August 18, 2025
We establish a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states, thereby extending the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Specifically, we consider the task of distinguishing whether a quantum system is prepared in a state from one of several convex, compact sets of quantum states, each of which may exhibit arbitrary correlations. Assuming their stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of a universal optimal test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal test for the most challenging states. We provide explicit characterizations of the universal optimal test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.
Hypothesis testing is a fundamental method for deciding between competing explanations of observed data. It provides a rigorous framework for making decisions under uncertainty and is central to statistics, information theory, and many applied fields such as signal processing, machine learning, and experimental physics [1]. The main goal of hypothesis testing is to determine which of several possible models or probability distributions best describes a set of observations, while keeping the probability of making an error as low as possible. In the classical setting, Chernoff showed that, when many independent samples are available, the probability of error decreases exponentially with the number of samples. The rate at which this error probability decays is given by the celebrated Chernoff bound [2], which quantifies the fundamental limit of distinguishability between different hypotheses.
The study of Chernoff bound has been generalized to the quantum domain, encompassing a variety of scenarios that can be broadly categorized as either simple or composite hypothesis testing. In the simple hypothesis setting, the state to be distinguished is specified by a single quantum state. For independent and identically distributed (i.i.d.) states, the optimal error exponent is given by the quantum Chernoff bound, as established by Audenaert et al. [3] and Nussbaum and Szkoła [4], with extensions to multiple hypotheses by Li [5]. For correlated quantum states, such as those arising in spin chains, the problem has been studied by Hiai et al. [6], [7]. In the composite hypothesis setting, the hypothesis is described by a set of quantum states. If each state in the set is i.i.d., this is referred to as the composite i.i.d. case, which has been investigated by Mosonyi et al. [8]. If the states in the set have a tensor product structure but are allowed to vary arbitrarily across subsystems, this leads to the composite arbitrarily varying case, also studied in [8]. Composite hypothesis testing with restricted measurements has also been studied by Brandão et al. [9].
However, in many practical scenarios, the states to be distinguished are not fully specified and may lack a simple tensor product structure. Instead, one only knows that the states belong to different sets of quantum states, which may exhibit arbitrary correlations and lack i.i.d. structure. Existing results do not directly apply to this general setting. Recent progress has been made in the asymmetric (Stein’s) regime, where the type-II error is minimized subject to a constraint on the type-I error [10]. In this work, we address the symmetric hypothesis testing scenario for discriminating between two (or more) sets of quantum states, thereby generalizing the quantum Chernoff bound to the general composite and correlated settings.
We establish a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states, allowing for arbitrary correlations and composite hypotheses. Specifically, we consider the task of distinguishing whether a quantum system is prepared in a state from one of several convex, compact sets of quantum states, each of which may exhibit arbitrary correlations and need not possess an i.i.d.structure. In Theorems 1 and 2, we show that, under the assumption of stability under tensor product, the optimal asymptotic error exponent for discrimination is given by the regularized Chernoff divergence between the sets. These results extend the classical and quantum Chernoff bounds to the general setting of composite and correlated hypotheses, unifying and generalizing several previously known results as special cases. Furthermore, we provide an explicit characterization of the optimal measurement for binary composite hypotheses in Theorem 3. Finally, in Theorem 4, we provide an operational interpretation of the maximum overlap between a pure state and a set of free states—a quantity that frequently arises in quantum resource theory—as the optimal error exponent in symmetric hypothesis testing. This, in turn, also yields explicit examples where the quantum Chernoff divergence between sets of quantum states is not additive, thereby highlighting the necessity of its regularization.
The remainder of this paper is structured as follows. Section 2 introduces the necessary notation, reviews relevant minimax theorems, and summarizes the quantum Chernoff bound for simple hypotheses. In Section 3, we develop the operational framework for quantum hypothesis testing between sets of quantum states, introduce the quantum Chernoff divergence for sets, and establish its key properties. We also present our main results: the quantum Chernoff bound for multiple composite hypotheses, including both the general and binary cases, together with detailed proofs. Section 4 provides an explicit construction of optimal tests for binary composite hypotheses, leveraging minimax optimization and saddle-point analysis. In Section 5, we offer an operational interpretation of the maximum overlap with free states in quantum resource theories and illustrate the necessity of regularization of quantum Chernoff divergence for sets. Finally, Section 6 concludes with a discussion of future research directions.
We adopt the following notational conventions throughout this work. Finite-dimensional Hilbert spaces are denoted by \({\cal H}\), with \(|{\cal H}|\) representing their dimension. The set of all linear operators on \({\cal H}\) is denoted by \({{\mathscr{L}}}({\cal H})\), while \(\mathscr{H}({\cal H})\) and \(\mathscr{H}_{{\scalebox{0.7}{+}}}({\cal H})\) denote the sets of Hermitian and positive semidefinite operators on \({\cal H}\), respectively. The set of density matrices (i.e., positive semidefinite operators with unit trace) on \({\cal H}\) is denoted by \(\mathscr{D}({\cal H})\). Calligraphic letters such as \({{\mathscr{A}}}\), \({{\mathscr{B}}}\), and \({{\mathscr{C}}}\) are used to denote sets of linear operators. Unless otherwise specified, all logarithms are taken to base two and denoted by \(\log(x)\). The positive semidefinite ordering is written as \(X \geq Y\) if and only if \(X - Y \geq 0\). The absolute value of an operator \(X\) is defined as \(|X| := (X^\dagger X)^{1/2}\). For a Hermitian operator \(X\) with spectral decomposition \(X = \sum_i x_i E_i\), the projection onto the non-negative eigenspaces is denoted by \(\{X \geq 0\} := \sum_{x_i \geq 0} E_{i}\).
The following lemma is a minimax theorem that account for the infinity values of the function. Let \(X\) be a convex set in a linear space. A function \(f: X \to (-\infty,-\infty]\) said to be convex, if \(f(px+(1-p)y) \leq pf(x)+(1-p)f(y)\), the multiplication \(0\cdot f(x)\) is interpreted as \(0\) and \(p\cdot +\infty=+\infty\) for \(p\neq 0\). Similar definiton holds for concave functions.
Lemma 1. [11]Let \(X\) be a compact, convex subset of a Hausdorff topological vector space and \(Y\) be a convex subset of the linear space. Let \(f : X \times Y \to (-\infty,+\infty]\) be lower semicontinuous on \(X\) for fixed \(y \in Y\), and assume that \(f\) is convex in the first and concave in the second variable. Then \[\begin{align} \sup_{y \in Y} \inf_{x \in X} f(x,y) = \inf_{x \in X} \sup_{y \in Y} f(x,y). \end{align}\]
Another minimax theorem is given by [12] or [13].
Lemma 2. Let \(X\) be a compact topological space, \(Y\) be an ordered set, and let \(f : X \times Y \to \mathbb{R} \cup \{\pm \infty\}\) be a function. Assume that (i) \(f(\cdot, y)\) is upper semicontinuous for every \(y \in Y\), and (ii) \(f(x, \cdot)\) is monotonic increasing (or decreasing) for every \(x \in X\). Then \[\begin{align} \sup_{x \in X} \inf_{y \in Y} f(x, y) = \inf_{y \in Y} \sup_{x \in X} f(x, y), \end{align}\] and the suprema can be replaced by maxima.
The following lemmas are standard results in mathematical analysis and will be used frequently in our proofs. For detailed proofs, see, e.g., [14].
Lemma 3. Let \(X\) be a nonempty compact topological space, and let \(f : X \to \overline{\mathbb{R}}\) be a function. Then if \(f\) is upper semicontinuous, it attains its maximum, meaning there is some \(x \in X\) such that for all \(x' \in X\), \(f(x') \leq f(x)\). Similarly, if \(f\) is lower semicontinuous, it attains its minimum.
Lemma 4. Let \(X\) be a topological space, let \(I\) be a set, and let \(\{f_i\}_{i \in I}\) be a collection of functions \(f_i : X \to \overline{\mathbb{R}}\). Then if each \(f_i\) is upper semicontinuous, the function \(f(x) = \inf_{i \in I} f_i(x)\) is also upper semicontinuous. Similarly, if each \(f_i\) is lower semicontinuous, the pointwise supremum is lower semicontinuous.
Consider the problem of distinguishing between two quantum hypotheses: the system is prepared either in state \(\rho_1\) (the null hypothesis) or in state \(\rho_2\) (the alternative hypothesis). In the Bayesian framework, each hypothesis is assigned a prior probability, denoted by \(\pi_1\) and \(\pi_2\), where \(\pi_1, \pi_2 > 0\) and \(\pi_1 + \pi_2 = 1\) (excluding the trivial case where either prior is zero).
Operationally, the discrimination is carried out using a two-outcome positive operator-valued measure (POVM) \(\{M, I-M\}\), with \(0 \leq M \leq I\). The average error probability for this measurement is given by \[\begin{align} P_e(M, \rho_1, \rho_2) := \pi_1 \operatorname{Tr}[(I-M) \rho_1] + \pi_2 \operatorname{Tr}[M \rho_2]. \end{align}\] The goal is to minimize the average error probability over all possible POVMs: \[\begin{align} P_{e, \min}(\rho_1, \rho_2) := \inf_{0 \leq M \leq I} P_e(M, \rho_1, \rho_2). \end{align}\] In a landmark result, Audenaert et al. [3] and Nussbaum and Szkoła [4] showed that, in the asymptotic regime, the optimal error probability decays exponentially with the number of copies, with the optimal exponent characterized by \[\begin{align} \label{eq:32chernoff32theorem} \lim_{n \to \infty} -\frac{1}{n} \log P_{e,\min}(\rho_1^{\otimes n}, \rho_2^{\otimes n}) = C(\rho_1 \| \rho_2), \end{align}\tag{1}\] where the quantum Chernoff divergence is defined as \[\begin{align} \label{eq:32definition32of32Chernoff32divergence} C(\rho_1 \| \rho_2) := \max_{0 \leq \alpha \leq 1} -\log Q_\alpha(\rho_1 \| \rho_2) , \quad \text{with} \quad Q_\alpha(\rho_1 \| \rho_2) := \operatorname{Tr}[\rho_1^\alpha \rho_2^{1-\alpha}]. \end{align}\tag{2}\] This fundamental result is known as the quantum Chernoff bound, which establishes the optimal asymptotic rate at which the error probability decays when discriminating between two quantum states in the i.i.d. setting.
