bibkey: khareletal2026pseudoentropy
authors: Pramod Kamal Kharel; Manghang Limbu; Nabaraj Khatri; Ashish Khanal; Kiran Adhikari
year: 2026
title: "Pseudo entropy and topological phases of matter"
doi: null
url: https://arxiv.org/abs/2606.29311v1
claim: "For pure states |ψ⟩, |φ⟩ with ⟨φ|ψ⟩ ≠ 0, the transition matrix is T = |ψ⟩⟨φ|/⟨φ|ψ⟩, the reduced transition matrix is T_A = Tr_B T, and the pseudo entropy is S(T_A) = −Tr(T_A log T_A) (Eqs. (1)–(3)); the averaged excess entropy is ∆S = Re S(T_A) − (S(ρ_A^ψ) + S(ρ_A^φ))/2 (Eq. (21)). For half-filled ground states of the periodic SSH Hamiltonian (Eq. (15)) at w = 1 in the same phase, the authors report that the conjectured non-positivity ∆S ≤ 0 holds and state in §VI that under periodic boundary conditions it is fully satisfied with no exceptions at all system sizes."
strata_touched:
  - D5/S3/Quantum/Fermionic/PeriodicSshExcessPseudoEntropyRefutation
license: citation-only
triage: anchor

Kharel, Limbu, Khatri, Khanal and Adhikari, pseudo entropy and topological phases of matter

P. K. Kharel, M. Limbu, N. Khatri, A. Khanal and K. Adhikari, Pseudo entropy and topological phases of matter, arXiv:2606.29311 [cond-mat.stat-mech] (v1 of 28 June 2026, the only version listed).

Verified locator

DOI: null

Source: https://arxiv.org/abs/2606.29311v1

No DOI or journal reference is listed in the arXiv metadata. The text was read from the PDF https://arxiv.org/pdf/2606.29311v1:

• p. 2, §II, Eqs. (1)–(5), the transition matrix and the pseudo entropy;
• pp. 3–4, Eqs. (13)–(17), the SSH Hamiltonian with open and periodic boundary conditions and its phases;
• p. 5, §IV, Eq. (21) and the setup of the non-positivity test;
• p. 9, §VI, the conclusion.

Source statements

Eqs. (1)–(3) (p. 2): “T^{ψ|φ} = |ψ⟩⟨φ| / ⟨φ|ψ⟩, (1) … the reduced transition matrix of system A is defined by tracing out the system B, T_A^{ψ|φ} ≡ Tr_B[T^{ψ|φ}] = Tr_B[|ψ⟩⟨φ| / ⟨φ|ψ⟩]. (2) and pseudo entropy is defined as: S(T_A^{ψ|φ}) = −Tr[T_A^{ψ|φ} log T_A^{ψ|φ}]. (3)”

Eq. (15) (p. 4): “For Periodic Boundary Condition (PBC), we connect the last unit cell to the first one, c_{N+1,A} ≡ c_{1,A}, and the Hamiltonian is: Ĥ_PBC = v Σ_{m=1}^{N} (c†{m,B} c{m,A} + h.c.) + w Σ_{m=1}^{N} (c†{m+1,A} c{m,B} + h.c.), (15)”

§IV (p. 5): “∆S(T_A^{ψ|φ}) = Re[S(T_A^{ψ|φ})] − ½ (S(ρ^ψ)_A + S(ρ^φ)_A) (21) … In [25, 26], it was conjectured that the averaged excess entropy is non-positive, ∆S(T_A^{ψ|φ}) ≤ 0, when the states |ψ⟩ and |φ⟩ are in the same phase. … We consider two SSH Hamiltonians (15) Ĥ(v1, w) and Ĥ(v2, w) with fixed w = 1, and denote their half-filled ground states by |Ψ1⟩ ≡ |Ψ(v1)⟩ and |Ψ2⟩ ≡ |Ψ(v2)⟩.”

§VI (p. 9): “Under periodic boundary conditions, the conjecture is fully satisfied with no exceptions at all system sizes and across all three phases: topological, trivial, and crossing.”

Reading used by the repository

The Fock space of the 2N modes (1,A), (1,B), …, (N,A), (N,B) carries Jordan–Wigner annihilation operators in this order, and Ĥ_PBC(v) = Σ h_ij c_i† c_j with h the single-particle matrix of Eq. (15) at w = 1. A half-filled ground state is a nonzero N-particle vector of minimal energy in the N-particle sector. A is the first L unit cells, and Tr_B is the partial trace over the remaining modes. The entropy of a matrix is −Σ λ log λ over its eigenvalues with multiplicity. The §VI assertion is read as: for every N ≥ 2, every L with 1 ≤ L < N, and all v1, v2 in the same phase, ∆S ≤ 0.