QuantumInfo.Finite.ResourceTheory.HypothesisTesting

Let $\rho$ be a quantum mixed state (represented by an element of `MState d`) and $S$ be a set of mixed states. For a given tolerance $\epsilon \in [0, 1]$, the optimal hypothesis testing rate $\beta_\epsilon(\rho \| S)$ is defined as the infimum of the maximum Type II error over all possible measurement operators $T$ (where $0 \leq T \leq I$) that satisfy a Type I error constraint. Specifically, it is given by: \[ \beta_\epsilon(\rho \| S) = \inf_{T} \left\{ \sup_{\sigma \in S} \text{Tr}(\sigma T) \right\} \] where the infimum is taken over all Hermitian matrices $T \in \text{Mat}_{d \times d}(\mathbb{C})$ such that $0 \leq T \leq I$ and $\text{Tr}(\rho(I - T)) \leq \epsilon$. Here, $T$ and $I-T$ represent the elements of a positive operator-valued measure (POVM) used to distinguish $\rho$ from the set $S$.

Let $\rho$ be a mixed state (represented as an `MState`), $\varepsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error, and $S$ be a set of mixed states. The notation $\beta_{\varepsilon}(\rho \Vert S)$ represents the **optimal hypothesis rate**. It is defined as the minimum Type II error probability: $$\beta_{\varepsilon}(\rho \Vert S) = \inf_{T} \left\{ \sup_{\sigma \in S} \text{Tr}(T\sigma) \right\}$$ where the optimization is performed over all measurement operators (POVM elements) $0 \le T \le I$ such that the Type I error satisfies $\text{Tr}((I - T)\rho) \le \varepsilon$. Here, $T$ is the operator associated with the decision that the state is $\rho$.

For a quantum state $\rho$ and an error tolerance $\varepsilon \in [0, 1]$, the set of measurement operators $T$ (represented by Hermitian matrices) such that $0 \le T \le 1$ and the Type I error probability $\text{Tr}(\rho(1 - T))$ is at most $\varepsilon$, is non-empty (is an inhabited type).

For a quantum state $\rho$ of dimension $d$ and a probability $\varepsilon \in [0, 1]$, the set of Hermitian operators $m$ satisfying the constraints $0 \le m \le 1$ and $\text{Tr}(\rho(1 - m)) \le \varepsilon$ is non-empty (Inhabited). Here, $\text{Tr}(\rho(1 - m))$ represents the Type I error (probability of incorrectly rejecting $\rho$) associated with the hypothesis test operator $m$.

Let $\rho$ be a quantum mixed state on a $d$-dimensional Hilbert space, and let $\epsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error. Consider the set of Hermitian operators $T$ representing measurement strategies (elements of a POVM) defined by: \[ \mathcal{T}_\epsilon = \{ T \in \text{Herm}(d, \mathbb{C}) \mid \text{Tr}((1 - T)\rho) \leq \epsilon, \text{ and } 0 \leq T \leq 1 \} \] where $0 \leq T \leq 1$ denotes that the eigenvalues of $T$ lie in the interval $[0, 1]$. Then the set $\mathcal{T}_\epsilon$ is a compact set in the space of Hermitian matrices.

For a quantum mixed state $\rho$ and an error tolerance $\epsilon \in [0, 1]$, the set of all measurement operators $m$ (represented by Hermitian matrices in $\text{Mat}_{d \times d}(\mathbb{C})$) that satisfy: 1. the Type I error condition $\text{Tr}((1 - m)\rho) \leq \epsilon$, and 2. the operator inequality bounds $0 \leq m \leq I$, is a convex set over the real numbers $\mathbb{R}$.

Let $\rho$ be a quantum mixed state in a $d$-dimensional Hilbert space and $\epsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error. If the set of alternative hypotheses $S$ is empty, then the optimal hypothesis testing rate $\beta_\epsilon(\rho \parallel \emptyset)$ is equal to $0$.

Let $\rho$ be a quantum mixed state of dimension $d$, $\varepsilon \in [0, 1]$ be a probability, and $S$ be a set of mixed states. For any Hermitian matrix $M \in \text{Mat}_{d \times d}(\mathbb{C})$ such that $0 \le M \le I$ and the condition $\text{Tr}((I - M)\rho) \le \varepsilon$ is satisfied, the optimal hypothesis rate $\beta_\varepsilon(\rho \Vert S)$ is bounded above by the supremum of the expectation values of $M$ over all states in $S$: \[ \beta_\varepsilon(\rho \Vert S) \le \sup_{\sigma \in S} \text{Tr}(M \sigma) \]

Let $\rho$ be a quantum mixed state of dimension $d$, and let $\varepsilon \in [0, 1]$ be a probability representing the maximum allowable Type I error. For any two sets of mixed states $S_1$ and $S_2$ such that $S_1 \subseteq S_2$, the optimal hypothesis testing error rate (Type II error) satisfies $\beta_\varepsilon(\rho \| S_1) \leq \beta_\varepsilon(\rho \| S_2)$.

