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The i.i.d. State Convertibility in the Resource Theory of Asymmetry for Finite Groups

Tomohiro Shitara, Yosuke Mitsuhashi, Hiroyasu Tajima

Abstract

We identify exact and approximate conversion rates between i.i.d. pure states under covariant operations in the resource theory of asymmetry for symmetries described by finite groups. We establish the formula for the exact conversion rate by completely specifying the relevant set of resource measures. The exact conversion is generally asymptotically irreversible due to the existence of multiple independent resource measures, and we find the necessary and sufficient condition for asymptotic reversibility. On the other hand, we show that the approximate conversion rates diverge or vanish, which implies that the asymmetry can be infinitely amplified if we allow a vanishingly small error. We reveal the underlying mechanism of such a counterintuitive phenomenon by utilizing the properties of maximally asymmetric states.

The i.i.d. State Convertibility in the Resource Theory of Asymmetry for Finite Groups

Abstract

We identify exact and approximate conversion rates between i.i.d. pure states under covariant operations in the resource theory of asymmetry for symmetries described by finite groups. We establish the formula for the exact conversion rate by completely specifying the relevant set of resource measures. The exact conversion is generally asymptotically irreversible due to the existence of multiple independent resource measures, and we find the necessary and sufficient condition for asymptotic reversibility. On the other hand, we show that the approximate conversion rates diverge or vanish, which implies that the asymmetry can be infinitely amplified if we allow a vanishingly small error. We reveal the underlying mechanism of such a counterintuitive phenomenon by utilizing the properties of maximally asymmetric states.
Paper Structure (4 sections, 15 theorems, 99 equations)

This paper contains 4 sections, 15 theorems, 99 equations.

Table of Contents

  1. Exact i.i.d. conversion
  2. Approximate i.i.d. conversion
  3. Reduction to non-projective unitary representations
  4. Properties of maximally asymmetric states

Key Result

Theorem 1

Let $G$ be a finite group, and $\psi$ and $\phi$ be pure states. Then, the exact conversion rate from $\psi$ to $\phi$ is given by where we define $c/0:=\infty$ when $0\leq c\leq \infty$ and $\infty/\infty:=\infty$ for the value of $L_{\psi}(g)/L_{\phi}(g)$ in extreme cases. Moreover, for any $r\in (0, R_{\mathrm{ex}}(\psi\to\phi))$, the conversion rate $r$ is exactly achievable if the number $N$

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Lemma S1
  • Lemma S2
  • Lemma S3
  • proof
  • Theorem S1
  • proof
  • Theorem S2
  • proof
  • ...and 16 more