The proof of the quantum Chernoff bound relies on two key lemmas, which will also play a central role in our analysis.
Lemma 5. For any \(V,W\in \mathscr{H}_{{\scalebox{0.7}{+}}}\), it holds that [15], [16] \[\begin{align} \inf_{0\leq M \leq I} \operatorname{Tr}[(I-M)V] + \operatorname{Tr}[MW] = \frac{1}{2} (\operatorname{Tr}[V+W] - \|V-W\|_1). \end{align}\]
Lemma 6. Let \(V,W \in \mathscr{H}_{{\scalebox{0.7}{+}}}\) and \(\alpha \in [0,1]\). It holds that [3] \[\begin{align} \operatorname{Tr}[V^\alpha W^{1-\alpha}] \geq \frac{1}{2} \operatorname{Tr}[V + W - |V- W|]. \end{align}\]
By its operational meaning, the quantum Chernoff divergence is additive under \(n\)-fold tensor product states, i.e., for any \(n \in {{\mathbb{N}}}\), \(C(\rho_1^{\otimes n}\|\rho_2^{\otimes n}) = n\, C(\rho_1\|\rho_2)\). More generally, the Chernoff divergence is subadditive under tensor products of different states: \[\begin{align} \label{eq:32subadditivity32of32Chernoff32divergence} C(\rho_1 \otimes \sigma_1\|\rho_2 \otimes \sigma_2) \leq C(\rho_1\|\rho_2) + C(\sigma_1\|\sigma_2). \end{align}\tag{3}\] This can be seen as follows: \[\begin{align} C(\rho_1 \otimes \sigma_1\|\rho_2 \otimes \sigma_2) & = \max_{\alpha \in [0,1]} - \log Q_\alpha (\rho_1 \otimes \sigma_1\|\rho_2 \otimes \sigma_2) \\ & = \max_{\alpha \in [0,1]} - \log Q_\alpha (\rho_1\|\rho_2) - \log Q_\alpha(\sigma_1\|\sigma_2) \\ & \leq \max_{\alpha \in [0,1]} - \log Q_\alpha (\rho_1\|\rho_2) + \max_{\alpha \in [0,1]} - \log Q_\alpha(\sigma_1\|\sigma_2) \\ & = C(\rho_1\|\rho_2) + C(\sigma_1\|\sigma_2), \end{align}\] where the second equality uses the multiplicativity of \(Q_\alpha\) under tensor products, and the inequality follows from splitting the maximization over \(\alpha\) for each term.
The quantum Chernoff bound for multiple hypotheses was established by Li in [5]. Consider \(r\) quantum states \(\{\rho_i\}_{i=1}^r\) with prior probabilities \(\{\pi_i\}_{i=1}^r\) such that \(\sum_{i=1}^r \pi_i = 1\). Let \(\{M_i\}_{i=1}^r\) be a POVM, i.e., a collection of positive semidefinite operators satisfying \(\sum_{i=1}^r M_i = I\). The average error probability for discriminating among these \(r\) quantum states is given by \[\begin{align} P_{e}(\{M_i\}_{i=1}^r, \{\rho_i\}_{i=1}^r) := \sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i (I-M_i)] = 1 - \sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i M_i]. \end{align}\] Optimizing over all POVMs, the minimum error probability is defined as \[\begin{align} \label{eq:32Pemin32multiple32states} P_{e,\min}(\{\rho_i\}_{i=1}^r) := \inf_{\{M_i\}_{i=1}^r} P_{e}(\{M_i\}_{i=1}^r, \{\rho_i\}_{i=1}^r). \end{align}\tag{4}\] It has been shown in [5] that \[\begin{align} \label{eq:32chernoff32theorem32for32multiple32states} \lim_{n \to \infty} -\frac{1}{n} \log P_{e,\min}(\{\rho_i^{\otimes n}\}_{i=1}^r) = C(\{\rho_i\}_{i=1}^r) := \min_{i \neq j} C(\rho_i, \rho_j). \end{align}\tag{5}\]
In this section, we generalize the quantum Chernoff bound to the discrimination of multiple sets of quantum states. We begin by formulating the operational framework for hypothesis testing among several sets, and then introduce the notion of the quantum Chernoff divergence between sets of quantum states, along with its key mathematical properties. We conclude by presenting our main result, which establishes the quantum Chernoff bound for multiple sets of quantum states under appropriate structural assumptions.
Consider the problem of discriminating among \(r\) sets of quantum states, denoted by \({{\mathscr{C}}}_i\) for \(i \in \{1, \ldots, r\}\), where each set is associated with a prior probability \(\pi_i\) such that \(\sum_{i=1}^r \pi_i = 1\). The task is to determine, via quantum measurement, from which set a given quantum state is drawn, without knowledge of the specific state within each set. Let \(\{M_i\}_{i=1}^r\) be a POVM, where each \(M_i\) corresponds to the decision that the state is drawn from set \({{\mathscr{C}}}_i\). For each \(i\), \(\operatorname{Tr}[\rho_i (I - M_i)]\) represents the probability of incorrectly rejecting set \({{\mathscr{C}}}_i\) when the true state is \(\rho_i \in {{\mathscr{C}}}_i\). Since the specific state within each set is unknown, we adopt a worst-case approach and define the error probability as the supremum, over all possible choices of states from each set, of the average probability of incorrectly identifying the set: \[\begin{align} P_{e}(\{M_i\}_{i=1}^r, \{{{\mathscr{C}}}_i\}_{i=1}^r) := \sup_{\forall i,\, \rho_i \in {{\mathscr{C}}}_i} \sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i (I - M_i)]. \end{align}\]
To characterize the fundamental limit of discrimination, we minimize the worst-case error probability over all possible POVMs: \[\begin{align} P_{e,\min}(\{{{\mathscr{C}}}_i\}_{i=1}^r) := \inf_{\{M_i\}_{i=1}^r} P_{e}(\{M_i\}_{i=1}^r, \{{{\mathscr{C}}}_i\}_{i=1}^r). \end{align}\] This quantity characterizes the optimal error probability for discriminating among multiple sets of quantum states, accounting for the worst-case selection of states from each set.
If each \({{\mathscr{C}}}_i = \{{{\mathscr{C}}}_{i,n}\}_{n\in {{\mathbb{N}}}}\) is a sequence of sets of quantum states,2 we are interested in the asymptotic behavior of the minimum error probability: \[\begin{align} \liminf_{n\to \infty} - \frac{1}{n} \log P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) = ? \end{align}\] As a special case, choosing the set to be the singleton \({{\mathscr{C}}}_{i,n} = \{\rho_i^{\otimes n}\}\), it recovers the quantum Chernoff bound for multiple i.i.d. states given in Eq. 5 .
We now introduce the quantum Chernoff divergence between two sets of quantum states, which generalizes the quantum Chernoff divergence defined in Eq. 2 . This quantity captures the fundamental distinguishability between two sets of quantum states and serves as a key tool in establishing the quantum Chernoff bound for composite correlated hypotheses. Building on this notion, we further extend the definition to encompass the case of multiple sets.
Definition 1 (Quantum Chernoff divergence between two sets of quantum states). Let \({\cal H}\) be a finite-dimensional Hilbert space, and let \({{\mathscr{C}}}_1, {{\mathscr{C}}}_2 \subseteq \mathscr{D}({\cal H})\) be two sets of quantum states. The quantum Chernoff divergence between these sets is defined as \[\begin{align} C({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) := \inf_{\substack{\rho_1 \in {{\mathscr{C}}}_1\\ \rho_2 \in {{\mathscr{C}}}_2}} C(\rho_1\|\rho_2). \end{align}\]
Moreover, let \({{\mathscr{C}}}_1 = \{{{\mathscr{C}}}_{1,n}\}_{n\in {{\mathbb{N}}}}\) and \({{\mathscr{C}}}_2 = \{{{\mathscr{C}}}_{2,n}\}_{n\in {{\mathbb{N}}}}\) be two sequences of sets of quantum states, where each \({{\mathscr{C}}}_{1,n}, {{\mathscr{C}}}_{2,n} \subseteq \mathscr{D}({\cal H}^{\otimes n})\). The regularized quantum Chernoff divergence between these sequences is defined as \[\begin{align} \underline{C}^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) & := \liminf_{n\to \infty} \frac{1}{n} C({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}),\\ \overline{C}^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) & := \limsup_{n\to \infty} \frac{1}{n} C({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}). \end{align}\] If the following limit exists, we define the regularized Chernoff divergence as \[\begin{align} C^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) := \lim_{n\to \infty} \frac{1}{n} C({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}). \end{align}\]
Remark 2. The quantum Chernoff divergence can be written in terms of the Petz Rényi divergence as \(C(\rho_1\|\rho_2) = \sup_{\alpha \in (0,1)} (1-\alpha) D_{{\scriptscriptstyle \rm P},\alpha}(\rho_1\|\rho_2)\). Since \(D_{{\scriptscriptstyle \rm P},\alpha}(\rho_1\|\rho_2)\) is lower semicontinuous in \((\rho_1,\rho_2) \in \mathscr{H}_{{\scalebox{0.7}{+}}{\scalebox{0.7}{+}}}\times \mathscr{H}_{{\scalebox{0.7}{+}}{\scalebox{0.7}{+}}}\) for any fixed \(\alpha\) [13], and \(C(\rho_1\|\rho_2)\) is the pointwise supremum of lower semicontinuous functions, it follows from Lemma 4 that \(C(\rho_1\|\rho_2)\) is also lower semicontinuous in \((\rho_1,\rho_2)\). Consequently, if the sets \({{\mathscr{C}}}_1\) and \({{\mathscr{C}}}_2\) are compact, we know from Lemma 3 that the infimum in the definition of \(C({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2)\) is achieved.
