QuantumInfo.ResourceTheory.SteinsLemma

Given a mixed state $\rho \in \text{MState}(H(i))$ and a tolerance probability $\epsilon \in [0, 1]$, the value $R_1(\rho, \epsilon)$ is defined as the limit inferior of the normalized negative logarithm of the optimal hypothesis testing rate as the number of copies $n$ tends to infinity: $$R_1(\rho, \epsilon) = \liminf_{n \to \infty} \frac{-\log \beta_\epsilon(\rho^{\otimes_r n} \| \text{IsFree})}{n}$$ where $\rho^{\otimes_r n}$ is the $n$-th tensor power of the state $\rho$ within the resource theory, and $\beta_\epsilon(\rho^{\otimes_r n} \| \text{IsFree})$ is the optimal Type II error rate for distinguishing $\rho^{\otimes_r n}$ from the set of free states $\text{IsFree}$ given a Type I error bound $\epsilon$. The result is an extended non-negative real number in $[0, \infty]$.

Given a mixed state $\rho \in \text{MState}(H(i))$ and a sequence of free states $\sigma = (\sigma_n)_{n \in \mathbb{N}}$ (where each $\sigma_n$ is a free state for the system index $i^n$), the value $R_2(\rho, \sigma)$ is defined as the limit inferior of the normalized quantum relative entropy between the $n$-th tensor power of $\rho$ and $\sigma_n$ as $n$ tends to infinity: $$R_2(\rho, \sigma) = \liminf_{n \to \infty} \frac{D(\rho^{\otimes_r n} \parallel \sigma_n)}{n}$$ where $D(\cdot \parallel \cdot)$ denotes the quantum relative entropy and $\rho^{\otimes_r n}$ denotes the $n$-th power of the state within the resource theory. The result is an extended non-negative real number in $[0, \infty]$.

The definition specifies $\sigma_1$ as a quantum mixed state on the finite-dimensional Hilbert space $H_i$. It is chosen (using the axiom of choice) to be a state that is both "free" within a specified resource theory and of full rank, based on the property provided by `FreeStateTheory.free_fullRank`.

For a mixed state $\sigma_1$ on the Hilbert space $H_i$, let its associated density operator be represented by a Hermitian matrix. This definition calculates the minimum eigenvalue of this matrix, expressed as the infimum $\inf_k \lambda_k$, where $\lambda_k$ are the eigenvalues of the density operator corresponding to $\sigma_1$.

For a natural number $n$, the sequence $c_n$ is defined as: $$c_n = \log \left( \frac{1}{\lambda_{\min}(\sigma_1)} \right) + \frac{\log 3}{\max(n, 1)}$$ where $\lambda_{\min}(\sigma_1)$ is the minimum eigenvalue of the quantum state $\sigma_1$. This definition corresponds to the sequence $c_n$ appearing in equation (S44) of Stein's Lemma, modified to handle the $n=0$ case by replacing $n$ with $\max(n, 1)$.

Let $\rho \in \text{MState}(H(i))$ be a mixed state and $\epsilon \in (0, 1)$ be a probability. Let $\sigma = (\sigma_n)_{n \in \mathbb{N}}$ be a sequence of free states where each $\sigma_n \in \text{IsFree}(i^n)$. If the asymptotic relative entropy rate $R_2(\rho, \sigma)$ is greater than or equal to the asymptotic optimal hypothesis testing rate $R_1(\rho, \epsilon)$, then for any probability $\epsilon' \in (0, \epsilon)$, there exists a sequence of free states $\sigma' = (\sigma'_n)_{n \in \mathbb{N}}$ such that $$R_2(\rho, \sigma') - R_1(\rho, \epsilon) \le (1 - \epsilon') (R_2(\rho, \sigma) - R_1(\rho, \epsilon)),$$ where $R_1(\rho, \epsilon)$ is the limit inferior of the normalized negative logarithm of the optimal Type II error rate, and $R_2(\rho, \sigma)$ is the limit inferior of the normalized quantum relative entropy $D(\rho^{\otimes n} \| \sigma_n)$.

Let $\rho \in \text{MState}(H(i))$ be a mixed state and $0 < \epsilon' < \epsilon < 1$ be probabilities. Given a sequence of free states $\sigma = (\sigma_n)_{n \in \mathbb{N}}$ where each $\sigma_n \in \text{IsFree}(i^n)$, the improver function returns a new sequence of free states $\sigma' = (\sigma'_n)_{n \in \mathbb{N}}$ defined as: 1. If $R_2(\rho, \sigma) \ge R_1(\rho, \epsilon)$, then $\sigma'$ is the sequence whose existence is guaranteed by Lemma 7, which satisfies the gap reduction inequality: $$R_2(\rho, \sigma') - R_1(\rho, \epsilon) \le (1 - \epsilon') (R_2(\rho, \sigma) - R_1(\rho, \epsilon)).$$ 2. If $R_2(\rho, \sigma) < R_1(\rho, \epsilon)$, the function returns the original sequence $\sigma$. Here, $R_1(\rho, \epsilon)$ denotes the asymptotic optimal hypothesis testing rate and $R_2(\rho, \sigma)$ denotes the asymptotic relative entropy rate.