A quantum channel describes the most general physical transformation that quantum states can undergo. It captures not only idealized processes like unitary evolutions but also non-ideal, noisy, or dissipative processes, making them crucial for understanding real-world quantum systems. The task of quantum channel discrimination is to identify which member of a given set of quantum channels governs the physical process in a black-box scenario. It is pivotal to various quantum information tasks [1]–[12] and gives insights into a wide range of quantum protocols and applications, from quantum foundations (e.g., exploring the quantum advantage of entanglement [13]–[16]) to quantum communication (e.g., estimating quantum channel capacity [17]–[22]), quantum sensing (e.g., quantum reading of classical data and quantum illumination of targets [23], [24]) and even quantum biology [25], [26].

In this work, we introduce an operationally-motivated adversarial framework for quantum channel discrimination, where a tester interacts with an untrusted quantum device that generates quantum states upon request. Such an adversarial setting arises naturally in other quantum information tasks such as quantum key distribution [27], [28], quantum adversarial learning [29], quantum interactive proofs [30] and quantum state verification [31], where ensuring reliable performance requires resilience against adversarial behavior. The task of adversarial discrimination was previously studied in the classical case by [32]. Here, we extend the investigation to the general quantum scenario ¹. Unlike the conventional channel discrimination model—where inputs are typically fixed or optimized by the tester to maximize distinguishability—the adversarial setting addresses the worst-case scenario where an unknown or malicious agent may manipulate the input to minimize the tester’s ability to distinguish between channels. This can be seen as the problem of discriminating between two quantum devices by observing the output on an untrusted input.

Technically, the discrimination task is formulated as a hypothesis testing problem, with the goal of finding discrimination strategies that give the optimal trade-off between two kinds of error probabilities, namely the probabilities of false detection (type-I error) and false rejection (type-II error) [33], [34]. The challenge in adversarial discrimination arises from the potential capabilities of the adversary, who may access the channel’s environmental system—the ancillary quantum system that interacts with the main system and can cause information leakage—or possess (potentially unbounded) quantum memory that stores partial information from previous rounds and uses it in subsequent rounds to enable adaptive strategies. This raises a central question: how effectively can the tester distinguish between two quantum channels while playing against such an adversary?

This work provides a complete answer to the above question in the context of asymmetric hypothesis testing. Specifically, we show that in adversarial quantum channel discrimination, the optimal type-II error decays exponentially at a rate characterized by a new notion of quantum channel divergence, termed the minimum output channel divergence, provided the type-I error remains within a fixed threshold. This result serves as a direct analog of the quantum Stein’s lemma in the adversarial channel discrimination and it is proved by new chain rules for quantum relative entropies. Notably, the optimal error exponent can be achieved via simple non-adaptive strategies by the adversary, and desirable mathematical properties such as computational efficiency and the strong converse property hold in general. Extending the adversarial discrimination framework and using the chain rules, we derive a relative entropy accumulation theorem, which states that the hypothesis testing relative entropy between two sequential processes of quantum channels can be lower bounded by the sum of the regularized minimum output channel divergences. A special case of this theorem recovers a version of the entropy accumulation theorem (a foundational tool in quantum cryptography [35]–[37]).

0.0.0.1 Minimum output channel divergence.—

Let \(\mathscr{D}(A)\), \(\mathscr{L}(A)\) and \(\mathscr{H}_{{\scalebox{0.7}{\rm +}}}(A)\) denote the set of density operators, linear operators and positive semidefinite operators on a finite-dimensional Hilbert space \({\cal H}_A\), respectively. The set of completely positive maps from \(\mathscr{L}(A)\) to \(\mathscr{L}(B)\) is denoted by \(\text{\rm CP}(A\!:\!B)\). A quantum channel \({\cal N}_{A\to B}\) is a linear map from \(\mathscr{L}(A)\) to \(\mathscr{L}(B)\) that is both completely positive and trace-preserving. The set of all such maps is denoted by \(\text{\rm CPTP}(A\!:\!B)\). A quantum divergence is a functional \({{\mathbb{D}}}:\mathscr{D}\times \mathscr{H}_{{\scalebox{0.7}{\rm +}}}\to {{\mathbb{R}}}\) that satisfies the data-processing inequality, characterizing the “distinguishability” or “distance” between two quantum states. For any subsets \({{\mathscr{A}}}\subseteq \mathscr{D}\) and \({{\mathscr{B}}}\subseteq \mathscr{H}_{{\scalebox{0.7}{\rm +}}}\), their divergence is defined as \({{\mathbb{D}}}({{\mathscr{A}}}\|{{\mathscr{B}}}):= \inf\{{{\mathbb{D}}}(\rho\|\sigma): \rho \in {{\mathscr{A}}}, \sigma \in {{\mathscr{B}}}\}\), which represents the minimum “distance” between the sets.

Let \({\cal N}\in \text{\rm CPTP}(A\!:\!B)\) and \({\cal M}\in \text{\rm CP}(A\!:\!B)\). We introduce the minimum output channel divergence as \[\begin{align} {{\mathbb{D}}}^{\inf}({\cal N}\|{\cal M}) := \inf \left\{ {{\mathbb{D}}}({\cal N}(\rho)\| {\cal M}(\sigma)): \rho, \sigma \in \mathscr{D}(A)\right\}, \end{align}\] capturing the worst-case distinguishability of the channels under optimized test states. It is also useful to see this as a divergence between two sets of quantum states, \({{\mathbb{D}}}^{\inf}({\cal N}\|{\cal M}) = {{\mathbb{D}}}({\cal N}(\mathscr{D})\|{\cal M}(\mathscr{D}))\), where \({\cal L}(\mathscr{D}) := \{{\cal L}(\rho): \rho \in \mathscr{D}\}\) denotes the image set of \(\mathscr{D}\) under a linear map \({\cal L}\). The regularized channel divergence is defined as \[\begin{align} {{\mathbb{D}}}^{\inf,\infty}({\cal N}\|{\cal M}) := \lim_{n \to \infty} \frac{1}{n} {{\mathbb{D}}}^{\inf}({\cal N}^{\otimes n}\|{\cal M}^{\otimes n}), \end{align}\] which accounts for the asymptotic behavior of the channel divergence over multiple uses.

0.0.0.2 Adversarial quantum channel discrimination.—

Consider a scenario where a tester interacts with an untrusted quantum device that generates quantum states upon request. The device guarantees that the states are produced by either a quantum channel \({\cal N}\) or a quantum channel \({\cal M}\). The tester is allowed to request multiple samples from the device and perform measurements to distinguish between the two cases. More formally, let \({\cal N}_{A\to B}\) and \({\cal M}_{A\to B}\) be the two quantum channels to distinguish, and let \(U_{A\to BE}\) and \(V_{A\to BE}\) be their Stinespring dilations, respectively, with \(E\) denoting the environmental system. These channels satisfy the relations that \({\cal N}_{A\to B} = \operatorname{Tr}_E \circ \,{\cal U}_{A\to BE}\) and \({\cal M}_{A\to B} = \operatorname{Tr}_E \circ \,{\cal V}_{A\to BE}\), where \({\cal U}_{A\to BE}(\cdot) := U (\cdot) U^\dagger\) and \({\cal V}_{A\to BE}(\cdot) := V (\cdot) V^\dagger\), and \(\operatorname{Tr}_E\) denotes the partial trace over the environment.

An adaptive strategy for adversarial quantum channel discrimination proceeds as follows (see Figure 1 (a)). Suppose the device operates according to the channel \({\cal N}\). Initially, the adversary prepares a quantum state using a channel \({\cal P}^1 \in \text{\rm CPTP}(R_0E_0\!:\!A_1R_1)\), where \(|R_0| = |E_0| = 1\), and sends system \(A_1\) through the channel \({\cal U}\), generating the output state \({\cal U}\circ {\cal P}^1\) and returning the system \(B_1\) to the tester. In the next round, the adversary performs an internal update \({\cal P}^2 \in \text{\rm CPTP}(E_1R_1\!:\!A_2R_2)\), leveraging information stored in the quantum memory \(R_1\) and the environmental system \(E_1\) from the previous round. The adversary sends system \(A_2\) through the channel, producing the output state \({\cal U}\circ {\cal P}^2 \circ {\cal U}\circ {\cal P}^1\) and returning the system \(B_2\) to the tester. This process is repeated for \(n\) rounds. Finally, the tester performs a quantum measurement \(\{M_n, I-M_n\}\) on the collected systems \(B_1 \cdots B_n\) in their possession to determine which channel was used to prepare the states.

After \(n\) rounds of state generation, the tester obtains an overall state on the systems \(B_1\cdots B_n\) as: \[\begin{align} \rho[\{{\cal P}^i\}_{i=1}^n] := \operatorname{Tr}_{R_nE_n} \prod_{i=1}^{n} {\cal U}_{A_i \to B_i E_i} \circ {\cal P}^i_{R_{i-1}E_{i-1} \to A_{i}R_{i}}. \end{align}\] Similarly, if the device is governed by \({\cal M}\) and the internal operations by the adversary are given by \({\cal Q}^i\), then the overall state on system \(B_1\cdots B_n\) is given by \[\begin{align} \sigma[\{{\cal Q}^i\}_{i=1}^n] := \operatorname{Tr}_{R_nE_n} \prod_{i=1}^{n} {\cal V}_{A_i \to B_i E_i} \circ {\cal Q}^i_{R_{i-1}E_{i-1} \to A_{i}R_{i}}. \end{align}\] Due to limited knowledge of the internal workings of the device, the tester only has access to partial information, knowing that their state belongs to either: \({{\mathscr{A}}}_n := \{\rho[\{{\cal P}^i\}_{i=1}^n] : {\cal P}^i \in \text{\rm CPTP}(R_{i-1}E_{i-1}\!:\!A_iR_i), \forall R_i, \forall i\}\) or \({{\mathscr{B}}}_n := \{\sigma[\{{\cal Q}^i\}_{i=1}^n] : {\cal Q}^i \in \text{\rm CPTP}(R_{i-1}E_{i-1}\!:\!A_iR_i), \forall R_i, \forall i\}\) where the adversary’s internal memory \(R_i\) can have arbitrarily large dimensions.