In many practical scenarios, the sequences of sets under consideration are not arbitrary but possess a structure that is compatible with tensor products. This property, known as stability (or closeness) under tensor product, is formalized as follows.
Definition 3 (Stable sequence). Let \({{\mathscr{C}}}_1 \subseteq \mathscr{H}_{{\scalebox{0.7}{+}}}({\cal H}_1)\), \({{\mathscr{C}}}_2 \subseteq \mathscr{H}_{{\scalebox{0.7}{+}}}({\cal H}_2)\), and \({{\mathscr{C}}}_{3} \subseteq \mathscr{H}_{{\scalebox{0.7}{+}}}({\cal H}_1 \otimes {\cal H}_2)\). We say that \(({{\mathscr{C}}}_1, {{\mathscr{C}}}_2, {{\mathscr{C}}}_{3})\) is stable under tensor product if, for any \(X_1 \in {{\mathscr{C}}}_1\) and \(X_2 \in {{\mathscr{C}}}_2\), it holds that \(X_1 \otimes X_2 \in {{\mathscr{C}}}_{3}\). In short, we write \({{\mathscr{C}}}_1 \otimes {{\mathscr{C}}}_2 \subseteq {{\mathscr{C}}}_{3}\). A sequence of sets \(\{{{\mathscr{C}}}_n\}_{n \in {{\mathbb{N}}}}\) with \({{\mathscr{C}}}_n \subseteq \mathscr{H}_{{\scalebox{0.7}{+}}}({\cal H}^{\otimes n})\) is called stable under tensor product if \({{\mathscr{C}}}_n \otimes {{\mathscr{C}}}_m \subseteq {{\mathscr{C}}}_{n+m}\) for all \(n, m \in {{\mathbb{N}}}\).
As a consequence of the subadditivity of the Chernoff divergence in Eq. 3 , its extension to sets of quantum states is also subadditive, provided that the sequences \({{\mathscr{C}}}_1 = \{{{\mathscr{C}}}_{1,n}\}_{n\in {{\mathbb{N}}}}\) and \({{\mathscr{C}}}_2 = \{{{\mathscr{C}}}_{2,n}\}_{n\in {{\mathbb{N}}}}\) are both stable under tensor product [10]. Therefore, the regularized quantum Chernoff divergence \(C^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2)\) exists and satisfies \[\begin{align} \label{eq:32multiplicativity32of32Chernoff32divergence} C^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) = \overline{C}^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) = \underline{C}^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) = \inf_{n \geq 1} \frac{1}{n} C({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,}). \end{align}\tag{6}\] In Section 5, we provide examples where the quantum Chernoff divergence between two sets is strictly subadditive, highlighting the necessity of regularization in general.
Analogous to the extension of the quantum Chernoff divergence to sets of quantum states, we also generalize the quantum Chernoff quasi-divergence to this setting.
Definition 4 (Quantum Chernoff quasi-divergence between two sets of quantum states). Let \(\alpha \in [0,1]\) and \({\cal H}\) be a finite-dimensional Hilbert space. Let \({{\mathscr{C}}}_1, {{\mathscr{C}}}_2 \subseteq \mathscr{D}({\cal H})\) be two sets of quantum states. The quantum Chernoff quasi-divergence between these sets is defined as \[\begin{align} Q_\alpha({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) := \sup_{\substack{\rho_1 \in {{\mathscr{C}}}_1\\ \rho_2 \in {{\mathscr{C}}}_2}} Q_\alpha(\rho_1\|\rho_2). \end{align}\]
Moreover, let \({{\mathscr{C}}}_1 = \{{{\mathscr{C}}}_{1,n}\}_{n\in {{\mathbb{N}}}}\) and \({{\mathscr{C}}}_2 = \{{{\mathscr{C}}}_{2,n}\}_{n\in {{\mathbb{N}}}}\) be two sequences of sets of quantum states, where each \({{\mathscr{C}}}_{1,n}, {{\mathscr{C}}}_{2,n} \subseteq \mathscr{D}({\cal H}^{\otimes n})\). The regularized quantum Chernoff quasi-divergence between these sequences is defined as \[\begin{align} \underline{Q}_\alpha^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) & := \liminf_{n\to \infty} [Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})]^\frac{1}{n},\\ \overline{Q}_\alpha^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) & := \limsup_{n\to \infty} [Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})]^\frac{1}{n}. \end{align}\] If the following limit exists, we define the regularized Chernoff quasi-divergence as \[\begin{align} Q_{\alpha}^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) := \lim_{n\to \infty} [Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})]^{\frac{1}{n}}. \end{align}\]
The multiplicativity of the quantum Chernoff quasi-divergence under tensor product implies that its extension to sets is supermultiplicative, provided that the sequences are stable under tensor product. Consequently, the regularized quasi-divergence \(Q_\alpha^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2)\) exists and satisfies \[\begin{align} \label{eq:32multiplicativity32of32Chernoff32quasi-divergence} Q_\alpha^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) = \overline{Q}_\alpha^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) = \underline{Q}_\alpha^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) = \sup_{n \geq 1} [Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})]^{\frac{1}{n}}. \end{align}\tag{7}\]
While the quantum Chernoff divergence and quasi-divergence between two states are directly related through their definitions, we have thus far extended these quantities for sets of quantum states independently. It is therefore natural to ask whether a relationship analogous to Eq. 2 holds between the Chernoff divergence and quasi-divergence when extended to sets of quantum states. The following result establishes this connection by applying minimax theorems.
Lemma 7. Let \({\cal H}\) be a finite-dimensional Hilbert space, and let \({{\mathscr{C}}}_1, {{\mathscr{C}}}_2 \subseteq \mathscr{D}({\cal H})\) be two convex sets of quantum states. Then it holds that \[\begin{align} \label{eq:32C32and32Q32between32two32sets} C({{\mathscr{C}}}_{1}\|{{\mathscr{C}}}_{2}) = \max_{\alpha \in [0,1]} -\log Q_\alpha({{\mathscr{C}}}_{1}\|{{\mathscr{C}}}_{2}). \end{align}\qquad{(1)}\] Moreover, let \({{\mathscr{C}}}_1 = \{{{\mathscr{C}}}_{1,n}\}_{n\in {{\mathbb{N}}}}\) and \({{\mathscr{C}}}_2 = \{{{\mathscr{C}}}_{2,n}\}_{n\in {{\mathbb{N}}}}\) be two stable sequences of convex sets of quantum states, where each \({{\mathscr{C}}}_{1,n}, {{\mathscr{C}}}_{2,n} \subseteq \mathscr{D}({\cal H}^{\otimes n})\). Then it holds that \[\begin{align} C^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) = \max_{\alpha \in [0,1]} -\log Q_\alpha^{\infty}({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2). \end{align}\]
Proof. We have the following chain of equalities: \[\begin{align} C({{\mathscr{C}}}_{1}\|{{\mathscr{C}}}_{2}) & = \inf_{\substack{\rho_1 \in {{\mathscr{C}}}_{1}\\\rho_2 \in {{\mathscr{C}}}_{2}}} C(\rho_1\|\rho_2)\\ & = \inf_{\substack{\rho_1 \in {{\mathscr{C}}}_{1}\\\rho_2 \in {{\mathscr{C}}}_{2}}} \max_{0\leq \alpha \leq 1} -\log Q_\alpha(\rho_1\|\rho_2)\\ & = -\log \sup_{\substack{\rho_1 \in {{\mathscr{C}}}_{1}\\\rho_2 \in {{\mathscr{C}}}_{2}}} \min_{0\leq \alpha \leq 1} Q_\alpha(\rho_1\|\rho_2)\\ & = -\log \min_{0\leq \alpha\leq 1} \sup_{\substack{\rho_1 \in {{\mathscr{C}}}_{1}\\\rho_2 \in {{\mathscr{C}}}_{2}}} Q_\alpha(\rho_1\|\rho_2)\\ & = \max_{0\leq \alpha\leq 1} \inf_{\substack{\rho_1 \in {{\mathscr{C}}}_{1}\\\rho_2 \in {{\mathscr{C}}}_{2}}} -\log Q_\alpha(\rho_1\|\rho_2)\\ & = \max_{0\leq \alpha\leq 1} -\log Q_\alpha({{\mathscr{C}}}_{1}\|{{\mathscr{C}}}_{2}),\label{eq:32C32set32minimax32tmp1} \end{align}\tag{8}\] where the first two equalities follow directly from the definitions. By Lieb’s concavity theorem, \(Q_\alpha(\rho_1\|\rho_2)\) is jointly concave in \((\rho_1, \rho_2)\) (see also [17]), and it is convex and continuous in \(\alpha\) [18]. These properties allow us to apply the minimax theorem in Lemma 1 to exchange the order of the supremum and infimum. The final equality then follows from the definition of \(Q_\alpha({{\mathscr{C}}}_{1}\|{{\mathscr{C}}}_{2})\).
