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Support-Projected Petz Monotone Geometry of Pure Two-Qubit Families: Universal Three-Channel Decomposition and Non-Reduction of Curvature Invariants

Gunhee Cho, Jeongwoo Jae

TL;DR

This work develops a support-projected Petz geometry for smooth pure two-qubit families by pulling back arbitrary Petz monotone metrics to the active spectral support and projecting onto it, unifying the SLD/Bures case with other metrics such as WY and BKM. It proves a universal three-channel decomposition of the reduced Petz QFI into population, coherence, and concurrence-derivative channels, with weights determined by the Morozova–Chentsov kernel; importantly, curvature invariants cannot be reduced to functions of concurrence or reduced entropy, showcasing a fundamental distinction between entanglement and curvature as quantum resources. An entanglement-orthogonal gauge is constructed to reveal how population and coherence derivatives influence curvature independently of entanglement, and explicit counterexamples in the SLD/Bures setting (via a four-parameter 2-HEA) demonstrate non-monotonicity and the failure of one-channel curvature formulas. The framework is leveraged to propose geometry-aware natural-gradient preconditioning for VQAs and is validated through VQE experiments, illustrating stable optimization on reduced manifolds despite rank deficiencies. The results have broad implications for quantum information geometry, Gaussian-state curvature, and the design of curvature-aware variational algorithms.

Abstract

We develop a support-projected Petz monotone geometry for pure two-qubit families, obtained by pulling back arbitrary Petz monotone quantum metrics to circuit-defined submanifolds and projecting onto the active spectral support of the associated quantum Fisher information tensor. This framework strictly generalizes the symmetric logarithmic derivative (SLD/Bures) case and includes, as special examples, the Wigner--Yanase and Bogoliubov--Kubo--Mori metrics among many others.

Support-Projected Petz Monotone Geometry of Pure Two-Qubit Families: Universal Three-Channel Decomposition and Non-Reduction of Curvature Invariants

TL;DR

This work develops a support-projected Petz geometry for smooth pure two-qubit families by pulling back arbitrary Petz monotone metrics to the active spectral support and projecting onto it, unifying the SLD/Bures case with other metrics such as WY and BKM. It proves a universal three-channel decomposition of the reduced Petz QFI into population, coherence, and concurrence-derivative channels, with weights determined by the Morozova–Chentsov kernel; importantly, curvature invariants cannot be reduced to functions of concurrence or reduced entropy, showcasing a fundamental distinction between entanglement and curvature as quantum resources. An entanglement-orthogonal gauge is constructed to reveal how population and coherence derivatives influence curvature independently of entanglement, and explicit counterexamples in the SLD/Bures setting (via a four-parameter 2-HEA) demonstrate non-monotonicity and the failure of one-channel curvature formulas. The framework is leveraged to propose geometry-aware natural-gradient preconditioning for VQAs and is validated through VQE experiments, illustrating stable optimization on reduced manifolds despite rank deficiencies. The results have broad implications for quantum information geometry, Gaussian-state curvature, and the design of curvature-aware variational algorithms.

Abstract

We develop a support-projected Petz monotone geometry for pure two-qubit families, obtained by pulling back arbitrary Petz monotone quantum metrics to circuit-defined submanifolds and projecting onto the active spectral support of the associated quantum Fisher information tensor. This framework strictly generalizes the symmetric logarithmic derivative (SLD/Bures) case and includes, as special examples, the Wigner--Yanase and Bogoliubov--Kubo--Mori metrics among many others.

Paper Structure

This paper contains 69 sections, 10 theorems, 178 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Quantum Fisher geometry and Petz monotone metrics
  3. Curvature as a proxy for quantum resources
  4. Main results
  5. Organization of the paper
  6. Petz Monotone Metrics for Pure Two-Qubit States
  7. Projective Hilbert space and pure-state QFIM
  8. Two-qubit reductions, Bloch variables, and concurrence
  9. Petz monotone metrics: operator-monotone functions and qubit Bloch form
  10. Support-projected Petz geometry and the regular set
  11. Universal Three-Channel Decomposition
  12. Statement of the three-channel identity
  13. Qubit Petz--Bloch representation
  14. Proof of the three-channel decomposition
  15. Physical interpretation of the three channels
  16. ...and 54 more sections

Key Result

Proposition 2.1

For pure states, the quantum Fisher information metric eq:pure-QFIM coincides with the Fubini--Study metric eq:FS-metric. Equivalently, the statistical distinguishability structure on $\mathbb{CP}^{2^n-1}$ is precisely the canonical Riemannian geometry described in Provost--Vallée ProvostVallee1980,

Figures (4)

  • Figure 1: VQE sensitivity to circuit design and schedules (instantiated with SLD/Bures). Deep ZZ/XX entanglers and particle-number symmetries lower final energy and smooth the curvature landscape; parity-only constraints can forbid the true ground state. Layer-wise growth reduces curvature-induced instabilities.
  • Figure 2: Support--projected Petz--NatGrad consistently accelerates early descent. Spectral shrinkage and $g^{(f)}$-norm step normalization stabilize updates, while TR+Armijo removes spikes from aggressive or partial-F updates.
  • Figure 3: When expressivity saturates (deep/strong $L{=}12$, or H$_2$ minimal), Euclidean and NatGrad converge to the same energy floor. Support projection keeps Petz--NatGrad well conditioned and unbiased.
  • Figure 4: Gaussian curvature $K(x,y)$ of the intrinsic metric $g$ for representative 2--HEA slices. All panels share the same colour bar and $100\times100$ sampling grid. (a)--(b): high positive curvature; (c)--(d): negative curvature domains.

Theorems & Definitions (26)

  • Proposition 2.1: Provost--Vallée; Braunstein--Caves; Bengtsson--Życzkowski
  • proof
  • Proposition 2.2: Bloch radius, concurrence, and reduced entropy
  • proof
  • Lemma 2.3: Radial form of Petz monotone qubit metrics
  • proof
  • Theorem 2.4: Intrinsic Petz metric on the regular support
  • proof
  • Theorem 3.1: Three-channel identity for pure two-qubit families
  • Remark 3.2
  • ...and 16 more