A non-adaptive strategies is a subclass of adversarial strategies that disregards the environmental systems \(E_i\) and performs no updates between rounds (see Figure 1 (b)). Here, the operations \({\cal P}_i, {\cal Q}_i\) (\(i \geq 2\)) are simply identity maps, with \(R_i = A_{i+1} \cdots A_n\). The sets of all possible outputs are denoted by \({{\mathscr{A}}}_n'\) and \({{\mathscr{B}}}_n'\), respectively, which also correspond to the images of the tensor product channels, \({{\mathscr{A}}}_n' = {\cal N}^{\otimes n}(\mathscr{D})\) and \({{\mathscr{B}}}_n' = {\cal M}^{\otimes n}(\mathscr{D})\).

The adversarial quantum channel discrimination is essentially to distinguish between the two sets \({{\mathscr{A}}}_n\) (\({{\mathscr{A}}}_n'\)) and \({{\mathscr{B}}}_n\) (\({{\mathscr{B}}}_n'\)). Define the type-I error \(\alpha({{\mathscr{A}}}_n, M_n) : = \sup_{\rho_n \in {{\mathscr{A}}}_n} \operatorname{Tr}[\rho_n (I-M_n)]\) and type-II error \(\beta({{\mathscr{B}}}_n, M_n) : = \sup_{\sigma_n \in {{\mathscr{B}}}_n} \operatorname{Tr}[\sigma_n M_n]\). Asymmetric hypothesis testing investigates the decay of the optimal exponent of the type-II error probability when the type-I error is within a constant threshold \(\varepsilon\), that is, to evaluate: \[\begin{align} \beta_{n, \varepsilon}({\cal N}\|{\cal M}) := \inf_{0\leq M_n\leq I}\left\{\beta({{\mathscr{B}}}_n, M_n): \alpha({{\mathscr{A}}}_n, M_n) \leq \varepsilon\right\}. \end{align}\]

Our main result, termed adversarial quantum Stein’s lemma in Theorem 1, establishes that this optimal exponent is precisely characterized by the regularized minimum output channel divergence induced by the Umegaki relative entropy [38] \(D(\rho\|\sigma):= \operatorname{Tr}[\rho(\log \rho - \log \sigma)]\) if \({\operatorname{supp}}(\rho) \subseteq {\operatorname{supp}}(\sigma)\) and \(+\infty\) otherwise.

The minimum output channel divergence \(D^{\inf}({\cal N}\|{\cal M})\) is not additive in general, necessitating the use of the regularized limit in the Stein’s exponent \(D^{\inf,\infty}({\cal N}\|{\cal M})\) [39]. While this could make its estimation challenging, the fact that the optimal exponent can be achieved by non-adaptive strategies simplifies the problem. As the sets \({{\mathscr{A}}}_n'\) and \({{\mathscr{B}}}_n'\) from non-adaptive strategies fall within the framework of the generalized asymptotic equipartition property (AEP) in [40], the Stein’s exponent \(D^{\inf,\infty}({\cal N}\|{\cal M})\) can be approximated within an additive error \(\delta\) by a quantum relative entropy program of size \(O((l+1)^k)\), where \(k = \max\{|A|^2, |B|^2\}\) is given by the channel dimensions, and \(l = \lceil \frac{8|B|^2}{\delta} \log \frac{|B|^2}{\delta} \rceil\) relates to the expected accuracy. Further details on this computational aspect can be found in the accompanying paper [39].

It is also worth noting that Eq. ?? established above universally applies to any \(\varepsilon\in (0,1)\), thereby demonstrating the “strong converse property”, a desirable mathematical property in information theory [41] that delineates a sharp boundary for the tradeoff between the Type-I and Type-II errors in the asymptotic regime. That is, any discrimination strategy achieving a Type-II error decay rate exceeding the Stein’s exponent \(D^{\inf,\infty}({\cal N}\|{\cal M})\) will necessarily result in the Type-I error converging to one in the asymptotic limit. While this property has been proven for the discrimination between two quantum states [34], it remains an open question for discriminating two quantum channels in the best-case scenario [22]. Our result gives an answer in the worst-case scenario and contains quantum states [33], [34] as a special case when the channels are taken as replacer channels.

0.0.0.3 Chain rules and proof outline for Theorem 1.—

The proof of Theorem 1 relies on new chain rules of quantum relative entropies. For classical probability distributions \(P_{XY}\) and \(Q_{XY}\), the chain rule [41] states that \(D(P_{XY}\|Q_{XY}) = D(P_X\|Q_X) + \sum_x P_X(x) D(P_{Y|X=x}\|Q_{Y|X=x})\), where \(D\) is the Kullback-Leibler divergence [42]. While no exact quantum analog exists, inequalities such as \(D(P_{XY}\|Q_{XY}) \leq D(P_X\|Q_X) + \max_{x} D(P_{Y|X=x}\|Q_{Y|X=x})\) have been extended to quantum settings for Umegaki [11] and Belavkin-Staszewski relative entropies [21]. Here, we establish a reverse inequality, providing the first quantum analog for \(D(P_{XY}\|Q_{XY}) \geq D(P_X\|Q_X) + \min_{x} D(P_{Y|X=x}\|Q_{Y|X=x})\).

The measured relative entropy is defined as [33], [43] \(D_{{\scriptscriptstyle \rm M}} (\rho\|\sigma) := \sup_{({\cal X},M)} D(P_{\rho,M}\|P_{\sigma,M})\), where the supremum is over all finite sets \({\cal X}\) and POVMs \(M = \{M_x\}_{x \in {\cal X}}\) satisfying \(M_x \geq 0\) and \(\sum_{x \in {\cal X}} M_x = I\), and \(P_{\rho,M}(x) := \operatorname{Tr}[M_x \rho]\). The measured Rényi divergence is [44] \(D_{{\scriptscriptstyle \rm M}, \alpha} (\rho\|\sigma) := \sup_{({\cal X},M)} D_{\alpha}(P_{\rho,M}\|P_{\sigma,M})\), where \(D_{\alpha}\) is the classical Rényi divergence. The sandwiched Rényi divergence is [45], [46] \(D_{{\scriptscriptstyle \rm S},\alpha}(\rho\|\sigma) := \frac{1}{\alpha-1}\log\operatorname{Tr}[\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}]^\alpha\), if \({\operatorname{supp}}(\rho) \subseteq {\operatorname{supp}}(\sigma)\), and \(+\infty\) otherwise. Then we have the chain rules as follows.

These chain rules can be seen as an enhancement of the data processing inequality under partial trace. Notably, they are also tight in the sense that \(\inf_{\rho,\sigma \in \mathscr{D}(RA)} \big[D_{{\scriptscriptstyle \rm S},\alpha}(\rho^{\cal N}_{RB}\|\sigma^{\cal M}_{RB}) - D_{{\scriptscriptstyle \rm S},\alpha}(\rho_R\|\sigma_R)\big] = D_{{\scriptscriptstyle \rm S},\alpha}^{\inf,\infty}({\cal N}\|{\cal M})\). That is, the amortized channel divergence coincides with the regularized divergence, serving as an analog to [11] in the worst-case scenario. Detailed proofs of the chain rules and their tightness are provided in the Supplemental Materials.

Let \(D_{{\scriptscriptstyle \rm H},\varepsilon}(\rho\|\sigma):=-\log\{\operatorname{Tr}[\sigma M]: \operatorname{Tr}[\rho(I-M)] \leq \varepsilon, 0 \leq M \leq I\}\) be the quantum hypothesis testing relative entropy. The proof for Theorem 1 then contains two parts. The converse part uses the chain rules to show that \(\liminf_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n) \geq D^{\inf,\infty}({\cal N}\|{\cal M})\) while the achievable part shows that \(\limsup_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n'\|{{\mathscr{B}}}_n') \leq D^{\inf,\infty}({\cal N}\|{\cal M})\) by applying the generalized AEP in [40] to \({{\mathscr{A}}}_n'\) and \({{\mathscr{B}}}_n'\).

1) Proof of the converse part. The first step is to lower bound \(D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n)\) by applying the chain rule recursively. Denote the joint states before the \(n\)-th use of the channel by \(\rho'_n := \operatorname{Tr}_{R_n}\circ\;{\cal P}^{n}\circ \prod_{i=1}^{n-1} {\cal U}\circ {\cal P}^i\) and \(\sigma'_n := \operatorname{Tr}_{R_n}\circ\;{\cal Q}^{n}\circ \prod_{i=1}^{n-1} {\cal V}\circ {\cal Q}^i\). Denote \(\rho_n := \rho[\{{\cal P}^i\}_{i=1}^n]\) and \(\sigma_n := \sigma[\{{\cal Q}^i\}_{i=1}^n]\). Then we have that \(\rho_n = \operatorname{Tr}_{R_nE_n} \circ \,{\cal U}(\rho'_n) = {\cal N}(\rho_n')\) and \(\sigma_n = \operatorname{Tr}_{R_nE_n} \circ \,{\cal V}(\sigma_n') = {\cal M}(\sigma_n')\). Note that for any \({\cal P}^n\) and \({\cal Q}^n\) we always have \(\operatorname{Tr}_{A_nR_n} \circ \, {\cal P}^{n} = \operatorname{Tr}_{E_{n-1}R_{n-1}}\) and \(\operatorname{Tr}_{A_nR_n} \circ \, {\cal Q}^{n} = \operatorname{Tr}_{E_{n-1}R_{n-1}}\). This gives the relations that \(\operatorname{Tr}_{A_n}(\rho_n') = \rho_{n-1}\) and \(\operatorname{Tr}_{A_n}(\sigma_n') = \sigma_{n-1}\). Applying the chain rule in Lemma 1 to \(\rho_n\) and \(\sigma_n\), we have \[\begin{align} D_{{\scriptscriptstyle \rm S},\alpha}(\rho_n\|\sigma_n)\geq D_{{\scriptscriptstyle \rm S},\alpha}(\rho_{n-1}\|\sigma_{n-1}) + D^{\inf,\infty}_{{\scriptscriptstyle \rm S},\alpha}({\cal N}\|{\cal M}). \end{align}\] Recursively applying this relation \(n\) times, we get \[\begin{align} D_{{\scriptscriptstyle \rm S},\alpha}(\rho_n\|\sigma_n) \geq n D^{\inf,\infty}_{{\scriptscriptstyle \rm S},\alpha}({\cal N}\|{\cal M}). \end{align}\] As this holds for any operations \({\cal P}^i\) and \({\cal Q}^i\), we get \[\begin{align} D_{{\scriptscriptstyle \rm S},\alpha}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n) \geq n D^{\inf,\infty}_{{\scriptscriptstyle \rm S},\alpha}({\cal N}\|{\cal M}). \end{align}\]