We now prove the second statement. Suppose \({{\mathscr{C}}}_1 = \{{{\mathscr{C}}}_{1,n}\}_{n\in {{\mathbb{N}}}}\) and \({{\mathscr{C}}}_2 = \{{{\mathscr{C}}}_{2,n}\}_{n\in {{\mathbb{N}}}}\) are stable sequences under tensor product. By Eq. 6 , the regularized Chernoff divergence exists and can be written as \[\begin{align} C^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) & = \inf_{n \geq 1} \frac{1}{n} C({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})\\ & = \inf_{n \geq 1} \frac{1}{n} \max_{\alpha \in [0,1]} -\log Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}) \\ & = \inf_{n \geq 1} \max_{\alpha \in [0,1]} - \frac{1}{n} \log Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}) , \end{align}\] where the second line follows from Eq. 8 established above. Define \[\begin{align} f(n,\alpha):= - \frac{1}{n} \log Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}). \end{align}\] Due to the supermultiplicativity of \(Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})\), the objective function \(f(n,\alpha)\) is monotone decreasing in \(n\) for each fixed \(\alpha\). Furthermore, since \(Q_{\alpha}(\rho_n\|\sigma_n)\) is continuous in \(\alpha \in [0,1]\), and \(Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})\) is defined as the pointwise supremum over lower semicontinuous functions, Lemma 4 ensures that \(Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})\) is also lower semicontinuous in \(\alpha\). Consequently, \(f(n,\alpha)\) is upper semicontinuous in \(\alpha\) for each \(n\). By applying the minimax theorem in Lemma 2, we can exchange the infimum and the maximum to obtain \[\begin{align} C^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) & = \max_{\alpha \in [0,1]} \inf_{n \geq 1} - \frac{1}{n} \log Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}) \\ & = \max_{\alpha \in [0,1]} - \log \sup_{n \geq 1} [Q_\alpha({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n})]^{\frac{1}{n}} \\ & = \max_{\alpha \in [0,1]} -\log Q_\alpha^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2) , \end{align}\] where the last line follows from Eq. 7 . This completes the proof. ◻
Remark 5 (Computability). For any fixed \(\alpha \in [0,1]\), the function \(Q_\alpha(\rho\|\sigma)\) is jointly concave in \((\rho,\sigma)\), so the quasi-divergence \(Q_{\alpha}({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2)\) can be efficiently computed using the QICS package [19] whenever \({{\mathscr{C}}}_1\) and \({{\mathscr{C}}}_2\) admit semidefinite representations. If the sets \({{\mathscr{C}}}_1\) and \({{\mathscr{C}}}_2\) possess additional symmetries, the computational complexity can be further reduced. Moreover, since \(Q_{\alpha}({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2)\) is convex and lower semicontinuous in \(\alpha\), the minimum over \(\alpha \in [0,1]\) is achieved at a unique optimal solution. Thus, the quantum Chernoff divergence \(C({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2)\) can be efficiently computed by scanning over \(\alpha\) (e.g., via a bisection method).
Motivated by the quantum Chernoff divergence for multiple quantum states in Eq. 5 , we now extend the notion of quantum Chernoff divergence from two sets to the case of multiple sets of quantum states, as formalized in the following definition.
Definition 6 (Quantum Chernoff divergence between multiple sets of quantum states). Let \({\cal H}\) be a finite-dimensional Hilbert space, and let \(\{{{\mathscr{C}}}_i\}_{i=1}^r\) be \(r\) sets of quantum states, where each \({{\mathscr{C}}}_i \subseteq \mathscr{D}({\cal H})\). The quantum Chernoff divergence between these sets is defined as \[\begin{align} C(\{{{\mathscr{C}}}_i\}_{i=1}^r) := \min_{i\neq j} C({{\mathscr{C}}}_i\| {{\mathscr{C}}}_j). \end{align}\]
Moreover, if each \({{\mathscr{C}}}_i:= \{{{\mathscr{C}}}_{i,n}\}_{n\in {{\mathbb{N}}}}\) is a sequence of sets of quantum states where \({{\mathscr{C}}}_{i,n} \subseteq \mathscr{D}({\cal H}^{\otimes n})\), then the regularized quantum Chernoff divergence between these sets is defined as \[\begin{align} \underline{C}^\infty(\{{{\mathscr{C}}}_i\}_{i=1}^r) & := \liminf_{n\to \infty} \frac{1}{n} C(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r),\\ \overline{C}^\infty(\{{{\mathscr{C}}}_i\}_{i=1}^r) & := \limsup_{n\to \infty} \frac{1}{n} C(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r). \end{align}\] If the following limit exists, we define the regularized Chernoff divergence as \[\begin{align} {C}^\infty(\{{{\mathscr{C}}}_i\}_{i=1}^r) & := \lim_{n\to \infty} \frac{1}{n} C(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r). \end{align}\]
We now present a fundamental relation for the minimum error probability, which follows directly from the minimax theorem. This result plays a central role in establishing the quantum Chernoff bound for sets of quantum states and in the subsequent discussion of universal optimal tests. Importantly, it shows that the problem of composite hypothesis testing can be reduced to discriminating between the worst-case states in the respective sets; that is, discriminating between sets of quantum states is no harder than discriminating between their most challenging elements.
Lemma 8. Let \({\cal H}\) be a finite-dimensional Hilbert space, and let \(\{{{\mathscr{C}}}_i\}_{i=1}^r\) be \(r\) convex sets of quantum states, where each \({{\mathscr{C}}}_i \subseteq \mathscr{D}({\cal H})\). Then it holds that \[\begin{align} \label{eq:32Pemin32minimax32multiple} P_{e,\min}(\{{{\mathscr{C}}}_i\}_{i=1}^r) = \sup_{\forall i,\, \rho_i\in {{\mathscr{C}}}_i} P_{e,\min}(\{\rho_i\}_{i=1}^r), \end{align}\qquad{(2)}\] where the suprema can be replaced by maxima if the sets are also compact.
Proof. By definition, we have \[\begin{align} P_{e,\min}(\{{{\mathscr{C}}}_i\}_{i=1}^r) = \inf_{\{M_i\}_{i=1}^r} \sup_{\forall i,\, \rho_i \in {{\mathscr{C}}}_i} \sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i (I - M_i)] \end{align}\] Due to the linearity of the error probability \(\sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i (I - M_i)]\) in each of its arguments \(M_i\) and \(\rho_i\), and the set of all POVMs is convex and compact, we can apply Lemma 1 to exchange the infimum and supremum and get \[\begin{align} \label{eq:32Pemin32minimax32tmp} P_{e,\min}(\{{{\mathscr{C}}}_i\}_{i=1}^r) = \sup_{\forall i,\, \rho_i \in {{\mathscr{C}}}_i} \inf_{\{M_i\}_{i=1}^r} \sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i (I - M_i)] & = \sup_{\forall i,\, \rho_i \in {{\mathscr{C}}}_i} P_{e,\min}(\{\rho_i\}_{i=1}^r), \end{align}\tag{9}\] where the second equality follows from Eq. 4 . Note that \(P_{e,\min}(\{\rho_i\}_{i=1}^r)\) is upper semicontinuous by Lemma 4. Therefore by Lemma 3 the supremum can be replaced by a maximum if the sets are compact. ◻
Remark 7 (Computability). For a fixed collection of \(r\) quantum states, the minimum error probability admits a semidefinite programming (SDP) formulation [5]: \[\begin{align} P_{e,\min}(\{\rho_i\}_{i=1}^r) = \max_{X\in \mathscr{H}} \left\{1-\operatorname{Tr}X: X \geq \pi_i \rho_i,\, \forall i = 1,\ldots, r \right\}. \end{align}\] Applying Lemma 8, we obtain \[\begin{align} P_{e,\min}(\{{{\mathscr{C}}}_i\}_{i=1}^r) = \sup_{\forall i,\, \rho_i \in {{\mathscr{C}}}_i} \max_{X\in \mathscr{H}} \left\{1-\operatorname{Tr}X: X \geq \pi_i \rho_i,\, \forall i = 1,\ldots, r \right\}, \end{align}\] which remains an SDP whenever the sets \({{\mathscr{C}}}_i\) admit SDP representations. In such cases, the optimal value can be efficiently computed.
Remark 8 (Symmetry reduction). Observe that \(\sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i (I - M_i)]\) is linear in \(\{\rho_i\}_{i=1}^r\). Consequently, \(P_{e,\min}(\{\rho_i\}_{i=1}^r) = \inf_{\{M_i\}_{i=1}^r} \sum_{i=1}^r \pi_i \operatorname{Tr}[\rho_i (I - M_i)]\) is concave in \(\{\rho_i\}_{i=1}^r\). Moreover, \(P_{e,\min}(\{\rho_i\}_{i=1}^r)\) is invariant under simultaneous application of the same unitary transformation to all states. Therefore, if each \({{\mathscr{C}}}_i\) is a convex, compact, and permutation-invariant 3 set, the supremum in \(\sup_{\forall i,\, \rho_i\in {{\mathscr{C}}}_i} P_{e,\min}(\{\rho_i\}_{i=1}^r)\) can be achieved by permutation-invariant states.
