strictly concave, unlike the bare Rényi-2 entropy

$$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$1

which is not concave in general. The additive constant $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$2 inside $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$3 is what “repairs” the concavity pathology and makes the quantity well behaved as an information measure (Chen et al., 13 May 2026).

3. Local reduced states and operational estimation

AAI is evaluated on the reduced state $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$4 of a local subsystem $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$5. Since the purity

$$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$6

measures how mixed the local state is, AAI increases as the subsystem becomes more mixed. The paper records two limiting cases: if $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$7 is pure, then $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$8 and $$Q_2(\rho_A)=\log\!\left(\frac{2}{1+\operatorname{Tr}(\rho_A^2)}\right),$$9; if $\mathcal{E}=\{p_i,\rho_i\},$0 is maximally mixed in large dimension, then $\mathcal{E}=\{p_i,\rho_i\},$1 and $\mathcal{E}=\{p_i,\rho_i\},$2 (Chen et al., 13 May 2026).

Experimentally, the quantity is not estimated by direct Haar integration. Instead, the paper uses the classical shadow protocol with single-qubit randomized Pauli measurements to estimate local purities and hence $\mathcal{E}=\{p_i,\rho_i\},$3 efficiently. The procedure is:

1. apply a random local unitary $\mathcal{E}=\{p_i,\rho_i\},$4 from a chosen ensemble,
2. measure in the computational basis,
3. reconstruct an unbiased single-shot estimator (“snapshot”) $\mathcal{E}=\{p_i,\rho_i\},$5 via the inverse measurement channel.

For randomized Pauli measurements, the snapshot has the explicit local form

$\mathcal{E}=\{p_i,\rho_i\},$6

The paper states that

$\mathcal{E}=\{p_i,\rho_i\},$7

shots are enough to estimate many $\mathcal{E}=\{p_i,\rho_i\},$8-local observables to additive precision $\mathcal{E}=\{p_i,\rho_i\},$9, independent of the full system size. This is the operational basis for treating AAI as a local and scalable scrambling diagnostic (Chen et al., 13 May 2026).

4. Role in scrambling diagnostics

The physical use of AAI is to track the information dynamics of local perturbations under many-body evolution. Because the final formula depends only on the local reduced density matrix purity, the diagnostic can be extracted across extended subsystems while remaining tied to randomized local probes rather than optimal decoding (Chen et al., 13 May 2026).

Many-body setting	AAI signature	Interpretation stated in the paper
Mixed-field Ising model	chaotic light cone; strong scrambling	confining potential prevents free domain-wall propagation
Transverse-field Ising model	ballistic spreading	dynamics dominated by free quasiparticle/domain-wall transport
PXP model	persistent revivals	weak ergodicity breaking and confinement to a special nonthermal subspace
Many-body localization	spatially localized AAI; suppression of transport	local integrals of motion (“l-bits”) trap information near its origin

In the mixed-field Ising model, AAI reveals a chaotic light cone and strong scrambling; the longitudinal field creates a confining potential, preventing free domain-wall propagation, and this leads to localized perturbation spreading that is consistent with genuine scrambling. In the transverse-field Ising model, AAI shows ballistic spreading of information, and the information loss is not due to strong chaos but to coherent propagation. In the PXP model, starting from the Néel state, AAI displays persistent revivals and faithfully tracks scar-induced oscillatory dynamics. In the many-body-localized regime, AAI remains spatially localized and shows strong suppression of transport; the comparison between the full system and a truncated local cage gives near-identical early-time AAI dynamics, confirming strict dynamical confinement (Chen et al., 13 May 2026).

5. Relation to accessible information, Holevo information, and adjacent concepts

AAI belongs to a larger family of information-extraction quantities, but it is not identical to the standard accessible information used in quantum information theory. In “Accessible Information and Informational Power of Quantum 2-designs” (Dall'Arno, 2014), the relevant quantities are the accessible information of an ensemble,

$$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$0

and the informational power of a POVM,

$$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$1

That work studies spherical quantum $$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$2-designs and proves the dimension-dependent bounds

$$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$3

with $$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$4 for all $$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$5. The paper does not define “Averaged Accessible Information” explicitly as a named quantity; rather, it studies standard accessible information, informational power, entropy bounds for $$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$6-designs, and the special cases of SICs and maximal sets of MUBs (Dall'Arno, 2014).

A different nearby notion appears in “Accessible information of a general quantum Gaussian ensemble” (Holevo, 2021), which computes the accessible information

$$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$7

for a broad class of quantum Gaussian ensembles under a threshold condition and shows that the maximizing measurement is Gaussian, namely a squeezed heterodyne measurement. Here again the object is standard accessible information rather than an averaged randomized-measurement functional (Holevo, 2021).

The distinction is sharper in “Robustness of Measurement, discrimination games and accessible information” (Skrzypczyk et al., 2018). That paper shows that the robustness of a measurement is the single-shot, min-entropy version of accessible information for the corresponding quantum-to-classical channel: $$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$8 It is therefore a one-shot accessible min-information identity rather than an averaged Shannon-type accessible information construction (Skrzypczyk et al., 2018).

Two additional lines of work clarify the broader terminology. “Non-Markovianity through accessible information” (Fanchini et al., 2014) uses the accessible information

$$\chi(t)=S\!\left(\sum_i p_i\rho_i(t)\right)-\sum_i p_i S(\rho_i(t)), \qquad S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$9

as “assisted knowledge” acquired by the environment about the system, but does not introduce a separately named AAI quantity (Fanchini et al., 2014). “Complementarity of quantum discord and classically accessible information” (Zwolak et al., 2013) likewise does not use the term AAI explicitly, but it does study the Holevo quantity

$\mathcal{M}=\{|\alpha_j\rangle\}$0

together with an averaging convention over all environment fragments of the same size. This is the closest fragment-averaged predecessor to later AAI language (Zwolak et al., 2013).

These neighboring constructions suggest that AAI occupies a specific niche: optimization over measurements is replaced by averaging over randomized local probes, while the output remains an operational information quantity rather than a purely geometric or entropic invariant.

6. Scope, assumptions, and interpretive limits

The clean formula

$\mathcal{M}=\{|\alpha_j\rangle\}$1

is derived under Haar-random measurement averaging. The experimental implementation replaces exact Haar sampling by randomized Pauli or Clifford measurements, which approximate the Haar result exactly when the ensemble is at least a $\mathcal{M}=\{|\alpha_j\rangle\}$2-design for purity-related quantities. The derivation also uses the annealed approximation $\mathcal{M}=\{|\alpha_j\rangle\}$3, justified by concentration of measure in high-dimensional settings (Chen et al., 13 May 2026).

AAI is therefore not the optimal accessible information of a subsystem. The paper states explicitly that it is a typical accessible information under randomized probes. This distinction matters conceptually: the standard accessible-information literature emphasizes optimization over measurements, whereas AAI emphasizes recoverability under blind randomized probing. The resulting shift is operational as well as methodological, and the paper characterizes it as a move from relying on optimal measurements to utilizing randomized local probes for the characterization of complex quantum information dynamics (Chen et al., 13 May 2026).

Within that scope, AAI has three defining properties. It is operational because it is tied to concrete measurements; it is experiment-friendly because it can be estimated with classical shadows and randomized Pauli bases; and it is local and scalable because it can be computed for many subsystems at once (Chen et al., 13 May 2026). A plausible implication is that AAI is most naturally interpreted not as a replacement for accessible information in the Shannon-theoretic sense, but as a purity-controlled randomized-measurement surrogate for local information recoverability in many-body systems.