Avg. Subsystem Bures Distance in Quantum Systems

• Average subsystem Bures distance is a quantitative metric defined via fidelity between reduced density matrices, measuring distinguishability in many-body quantum systems.
• It employs algorithmic and analytical techniques, such as mapping to free fermions and using Majorana correlation matrices, to compute state differences efficiently.
• The metric distinguishes integrable from chaotic systems by exhibiting linear scaling in integrable regimes and vanishing behavior in chaotic settings.

The average subsystem Bures distance is a quantitative metric for distinguishing quantum states based on the average Bures distance between reduced density matrices (“subsystems”) across different partitions or pairs of eigenstates in many-body quantum systems. It plays a central role in quantum information theory and statistical physics as a tool for probing complex correlations, diagnosing integrability versus chaos, and benchmarking quantum state distinguishability. The Bures distance, being Riemannian and monotonic, is structurally adapted to operational and geometric features in quantum mechanics, providing robustness under physically relevant operations such as partial trace and quantum channels.

1. Definition and Formalism

For two density matrices $\rho$ and $\sigma$ (possibly reduced from pure or mixed global states), the Bures distance is defined as

$$B(\rho, \sigma) = \sqrt{2 \left(1 - F(\rho, \sigma)\right)}$$

where the fidelity is

$$F(\rho, \sigma) = \operatorname{tr}\left[\sqrt{ \sqrt{\rho}\ \sigma\ \sqrt{\rho} }\,\right].$$

Given an orthonormal basis $\{|\psi_i\rangle\}$ of eigenstates (possibly ordered with respect to energy or local conserved charges), and a subsystem $A$ defined by tracing out degrees of freedom from the complement, the reduced density matrices are

$$\rho_{A,i} = \operatorname{tr}_{\bar{A}} |\psi_i\rangle\langle\psi_i|.$$

The average subsystem Bures distance for subsystem size $\ell$ is

$$\langle B_A \rangle = \frac{1}{d-1} \sum_{i=1}^{d-1} B(\rho_{A,i}, \rho_{A,i+1}),$$

with $d$ the number of states scanned (typically all eigenstates within a given symmetry sector) (Guo et al., 13 Aug 2025).

This average quantifies the typical distinguishability between “neighboring” reduced states in the spectrum and can similarly be extended to averages over random pairs or ensembles.

2. Algorithmic and Analytical Computation

Analysis of the average subsystem Bures distance in large many-body systems, such as the transverse-field Ising chain, crucially leverages the structure of the eigenstates. For quadratic (integrable) models, all eigenstates are Gaussian, enabling an efficient algorithmic procedure:

• Map the system (e.g., via the Jordan–Wigner transformation) to free fermions;
• Construct $2\ell \times 2\ell$ Majorana correlation matrices $\Gamma$ for subsystem size $\ell$,

$$\Gamma_{m n} = \operatorname{tr}(\rho_{\text{G}} d_m d_n) - \delta_{m n}$$

where $d_m$ are Majorana operators.

• For two Gaussian states $\rho_{\Gamma_1}, \rho_{\Gamma_2}$, the fidelity is given by (Guo et al., 13 Aug 2025): $$F(\rho_{\Gamma_1}, \rho_{\Gamma_2}) = \left[ \det \frac{1 + \Gamma_1}{2} \right]^{1/4} \left[ \det \frac{1 + \Gamma_2}{2} \right]^{1/4} \left[ \det \left( 1 + \sqrt{ \frac{1 - \Gamma_1}{1 + \Gamma_1} \frac{1 - \Gamma_2}{1 + \Gamma_2} } \right) \right]^{1/2}.$$
• In cases with numerical instabilities (e.g., pure state blocks where eigenvalues of $\Gamma$ are $\pm1$), an iterative rank reduction is performed, effectively restricting the problem to the lower-dimensional mixed sector.

This pipeline enables tractable evaluation of $\langle B_A \rangle$ for system sizes substantially larger than what is possible with generic density matrices.

3. Ordering and Integrability Structure

In integrable systems, degeneracies proliferate due to nontrivial sets of local conserved charges. The order in which eigenstates are paired affects $\langle B_A \rangle$:

• In the transverse-field Ising chain, each eigenstate is labeled not only by its energy (eigenvalue of $Q_0$) but also by eigenvalues of higher ($Q_1, Q_2, ...$) local charges (Guo et al., 13 Aug 2025).
• Systematic ordering is accomplished by lexicographically sorting eigenstates first by $Q_0$, then $Q_1$, and so on, to uniquely pair degenerate eigenstates and meaningfully define “neighboring state” distances.

Numerical experiments reveal that this procedure almost entirely lifts degeneracies for each total system size and field value. For random pure Gaussian states (generated by Haar-random orthogonal rotations), which lack the organizing principle of integrability, the same linear scaling is not observed.