We are now prepared to state and prove the quantum Chernoff bound for multiple sets of quantum states. This result demonstrates that the minimum error probability decays exponentially at a rate determined by the regularized Chernoff divergence between the sets, thereby extending the quantum Chernoff bound to the composite correlated setting.
Theorem 1 (Quantum Chernoff bound for multiple composite hypotheses). Let \({\cal H}\) be a finite-dimensional Hilbert space, and let \({{\mathscr{C}}}_i = \{{{\mathscr{C}}}_{i,n}\}_{n\in {{\mathbb{N}}}}\) for \(i \in \{1, \ldots, r\}\) be \(r\) stable sequences of convex, compact, permutation-invariant sets of quantum states, where each \({{\mathscr{C}}}_{i,n} \subseteq \mathscr{D}({\cal H}^{\otimes n})\). Moreover, let \(\{\pi_i\}_{i=1}^r\) be the prior probability. Then, \[\begin{align} \liminf_{n\to \infty} - \frac{1}{n} \log P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) = \underline{C}^{\infty}(\{{{\mathscr{C}}}_{i}\}_{i=1}^r). \end{align}\]
Proof. 1) proof of the achievable part: Let \(\{\rho_i\}_{i=1}^r\) be \(r\) quantum states with a prior probability \(\{\pi_i\}_{i=1}^r\). Then it holds that [5] \[\begin{align} P_{e,\min}(\{\rho_i\}_{i=1}^r) \leq 10(r-1)^2 T(\{\rho_i\}_{i=1}^r)^2 \sum_{i<j} \min_{\alpha \in [0,1]} \pi_i^\alpha \pi_j^{1-\alpha}Q_\alpha(\rho_i\|\rho_j), \end{align}\] where \(T(\{\rho_i\}_{i=1}^r) = \max_{i} \{|\text{\rm spec}(\rho_i)|\}\) and \(|\text{\rm spec}(X)|\) is the number of mutually different eigenvalues of \(X\). Therefore, we have for any \(\rho_{i,n} \in {{\mathscr{C}}}_{i,n}\) that \[\begin{align} P_{e,\min}(\{\rho_{i,n}\}_{i=1}^r) & \leq 10(r-1)^2 T(\{\rho_{i,n}\}_{i=1}^r)^2 \sum_{i<j} \min_{\alpha \in [0,1]} Q_\alpha(\rho_{i,n}\|\rho_{j,n})\\ & \leq 10(r-1)^2 T(\{\rho_{i,n}\}_{i=1}^r)^2 C_r^2 \max_{i\neq j} \min_{\alpha \in [0,1]} Q_\alpha(\rho_{i,n}\|\rho_{j,n})\label{eq:32Chernoff32multiple32proof32tmp1}, \end{align}\tag{10}\] where we use the fact that \(\pi_i^\alpha \pi_j^{1-\alpha} \leq 1\) and then relax the summation on the right hand side to the maximum value times the number of terms \(C_r^2 = \frac{r(r-1)}{2}\). Let \(\rho_{i,n}^{*}\) be the maximizer of Eq. ?? , which can be choosen as permutation-invariant state due to Remark 8. We have \[\begin{align} P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) & = P_{e,\min}(\{\rho_{i,n}^{*}\}_{i=1}^r)\\ & \leq 10(r-1)^2 T(\{\rho_{i,n}^{*}\}_{i=1}^r)^2 C_r^2 \max_{i\neq j} \min_{\alpha \in [0,1]} Q_\alpha(\rho_{i,n}^*\|\rho_{j,n}^*)\\ & \leq 10(r-1)^2 T(\{\rho_{i,n}^{*}\}_{i=1}^r)^2 C_r^2 \max_{i\neq j} \min_{\alpha \in [0,1]} Q_\alpha({{\mathscr{C}}}_{i,n}\|{{\mathscr{C}}}_{j,n}), \end{align}\] where the first inequality follows from Eq. 10 and \(T(\{\rho_{i,n}^{*}\}_{i=1}^r) \leq (n+1)^{|{\cal H}|}\) by the permutation invariance of each states and the second inequality follows from the fact that \(\rho_{i,n}^* \in {{\mathscr{C}}}_{i,n}\) for all \(i\) is a specific choice in the sets. This gives \[\begin{align} - \log P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) \geq -\log [10(r-1)^2 T(\{\rho_{i,n}^{*}\}_{i=1}^r)^2 C_r^2] + \min_{i\neq j} C({{\mathscr{C}}}_{i,n}\|{{\mathscr{C}}}_{j,n}), \end{align}\] where we use Eq. ?? in Lemma 7. This implies \[\begin{align} \liminf_{n\to \infty} - \frac{1}{n}\log P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) & \geq \liminf_{n\to \infty} \frac{1}{n} \min_{i\neq j} C({{\mathscr{C}}}_{i,n}\|{{\mathscr{C}}}_{j,n}) = \underline{C}^\infty(\{{{\mathscr{C}}}_{i}\}_{i=1}^r), \end{align}\] the equality follows by definition.
2) proof of the converse part: For any fixed \(m \in {{\mathbb{N}}}\) and \(\rho_{i,m} \in {{\mathscr{C}}}_{i,m}\), we have \[\begin{align} \liminf_{n\to \infty} -\frac{1}{n} \log P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) & \leq \liminf_{n\to \infty} -\frac{1}{nm} \log P_{e,\min}(\{{{\mathscr{C}}}_{i,mn}\}_{i=1}^r)\\ & = \liminf_{n\to \infty} -\frac{1}{nm} \log \sup_{\forall i,\, \rho_{i,mn} \in {{\mathscr{C}}}_{i,mn}}P_{e,\min}(\{\rho_{i,mn}\}_{i=1}^r)\\ & \leq \liminf_{n\to \infty} -\frac{1}{nm} \log P_{e,\min}(\{(\rho_{i,m})^{\otimes n}\}_{i=1}^r)\\ & = \frac{1}{m} \min_{i\neq j} C(\rho_{i,m}\|\rho_{j,m}), \end{align}\] where the first inequality follows as the lower limit of a subsequence is no smaller than the lower limit of the sequence, the first equality follows from Lemma 8, the second inequality follows by taking a particular feasible solution and the stability of the sequences, the second equality follows from the quantum Chernoff bound for multiple i.i.d. quantum states (see Eq. 5 ). As this holds for any \(\rho_{i,m} \in {{\mathscr{C}}}_{i,m}\), we have \[\begin{align} \liminf_{n\to \infty} -\frac{1}{n} \log P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) & \leq \inf_{\forall i\, \rho_{i,m} \in {{\mathscr{C}}}_{i,m}}\frac{1}{m} \min_{i\neq j} C(\rho_{i,m}\|\rho_{j,m})\\ & = \min_{i\neq j} \inf_{\forall i\, \rho_{i,m} \in {{\mathscr{C}}}_{i,m}}\frac{1}{m} C(\rho_{i,m}\|\rho_{j,m})\\ & = \min_{i\neq j} \frac{1}{m} C({{\mathscr{C}}}_{i,m}\|{{\mathscr{C}}}_{j,m}), \end{align}\] where in the second line we exchange the order of the minimum and infimum, and in the last line we use the definition of \(C({{\mathscr{C}}}_{i,m}\|{{\mathscr{C}}}_{j,m})\). As this holds for any \(m \in {{\mathbb{N}}}\), we have \[\begin{align} \liminf_{n\to \infty} -\frac{1}{n} \log P_{e,\min}(\{{{\mathscr{C}}}_{i,n}\}_{i=1}^r) & \leq \liminf_{m \to \infty} \min_{i\neq j} \frac{1}{m} C({{\mathscr{C}}}_{i,m}\|{{\mathscr{C}}}_{j,m}) = \underline{C}^\infty(\{{{\mathscr{C}}}_{i}\}_{i=1}^r). \end{align}\] This completes the proof. ◻
Remark 9. If we choose \({{\mathscr{C}}}_{i,n}=\{\rho_i^{\otimes n}\}\) as the singleton i.i.d. quantum states, then we can recover the quantum Chernoff bound for multiple quantum states in [5].
For the case of multiple sets of quantum states, the above theorem requires permutation invariance of each set in the proof of the achievable part. However, as we show below, this assumption is not necessary when considering only two sequences of sets of quantum states.
