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arxiv: 2605.23250 · v1 · pith:6QY6GQ2Snew · submitted 2026-05-22 · 🪐 quant-ph

A General Quantum Speed Limit for Non-Hermitian Systems

Zhanxi Wang , Xiaozhe Hao , X. X. Yi This is my paper

Pith reviewed 2026-05-25 04:43 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum speed limitnon-Hermitian systemsbiorthogonal basisMandelstam-Tamm boundMargolus-Levitin boundfastest initial statesopen quantum systems
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The pith

Two tighter quantum speed limits for non-Hermitian systems are derived from biorthogonal geometry and shown to be attainable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes two new bounds on the quantum speed limit that apply when a system evolves under a non-Hermitian Hamiltonian. These bounds are constructed by extending the Mandelstam-Tamm and Margolus-Levitin approaches through the use of a complete biorthogonal basis rather than an orthogonal one. The resulting expressions are tighter than prior non-Hermitian proposals and are saturated when the system starts in specially chosen fastest initial states. The work also supplies a practical bound for other initial states and verifies the results on a minimal non-Hermitian example. Readers care because non-Hermitian descriptions are standard for open or PT-symmetric systems where ordinary Hermitian speed limits cannot be used directly.

Core claim

Based on the biorthogonal basis theory we derive two distinct and tighter bounds on the QSL for non-Hermitian systems, which correspond to the MT and ML bounds for Hermitian systems. We show that the shortest evolution time corresponding to the two bounds of the non-Hermitian system can be attained by certain initial states, showing the compactness and tightness of our bounds. These initial states dubbed fastest initial states (FIS) are different from that in Hermitian systems. A bound close to QSL for non-FIS is presented and comparison of our bound with others in literature is performed. To illustrate our results, we present a minimal non-Hermitian system to show QSL, and the condition for

What carries the argument

Biorthogonal basis used to define the time-evolution operator and the geometric quantities that enter the speed-limit expressions.

If this is right

  • The MT-type and ML-type bounds are saturated by distinct fastest initial states (FIS).
  • A separate, looser bound holds for initial states that are not FIS.
  • The new bounds are tighter than earlier non-Hermitian proposals when compared on the same example.
  • An analytic condition for the shortest evolution time is obtained for a minimal non-Hermitian model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The FIS construction may supply a design principle for preparing states that evolve at the ultimate speed allowed by a given non-Hermitian generator.
  • Because many open-system master equations can be recast as non-Hermitian Schrödinger equations, the bounds could constrain the speed of information processing or decoherence in lossy platforms.
  • If the biorthogonal completeness assumption is relaxed, the same geometric construction might be generalized to defective Hamiltonians that require Jordan blocks.

Load-bearing premise

The non-Hermitian Hamiltonian admits a complete biorthogonal basis that lets the evolution operator and geometric quantities be defined exactly as in the Hermitian case.

What would settle it

Measurement of an evolution time to an orthogonal state in a controlled non-Hermitian system that is shorter than either of the two derived bounds.

Figures

Figures reproduced from arXiv: 2605.23250 by Xiaozhe Hao, X. X. Yi, Zhanxi Wang.

Figure 2
Figure 2. Figure 2: FIG. 2: (a), [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Upper and lower bound of the evolution time needed for the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

The quantum speed limit (QSL) refers to the maximum speed of a quantum system to evolve from an initial state to its orthogonal states. The bound on the QSL for Hermitian systems, for example the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) as well as Sun-Zheng(SZ) bound, was studied respectively from the perspectives of average value and variance of the system Hamiltonian as well as the geometry of the system. While the compactness of the MT-type, ML-type and SZ-type bounds has been examined well for Hermitian systems, a compact QSL for non-Hermitian systems has not been well studied. In this work, based on the biorthogonal basis theory we derive two distinct and tighter bounds on the QSL for non-Hermitian systems, which correspond to the MT and ML bounds for Hermitian systems. We show that the shortest evolution time corresponding to the two bounds of the non-Hermitian system can be attained by certain initial states, showing the compactness and tightness of our bounds. These initial states dubbed fastest initial states(FIS) are different from that in Hermitian systems. A bound close to QSL for non-FIS is presented and comparison of our bound with others in literature is performed. To illustrate our results, we present a minimal non-Hermitian system to show QSL, and the condition for the shortest evolution time is derived analytically using the present theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper derives two tighter quantum speed limit (QSL) bounds for non-Hermitian systems using biorthogonal basis theory, analogous to the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) bounds for Hermitian systems. It claims these bounds are compact and tight because certain fastest initial states (FIS) attain the minimal evolution time to orthogonality, provides a bound for non-FIS, compares with existing literature, and illustrates with a minimal non-Hermitian example where the condition for shortest time is derived analytically.

Significance. If the derivation holds under its stated assumptions, the work would address an understudied area by extending compact MT- and ML-type QSL bounds to non-Hermitian systems and explicitly constructing attaining states (FIS), with the analytical example providing a concrete test case.

major comments (1)
  1. [Derivation section (opening)] Derivation section (opening paragraphs on biorthogonal expansion): the central construction of the time-evolution operator, overlap, and variance quantities assumes a complete biorthogonal basis of left and right eigenvectors satisfying the biorthogonality relation. This assumption is load-bearing for both the explicit bound formulas and the tightness claim via FIS attainment, yet the manuscript presents the result as general without restricting to diagonalizable Hamiltonians. At exceptional points the algebraic and geometric multiplicities differ, Jordan blocks appear, the basis is incomplete, and the spectral decomposition used does not hold, so the derived bounds and compactness statement do not apply.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. The concern about the scope of the biorthogonal assumption is valid and we address it directly below.

read point-by-point responses

  1. Referee: Derivation section (opening paragraphs on biorthogonal expansion): the central construction of the time-evolution operator, overlap, and variance quantities assumes a complete biorthogonal basis of left and right eigenvectors satisfying the biorthogonality relation. This assumption is load-bearing for both the explicit bound formulas and the tightness claim via FIS attainment, yet the manuscript presents the result as general without restricting to diagonalizable Hamiltonians. At exceptional points the algebraic and geometric multiplicities differ, Jordan blocks appear, the basis is incomplete, and the spectral decomposition used does not hold, so the derived bounds and compactness statement do not apply.

    Authors: We agree that the derivation relies on a complete biorthogonal basis of left and right eigenvectors, which requires the non-Hermitian Hamiltonian to be diagonalizable. The manuscript implicitly adopts this standard assumption of biorthogonal quantum mechanics but does not state it explicitly. The bounds and FIS attainment therefore do not apply at exceptional points. We will revise the manuscript to add an explicit statement of the diagonalizability assumption in the abstract, introduction, and derivation section, together with a brief remark that at exceptional points the spectral decomposition fails and other techniques (e.g., Jordan-form methods) are required. This clarification will be made without altering the technical content of the bounds under the stated assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation follows from external biorthogonal theory

full rationale

The paper presents its MT- and ML-type bounds for non-Hermitian systems as derived directly from the standard biorthogonal expansion of the time-evolution operator and associated geometric quantities (overlap, variance). No step reduces by construction to a fitted parameter renamed as prediction, a self-defined quantity, or a load-bearing self-citation chain. The completeness assumption for the biorthogonal basis is an explicit modeling choice (not smuggled via citation), and the tightness claim is shown by explicit construction of fastest initial states within that framework. This is the normal case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard biorthogonal formalism for non-Hermitian operators; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Non-Hermitian Hamiltonians admit a complete biorthogonal basis allowing definition of time evolution and geometric quantities analogous to the Hermitian case
    Invoked at the start of the derivation to obtain the two bounds

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