Using the relation of hypothesis testing relative entropy and the Petz Rényi divergence in [47] and the fact that Petz Rényi divergence is no smaller than the sandwiched Rényi divergence [48], we have for any \(\alpha \in [1/2,1)\) and \(\varepsilon\in (0,1)\), that \[\begin{align} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n) \geq D_{{\scriptscriptstyle \rm S},\alpha}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n) + (\log (1/\varepsilon))\alpha/(\alpha-1). \end{align}\] Combining the above relations, we have \[\begin{align} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n) \geq n D^{\inf,\infty}_{{\scriptscriptstyle \rm S},\alpha}({\cal N}\|{\cal M}) + (\log (1/\varepsilon))\alpha/(\alpha-1). \end{align}\] Taking the limits on both sides, we get \[\begin{align} \liminf_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n) \geq \sup_{\alpha \in [1/2,1)}D^{\inf,\infty}_{{\scriptscriptstyle \rm S},\alpha}({\cal N}\|{\cal M}). \end{align}\] Finally, note that \(\sup_{\alpha \in [1/2,1)} D^{\inf,\infty}_{{\scriptscriptstyle \rm S},\alpha}({\cal N}\|{\cal M}) \geq \sup_{\alpha \in [1/2,1)} D^{\inf,\infty}_{{\scriptscriptstyle \rm M},\alpha}({\cal N}\|{\cal M}) = D^{\inf,\infty}({\cal N}\|{\cal M})\), where the inequality follows as \(D_{{\scriptscriptstyle \rm M},\alpha}(\rho\|\sigma) \leq D_{{\scriptscriptstyle \rm S},\alpha}(\rho\|\sigma)\) for \(\alpha \in [1/2,1)\), and the equality is a consequence of [40], applied to the image sets of the channels. This concludes the proof of the converse part.

2) Proof of the achievable part. We check that the sets \(\{{{\mathscr{A}}}_n'\}_n\) and \(\{{{\mathscr{B}}}_n'\}_n\) meet all the structural assumptions to apply the generalized AEP in [40]. First, the set of all density matrices \(\mathscr{D}\) is convex and compact, so \({{\mathscr{A}}}_n'\) is also convex and compact. Since \({\cal N}^{\otimes n}\) and \(\mathscr{D}\) are permutation invariant, we know that \({{\mathscr{A}}}_n'\) is also permutation invariant. For any \({\cal N}^{\otimes m}(\rho_m) \in {{\mathscr{A}}}_m'\) and \({\cal N}^{\otimes k}(\rho_k) \in {{\mathscr{A}}}_k'\), we have \({\cal N}^{\otimes m}(\rho_m) \otimes{\cal N}^{\otimes k}(\rho_k) = {\cal N}^{\otimes(m+k)}(\rho_m\otimes\rho_k)\in {{\mathscr{A}}}_{m+k}'\). This implies \({{\mathscr{A}}}_m' \otimes{{\mathscr{A}}}_k' \subseteq {{\mathscr{A}}}_{m+k}'\). The support function of \({{\mathscr{A}}}_n'\) is given by \(h_{{{\mathscr{A}}}'_n}(X_n) := \sup_{\rho_n \in \mathscr{D}} \operatorname{Tr}\left[X_n {\cal N}^{\otimes n}(\rho_n) \right]= \sup_{\rho_n \in \mathscr{D}} \operatorname{Tr}\left[({\cal N}^{\otimes n})^\dagger (X_n) \rho_n \right] = \lambda_{\max}\left(({\cal N}^{\otimes n})^\dagger (X_n)\right)\) where \(\lambda_{\max}(\cdot)\) denotes the maximum eigenvalue. Therefore, for any \(X_m \in \mathscr{H}_{{\scalebox{0.7}{\rm +}}}\) and \(X_k\in \mathscr{H}_{{\scalebox{0.7}{\rm +}}}\), we have \(h_{{{\mathscr{A}}}'_{m+k}}(X_m \otimes X_k) = h_{{{\mathscr{A}}}'_{m}}(X_m)h_{{{\mathscr{A}}}'_{k}}(X_k)\), by the multiplicativity of the maximum eigenvalue under tensor product. This proves that \(\{{{\mathscr{A}}}'_n\}_{n\in {{\mathbb{N}}}}\) satisfy all the required assumptions, and the same argument works for \(\{{{\mathscr{B}}}_n'\}_{n\in {{\mathbb{N}}}}\). This gives \(\limsup_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n'\|{{\mathscr{B}}}_n') = D^{\inf,\infty}({\cal N}\|{\cal M})\) from [40] and proves the achievable part.

0.0.0.4 Relative entropy accumulation.—

The entropy accumulation theorem [49], [50] is a technique to find bounds on the operationally relevant uncertainty present in the outputs of a sequential process as a sum of the worst case uncertainties of each step. It has been widely used in quantum cryptography [35]–[37]. More specifically, a variant for the max-entropy \(H_{\max}^{\varepsilon}\) states that [50] for channels \({\cal N}_{i} \in \text{\rm CPTP}(Y_{i-1}\!:\! Y_{i} S_i C_i)\), we have \(H_{\max}^{\varepsilon}(S_1 \dots S_n | C_1 \dots C_n)_{{\cal N}_n \circ \cdots \circ {\cal N}_1(\rho_{Y_0})} \leq \sum_{i=1}^n \sup_{\omega_{Y_{i-1}}} H(S_i |C_i)_{{\cal N}_i(\omega)} + O(\sqrt{n})\), where \(H_{\max}^{\varepsilon}(S|C)_{\rho} \approx -\inf_{\sigma \in \mathscr{D}(C)} D_{{\scriptscriptstyle \rm H}, \varepsilon}(\rho_{SC} \| I_{S} \otimes \sigma_C)\)² and \(H(S|C)_{\rho} = -\inf_{\sigma \in \mathscr{D}(C)} D(\rho_{SC} \| I_{S} \otimes \sigma_C)\). This naturally raises the question of whether such a statement can be generalized to divergences between arbitrary sequential processes of channels, rather than being restricted only to entropies (which corresponds to choosing \({\cal M}\) in Figure 1 to be a replacer channel). This was first asked as an open question in [50] for the max-relative entropy. By extending the adversarial channel discrimination framework in Figure 1 to allow different channels \({\cal N}_i\) and \({\cal M}_i\) at each round, we can unify the relative entropy accumulation within this broader framework. This new perspective enables us to establish a relative entropy accumulation theorem for \(D_{{\scriptscriptstyle \rm H}, \varepsilon}\) (a smoothed form of the min-relative entropy), giving an answer to the dual formulation of this open question. Specifically, we generalize the converse part of Theorem 1 in two ways: we allow the channels applied at different steps to vary, and we compute explicit finite-size bounds.

The proof makes important use of the chain rules in Lemma 1. Moreover, choosing the channels \({\cal M}_i\) to be replacer channels, we recover a slightly weaker version of the \(H_{\max}^{\varepsilon}\) entropy accumulation statement previously mentioned. We leave it as an open question for future work whether this new proof technique can lead to better entropy accumulation theorems.

0.0.0.5 Discussion.—

We introduced the adversarial quantum channel discrimination and established a direct analog of quantum Stein’s lemma in this new setting. Notably, the optimal error exponent can be efficiently computed despite its regularization, and the strong converse property holds in general. Moreover, the equivalence of adaptive and non-adaptive strategies by the adversary provides significant insight for the tester, as their optimal measurements can be performed as if playing against a much weaker non-adaptive adversary. We also derived a relative entropy accumulation theorem, extending the existing entropy accumulation theorem from entropy to divergence and providing a solution to the dual formulation of the open problem presented in [50]. On the technical side, we introduced the minimum output channel divergence and established its chain rules. This tool has recently been used in [51] for the security analysis of quantum cryptographic protocols. Given the fundamental importance of quantum channel discrimination in quantum information, we anticipate that the new framework and tools developed in this work will open new directions for future investigations.

In this Supplemental Material, we provide more detailed expositions, proofs and discussions of the results in the main text. We may reiterate some of the steps to ensure that the Supplemental Material are explicit and self-contained.

1 Preliminaries↩︎

1.1 Notations and quantum divergences↩︎

Let \(\mathscr{D}(A)\) denote the set of all density operators on a finite-dimensional Hilbert space \({\cal H}_A\). Let \(\mathscr{L}(A)\) represent the set of all linear operators. Let \(\mathscr{H}(A)\), \(\mathscr{H}_{{\scalebox{0.7}{\rm +}}}(A)\) and \(\mathscr{H}_{{\scalebox{0.7}{\rm +}}{\scalebox{0.7}{\rm +}}}(A)\) be the set of all hermitian operators, positive semidefinite operators and positive definite operators on \({\cal H}_A\), respectively. The set of completely positive maps from \(\mathscr{L}(A)\) to \(\mathscr{L}(B)\) is denoted by \(\text{\rm CP}(A\!:\!B)\). A quantum channel \({\cal N}_{A\to B}\) is a linear map from \(\mathscr{L}(A)\) to \(\mathscr{L}(B)\) that is both completely positive and trace-preserving. The set of all such maps is denoted by \(\text{\rm CPTP}(A\!:\!B)\). A quantum divergence is a functional \({{\mathbb{D}}}:\mathscr{D}\times \mathscr{H}_{{\scalebox{0.7}{\rm +}}}\to {{\mathbb{R}}}\) that satisfies the data-processing inequality, characterizing the “distinguishability” or “distance” between two quantum states. There are a few quantum divergences used throughout this work.

When \(\alpha \to 1\), \(D_{{\scriptscriptstyle \rm S},\alpha}\) converge to the Umegaki relative entropy [45], [46], \[\begin{align} \label{eq:32state32Renyi32continuous} \lim_{\alpha \to 1} D_{{\scriptscriptstyle \rm S},\alpha}(\rho\|\sigma) = D(\rho\|\sigma). \end{align}\tag{1}\] When \(\alpha \to \infty\), the sandwiched Rényi divergence converges to the max-relative entropy [52], [53], \[\begin{align} \label{eq:32definition32of32Dmax} \lim_{\alpha \to \infty} D_{{\scriptscriptstyle \rm S},\alpha}(\rho\|\sigma) = D_{\max}(\rho\|\sigma):= \log\inf\big\{t \in {{\mathbb{R}}}\;:\; \rho \leq t\sigma \big\}\;, \end{align}\tag{2}\] if \({\operatorname{supp}}(\rho) \subseteq {\operatorname{supp}}(\sigma)\) and \(+\infty\) otherwise.