Theorem 2 (Quantum Chernoff bound for binary composite hypotheses). Let \({\cal H}\) be a finite-dimensional Hilbert space, and let \({{\mathscr{C}}}_i = \{{{\mathscr{C}}}_{i,n}\}_{n\in {{\mathbb{N}}}}\) for \(i \in \{1, 2\}\) be two stable sequences of convex, compact sets of quantum states, where each \({{\mathscr{C}}}_{i,n} \subseteq \mathscr{D}({\cal H}^{\otimes n})\). Moreover, let \(\{\pi_1,\pi_2\}\) be the prior probability. Then it holds that \[\begin{align} \liminf_{n\to \infty} - \frac{1}{n} \log P_{e,\min}({{\mathscr{C}}}_{1,n}, {{\mathscr{C}}}_{2,n}) = C^\infty({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2). \end{align}\]
Proof. 1) proof of the achievable part: For any \(\alpha \in [0,1]\), we have that \[\begin{align} P_{e, \min}(\rho_{1,n},\rho_{2,n}) & = \frac{1}{2} [\operatorname{Tr}(\pi_2 \rho_{2,n} + \pi_1 \rho_{1,n}) - \|\pi_1 \rho_{1,n} - \pi_2 \rho_{2,n}\|_1] \\ & \leq \pi_1^\alpha \pi_2^{1-\alpha} Q_{\alpha}(\rho_{1,n}\|\rho_{2,n}), \end{align}\] where the equality follows from Lemma 5 and the inequality follows from Lemma 6. As this holds for any \(\rho_{1,n} \in {{\mathscr{C}}}_{1,n}\) and \(\rho_{2,n} \in {{\mathscr{C}}}_{2,n}\), we have from Lemma 8 that \[\begin{align} P_{e,\min}({{\mathscr{C}}}_{1,n},{{\mathscr{C}}}_{2,n}) \leq (\pi_1^\alpha \pi_2^{1-\alpha}) Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}) \leq Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}), \end{align}\] where the second inequality follows from the fact that \(0 \leq \pi_1^\alpha \pi_2^{1-\alpha} \leq \alpha \pi_1 + (1-\alpha) \pi_2 \leq 1\) for any \(\alpha \in [0,1]\). This gives \[\begin{align} - \log P_{e,\min}({{\mathscr{C}}}_{1,n},{{\mathscr{C}}}_{2,n}) \geq - \log Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}). \end{align}\] As this holds for any \(\alpha \in [0,1]\), we have \[\begin{align} - \log P_{e,\min}({{\mathscr{C}}}_{1,n},{{\mathscr{C}}}_{2,n}) & \geq \max_{\alpha \in [0,1]}- \log Q_{\alpha}({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}) = C({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}), \end{align}\] where the equality follows Lemma 7. This implies \[\begin{align} \liminf_{n\to \infty} - \frac{1}{n}\log P_{e,\min}({{\mathscr{C}}}_{1,n},{{\mathscr{C}}}_{2,n}) \geq \liminf_{n\to \infty} \frac{1}{n} C({{\mathscr{C}}}_{1,n}\|{{\mathscr{C}}}_{2,n}) = {C}^\infty({{\mathscr{A}}}\|{{\mathscr{B}}}), \end{align}\] where the equality follows from the stability assumption of the sequences and Eq. 6 .
2) proof of the converse part: For any fixed \(m \in {{\mathbb{N}}}\) and any \(\rho_{1,m} \in {{\mathscr{C}}}_{1,m}, \rho_{2,m} \in {{\mathscr{C}}}_{2,m}\), we have the following chain of inequalities: \[\begin{align} \liminf_{n\to \infty} -\frac{1}{n} \log P_{e,\min}({{\mathscr{C}}}_{1,n},{{\mathscr{C}}}_{2,n}) & \leq \liminf_{n\to \infty} -\frac{1}{nm} \log P_{e,\min}({{\mathscr{C}}}_{1,nm},{{\mathscr{C}}}_{2,nm})\\ & = \liminf_{n\to \infty} -\frac{1}{nm} \log \sup_{\substack{\rho_{1,mn} \in {{\mathscr{C}}}_{1,mn}\\ \rho_{2,mn} \in {{\mathscr{C}}}_{2,mn}}}P_{e,\min}(\rho_{1,nm},\rho_{2,nm})\\ & \leq \liminf_{n\to \infty} -\frac{1}{nm} \log P_{e,\min}((\rho_{1,m})^{\otimes n},(\rho_{2,m})^{\otimes n})\\ & = \frac{1}{m} C(\rho_{1,m}\|\rho_{2,m}), \end{align}\] where the first inequality follows as the lower limit of a subsequence is no smaller than the lower limit of the sequence, the first equality follows from Lemma 8, the second inequality follows by taking a particular feasible solution and the stability of the sequences, the second equality follows from the quantum Chernoff bound between two quantum states (see Eq. 1 ). As this holds for any \(\rho_{1,m} \in {{\mathscr{C}}}_{1,m}\) and \(\rho_{2,m} \in {{\mathscr{C}}}_{2,m}\), we have \[\begin{align} \liminf_{n\to \infty} -\frac{1}{n} \log P_{e,\min}({{\mathscr{C}}}_{1,n},{{\mathscr{C}}}_{2,n}) \leq \frac{1}{m} C({{\mathscr{C}}}_{1,m}\|{{\mathscr{C}}}_{2,m}). \end{align}\] As this holds for any \(m \in {{\mathbb{N}}}\), we have \[\begin{align} \liminf_{n\to \infty} -\frac{1}{n} \log P_{e,\min}({{\mathscr{C}}}_{1,n},{{\mathscr{C}}}_{2,n}) \leq \liminf_{m \to \infty} \frac{1}{m} C({{\mathscr{C}}}_{1,m}\|{{\mathscr{C}}}_{2,m}) = C^{\infty}({{\mathscr{C}}}_1\|{{\mathscr{C}}}_2), \end{align}\] where the equality follows from the stability assumption of the sequences and Eq. 6 . ◻
Remark 10. If we choose \({{\mathscr{C}}}_{i,n}=\{\rho_i^{\otimes n}\}\) as the singleton i.i.d. quantum states, then we can recover the quantum Chernoff bound for two quantum states in [3], [4].
In quantum hypothesis testing between two quantum states \(\rho_1\) and \(\rho_2\) with prior probabilities \(\pi_1\) and \(\pi_2\), the optimal measurement is given by the Holevo-Helstrom test \(\{\pi_1 \rho_1 - \pi_2 \rho_2 \geq 0\}\) [15], [16]. However, in the composite hypothesis setting—where only partial information about the states is available—the problem becomes more subtle. Here, one must design a test that universally minimizes the average error probability for all possible states within the specified sets. Lemma 8 guarantees the existence of such a universal optimal test: one that achieves the minimum error probability for all states in the sets, matching the performance of the optimal test for the most challenging (worst-case) states. While the minimax theorem ensures the existence of a universal optimal test, it does not provide an explicit construction.
In this section, we explicitly construct a universal optimal test for binary composite hypotheses by analyzing the saddle point (or equilibrium point) of the minimax problem in more detail.
Before the main result, we first recall the minimax inequality and saddle points in the context of minimax optimization.
Consider a function \(f: X \times Y \to {{\mathbb{R}}}\) where \(X,Y \subseteq {{\mathscr{L}}}\) are nonempty subsets of linear operators, respectively. We always have the minimax inequalty \[\begin{align} \sup_{y \in Y} \inf_{x \in X} f(x,y) \leq \inf_{x \in X} \sup_{y \in Y} f(x,y). \end{align}\] If the equality holds, we call it minimax equality and the value on both sides are minimax value. The minimax equality is a very important property in optimization and game theory.
Definition 11. A pair of solutions \(x^* \in X\) and \(y^* \in Y\) is called a saddle point of \(f\) if \[\begin{align} f(x^*, y) \leq f(x^*, y^*) \leq f(x, y^*), \quad \forall x \in X, y \in Y. \end{align}\]
Remark 12. From the definition, it is clear that \((x^*, y^*)\) is a saddle point of \(f\) if and only if \(x^* \in X\), \(y^* \in Y\), and \[\begin{align} \label{eq:32saddle32point32minimax32remark} \sup_{y \in Y} f(x^*, y) = f(x^*, y^*) = \inf_{x \in X} f(x, y^*). \end{align}\qquad{(3)}\] That is, \(x^*\) minimizes against \(y^*\) and \(y^*\) maximizes against \(x^*\).
We also recall a standard characterization of saddle points from convex optimization; see, for example, [20]. We restate it here and present the proof for completeness.
Lemma 9. A pair of solutions \((x^*, y^*)\) is a saddle point of \(f\) if and only if the minimax equality holds and \(x^*\) is an optimal solution of the problem \[\begin{align} \label{eq:32saddle32point32minimax32tmp1} \min_{x \in X} \left(\sup_{y \in Y} f(x,y)\right), \end{align}\qquad{(4)}\] while \(y^*\) is an optimal solution of the problem \[\begin{align} \label{eq:32saddle32point32minimax32tmp2} \max_{y \in Y} \left(\inf_{x \in X} f(x,y)\right). \end{align}\qquad{(5)}\]
Proof. Suppose that \(x^*\) is an optimal solution of the problem ?? and \(y^*\) is an optimal solution of the problem ?? . Then we have \[\begin{align} \label{eq:32saddle32point32minimax32tmp3} \sup_{y \in Y} \inf_{x \in X} f(x,y) = \inf_{x \in X} f(x, y^*)\leq f(x^*,y^*) \leq \sup_{y \in Y} f(x^*,y) = \inf_{x \in X} \sup_{y \in Y} f(x,y), \end{align}\tag{11}\] where the two equalities follow from the optimality of \(x^*\) and \(y^*\). Therefore, if the minimax equality holds, then equality holds throughtout above, so that \[\begin{align} \sup_{y \in Y} f(x^*, y) = f(x^*, y^*) = \inf_{x \in X} f(x, y^*). \end{align}\] From Remark 12, we know that \((x^*, y^*)\) is a saddle point of \(f\).
Conversely, if \((x^*, y^*)\) is a saddle point of \(f\), then we have from Eq. ?? that \[\begin{align} \label{eq:32saddle32point32minimax32tmp4} \inf_{x \in X} \sup_{y \in Y} f(x,y) \leq \sup_{y \in Y} f(x^*, y) = f(x^*, y^*) = \inf_{x \in X} f(x, y^*) \leq \sup_{y \in Y} \inf_{x \in X} f(x,y). \end{align}\tag{12}\] Combined with the minimax inequality, we conclude the minimax equality. Therefore, equality holds throughtout above, so that \(x^*\) is an optimal solution of the problem ?? and \(y^*\) is an optimal solution of the problem ?? . ◻
To obtain saddle points, we may calculate the inner “sup” and “inf” functions appearing in the lemma, then minimize and maximize them, respectively, and obtain the corresponding sets of minima \(X^*\) and maxima \(Y^*\). If the optimal values are equal (i.e., minimax equality holds), the set of saddle points is \(X^* \times Y^*\). Otherwise, there are no saddle points [21].