Let $\rho$ and $\sigma$ be two quantum mixed states of dimension $d$, and let $\epsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error. The optimal hypothesis rate $\beta_\epsilon(\rho \| \{\sigma\})$ for distinguishing $\rho$ from the singleton set $\{\sigma\}$ is given by the infimum of the expectation value $\text{Tr}(\sigma T)$ over all Hermitian operators $T$ that satisfy $0 \le T \le I$ and the Type I error constraint $\text{Tr}(\rho(I - T)) \le \epsilon$: \[ \beta_\epsilon(\rho \| \{\sigma\}) = \inf \{ \text{Tr}(\sigma T) \mid T \in \text{Herm}(d), 0 \le T \le I, \text{Tr}(\rho(I - T)) \le \epsilon \} \] where $T$ represents the measurement effect for accepting the hypothesis that the state is $\rho$.

Let $\rho$ be a mixed state of dimension $d$, $\varepsilon$ be a probability representing the maximum Type I error, and $S$ be a set of mixed states. For any state $\sigma$ such that $\sigma \in S$, the negative logarithm of the optimal hypothesis testing rate $\beta_{\varepsilon}(\rho \Vert S)$ is less than or equal to the negative logarithm of the optimal hypothesis testing rate $\beta_{\varepsilon}(\rho \Vert \{\sigma\})$, that is, $-\log \beta_{\varepsilon}(\rho \Vert S) \le -\log \beta_{\varepsilon}(\rho \Vert \{\sigma\})$.

Let $\rho$ and $\sigma$ be quantum mixed states of dimension $d$, and let $\varepsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error. Let $M$ be a Hermitian matrix satisfying the operator inequality $0 \le M \le I$, such that the expectation value of $I - M$ in state $\rho$ satisfies $\text{Tr}(\rho(I - M)) \le \varepsilon$. Then the optimal hypothesis rate for distinguishing $\rho$ from the singleton set $\{\sigma\}$, denoted $\beta_\varepsilon(\rho \| \{\sigma\})$, is less than or equal to the expectation value of $M$ in state $\sigma$, $\text{Tr}(\sigma M)$.

Let $\rho$ be a quantum mixed state of dimension $d$, $\varepsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error, and $S$ be a set of mixed states. There exists an optimal Hermitian operator $T$ satisfying $0 \le T \le I$ and $\text{Tr}(\rho(I - T)) \le \varepsilon$ (i.e., the probability of correctly identifying $\rho$ is at least $1 - \varepsilon$) such that: 1. The maximum Type II error over the set $S$, given by $\sup_{\sigma \in S} \text{Tr}(\sigma T)$, achieves the optimal hypothesis testing rate $\beta_\varepsilon(\rho \| S)$. 2. The probability of correctly identifying $\rho$ satisfies $\text{Tr}(\rho T) \ge 1 - \varepsilon$.

Let $\rho$ be a quantum mixed state of dimension $d$, $\varepsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error, and $S$ be a set of mixed states. There exists an optimal measurement operator $T$ (a Hermitian matrix satisfying $0 \le T \le I$) such that the Type I error satisfies $\text{Tr}(\rho(I - T)) \le \varepsilon$ and the expectation value of $T$ in state $\rho$ is exactly $\text{Tr}(\rho T) = 1 - \varepsilon$. This operator $T$ achieves the optimal hypothesis testing rate $\beta_\varepsilon(\rho \| S)$, such that: \[ \sup_{\sigma \in S} \text{Tr}(\sigma T) = \beta_\varepsilon(\rho \| S) \] where $\beta_\varepsilon(\rho \| S)$ is the minimum possible worst-case Type II error.

Let $\rho$ be a mixed state on a $d$-dimensional complex Hilbert space and $S$ be a set of mixed states. Suppose there exists some state $\sigma \in S$ such that the kernel of its density matrix $M_\sigma$ is contained within the kernel of the density matrix $M_\rho$ (i.e., $\ker(M_\sigma) \subseteq \ker(M_\rho)$). For any allowed Type I error probability $\epsilon < 1$, the optimal hypothesis rate $\beta_\epsilon(\rho \| S)$—the minimum possible Type II error—is strictly positive: \[ \beta_\epsilon(\rho \| S) > 0 \] This implies that under these conditions, type II errors cannot be completely eliminated.