When \(\alpha \to 1\), the measured Rényi divergence converges to the measured relative entropy.

1.2 Minimum output channel divergence↩︎

The quantum divergence between two quantum states can be naturally extended to two sets of quantum states.

It is also useful to see this as a divergence between two sets of quantum states, expressed as \[\begin{align} {{\mathbb{D}}}^{\inf}({\cal N}\|{\cal M}) = {{\mathbb{D}}}({\cal N}(\mathscr{D})\|{\cal M}(\mathscr{D})), \end{align}\] where \({\cal L}(\mathscr{D}) := \{{\cal L}(\rho): \rho \in \mathscr{D}\}\) denotes the image set of \(\mathscr{D}\) under the linear map \({\cal L}\). We also define the regularized channel divergence as \[\begin{align} {{\mathbb{D}}}^{\inf,\infty}({\cal N}\|{\cal M}) := \lim_{n \to \infty} \frac{1}{n} {{\mathbb{D}}}^{\inf}({\cal N}^{\otimes n}\|{\cal M}^{\otimes n}), \end{align}\] which accounts for the asymptotic behavior of the channel divergence over multiple uses.

2 Chain rules↩︎

2.1 Proof of the chain rules↩︎

We have the following chain rules for measured Rényi divergences and sandwiched Rényi divergences, which can also be seen as an enhancement of the data processing inequality under partial trace. The proof of these chain rules requires the notion of (reverse) polar sets.

Proof.  The proof utilizes the superadditivity of the divergence between two sets of quantum states, as established in [40]. To this end, we consider the following sets: \[\begin{align} {3} {{\mathscr{A}}}_1 & = \{\rho_R\}, \quad {{\mathscr{A}}}_2 &= {\cal N}(\mathscr{D}), \quad {{\mathscr{A}}}_3 & = \{\rho_{RB}^{{\cal N}}\},\\ {{\mathscr{B}}}_1 & = \{\sigma_R\}, \quad {{\mathscr{B}}}_2 & = {\cal M}(\mathscr{D}), \quad {{\mathscr{B}}}_3 & = \{\sigma_{RB}^{\cal M}\}, \end{align}\] and verify that they meet the required assumptions. We do this for \(\{{{\mathscr{A}}}_1, {{\mathscr{A}}}_2, {{\mathscr{A}}}_3\}\) and the same argument works for \(\{{{\mathscr{B}}}_1, {{\mathscr{B}}}_2, {{\mathscr{B}}}_3\}\). For any \(Y_B \in {({{\mathscr{A}}}_{2})}_{{\scalebox{0.7}{\rm +}}}^{\circ}\), we have \(\operatorname{Tr}[Y_B {\cal N}(\rho)] \leq 1\) for any \(\rho \in \mathscr{D}(A)\). This implies that \({\cal N}^\dagger(Y_B) \leq I_A\), with \({\cal N}^\dagger\) being the adjoint map of \({\cal N}\). Therefore, for any \(X_R \in {({{\mathscr{A}}}_{1})}_{{\scalebox{0.7}{\rm +}}}^{\circ}\) and \(Y_B \in {({{\mathscr{A}}}_{2})}_{{\scalebox{0.7}{\rm +}}}^{\circ}\), we have the following equations, \[\begin{align} \operatorname{Tr}[(X_R \otimes Y_B) {\cal N}_{A\to B}(\rho_{RA})] & = \operatorname{Tr}[(X_R \otimes{\cal N}^\dagger(Y_B)) \rho_{RA}]\\ & \leq \operatorname{Tr}[(X_R \otimes I_A) (\rho_{RA})]\\ & = \operatorname{Tr}[X_R \rho_R]\\ & \leq 1. \end{align}\] This implies that \(X_R \otimes Y_B \in {({{\mathscr{A}}}_3)}_{{\scalebox{0.7}{\rm +}}}^{\circ}\) and therefore \({({{\mathscr{A}}}_{1})}_{{\scalebox{0.7}{\rm +}}}^{\circ} \otimes{({{\mathscr{A}}}_{2})}_{{\scalebox{0.7}{\rm +}}}^{\circ} \subseteq {({{\mathscr{A}}}_3)}_{{\scalebox{0.7}{\rm +}}}^{\circ}\). Applying the superadditivity in  [40], we have the asserted result in Eq. ?? for \(\alpha \in (0,1]\). The proof for \(\alpha \in (1,+\infty)\) follows in the same way. For \(\alpha \in (1,\infty)\), we work with the reverse polar sets instead: if \(Y_B \in ({{\mathscr{A}}}_2)^{\star}_{{\scalebox{0.7}{\rm +}}}\), then \({\cal N}^{\dagger}(Y_B) \geq I_{A}\) and as a result, if \(X_R \in ({{\mathscr{A}}}_1)^{\star}_{{\scalebox{0.7}{\rm +}}}\), we have \(X_R \otimes Y_B \in ({{\mathscr{A}}}_3)^{\star}_{{\scalebox{0.7}{\rm +}}}\) and we can similarly apply [40]. The result in Eq. ?? is a direct consequence of Eq. ?? . More specifically, we have that for any \(\alpha \in [1/2,\infty)\), \[\begin{align} D_{{\scriptscriptstyle \rm S},\alpha}(\rho^{\cal N}_{RB}\|\sigma^{\cal M}_{RB}) & = \lim_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm M},\alpha}((\rho^{\cal N}_{RB})^{\otimes n}\|(\sigma^{\cal M}_{RB})^{\otimes n})\\ & \geq \lim_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm M},\alpha}((\rho_R)^{\otimes n}\|(\sigma_{B})^{\otimes n}) + \lim_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm M},\alpha}^{\inf}({\cal N}^{\otimes n}\|{\cal M}^{\otimes n})\\ & = D_{{\scriptscriptstyle \rm S},\alpha}(\rho_R\|\sigma_R) + D_{{\scriptscriptstyle \rm S},\alpha}^{\inf,\infty}({\cal N}\|{\cal M}), \end{align}\] where the first line follows from [40], the second line follows from Eq. ?? and the last line follows from [40]. \(\square\)

2.2 Tightness of the chain rules↩︎

In the following, we introduce the notion of the amortized minimum output channel divergence and show that it coincides with the regularized divergence, being an analog result for the best-case channel divergence [11]. This, in turn, demonstrates the tightness of our chain rule properties.

Similar to the amortized channel divergence used in the existing literature [6], we can define the minimum output version as follows. Let \({{\mathbb{D}}}\) be a quantum divergence between states. Let \({\cal N}\in \text{\rm CPTP}(A\!:\!B)\) and \({\cal M}\in \text{\rm CP}(A\!:\!B)\). Then the amortized minimum output channel divergence is defined by \[\begin{align} {{\mathbb{D}}}^{\inf, \text{\rm amo}}({\cal N}\|{\cal M}):= \inf_{\substack{\rho\in \mathscr{D}(RA)\\ \sigma \in \mathscr{D}(RA)}} {{\mathbb{D}}}({\cal N}_{A\to B}(\rho_{RA})\|{\cal M}_{A\to B}(\sigma_{RA})) - {{\mathbb{D}}}(\rho_R\|\sigma_R). \end{align}\]

Proof.  We prove the result for the quantum relative entropy (\(\alpha = 1\)) and the same argument works for the sandwiched Rényi divergence of order \(\alpha \neq 1\) as well. Note that the chain rule property in Lemma 2 is equivalent to \(D^{\inf, \text{\rm amo}}({\cal N}\|{\cal M}) \geq D^{\inf,\infty}({\cal N}\|{\cal M})\). Now we prove the reverse direction. For this, we will first show the superadditivity of the amortized divergence under tensor product. That is, \[\begin{align} \label{eq:32amortized32worst32case32superadditivity} D^{\inf, \text{\rm amo}}({\cal N}_1 \otimes{\cal N}_2 \|{\cal M}_1\otimes{\cal M}_2) \geq D^{\inf, \text{\rm amo}}({\cal N}_1\|{\cal M}_1) + D^{\inf, \text{\rm amo}}({\cal N}_2\|{\cal M}_2), \end{align}\tag{3}\] for any quantum channels \({\cal N}_1 \in \text{\rm CPTP}(A_1\!:\!B_1)\), \({\cal N}_2\in \text{\rm CPTP}(A_2\!:\!B_2)\), CP maps \({\cal M}_1 \in \text{\rm CP}(A_1\!:\!B_1)\), \({\cal M}_2\in \text{\rm CP}(A_2\!:\!B_2)\). To see this, let \((\rho_{RA_1A_2}, \sigma_{RA_1A_2})\) be any feasible solution to the optimization on the left-hand side of Eq. 3 and denote its corresponding objective value by \[\begin{align} \delta_{12} := D({\cal N}_1 \otimes{\cal N}_2(\rho_{RA_1A_2})\|{\cal M}_1 \otimes{\cal M}_2(\sigma_{RA_1A_2})) - D(\rho_R\|\sigma_R). \end{align}\] Let \(\omega_{RA_2B_1} = {\cal N}_1(\rho_{RA_1A_2})\) and \(\gamma_{RA_2B_1} = {\cal M}_1(\sigma_{RA_1A_2})\). We can check that \((\rho_{RA_1},\sigma_{RA_1})\) and \((\omega_{RA_2B_1}, \gamma_{RA_2B_1})\) are feasible solutions to the amortized divergence on the right-hand side of Eq. 3 , respectively, with the corresponding objective values by \[\begin{align} \delta_1 := & D({\cal N}_1(\rho_{RA_1})\|{\cal M}_1(\sigma_{RA_1})) - D(\rho_R\|\sigma_R) \geq D^{\inf, \text{\rm amo}}({\cal N}_1\|{\cal M}_1)\\ \delta_2 := & D({\cal N}_2(\omega_{RA_2B_1})\|{\cal M}_2(\gamma_{RA_2B_1})) - D(\omega_{RB_1}\|\gamma_{RB_1}) \geq D^{\inf, \text{\rm amo}}({\cal N}_2\|{\cal M}_2). \end{align}\] Noting that \(\omega_{RB_1} = {\cal N}_1(\rho_{RA_1})\) and \(\gamma_{RB_1} = {\cal M}_1(\sigma_{RA_1})\), we have \(\delta_{12} = \delta_1 + \delta_2\). This implies \[\begin{align} \delta_{12} \geq D^{\inf, \text{\rm amo}}({\cal N}_1\|{\cal M}_1) + D^{\inf, \text{\rm amo}}({\cal N}_2\|{\cal M}_2). \end{align}\] As this holds for any feasible solution \((\rho_{RA_1A_2}, \sigma_{RA_1A_2})\), we have the asserted result in Eq. 3 .