4. Scaling Behaviors and Diagnostic Power

The average subsystem Bures distance displays sharply contrasting scaling behaviors depending on the global system’s integrability or chaos:

• Chaotic Systems (e.g., non-integrable or thermalizing Hamiltonians):
  • As system size $L \rightarrow \infty$ with fixed subsystem fraction $x = \ell/L < 1/2$, $\langle B_A \rangle$ (and similarly the average subsystem trace distance) vanishes (Miranda et al., 2022, Guo et al., 13 Aug 2025).
  • This is consistent with the Eigenstate Thermalization Hypothesis: locally, adjacent eigenstates become nearly indistinguishable as the system becomes large.
• Integrable Systems (with many local charges):
  • $\langle B_A \rangle$ grows linearly with subsystem size for $x < 1/2$.
  • Quantitatively, the data are consistent with $f(x) \approx 2x$ (piecewise linear up to $x = 1/2$, saturating at unity for $x \geq 1/2$).
  • This scaling is attributed to abrupt discontinuities in local conserved charges across the spectrum, leading to sharp differences in the reduced states despite similar energies.
• Random States and Ensembles:
  • For random pure Gaussian states or random Haar states, the average subsystem Bures distance does not exhibit linear scaling with subsystem size, but instead saturates rapidly at a finite value less than unity (Guo et al., 13 Aug 2025, Laha et al., 2021, Laha et al., 2022).

These results are corroborated by both analytic calculations and large-scale numerics.

5. Relation to Other Distances and Statistical Properties

The Bures distance is tightly related to other notions of distinguishability:

• Trace Distance: $\| \rho - \sigma \|_1 / 2$, whose average subsystem version shows similar scaling and, due to the Fuchs–van de Graaf inequalities, is closely bounded by a function of the Bures distance (Miranda et al., 2022). In the thermodynamic limit, both measures exhibit a sharp transition (first-order) in random pure states as the measured subsystem reaches half the total system size.
• Statistical Distributions: For ensembles of random states, the distribution of subsystem distances (including purity, entanglement entropy, and the Bures distance) admits analytic characterization via random matrix theory, e.g., using the Bures-Hall ensemble for entanglement entropies (Wei, 2020, Wei, 2020), with Gaussian approximations valid in the large-dimension limit.

6. Physical and Operational Implications

The average subsystem Bures distance serves as a probe for:

• Quantum Integrability and Chaos: Linear scaling in integrable systems versus vanishing behavior in chaotic (thermalizing) systems provides a diagnostic for the presence of extensive sets of local conserved charges (Guo et al., 13 Aug 2025).
• Subystem State Fluctuation: Provides a universal benchmark for the “typical” distance between subsystems, relevant for understanding local distinguishability and the spread of local observables across eigenstates.
• Statistical State Characterization: In high-dimensional systems, the mean and variance of the Bures distance capture most of the information about subsystem fluctuations, with the full distribution approximated by a gamma or Gaussian law in many instances (Wei, 2020, Laha et al., 2022).
• Quantum Estimation and Information Processing: In tasks such as state and parameter estimation, the Bures distance, due to its operational meaning via quantum Fisher information and contractive properties, is more appropriate for error certification than the trace-norm in many contexts (Holevo, 6 Dec 2024, Bhatia et al., 2017).

7. Extensions and Open Problems

Recent algorithmic advances enable the efficient computation of subsystem Bures distances for large lattices of free fermions, paving the way for the systematic exploration of many-body integrable and chaotic regimes (Guo et al., 13 Aug 2025). The disordering and symmetry structure (e.g., presence of local or global conservation laws) crucially affect scaling behavior. For more general systems (including interacting non-quadratic models), tractable computation of Bures distances remains a significant challenge.

Analytically, much of the statistical machinery (Bures measure, Bures–Hall ensemble) relies on connections to random matrix theory and, in the thermodynamic limit, allows explicit computation of means, variances, and even distributions for distance-related observables (Forrester et al., 2014, Wei, 2020, Wei, 2020, Laha et al., 2021, Laha et al., 2022).

Finally, the theoretical identification of the sharp transition at half-partition size and its robustness under conservation law constraints has ramifications for information-theoretic questions (e.g., questions related to black hole information paradox, many-body localization, and quantum error correction), where subsystem state distinguishability is a central concept (Miranda et al., 2022).

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Summary Table: Key Features of Average Subsystem Bures Distance

Property	Chaotic (ergodic) systems	Integrable systems
Scaling of $\langle B_A \rangle$ with $\ell$	Vanishes for $x<1/2$	$2x$ for $x<1/2$, saturates at 1
Eigenstate organization required?	No (energy ordering suffices)	Yes (must sort by conserved charges)
Transition behavior ($\ell/L = 1/2$)	Sharp first-order	Sharp, but charge-dependent
Effect of random/chaotic eigenstates	No linear increase	Not observed in random Gaussian

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The average subsystem Bures distance is thus a robust, physically meaningful, and computationally tractable metric for the global-to-local structure of quantum correlations and eigenstate organization, with clear diagnostic power for chaos, integrability, and subsystem state fluctuation regimes in quantum many-body systems (Guo et al., 13 Aug 2025, Miranda et al., 2022, Bhatia et al., 2017, Forrester et al., 2014).