Let $\rho$ be a mixed state in a $d$-dimensional Hilbert space and $\epsilon \in [0, 1]$ be the maximum allowed Type I error probability. Let $S$ be a set of mixed states such that $S$ is compact and the set of its corresponding density matrices $M(S) = \{M_\sigma \mid \sigma \in S\}$ is convex. Then the optimal Type II error rate for distinguishing $\rho$ from the set $S$ is equal to the supremum of the optimal Type II error rates for distinguishing $\rho$ from each individual state $\sigma \in S$. Mathematically: \[ \beta_\epsilon(\rho \| S) = \sup_{\sigma \in S} \beta_\epsilon(\rho \| \{\sigma\}) \] where $\beta_\epsilon(\rho \| S) = \inf \{ \max_{\sigma \in S} \text{Tr}(T M_\sigma) \mid 0 \le T \le I, \text{Tr}((I-T)M_\rho) \le \epsilon \}$.

Let $q$ be a probability such that $0 < q < 1$. Consider the diagonal Hermitian matrix formed by the probability distribution of a biased coin with parameter $q$ (representing the probabilities $q$ and $1-q$). The kernel of this diagonal matrix is trivial, i.e., $\ker(\text{diag}(q, 1-q)) = \{0\}$.

Let $\rho$ and $\sigma$ be two quantum mixed states of dimension $d$, and let $\mathcal{E}$ be a completely positive trace-preserving (CPTP) map from dimension $d$ to $d_2$. For any error probability $\epsilon \in [0, 1]$, the optimal hypothesis rate for distinguishing $\rho$ and $\sigma$ is less than or equal to the optimal hypothesis rate for distinguishing their images under the map $\mathcal{E}$, i.e., \[ \beta_\epsilon(\rho \| \{\sigma\}) \le \beta_\epsilon(\mathcal{E}(\rho) \| \{\mathcal{E}(\sigma)\}) \] where $\beta_\epsilon(\rho \| S)$ denotes the minimum Type II error given a maximum Type I error $\epsilon$.

Let $\rho$ and $\sigma$ be mixed states of dimension $d$. For any error probability $\varepsilon \in [0, 1)$ and any real parameter $\alpha > 1$, the optimal hypothesis testing error rate $\beta_\varepsilon(\rho \| \sigma)$ and the sandwiched Rényi relative entropy $\tilde{D}_\alpha(\rho \| \sigma)$ satisfy the following inequality: \[ -\log \beta_\varepsilon(\rho \| \sigma) \le \tilde{D}_\alpha(\rho \| \sigma) + \frac{\alpha}{\alpha - 1} \log \left( \frac{1}{1 - \varepsilon} \right) \] In the notation of the formal statement, the right-hand side is expressed as $\tilde{D}_\alpha(\rho \| \sigma) + (-\log(1 - \varepsilon)) \cdot \alpha / (\alpha - 1)$, where $-\log(1-\varepsilon)$ represents the information gain associated with the success probability $1-\varepsilon$.

Let $\rho$, $\sigma_1$, and $\sigma_2$ be quantum mixed states in a finite-dimensional Hilbert space of dimension $d$. Let $\epsilon \in [0, 1]$ be a probability representing the maximum allowed Type I error. Suppose that the optimal hypothesis testing rate (Type II error) for distinguishing $\rho$ from the singleton set $\{\sigma_2\}$, denoted as $\beta_\epsilon(\rho \| \{\sigma_2\})$, is strictly positive. If there exists a positive real constant $c > 0$ such that $c \sigma_2 \le \sigma_1$ (in the sense of the Loewner partial order on density matrices), then the optimal hypothesis testing rate for $\rho$ against $\sigma_1$, $\beta_\epsilon(\rho \| \{\sigma_1\})$, is also strictly positive.

For a fixed quantum mixed state $\rho$ and a fixed error threshold $\varepsilon \in [0, 1]$, the optimal hypothesis testing rate $\beta_\varepsilon(\rho \| \{\sigma\})$ (the minimum Type II error) is a continuous function with respect to the mixed state $\sigma$. Specifically, the mapping $\sigma \mapsto \beta_\varepsilon(\rho \| \{\sigma\})$ is continuous from the space of $d$-dimensional mixed states $\text{MState } d$ to the interval $[0, 1]$, where $\beta_\varepsilon(\rho \| \{\sigma\})$ is defined as: \[ \beta_\varepsilon(\rho \| \{\sigma\}) = \inf \{ \text{Tr}(\sigma T) \mid T \in \text{HermitianMat } d, \ 0 \le T \le I, \ \text{Tr}(\rho(I - T)) \le \varepsilon \} \]

Let $\rho$ and $\sigma$ be mixed states acting on a 1-dimensional Hilbert space (where the underlying basis type $d$ has unique cardinality). For any error probability $\varepsilon \in [0, 1]$, the optimal hypothesis testing rate $\beta_\varepsilon(\rho \parallel \{\sigma\})$ is equal to $1 - \varepsilon$.