QuantumInfo.Finite.ResourceTheory.SteinsLemma

Given a positive integer $m$, a mixed state $\sigma_f$ on a Hilbert space $H(i)$, and a mixed state $\sigma_m$ on the tensor product space $H(i^m)$, this function defines a sequence of mixed states $\tilde{\sigma}_n$ on $H(i^n)$ for each $n \in \mathbb{N}$. The state is constructed by taking the tensor product of $\lfloor n/m \rfloor$ copies of $\sigma_m$ and $(n \pmod m)$ copies of $\sigma_f$: $$\tilde{\sigma}_n = \text{relabel} \left( \sigma_m^{\otimes \lfloor n/m \rfloor} \otimes \sigma_f^{\otimes (n \pmod m)} \right)$$ The `relabel` operation maps the state from the space $H(i^m)^{\lfloor n/m \rfloor} \otimes H(i)^{(n \pmod m)}$ to the target space $H(i^n)$ using the property that $m \cdot \lfloor n/m \rfloor + (n \pmod m) = n$. This matches the definition of $\tilde{\sigma}_n$ provided in Lemma 6 and equation (S40) of Stein's Lemma.

Let $\sigma_1$ be a mixed state on the Hilbert space $H(i)$ and let $\sigma_m$ be a sequence of mixed states on the Hilbert spaces $H(i^m)$ for each $m \in \mathbb{N}$. If $\sigma_1$ is a free state and $\sigma_m$ is a free state for every $m \in \mathbb{N}$, then for any $m, n \in \mathbb{N}$, the state constructed via `Lemma6_σn` (using $\sigma_1$ and $\sigma_m$ as parameters) is also a free state.

Let $\epsilon \in [0, 1]$ be a probability, and let $\delta > 0$ be a non-negative real constant. Consider a sequence of finite-dimensional Hilbert spaces with dimensions $d_n$ indexed by $n \in \mathbb{N}$, and let $\rho_n$ and $\sigma_n$ be sequences of quantum states (density matrices) in these spaces. Let $R$ be a non-negative real number such that $R$ is greater than or equal to the limit inferior of the sequence of normalized optimal type-II error rates, specifically: \[ R \ge \liminf_{n \to \infty} \frac{-\log \beta_\epsilon(\rho_n \| \{\sigma_n\})}{n} \] where $\beta_\epsilon(\rho_n \| \{\sigma_n\})$ denotes the optimal hypothesis testing error rate (Type-II error) for a fixed Type-I error $\epsilon$. Then it holds that: \[ \liminf_{n \to \infty} \text{Tr}\left( \Pi_n M_{\rho_n} \right) \le 1 - \epsilon \] where $\Pi_n$ is the projector onto the positive subspace of $M_{\rho_n} - e^{n(R + \delta)} M_{\sigma_n}$ (the Neyman-Pearson measurement operator), and $M_{\rho_n}, M_{\sigma_n}$ are the density matrices of the states.

Let $\epsilon_3 \in [0, 1]$ be a probability and $\epsilon_4 > 0$ be a non-negative real number. For a sequence of finite-dimensional Hilbert spaces of dimensions $d_n$, let $\rho_n$ and $\sigma_n$ be sequences of quantum mixed states. Let $R_{\sup}$ be a non-negative real number such that \[ R_{\sup} \geq \limsup_{n \to \infty} \frac{-\log \beta_{\epsilon_3}(\rho_n \| \{\sigma_n\})}{n} \] where $\beta_{\epsilon_3}(\rho_n \| \{\sigma_n\})$ denotes the optimal hypothesis testing rate (type-II error probability) for the state $\rho_n$ against the singleton set $\{\sigma_n\}$ with a constraint $\epsilon_3$ on the type-I error. Then it holds that \[ \limsup_{n \to \infty} \text{Tr}(\rho_n \Pi_n) \leq 1 - \epsilon_3 \] where $\Pi_n$ is the projector onto the positive subspace of the operator $\rho_n - e^{n(R_{\sup} + \epsilon_4)} \sigma_n$ (denoted by the spectral projection $\{\rho_n \geq_p e^{n(R_{\sup} + \epsilon_4)} \sigma_n\}$), and the term $\langle \Pi_n, \rho_n \rangle$ represents the expectation value $\text{Tr}(\rho_n \Pi_n)$.

Let $\sigma_{1,c}(i, n)$ be a sequence indexed by $n \in \mathbb{N}$ (with a fixed parameter $i$). Then the expression $\sigma_{1,c}(i, n) + \log 3$ is little-o of $n$ as $n \to \infty$. That is, $$\sigma_{1,c}(i, n) + \log 3 = o(n) \quad \text{as } n \to \infty.$$

Let $\rho \in \text{MState}(\mathcal{H}_i)$ be a mixed state, and let $\epsilon, \epsilon' \in [0, 1]$ be probabilities such that $0 < \epsilon < 1$ and $0 < \epsilon' < \epsilon$. For any state $\sigma$, let $\sigma'$ be the improved state defined by the construction $\text{Lemma7\_improver}(\rho, \epsilon, \epsilon', \sigma)$. Then the gap between the quantities $R_2$ and $R_1$ satisfies the inequality: $$R_2(\rho, \sigma') - R_1(\rho, \epsilon) \leq (1 - \epsilon') \cdot (R_2(\rho, \sigma) - R_1(\rho, \epsilon))$$ where the coefficient $(1 - \epsilon')$ serves as a contraction factor that reduces the gap.

Let $\rho$ be a mixed state in a quantum system with Hilbert space $H(i)$. For any error probability $\epsilon \in (0, 1)$, the regularized optimal type-II error rate for testing the $n$-fold tensor power state $\rho^{\otimes n}$ against the set of free states $\mathcal{F}$ satisfies: $$\lim_{n \to \infty} \frac{-\log \beta_\epsilon(\rho^{\otimes n} \| \mathcal{F})}{n} = R_{\mathcal{R}}^\infty(\rho)$$ where $\beta_\epsilon(\rho \| S)$ denotes the optimal hypothesis testing power (the minimum type-II error probability given a constraint $\epsilon$ on the type-I error), and $R_{\mathcal{R}}^\infty(\rho)$ is the regularized relative entropy of resource defined as $\lim_{n \to \infty} \frac{1}{n} \inf_{\sigma \in \mathcal{F}} D(\rho^{\otimes n} \| \sigma)$.

Let $\rho$ be a quantum state in the state space $\mathcal{M}(H_i)$. For any error probability $\epsilon \in (0, 1)$, there exists a non-negative real constant $d$ (specifically the regularized relative entropy of resource $R_r^\infty(\rho)$) such that two limits coincide: 1. The limit of the normalized optimal hypothesis testing rate as the number of copies $n$ tends to infinity: $$\lim_{n \to \infty} \frac{-\log \beta_\epsilon(\rho^{\otimes n} \| \text{IsFree})}{n} = d$$ where $\beta_\epsilon(\rho^{\otimes n} \| \text{IsFree})$ denotes the optimal type-II error probability for testing the $n$-copy state $\rho^{\otimes n}$ against the set of free states. 2. The limit of the normalized minimum relative entropy between the $n$-copy state and the set of free states: $$\lim_{n \to \infty} \frac{\inf_{\sigma \in \text{IsFree}} D(\rho^{\otimes n} \| \sigma)}{n} = d$$ where $D$ is the quantum relative entropy.