While the standard approach to finding saddle points involves evaluating both sides of the minimax inequality, we provide an alternative method in the following. It allows for the construction of saddle points by optimizing only one side of the minimax problem, which will prove useful in the explicit construction of optimal tests in subsequent sections.
Lemma 10. Let \(f: X \times Y \to {{\mathbb{R}}}\) and \(g(y) = \inf_{x \in X} f(x,y)\). Suppose \(y^*\in Y\) is a maximizer of \(g(y)\), i.e., \(g(y^*) = \sup_{y \in Y} g(y)\), and \(x^* \in X\) is a minimizer of \(\inf_{x \in X} f(x,y^*)\), i.e., \(f(x^*,y^*) = \inf_{x \in X} f(x,y^*)\). If the minimax equality holds for function \(f\) and the optimization \(\inf_{x\in X}f(x,y^*)\) has a unique minimizer, then \((x^*, y^*)\) is a saddle point. Consequently, \(x^*\) is a minimizer of the optimization \(\inf_{x\in X}[\sup_{y \in Y} f(x,y)]\).
Proof. Let \(x^{**} \in X\) be a minimizer of \(\inf_{x \in X} [\sup_{y\in Y} f(x,y)]\). Then we know from Lemma 9 that \((x^{**}, y^*)\) is a saddle point of \(f\). By Eq. 12 (note that all equalities holds), any saddle point \((x^*, y^*)\) gives the same minimax value. So we have \[\begin{align} f(x^{**}, y^*) = f(x^*, y^*) = \inf_{x \in X} f(x, y^*). \end{align}\] As we assume that \(\inf_{x\in X} f(x,y^*)\) has a unique minimizer, we have \(x^{**} = x^*\). Therefore, \((x^*, y^*)\) is a saddle point of \(f\). ◻
We note that the uniqueness of the minimizer in \(\inf_{x \in X} f(x,y^*)\) is essential for Lemma 10 to hold. If the minimizer of \(\inf_{x \in X} f(x,y^*)\) is not unique, it is generally unclear whether any minimizer will yield a saddle point. In particular, it is not guaranteed that every minimizer of \(\inf_{x \in X} f(x,y^*)\) is also a minimizer of \(\inf_{x \in X}[\sup_{y \in Y} f(x,y)]\).
Lemma 11. Let \(X \in \mathscr{H}\) be a full rank Hermitian operator. Then the optimal solution to the semidefinite program \(\max_{0\leq M \leq I} \operatorname{Tr}[XM]\) is unique and is given by \(\{X \geq 0\}\).
Proof. Let \(X = \sum_{i=1}^d \lambda_i |\psi_i\rangle\langle\psi_i|\) be the spectral decomposition of \(X\), where \(\lambda_i > 0\) for \(i\in \{1,\cdots,k\}\) and \(\lambda_i < 0\) for \(i\in \{k+1,\cdots,d\}\). For any \(M\), let \(m_{ij} = \langle\psi_i|M|\psi_j\rangle\). Then we have \[\begin{align} \operatorname{Tr}[XM] = \sum_{i=1}^d \lambda_i \operatorname{Tr}[|\psi_i\rangle\langle\psi_i|M] = \sum_{i=1}^d \lambda_i m_{ii} = \sum_{i=1}^k \lambda_i m_{ii} + \sum_{i=k+1}^d \lambda_i m_{ii} \leq \sum_{i=1}^k \lambda_i, \end{align}\] where the equality holds if and only if \(m_{ii} = 1\) for \(i\in \{1,\cdots,k\}\) and \(m_{ii} = 0\) for \(i\in \{k+1,\cdots,d\}\). Denote the matrix form of \(M\) in the basis of \(\{|\psi_i\rangle\}_{i=1}^d\) as \[\begin{align} \begin{pmatrix} M_{11} & M_{12}\\ M_{21} & M_{22} \end{pmatrix} \end{align}\] where \(M_{11}\) is of size \(k\times k\) and \(M_{22}\) is of size \((d-k)\times (d-k)\). As \(M \geq 0\), we have \(M_{22} \geq 0\). Since the diagonal elements of \(M_{22}\) are all zeros, we have \(M_{22}=0\). This further implies that \(M_{12} = 0\) and \(M_{21} = 0\). So any optimal \(M\) is in the form of block diagonal matrix \[\begin{align} \begin{pmatrix} M_{11} & 0\\ 0 & 0 \end{pmatrix}, \end{align}\] where \(0\) above represents the zero matrix of suitable size. Since \(M \leq I\), we have \(M_{11}\leq I\). Moreover, since the dignoal elements of \(M_{11}\) are all ones, we can conclude that any offdiagonal elements of \(M_{11}\) are zeros. This is because any principle submatrix \[\begin{align} \begin{pmatrix} 1 & x \\ x^* & 1 \end{pmatrix} \leq \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \end{align}\] implies that \(x = 0\). This shows that the optimal solution is unique and is given by \(\sum_{i=1}^k |\psi_i\rangle\langle\psi_i| = \{X \geq 0\}\). This concludes the proof. ◻
With Lemma 10 and Lemma 11 in place, we are now prepared to explicitly construct the optimal test. The following result shows that, for binary composite hypotheses, the universal optimal measurement is given by the Holevo-Helstrom test corresponding to the pair of states from the two sets that achieve the minimal trace distance.
Theorem 3 (Optimal test for binary composite hypotheses). Let \({\cal H}\) be a finite-dimensional Hilbert space. Let \({{\mathscr{C}}}_1, {{\mathscr{C}}}_2 \subseteq \mathscr{D}({\cal H})\) be two convex, compact sets of quantum states and \(\{\pi_1, \pi_2\}\) be the prior probabilities. Suppose \((\rho_1^*, \rho_2^*) \in {{\mathscr{C}}}_1 \times {{\mathscr{C}}}_2\) is a minimizer of the convex optimization problem \(\inf_{\rho_1 \in {{\mathscr{C}}}_1,\, \rho_2 \in {{\mathscr{C}}}_2} \|\pi_1 \rho_1 - \pi_2 \rho_2\|_1\). Then, the optimal test for discriminating between \({{\mathscr{C}}}_1\) and \({{\mathscr{C}}}_2\) is given by the projection \(\{\pi_1 \rho_1^* - \pi_2 \rho_2^* \geq 0\}\), provided that \(\pi_1 \rho_1^* - \pi_2 \rho_2^*\) is full rank.
Proof. By Lemma 8, we have the minimax equality: \[\begin{align} \inf_{0\leq M \leq I} \sup_{\substack{\rho_1 \in {{\mathscr{C}}}_1\\ \rho_2 \in {{\mathscr{C}}}_2}}P_e(M, \rho_1, \rho_2) = \sup_{\substack{\rho_1 \in {{\mathscr{C}}}_1\\ \rho_2 \in {{\mathscr{C}}}_2}} \inf_{0\leq M \leq I} P_e(M, \rho_1, \rho_2). \end{align}\] Lemma 5 states that for any \(\rho_1, \rho_2\), \[\begin{align} \inf_{0\leq M \leq I} P_e(M, \rho_1, \rho_2) = \frac{1}{2} \left( 1 - \frac{1}{2} \|\pi_1 \rho_1 - \pi_2 \rho_2\|_1 \right). \end{align}\] Since \((\rho_1^*, \rho_2^*)\) is the minimizer of the convex optimization problem \(\inf_{\rho_1 \in {{\mathscr{C}}}_1,\, \rho_2 \in {{\mathscr{C}}}_2} \|\pi_1 \rho_1 - \pi_2 \rho_2\|_1\), it also maximizes \(\sup_{{\rho_1 \in {{\mathscr{C}}}_1, \rho_2 \in {{\mathscr{C}}}_2}} [\inf_{0\leq M \leq I} P_e(M, \rho_1, \rho_2)]\). Fixing \(\rho_1^*, \rho_2^*\), we have \[\begin{align} \inf_{0\leq M \leq I} P_e(M, \rho_1^*, \rho_2^*) = \pi_1 - \sup_{0\leq M \leq I} \operatorname{Tr}M [\pi_1 \rho_1^* - \pi_2 \rho_2^*]. \end{align}\] Since \(\pi_1 \rho_1^* - \pi_2 \rho_2^*\) is assumed to be full rank, Lemma 11 ensures that \(\{\pi_1 \rho_1^* - \pi_2 \rho_2^* \geq 0\}\) is the unique maximizer of \(\sup_{0\leq M \leq I} \operatorname{Tr}[M(\pi_1 \rho_1^* - \pi_2 \rho_2^*)]\). Therefore, it is also the unique minimizer of \(\inf_{0\leq M \leq I} P_e(M, \rho_1^*, \rho_2^*)\). Finally, by Lemma 10, \(\{\pi_1 \rho_1^* - \pi_2 \rho_2^* \geq 0\}\) is the minimizer of \(\inf_{0\leq M \leq I} [\sup_{{\rho_1 \in {{\mathscr{C}}}_1, \rho_2 \in {{\mathscr{C}}}_2}}P_e(M, \rho_1, \rho_2)]\). Therefore, it is the optimal test for \({{\mathscr{C}}}_1\) and \({{\mathscr{C}}}_2\). ◻
The maximum overlap between a pure state \(|\psi\rangle\) and a set of free states \({{\mathscr{F}}}\), \[\begin{align} O_{{{\mathscr{F}}}}(\psi):= \sup_{\sigma \in {{\mathscr{F}}}} \langle\psi| \sigma |\psi\rangle, \end{align}\] is a technical quantity that appears frequently in quantum resource theory [22]–[24]. Here, we provide an operational interpretation of this quantity as the optimal error exponent in symmetric hypothesis testing. This also, in turns, provides explicit examples where the quantum Chernoff divergence between sets of quantum states is not additive, thereby justifying the need for regularization of the quantum Chernoff divergence.