By trivializing the reference system in the amortized divergence, we get \(D^{\inf, \text{\rm amo}}({\cal N}\|{\cal M}) \leq D^{\inf}({\cal N}\|{\cal M})\). Then we have \[\begin{align} D^{\inf, \text{\rm amo}}({\cal N}\|{\cal M}) \leq \frac{1}{n} D^{\inf, \text{\rm amo}}({\cal N}^{\otimes n}\|{\cal M}^{\otimes n}) \leq \frac{1}{n} D^{\inf}({\cal N}^{\otimes n}\|{\cal M}^{\otimes n}), \end{align}\] where the first inequality follows from Eq. 3 . As the above holds for any \(n\), we can take \(n\to \infty\) on the right-hand side and conclude that \(D^{\inf, \text{\rm amo}}({\cal N}\|{\cal M}) \leq D^{\inf,\infty}({\cal N}\|{\cal M})\). This completes the proof. \(\square\)

2.3 Counter-example to a potential improvement of the chain rule↩︎

The quantum channel divergence studied in most existing literatures [56] use the same test states for both channels. So it may be expected that we can enhance the chain rules by using the same test states as well. However, we show here that this is not possible by giving a counter-example. That is, the chain rule cannot be enhanced to \[\begin{align} \label{eq:32chain32rule32enhancement} D_{{\scriptscriptstyle \rm M}}({\cal N}_{A\to B}(\rho_{RA})\|{\cal M}_{A\to B}(\sigma_{RA})) \geq D_{{\scriptscriptstyle \rm M}}(\rho_R\|\sigma_R) + D_{{\scriptscriptstyle \rm M}}^{\inf'}({\cal N}\|{\cal M}) \end{align}\tag{4}\] where the channel divergence takes the same input state \[\begin{align} D_{{\scriptscriptstyle \rm M}}^{\inf'}({\cal N}\|{\cal M}):= \inf_{\rho\in \mathscr{D}(A)} D_{{\scriptscriptstyle \rm M}}({\cal N}_{A\to B}(\rho_{A})\| {\cal M}_{A \to B}(\rho_{A})). \end{align}\] To see this, consider the generalized amplitude damping (GAD) channel, which is defined as \[\begin{align} {\cal A}_{\gamma, N}(\rho) = \sum_{i=1}^4 A_i \rho A_i^\dagger, \quad \gamma, N \in [0,1], \end{align}\] with the Kraus operator \[\begin{align} {2} A_1 & = \sqrt{1-N}(|0\rangle\langle 0| + \sqrt{1-\gamma}|1\rangle\langle 1|), \qquad && A_2 = \sqrt{\gamma(1-N)} |0\rangle\langle 1|,\\ A_3 & = \sqrt{N}(\sqrt{1-\gamma} |0\rangle\langle 0| + |1\rangle\langle 1|), \qquad && A_4 = \sqrt{\gamma N} |1\rangle\langle 0|. \end{align}\]

Using convex optimization, we can numerically evaluate each terms \(D_{{\scriptscriptstyle \rm M}}({\cal N}_{A\to B}(\rho_{RA})\|{\cal M}_{A\to B}(\sigma_{RA}))\), \(D_{{\scriptscriptstyle \rm M}}(\rho_R\|\sigma_R)\) and \(D_{{\scriptscriptstyle \rm M}}^{\inf'}({\cal N}\|{\cal M})\). Then in Figure 2 (a), we show that the channel divergence \(D_{{\scriptscriptstyle \rm M}}^{\inf'}\) is subadditive under tensor product of channels. That is, it does not inherit the properties of the state divergence, making it not a suitable channel extension. Moreover, in Figure 2 (b), we show that the chain rule property in Eq. 4 does not hold, as there are cases such that \(y < x_2\) in the plot.

3 Proof of the adversarial quantum Stein’s lemma↩︎

Since non-adaptive strategies are a specific type of adaptive strategy, we have the inclusions that \({{\mathscr{A}}}_n' \subseteq {{\mathscr{A}}}_n \quad \text{and} \quad {{\mathscr{B}}}_n' \subseteq {{\mathscr{B}}}_n\). This gives the relations for the type-I and type-II errors by \(\alpha({{\mathscr{A}}}_n', M_n) \leq \alpha({{\mathscr{A}}}_n, M_n)\) and \(\beta({{\mathscr{B}}}_n', M_n) \leq \beta({{\mathscr{B}}}_n, M_n)\). So we have the general relation that \(\beta_{n,\varepsilon}'({\cal N}\|{\cal M}) := \inf \{\beta({{\mathscr{B}}}_n', M_n): 0\leq M_n\leq I, \alpha({{\mathscr{A}}}_n', M_n) \leq \varepsilon\} \leq \beta_{n,\varepsilon}({\cal N}\|{\cal M})\) where the inequality holds because the right-hand side represents the infimum of a larger objective value over a smaller feasible set. We can also rewrite the optimal type-II errors as the hypothesis testing relative entropy between two sets, \[\begin{align} -\log \beta_{n,\varepsilon}({\cal N}\|{\cal M}) & = D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n),\\ -\log \beta_{n,\varepsilon}'({\cal N}\|{\cal M}) & = D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n'\|{{\mathscr{B}}}_n'), \end{align}\] where \(D_{{\scriptscriptstyle \rm H},\varepsilon}(\rho\|\sigma):=-\log\{\operatorname{Tr}[\sigma M]: \operatorname{Tr}[\rho(I-M)] \leq \varepsilon, 0 \leq M \leq I\}\). This can be done by showing that \({{\mathscr{A}}}_n ({{\mathscr{A}}}_n')\) and \({{\mathscr{B}}}_n ({{\mathscr{B}}}_n')\) are convex sets (see Lemma 4 below) and applying [40]. After this, the proof for the adversarial quantum Stein’s lemma then contains two parts, as shown in the main text. The converse part makes use of the chain rule property to show that \[\begin{align} \liminf_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n\|{{\mathscr{B}}}_n) \geq D^{\inf,\infty}({\cal N}\|{\cal M}), \end{align}\] while the achievable part aims to show that \[\begin{align} \limsup_{n\to \infty} \frac{1}{n} D_{{\scriptscriptstyle \rm H},\varepsilon}({{\mathscr{A}}}_n'\|{{\mathscr{B}}}_n') \leq D^{\inf,\infty}({\cal N}\|{\cal M}), \end{align}\] by applying the generalized quantum asymptotic equipartition (AEP) property in [40] to \({{\mathscr{A}}}_n'\) and \({{\mathscr{B}}}_n'\).

Proof.  We prove the assumptions for \(\{{{\mathscr{A}}}_n\}_{n\in {{\mathbb{N}}}}\) and the same reasoning works for \(\{{{\mathscr{B}}}_n\}_{n\in {{\mathbb{N}}}}\) as well. Let \(\{{\cal P}^i\}_{i=1}^n\) with systems \(R_1, \dots, R_n\) and \(\{\bar{{\cal P}}^i\}_{i=1}^n\) with systems \(R_1, \dots, R_n\) be two strategies and \(\lambda \in [0,1]\). Note that we may assume both strategies have the same systems \(R_i\) as we can always increase the dimension of the systems \(R_i\) by extending the action of the channel in an arbitrary way without affecting the output. Let us now define another strategy \(\{\bar{\bar{{\cal P}}}^i\}_{i=1}^n\) as follows. Let \(\bar{\bar{R}}_i = R_i \otimes C\) for all \(i=1,\dots,n\) where \(C\) is a two-dimensional system. Then define \[\begin{align} \bar{\bar{{\cal P}}}^1(\cdot) = \lambda {\cal P}^1(\cdot) \otimes| 0\rangle\!\langle 0 |_C + (1-\lambda) \bar{{\cal P}}^1(\cdot) \otimes| 1\rangle\!\langle 1 |_C \end{align}\] and for \(i \geq 2\), define \[\begin{align} \bar{\bar{{\cal P}}}^i(X) = ({\cal P}^i \circ {\cal C}_0(X)) \otimes| 0\rangle\!\langle 0 |_C + (\bar{{\cal P}}^i \circ {\cal C}_1(X)) \otimes| 1\rangle\!\langle 1 |_C \end{align}\] where \({\cal C}_0(X) = \langle 0|_C X |0\rangle_C\) and \({\cal C}_1(X) = \langle 1|_C X |1\rangle_C\). Since \({\cal C}_0, {\cal C}_1, {\cal P}^i, \bar{{\cal P}}^i\) are all CP maps, we know that \(\bar{\bar{{\cal P}}}^i\) is also a CP map. Moreover, if \({\cal P}^i, \bar{{\cal P}}^i\) are trace-preserving, then \(\bar{\bar{{\cal P}}}^i\) is also trace-preserving. It is easy to check the following relations, \[\begin{align} {3} {\cal C}_0 \circ \bar{\bar{{\cal P}}}^1 & = \lambda {\cal P}^1 \quad & \text{and} \quad {\cal C}_1 \circ \bar{\bar{{\cal P}}}^1 & = (1-\lambda) \bar{{\cal P}}^1,\tag{5}\\ {\cal C}_0 \circ \bar{\bar{{\cal P}}}^i & = {\cal P}^i \circ {\cal C}_0 \quad & \text{and} \quad {\cal C}_1 \circ \bar{\bar{{\cal P}}}^i & = \bar{{\cal P}}^i \circ {\cal C}_1, \quad \forall i \geq 2.\tag{6} \end{align}\] Noting that \(\operatorname{Tr}_C\) commutes with \({\cal U}\) as they are acting on different systems, we have \[\begin{align} \label{eq:32ad32convex32tmp3} \operatorname{Tr}_C \circ \prod_{i=1}^n {\cal U}\circ \bar{\bar{{\cal P}}}^i & = {\cal U}\circ {\cal P}^n \circ {\cal C}_0 \circ \prod_{i=1}^{n-1} {\cal U}\circ \bar{\bar{{\cal P}}}^i + {\cal U}\circ \bar{{\cal P}}^n \circ {\cal C}_1 \circ \prod_{i=1}^{n-1} {\cal U}\circ \bar{\bar{{\cal P}}}^i. \end{align}\tag{7}\] Also noting that \({\cal C}_0\) and \({\cal C}_1\) both commute with \({\cal U}\) as they are acting on different systems and using the relations in Eqs. 5 and 6 , we have \[\begin{align} {\cal U}\circ {\cal P}^n \circ {\cal C}_0 \circ \prod_{i=1}^{n-1} {\cal U}\circ \bar{\bar{{\cal P}}}^i & = \lambda\, {\cal U}\circ {\cal P}^n \circ \prod_{i=1}^{n-1} {\cal U}\circ {{{\cal P}}}^i = \lambda\, \prod_{i=1}^{n} {\cal U}\circ {{{\cal P}}}^i,\\ {\cal U}\circ \bar{{\cal P}}^n \circ {\cal C}_1 \circ \prod_{i=1}^{n-1} {\cal U}\circ \bar{\bar{{\cal P}}}^i & = (1-\lambda)\, {\cal U}\circ {\cal P}^n \circ \prod_{i=1}^{n-1} {\cal U}\circ {\bar{{\cal P}}}^i = (1-\lambda)\, \prod_{i=1}^{n} {\cal U}\circ {\bar{{\cal P}}}^i. \end{align}\] Taking these into Eq. 7 , we have \[\begin{align} \operatorname{Tr}_{\bar{\bar{R}}_n E_n} \circ \prod_{i=1}^n {\cal U}\circ \bar{\bar{{\cal P}}}^i = \lambda\, \operatorname{Tr}_{R_n E_n} \circ \prod_{i=1}^{n} {\cal U}\circ {{{\cal P}}}^i + (1-\lambda)\, \operatorname{Tr}_{R_n E_n} \circ \prod_{i=1}^{n} {\cal U}\circ {\bar{{\cal P}}}^i \end{align}\] This shows that any mixture of the reduced states on \(B_1\cdots B_n\) by the strategies \(\{{\cal P}^i\}_{i=1}^n\) and \(\{\bar{{\cal P}}^i\}_{i=1}^n\) is also given by the reduced state of another strategy \(\{\bar{\bar{{\cal P}}}^i\}_{i=1}^{n}\) which proves the convexity of \({{\mathscr{A}}}_n\). \(\square\)

