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Instability as a Quantum Resource

Goni Yoeli, Gilad Gour

Abstract

We consolidate coherence, athermality, and nonuniformity as sub-resources within an underlying quantum resource theory: instability. We formulate instability axiomatically as the transient information within a decaying physical system. Specifying a decay mechanism (e.g., dephasing, thermalization) recovers these familiar resources as specific manifestations of instability. We compute the one-shot distillation yield and dilution cost in various operational paradigms, and use them to pin down the extremal additive monotones. In the asymptotic regime, we show that all conversion rates are governed by a single additive monotone, and thereby we establish a universal second law for instability.

Instability as a Quantum Resource

Abstract

We consolidate coherence, athermality, and nonuniformity as sub-resources within an underlying quantum resource theory: instability. We formulate instability axiomatically as the transient information within a decaying physical system. Specifying a decay mechanism (e.g., dephasing, thermalization) recovers these familiar resources as specific manifestations of instability. We compute the one-shot distillation yield and dilution cost in various operational paradigms, and use them to pin down the extremal additive monotones. In the asymptotic regime, we show that all conversion rates are governed by a single additive monotone, and thereby we establish a universal second law for instability.
Paper Structure (16 sections, 15 theorems, 109 equations, 2 figures, 1 table)

This paper contains 16 sections, 15 theorems, 109 equations, 2 figures, 1 table.

Table of Contents

  1. Conditional Expectations
  2. Characterizing Conditional Expectations
  3. The Structure of a Faithful Conditional Expectation
  4. The Quantum Resource Theory of Instability
  5. Monotones
  6. Intermediate Monotone $\mathbb D$-Extensions
  7. Optimizing Trace Functionals
  8. Rényi Monotones
  9. Extremal Additive Monotones
  10. The Currency of Instability
  11. The Dilution Cost
  12. Calculation of the One-Shot Dilution Cost
  13. Bounds of the One-Shot Cost and the Asymptotic Cost
  14. The Distillable Instability Yield
  15. Calculation of the One-Shot Distillable Yield
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\epsilon \in [0,1]$ and $\rho \in \mathfrak{D}$. Then,

Figures (2)

  • Figure 1: Athermality and coherence are now formally comparable as forms of instability.
  • Figure 2: Venn diagram of sub-resources (forms) of instability. Conditioning on the trivial system ($\mathbb{C}$) recovers the unconditional forms of nonuniformity and athermality from their conditional counterparts. The sub-theory of instability on which $\Delta$ is self-adjoint includes coherence and (conditional) nonuniformity; it was studied in GR2024 as 'coherence relative to subalgebra'. (Conditional) athermality is self-adjoint exactly when the Gibbs state is uniform $\mathbf{u} = I/d$ (infinite temperature), in which case it reduces to nonuniformity.

Theorems & Definitions (53)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark
  • Claim A.1
  • proof
  • Definition A.1
  • Claim A.2
  • ...and 43 more