Theorem 4. Let \(|\psi\rangle\langle\psi|\) be a pure state and \({{\mathscr{F}}}\subseteq \mathscr{D}\) be a convex set of quantum states. Then \[\begin{align} C(|\psi\rangle\langle\psi|\|{{\mathscr{F}}}) = -\log O_{{{\mathscr{F}}}}(\psi). \end{align}\]
Proof. We have the following chain of equalities: \[\begin{align} C(|\psi\rangle\langle\psi|\|{{\mathscr{F}}}) & = \inf_{\sigma \in {{\mathscr{F}}}} C(|\psi\rangle\langle\psi|\|\sigma)\\ & = -\log \sup_{\sigma \in {{\mathscr{F}}}} \inf_{\alpha \in [0,1]} \operatorname{Tr}[|\psi\rangle\langle\psi|^{\alpha} \sigma^{1-\alpha}]\\ & = -\log \sup_{\sigma \in {{\mathscr{F}}}} \inf_{\alpha \in [0,1]} \operatorname{Tr}[|\psi\rangle\langle\psi|\sigma^{1-\alpha}]\\ & = -\log \sup_{\sigma \in {{\mathscr{F}}}} \operatorname{Tr}[|\psi\rangle\langle\psi|\sigma]\\ & = -\log O_{{{\mathscr{F}}}}(\psi), \end{align}\] where the first, second and last equalities follow from definitions, the fourth equality follows from the fact that \(\sigma^\alpha \geq \sigma^\beta\) if \(\alpha \leq \beta\). This concludes the proof. ◻
We now present explicit examples from several resource theories in which the maximum overlap with free states can be computed exactly. These values directly determine the optimal error exponent for quantum hypothesis testing between a pure state and a set of free states.
The maximum overlap with free states is a fundamental quantity in the resource theory of magic [25]. In this context, the set of free states \({{\mathscr{F}}}\) is typically chosen as the set of stabilizer states on \(n\) qubits, denoted \(\text{\rm STAB}_n\), which is stable under tensor product. The maximum overlap \(O_{\text{\rm STAB}}(\psi)\) is closely related to the stabilizer rank and stabilizer extent—key quantities in fault-tolerant quantum computation.4 For certain states, such as the magic T-state \(|T\rangle:= (|0\rangle + e^{i\pi/4}|1\rangle)/\sqrt{2}\), the maximum overlap can be computed explicitly [25]: \[\begin{align} O_{\text{\rm STAB}}(|T\rangle\langle T|^{\otimes m}) = (4-2\sqrt{2})^{-m}. \end{align}\] This leads to the quantum Chernoff divergence, \[\begin{align} C(|T\rangle\langle T|\|\text{\rm STAB}_1) = C^\infty(\{|T\rangle\langle T|^{\otimes n}\}_{n\in {{\mathbb{N}}}}\|\{\text{\rm STAB}_n\}_{n\in {{\mathbb{N}}}}) = \log(4-2\sqrt{2}). \end{align}\] It is also known from [25] that there exist quantum states for which the maximum overlap with stabilizer states is not multiplicative. Consequently, the quantum Chernoff divergence between two sets is not additive in general.
In the resource theory of coherence, the set of free states is the set of incoherent states \({\cal I}_n = \{\rho \in \mathscr{D}({\cal H}^{\otimes n}): \rho = \text{diag}(\rho)\}\), i.e., states diagonal in a fixed basis [26]–[28]. Let \(|\psi\rangle = \sum_{i=1}^d a_i |i\rangle\) and \(|\psi\rangle\langle\psi| \in \mathscr{D}({\cal H})\). Then, \[\begin{align} O_{{\cal I}_1}(|\psi\rangle\langle\psi|) & = \max_{\sigma \in {\cal I}_1} \langle\psi| \sigma |\psi\rangle = \max_{\sigma \in {\cal I}_1} \sum_{i=1}^d |a_i|^2 \sigma_{i} \end{align}\] where \(\sigma = \sum_{i=1}^d \sigma_{i} |i\rangle\langle i|\), \(\sigma_{i} \geq 0\), and \(\sum_{i=1}^d \sigma_{i} = 1\). Note that the objective function is an average of the probability vector \((|a_1|^2, |a_2|^2, \ldots, |a_d|^2)\). So it is no greater than \(\max_i |a_i|^2\). This value can be achieved by choosing \(\sigma_{i_{\max}}=1\) for \(i_{\max}=\arg\max_i |a_i|^2\) and \(\sigma_i=0\) otherwise. Therefore, we have \[\begin{align} O_{{\cal I}_1}(|\psi\rangle\langle\psi|) = \max_{i} |a_i|^2, \end{align}\] which is the infinity norm of the probability vector \((|a_1|^2, |a_2|^2, \ldots, |a_d|^2)\). This leads to the quantum Chernoff divergence \[\begin{align} C(|\psi\rangle\langle\psi|\|{\cal I}_1) = C^\infty(\{|\psi\rangle\langle\psi|^{\otimes n}\}_{n\in {{\mathbb{N}}}}\|\{{\cal I}_n\}_{n\in {{\mathbb{N}}}}) = -\log \max_{i} |a_i|^2. \end{align}\]
In the resource theory of entanglement, the standard resource is the maximally entangled state \(|\Phi_m\rangle := \frac{1}{\sqrt{m}} \sum_{i=1}^m |ii\rangle\) [29], [30]. The standard sets of free states are the separable states \(\text{\rm SEP}\) and the positively partial transpose states \(\text{\rm PPT}\), with the inclusion \(\text{\rm SEP}(A:B) \subseteq \text{\rm PPT}(A:B)\). The maximum overlap of \(|\Phi_m\rangle\) with these sets is given by [31]: \[\begin{align} O_\text{\rm SEP}(\Phi_m) = O_\text{\rm PPT}(\Phi_m) = \frac{1}{m}, \end{align}\] where the maximum is achieved, for example, by the product state \(|0\rangle\langle 0|_A \otimes |0\rangle\langle 0|_B\). Consequently, the quantum Chernoff divergence is \[\begin{align} C(\Phi_m\|\text{\rm SEP}(A:B)) & = C^\infty(\{\Phi_m^{\otimes n}\}_{n\in {{\mathbb{N}}}}\|\{\text{\rm SEP}(A^n:B^n)\}_{n\in {{\mathbb{N}}}}) = \log m,\\ C(\Phi_m\|\text{\rm PPT}(A:B)) & = C^\infty(\{\Phi_m^{\otimes n}\}_{n\in {{\mathbb{N}}}}\|\{\text{\rm PPT}(A^n:B^n)\}_{n\in {{\mathbb{N}}}}) = \log m. \end{align}\]
We have established a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states, thereby extending the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypothesis testing. Our main results demonstrate that the optimal asymptotic error exponent for discriminating between stable sequences of sets of quantum states is precisely characterized by the regularized Chernoff divergence between the sets. Additionally, we have provided explicit constructions of the optimal measurement for binary composite hypotheses and offered an operational interpretation of the maximum overlap with free states in quantum resource theories.
Several open questions and future directions remain. While the optimal exponent in the asymmetric (Stein’s) setting can be efficiently computed despite the need for regularization [32], it remains open whether efficient algorithms exist for computing the regularized Chernoff divergence in the symmetric setting. As noted in Remark 5, the Chernoff divergence can be efficiently computed for fixed \(n\), but the rate of convergence of the regularized Chernoff divergence as \(n\) increases is not well understood. Additionally, our construction of the optimal test for binary composite hypotheses assumes that the difference between the closest states is full rank. It would be interesting to determine whether this assumption can be relaxed, perhaps by appealing to continuity arguments or alternative analytical techniques.
Another direction concerns the equivalence between adaptive and nonadaptive strategies in adversarial quantum channel discrimination. It was shown in [33] that, in the asymmetric hypothesis testing setting, adaptive and nonadaptive strategies yield the same optimal error exponent. Whether this equivalence persists in the symmetric setting remains an open question. In particular, based on the quantum Chernoff bound established in this work, this question reduces to whether the regularized Chernoff divergence between sequences of sets of quantum states generated by adaptive and nonadaptive strategies coincide.
We thank Li Gao, Masahito Hayashi, Andre Milzarek, Milan Mosonyi and Ryuji Takagi for helpful discussions. K.F. is supported by the National Natural Science Foundation of China (grant No. 92470113 and 12404569), the Shenzhen Science and Technology Program (grant No. JCYJ20240813113519025), the Shenzhen Fundamental Research Program (grant No. JCYJ20241202124023031), the 1+1+1 CUHK-CUHK(SZ)-GDST Joint Collaboration Fund (grant No. GRDP2025-022), and the University Development Fund (grant No. UDF01003565).
kunfang@cuhk.edu.cn↩︎
We abuse the notation \({{\mathscr{C}}}_i\) to refer both to sets of quantum states and to sequences of such sets, depending on the context.↩︎
That is, any quantum state in the set remains in the set under any permutation of subsystems.↩︎
In [25], the maximum overlap is defined with respect to pure stabilizer states. However, since the objective function is linear, maximizing over the convex hull (i.e., all stabilizer states) yields the same value.↩︎