4 Relative entropy accumulation↩︎

The entropy accumulation theorem [49], [50] is a technique to find bounds on the operationally relevant uncertainty (entropy) present in the outputs of a sequential process as a sum of the worst case uncertainties (entropies) of each step. It has been widely used in quantum cryptography [35]–[37]. This naturally raises the question of whether such a statement can be generalized to divergences between arbitrary sequential processes of channels, rather than being restricted only to entropies. This was first asked as an open question in [50] for the max-relative entropy.

More specifically, the operational setting for relative entropy accumulation is depicted in Figure 3. Consider two states \(\rho_{A_1}\) and \(\sigma_{A_1}\) and quantum channels \({\cal N}_i \in \text{\rm CPTP}(A_i\!:\!A_{i+1} B_i)\) and \({\cal M}_i \in \text{\rm CP}(A_i\!:\!A_{i+1} B_i)\) that are applied sequentially from \(i=1\) to \(i=n\) and generating the systems \(B_i\). The systems \(A_i\) should be seen as an internal memory system that we do not control. The key question in the relative entropy accumulation asks: Can we bound the operationally relevant divergence between the obtained states as the sum of the contributions of each step?

Similar to the adversarial channel discrimination in the main text, we denote the Stinespring dilations of \({\cal N}_i\) and \({\cal M}_i\) as \(U_i\) and \(V_i\), respectively, with the environmental system denoting as \(T_i\). Denote also the corresponding channel as \({\cal U}_i(\cdot)=U_i(\cdot)U_i^\dagger\) and \({\cal V}_i(\cdot) = V_i(\cdot)V_i^\dagger\). Note that \(U_i\) is an isometry because \({\cal N}_i\) is trace-preserving, but this is not necessarily the case for \(V_i\). Then the setting in Figure 3 is equivalent to the diagram in Figure 4.

By extending the adversarial channel discrimination framework to allow different channels \({\cal N}_i\) and \({\cal M}_i\) at each round, we can unify the relative entropy accumulation within this broader framework. More specifically, using a similar notation in the main text, we denote the Stinespring dilation of \({\cal N}_i\) and \({\cal M}_i\) as \(U_i\) and \(V_i\), respectively. Denote also the corresponding channel as \({\cal U}_i(\cdot)=U_i(\cdot)U_i^\dagger\) and \({\cal V}_i(\cdot) = V_i(\cdot)V_i^\dagger\). An illustrative figure is given in Figure 5. Then, taking a particular choice, \[\begin{align} \dim R_i & = 1,\\ E_i & = A_{i+1}T_i\\ {\cal P}^1 & = \rho_{A_1},\\ {\cal Q}^1 & = \sigma_{A_1},\\ {\cal P}^i & = {\cal Q}^i = \operatorname{Tr}_{T_{i-1}}, \quad \forall i \geq 2, \end{align}\] the adversarial discrimination framework in Figure 5 reduces to the relative entropy accumulation in Figure 4.

This new perspective enables us to establish a relative entropy accumulation theorem for \(D_{{\scriptscriptstyle \rm H}, \varepsilon}\) (a smoothed form of the min-relative entropy), giving an answer to the dual formulation of the open question in [50]. Specifically, we generalize the converse part of the adversarial quantum Stein’s lemma in two ways: we allow the channels applied at different steps to vary, and we compute explicit finite-size bounds.

We state the relative entropy accumulation theorem in this more general setting depicted in Figure 5. From the above discussion, it is clear that the version stated in the main text is a special case.

Proof.  The proof below is basically the same as the converse part of the adversarial quantum Stein’s lemma in the main text. We start as usual by bounding the hypothesis testing relative entropy with a Rényi divergence of order \(\alpha \in (0,1)\): \[\begin{align} D_{{\scriptscriptstyle \rm H}, \varepsilon}\left(\rho_n \bigg\| \sigma_n \right) \geq D_{{\scriptscriptstyle \rm S},\alpha}\left(\rho_n \bigg\| \sigma_n \right) + \frac{\alpha}{\alpha-1}\log(1/\varepsilon). \end{align}\] We can now introduce \(\rho'_n = \operatorname{Tr}_{R_n} \circ {\cal P}^n \circ \prod_{i=1}^{n-1} {\cal U}_i \circ {\cal P}^i\) and \(\sigma'_n = \operatorname{Tr}_{R_n} \circ {\cal Q}^n \circ \prod_{i=1}^{n-1} {\cal V}_i \circ {\cal Q}^i\) the joint states before the \(n\)-th use the channel \({\cal U}_{i}/{\cal V}_{i}\). We have \(\rho_n = (\operatorname{Tr}_{E_{n}} \circ {\cal U}_{n})(\rho'_n)\) We use the chain rule for sandwiched relative entropy to bound \[\begin{align} D_{{\scriptscriptstyle \rm S},\alpha}\left( (\operatorname{Tr}_{E_n} \circ {\cal U}_{n})(\rho'_n) \bigg\| (\operatorname{Tr}_{E_n} \circ {\cal V}_n)(\sigma'_n)\right) &\geq D_{{\scriptscriptstyle \rm S},\alpha}\left( \operatorname{Tr}_{A_n} \rho'_n \bigg\| \operatorname{Tr}_{A_n} \sigma'_n \right) + D^{\inf, \infty}_{{\scriptscriptstyle \rm S},\alpha}(\operatorname{Tr}_{E_n} \circ {\cal U}_{n} \| \operatorname{Tr}_{E_n} \circ {\cal V}_n). \end{align}\] Now note that \(\operatorname{Tr}_{A_n} \rho'_n = \rho_{n-1}\) where \(\rho_{n-1} = \operatorname{Tr}_{R_{n-1}E_{n-1}} \prod_{i=1}^{n-1} {\cal U}_{i} \circ {\cal P}^{i}\) and similarly for \(\sigma\). As a result, applying the chain rule \(n-1\) times, we get \[\begin{align} D_{{\scriptscriptstyle \rm S},\alpha}\left( \rho_n \bigg\| \sigma_n \right) &\geq \sum_{i=1}^n D^{\inf,\infty}_{{\scriptscriptstyle \rm S},\alpha}(\operatorname{Tr}_{E_{i}} \circ {\cal U}_i \| \operatorname{Tr}_{E_{i}} \circ {\cal V}_i) \\ &= \sum_{i=1}^n D^{\inf,\infty}_{{\scriptscriptstyle \rm M},\alpha}(\operatorname{Tr}_{E_{i}} \circ {\cal U}_i \| \operatorname{Tr}_{E_{i}} \circ {\cal V}_i) \\ &\geq \sum_{i=1}^n \frac{1}{m} D^{\inf}_{{\scriptscriptstyle \rm M},\alpha}((\operatorname{Tr}_{E_{i}} \circ {\cal U}_i)^{\otimes m} \| (\operatorname{Tr}_{E_{i}} \circ {\cal V}_i)^{\otimes m}) \end{align}\] where \(m \geq 2\) and we used [40] for the equality and the superadditivity of \(D_{{\scriptscriptstyle \rm M}, \alpha}\) in [40] for the last inequality. Note that both of these results were applied for the family of states \({{\mathscr{A}}}_m'' = (\operatorname{Tr}_{E_{i}} \circ {\cal U}_i)^{\otimes m}(\mathscr{D})\) and \({{\mathscr{B}}}_m'' = (\operatorname{Tr}_{E_{i}} \circ {\cal V}_i)^{\otimes m}(\mathscr{D})\) which satisfies [40], as shown in the proof of the adversarial quantum Stein’s lemma.

Observe that assumption ?? implies assumption (\(*\)) in [40] as \(\log \operatorname{Tr}({\cal V}_i^{\otimes m}(\sigma)) \leq m \log \lambda_{\max}({\cal V}_i^{\dagger}(I)) \leq \frac{C}{4} m\). As a result, [40] gives \[\begin{align} \frac{1}{m} D^{\inf}_{{\scriptscriptstyle \rm M},\alpha}&((\operatorname{Tr}_{E_{i}} \circ {\cal U}_i)^{\otimes m} \| (\operatorname{Tr}_{E_{i}} \circ {\cal V}_i)^{\otimes m})\notag \\ &\geq D^{\inf, \infty}(\operatorname{Tr}_{E_{i}} \circ {\cal U}_i \| \operatorname{Tr}_{E_{i}} \circ {\cal V}_i) - (1-\alpha) (2+C)^2 m - \frac{2(d^2+d) \log(m+d)}{m}, \end{align}\] for \(1-\frac{1}{(2+C)m} < \alpha < 1\). Let us now choose \(1-\alpha = \frac{8d^2 \log m}{(2+C)^2 m^2}\) and assume that \(m \geq \max\left(d,\left(\frac{16 d^2}{2+C}\right)^2\right)\) so that the condition \(\alpha \geq 1 - \frac{1}{(2+C)m}\) is satisfied and \(\log(m+d) \leq 2 \log m\).

Putting everything together, we get \[\begin{align} D_{{\scriptscriptstyle \rm H}, \varepsilon} \left(\rho_n \bigg\| \sigma_n \right) \geq \sum_{i=1}^n D^{\inf,\infty}(\operatorname{Tr}_{E_{i}} \circ {\cal U}_{i} \| \operatorname{Tr}_{E_{i}} \circ {\cal V}_{i}) - n \frac{16d^2 \log m}{m} - \frac{(2+C)^2 m^2}{8d^2\log m}\log \frac{1}{\varepsilon}. \end{align}\] We now choose \(m = \left(\frac{64 d^4 n }{(2+C)^2 \log \frac{1}{\varepsilon} }\right)^{1/3}\). With this choice \[\begin{align} D_{{\scriptscriptstyle \rm H}, \varepsilon}\left(\rho_n \bigg\| \sigma_n \right) &\geq \sum_{i=1}^n D^{\inf,\infty}(\operatorname{Tr}_{E_{i}} \circ {\cal U}_{i} \| \operatorname{Tr}_{E_{i}} \circ {\cal V}_{i}) - C' n^{2/3} \log n \log^{1/3} \frac{1}{\varepsilon}, \end{align}\] for a constant \(C'\) that only depends on \(C\) and \(d\). \(\square\)

As a corollary, we get a weak form of the entropy accumulation statement for the smoothed max-entropy obtained in [50]. We recall that the smoothed max-entropy is defined as \[\begin{align} H_{\max}^{\varepsilon}(B|C)_{\rho} = \log \inf_{\substack{\tilde{\rho}_{BC} \in \mathscr{H}_{{\scalebox{0.7}{\rm +}}}(BC) \\ \operatorname{Tr}(\tilde{\rho}) \leq 1 \\ P(\rho, \tilde{\rho}) \leq \varepsilon}} \sup_{\sigma_C \in \mathscr{D}(C)} \left\| \tilde{\rho}_{BC}^{\frac{1}{2}} I_{B} \otimes \sigma_C^{\frac{1}{2}} \right\|_1^2, \end{align}\] where \(P(\rho,\sigma) = \sqrt{1-F(\rho,\sigma)^2}\) is the purifided distance with \(F(\rho,\sigma) := \|\sqrt{\rho}\sqrt{\sigma}\|_1 + \sqrt{(1-\operatorname{Tr}\rho)(1-\operatorname{Tr}\sigma)}\).

Proof.  We apply the relative entropy accumulation theorem as stated in the main text with the following replacements: letting \(C'_i\) be isomorphic to \(C_i\), we set \(A_{i+1} \leftarrow Y_{i} C'_{i+1} \dots C'_n\) for \(i=0\) to \(n-1\), \(A_{n+1} \leftarrow Y_n\), \(B_i \leftarrow S_i C_i\) for \(i=1\) to \(n\), \({\cal N}_i \leftarrow {\cal N}_{i}' \otimes \operatorname{Tr}_{C'_{i}} \otimes \operatorname{id}_{C'_{i+1} \dots C'_n}\) (here, \(\operatorname{id}\) refers to the identity map) and finally \({\cal M}_i\) is defined as \({\cal M}_i(X_{Y_{i-1}C'_i \dots C'_n}) = I_{S_i} \otimes X_{C_i C'_{i+1} \dots C'_{n}} \otimes | 0\rangle\!\langle 0 |_{Y_i}\), where \(| 0\rangle\!\langle 0 |_{Y_i}\) is an arbitrary fixed state in \(\mathscr{D}(Y_i)\). An illustrative figure is given in Figure 6.

Theorem 3 gives for the left hand side of ?? and any \(\sigma \in \mathscr{D}(C_1' \dots C_n')\): \[\begin{align} D_{{\scriptscriptstyle \rm H}, \varepsilon}&\left(\operatorname{Tr}_{A_{n+1}} \circ \prod_{i=1}^n {\cal N}_i (\rho_{Y_0} \otimes \sigma_{C'_1 \dots C'_n}) \bigg\| \operatorname{Tr}_{A_{n+1}} \circ \prod_{i=1}^n {\cal M}_i (| 0\rangle\!\langle 0 |_{Y_0} \otimes \sigma_{C'_1 \dots C'_n})\right)\notag \\ &= D_{{\scriptscriptstyle \rm H}, \varepsilon}(\rho_{S_1 \dots S_n C_1 \dots C_n} \| I_{S_1 \dots S_n} \otimes \sigma_{C_1 \dots C_n}). \end{align}\] As the right hand side of ?? does not depend on \(\sigma\), we can take an infimum over \(\sigma\) and use Proposition 1 to get the following \[\begin{align} \inf_{\sigma} D_{{\scriptscriptstyle \rm H}, \varepsilon}&\left(\operatorname{Tr}_{A_{n+1}} \circ \prod_{i=1}^n {\cal N}_i (\rho_{Y_0} \otimes \sigma_{C'_1 \dots C'_n}) \bigg\| \operatorname{Tr}_{A_{n+1}} \circ \prod_{i=1}^n {\cal M}_i (| 0\rangle\!\langle 0 |_{Y_0} \otimes \sigma_{C'_1 \dots C'_n})\right) \notag\\ &\leq - H_{\max}^{\sqrt{2\varepsilon}}(S_1 \dots S_n | C_1 \dots C_n)_{\rho}. \end{align}\]

On the right hand side of ?? , we have terms of the form \(D^{\inf,\infty}(\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i \| \operatorname{Tr}_{A_{i+1}} \circ {\cal M}_i)\). Note that for \(\omega \in \mathscr{D}(Y_{i-1}C'_{i} \dots C'_n)\), we have \((\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i)(\omega) = \operatorname{Tr}_{Y_i}{\cal N}_i'(\omega_{Y_{i-1}})\) and \((\operatorname{Tr}_{A_{i+1}} \circ {\cal M}_i)(\omega) = I_{S_i} \otimes \omega_{C_i}\). As a result, \(D^{\inf}(\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i \| \operatorname{Tr}_{A_{i+1}} \circ {\cal M}_i) = \inf_{\omega \in \mathscr{D}(Y_{i-1})} - H(S_i | C_i)_{{\cal N}_i'(\omega)}\), where we used the fact that \(-H(B|C)_{\rho} = \inf_{\sigma \in \mathscr{D}(C)} D(\rho_{BC} \| I_{B} \otimes \sigma_{C})\). We now need to evaluate the regularization: \[\begin{align} D^{\inf,\infty}&(\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i \| \operatorname{Tr}_{A_{i+1}} \circ {\cal M}_i) \notag \\ &= \inf_{m \geq 1} \frac{1}{m} \inf_{\substack{\omega \in \mathscr{D}((Y_{i-1})^{\otimes m}) \\ \sigma \in \mathscr{D}((C_{i} \dots C_n)^{\otimes m})}} D((\operatorname{Tr}_{Y_i}\circ{\cal N}_i')^{\otimes m}(\omega) \| I_{S_{i,1} \dots S_{1,m}} \otimes \sigma_{C_{i,1} \dots C_{i,m}}) \\ &= - \sup_{m \geq 1} \sup_{\omega} \frac{1}{m} H(S_{i,1} \dots S_{i,m} | C_{i,1} \dots C_{i,m})_{{\cal N}_i'^{\otimes m}(\omega)} \\ &= - \sup_{m \geq 1} \sup_{\omega} \frac{1}{m} \sum_{j=1}^m H(S_{i,j} | C_{i,1} \dots C_{i,m} S_{i,1} \dots S_{i,j-1})_{{\cal N}_i'^{\otimes m}(\omega)} \\ &\geq - \sup_{m \geq 1} \sup_{\omega} \frac{1}{m} \sum_{j=1}^m H(S_{i,j} | C_{i,j} )_{{\cal N}_i'(\omega_{Y_{i,j}})}, \end{align}\] where we used the chain rule and then strong subadditivity of the von Neumann entropy. Now note that each term \(H(S_{i,j} | C_{i,j} )_{{\cal N}_i'(\omega_{Y_{i,j}})} \leq \sup_{\omega} H(S_i | C_i)_{{\cal N}'_i(\omega)}\). As a result, \(D^{\inf,\infty}(\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i \| \operatorname{Tr}_{A_{i+1}} \circ {\cal M}_i) \geq - \sup_{\omega \in \mathscr{D}(Y_{i-1})} H(S_i | C_i)_{{\cal N}_i'(\omega)}\) which shows that in this case, the regularization is not needed. In order to apply Theorem 3, we need to check condition ?? . First, we have for any \(\omega \in \mathscr{D}(Y_i)\), \(\operatorname{Tr}({\cal M}_i(\omega)) = \dim S_i\). In addition, for \(\alpha \in [1/2,1]\), we have \[\begin{align} D^{\inf}_{{\scriptscriptstyle \rm P},\alpha}&((\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i)^{\otimes m} \| (\operatorname{Tr}_{A_{i+1}} \circ {\cal M}_i)^{\otimes m}) \notag\\ &=\inf_{\substack{\omega \in \mathscr{D}((Y_{i-1})^{\otimes m}) \\ \sigma \in \mathscr{D}((C_{i} \dots C_n)^{\otimes m})}} D_{{\scriptscriptstyle \rm P}, \alpha}((\operatorname{Tr}_{Y_i}{\cal N}_i')^{\otimes m}(\omega_{Y_{i-1,1} \dots Y_{i-1,m}}) \| I_{S_{i,1} \dots S_{1,m}} \otimes \sigma_{C_{i,1} \dots C_{i,m}}) \label{eq:expression-dalpha-opt} \\ &= - \sup_{\omega} H^{\uparrow}_{{\scriptscriptstyle \rm P}, \alpha}(S_{i,1} \dots S_{i,m} | C_{i,1} \dots C_{i,m})_{{\cal N}_i'^{\otimes m}(\omega)}, \end{align}\tag{8}\] using the notation \(H^{\uparrow}_{{\scriptscriptstyle \rm P}, \alpha}(D| E)_{\rho} = -\inf_{\sigma \in \mathscr{D}(E)}D_{{\scriptscriptstyle \rm P},\alpha}(\rho_{DE} \| I_{D} \otimes \sigma_E)\) from [57]. It is shown in [57], that an explicit choice of \(\sigma\) achieves this infimum namely \(\sigma^{(\alpha)}_{E} = \frac{(\operatorname{Tr}_D \rho_{DE}^{\alpha})^{1/\alpha}}{\operatorname{Tr}(\operatorname{Tr}_D \rho_{DE}^{\alpha})^{1/\alpha}}\). But in Lemma 5, we showed that for this choice \(D_{{\scriptscriptstyle \rm P},3/2}(\rho_{DE} \| I_{D} \otimes \sigma^{(\alpha)}_{E}) \leq 4 \log \dim D\). Applying this to the state \(\rho = (\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i)^{\otimes m}(\omega)\), an optimal choice for \(\sigma\) in 8 is given by \(\sigma_{m}^{(\alpha)} = \frac{(\operatorname{Tr}_{B_{i}^{\otimes m}} \rho^{\alpha})^{1/\alpha}}{\operatorname{Tr}(\operatorname{Tr}_{B_{i}^{\otimes m}} \rho^{\alpha})^{1/\alpha}}\). We get that for any \(\omega \in \mathscr{D}(R_{i-1}^{\otimes m})\), \[\begin{align} D_{{\scriptscriptstyle \rm P},3/2}((\operatorname{Tr}_{A_{i+1}} \circ {\cal N}_i)^{\otimes m}(\omega) \| I_{S_i}^{\otimes m} \otimes \sigma^{(\alpha)}_{m}) \leq 4 m \log \dim S_i. \end{align}\] This means that choosing \(C = 16 \max_{i} \log \dim S_i\) satisfies condition ?? . \(\square\)

Note that the second order term we achieve with our proof technique is worse than the one achieved in [49] and [50]. In addition, the statement of [50] is stronger in that it also includes conditioning on the system \(Y_i\) provided a non-signalling assumption is satisfied. Nevertheless, we believe that our new proof technique, which is more naturally adapted to find upper bounds on the max-entropy \(H_{\max}\) (as opposed to the techniques of [50] which naturally apply to the min-entropy \(H_{\min}\)) could lead to insights and improvements for the applications of entropy accumulation. However, this is outside the scope of this paper and we leave it for future work.

5 Useful properties↩︎

Note that the choice of the parameter \(\frac{3}{2}\) is not arbitrary and this lemma does not hold for higher values. See the discussion in the proof of  [58].

Proof.  We have \[\begin{align} \operatorname{Tr}\left(\rho_{AB}^{3/2} (I_{A} \otimes \sigma_{B}^{(\alpha)})^{-1/2}\right) &= Z^{1/2} \operatorname{Tr}\left(\rho_{AB}^{3/2} I_{A} \otimes (\operatorname{Tr}_{A} \rho_{AB}^{\alpha})^{-\frac{1}{2\alpha}}\right). \end{align}\] We start by showing that \(Z \leq d_{A}\). In fact, we use the operator Jensen inequality for the operator concave function \(x \mapsto x^{\alpha}\) as follows: \[\begin{align} \operatorname{Tr}_{A}(\rho_{AB}^{\alpha}) &= d_A \sum_{a} \frac{\langle a|}{\sqrt{d_A}} \rho_{AB}^{\alpha} \frac{|a\rangle}{\sqrt{d_A}} \\ &\leq d_A \left(\sum_{a} \frac{\langle a|}{\sqrt{d_A}} \rho_{AB} \frac{|a\rangle}{\sqrt{d_A}}\right)^{\alpha} \\ &= d_A^{1-\alpha} \rho_B^{\alpha}. \end{align}\] As a result, we get \[\begin{align} Z &\leq \operatorname{Tr}( (d_A^{1-\alpha} \rho_B^{\alpha})^{\frac{1}{\alpha}}) = d_{A}^{\frac{1-\alpha}{\alpha}} \leq d_A. \end{align}\] For the inequality, we used the fact that the function \(x \mapsto x^{1/\alpha}\) is monotone and continuous and thus \(X \mapsto \operatorname{Tr}(X^{\frac{1}{\alpha}})\) is monotone (see e.g., [59]).

Now let us consider \[\begin{align} \operatorname{Tr}\left(\rho_{AB}^{3/2} I_{A} \otimes (\operatorname{Tr}_{A} \rho_{AB}^{\alpha})^{\frac{1}{\alpha}}\right) &\leq \operatorname{Tr}\left(\rho_{AB} I_{A} \otimes (\operatorname{Tr}_{A} \rho_{AB})^{-\frac{1}{2\alpha}}\right) \\ &\leq \operatorname{Tr}(\rho_{AB} \rho_{B}^{-1}) \\ &\leq d_A, \end{align}\] where we used for the first inequality the fact that \(\rho_{AB}^{3/2} \leq \rho_{AB}\), \(\rho_{AB}^{\alpha} \geq \rho_{AB}\) and the operator anti-monotonicity of the function \(x \mapsto x^{-\frac{1}{2\alpha}}\), and for the second inequality the fact that \(\rho_{B}^{\frac{1}{2\alpha}} \geq \rho_{B}\). As a result, \[\begin{align} D_{{\scriptscriptstyle \rm P},\frac{3}{2}}(\rho_{AB} \| I_{A} \otimes \sigma_B^{(\alpha)}) \leq 2 \log (d_A^{1/2} d_A) \leq 4 \log d_A. \end{align}\] \(\square\)

Proof.  Using [40] with \({{\mathscr{A}}}= \{\rho_{BC}\}\) and \({{\mathscr{B}}}= \{I_{B} \otimes \sigma_C : \sigma_{C} \in \mathscr{D}(C)\}\), we can write \[\begin{align} -\inf_{\sigma_{C} \in \mathscr{D}(C)} D_{{\scriptscriptstyle \rm H}, \varepsilon}(\rho_{BC} \| I_{B} \otimes \sigma_C) = \log \inf_{0 \leq M \leq I} \left\{ \sup_{\sigma_{C} \in \mathscr{D}(C)} \{\operatorname{Tr}(M_{BC} I_B \otimes \sigma_C) : \operatorname{Tr}(M_{BC} \rho_{BC}) \geq 1- \varepsilon\right\}. \end{align}\] Let \(M_{BC}\) be such that \(\operatorname{Tr}(M_{BC}\rho_{BC}) \geq 1-\varepsilon\), define \(\tilde{\rho}_{BC} = \sqrt{M_{BC}} \rho_{BC} \sqrt{M_{BC}}\). Then, by the gentle measurement lemma, we have \(P(\rho, \tilde{\rho}) \leq \sqrt{2\varepsilon}\) (see e.g., [60]). Then, using Lemma 6, we get \[\begin{align} \sup_{\sigma_{C}\in \mathscr{D}(C)} \left\| (\sqrt{M_{BC}} \rho_{BC} \sqrt{M_{BC}})^{\frac{1}{2}} I_B \otimes \sigma_C^{\frac{1}{2}} \right\|_1^2 \leq \sup_{\sigma_C} \operatorname{Tr}(M_{BC} I_B \otimes \sigma_C) \end{align}\] As such, \[\begin{align} -\inf_{\sigma_{C} \in \mathscr{D}(C)} & D_{{\scriptscriptstyle \rm H}, \varepsilon}(\rho_{BC} \| I_{B} \otimes \sigma_C) \notag \\ &\geq \log \inf_{0 \leq M \leq I} \sup_{\sigma_{C}} \left\{ \left\| (\sqrt{M_{BC}} \rho_{BC} \sqrt{M_{BC}})^{\frac{1}{2}} I_B \otimes \sigma_C^{\frac{1}{2}} \right\|_1^2 : \operatorname{Tr}(M_{BC} \rho_{BC}) \geq 1- \varepsilon\right\} \\ &\geq \log \inf_{\substack{\tilde{\rho}_{BC} \in \mathscr{H}_{{\scalebox{0.7}{\rm +}}}(BC) \\ \operatorname{Tr}(\tilde{\rho}) \leq 1 \\ P(\rho, \tilde{\rho}) \leq \sqrt{2\varepsilon}}} \sup_{\sigma_C \in \mathscr{D}(C)} \left\| \tilde{\rho}_{BC}^{\frac{1}{2}} I_{B} \otimes \sigma_C^{\frac{1}{2}} \right\|_1^2\\ &= H_{\max}^{\sqrt{2\varepsilon}}(B|C)_{\rho}. \end{align}\] \(\square\)

Proof.  This fact is used in [60]. It uses the semidefinite program for the fidelity \(\| \sqrt{\omega} \sqrt{\theta} \|_{1}^2 = \min\{ \operatorname{Tr}(Z \theta) : \omega_{AE} \leq Z \otimes I_E, Z \geq 0\}\) [61], where \(\omega_{AE}\) is a purification of \(\omega\). Let \(\rho_{AE}\) be a purification of \(\rho_A\). Then we have \(\sqrt{M_A} \rho_{AE} \sqrt{M_A} \leq M_A \otimes I_{E}\) and \(\sqrt{M_A} \rho_{AE} \sqrt{M_A}\) is a purification of \(\sqrt{M} \rho_A \sqrt{M}\). As a result \(Z = M\) is feasible for the semidefinite program above and we get the desired result. \(\square\)

References↩︎