1 Introduction↩︎

Quantum information decoupling addresses the challenge of eliminating correlations between a local system and its environment through quantum evolution. This task involves transforming a bipartite quantum state \(\rho_{\mathsf{A}\mathsf{E}}\) by applying a unitary operation on system \(\mathsf{A}\), followed by a decoupling map \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\), such that the the resulting system \(\mathsf{C}\) becomes independent to the environment \(\mathsf{E}\). A prominent special case of this theory is standard quantum information decoupling, when \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\) is given by the partial trace over a subsystem of \(\mathsf{A}\). As a fundamental structural pillar in quantum information theory, decoupling provides the theoretical foundation for numerous landmark results. Its applications are widespread, ranging from quantum state merging [1]–[3], quantum channel simulation [4]–[6], entanglement distillation [7], [8], to quantum channel coding [9]–[11].

In many physically relevant scenarios, decoupling is considered in the absence of auxiliary resources, where the unitary operation is drawn from the Haar measure and no additional catalytic systems are available. The performance of such a scheme is quantified by a divergence measuring the residual correlations between \(\mathsf{C}\) and \(\mathsf{E}\). A substantial body of work has been devoted to the trace distance and purified distance criteria. In particular, one-shot upper bounds on the decoupling error under the trace distance were derived for general maps \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\), expressed in terms of smooth conditional min-entropies [12]. While the analysis suffices for proving standard coding theorems, these smooth-entropy-based bounds necessarily involve non-negligible fudge terms. As a consequence, they do not yield exact non-asymptotic error exponent characterizations and are primarily tailored to first- and second-order asymptotics. To address this issue, Cheng et al. [13] recently strengthened the one-shot bound by expressing it in terms of sandwiched conditional Rényi entropies, thereby clarifying the associated achievable error exponents. However, even in this refined form, the analysis remains tied to the trace distance and does not provide a sharp characterization under quantum relative entropy (or purified distance).

Quantum relative entropy plays a central role in quantum information theory. Beyond its operational significance, bounds formulated under relative entropy can be converted into purified-distance statements via standard entropy-fidelity inequalities, thereby enabling Uhlmann-type arguments that are indispensable across a wide range of fully quantum applications. Consequently, for tasks such as quantum state merging and quantum channel coding, relative entropy thus provides a robust performance criterion. Despite its foundational importance, quantum information decoupling under the relative entropy has remained comparatively unexplored. In particular, general one-shot bounds — analogous to the well-established results for trace distance — have been notably absent from the literature. This gap constitutes a genuine technical bottleneck: Without a sharp one-shot bound formulated directly in terms of the relative entropy, a precise characterization of exact finite-blocklength error exponents remained out of reach.

In this work, we overcome this bottleneck by leveraging a recently developed trace inequality [14] to establish a general one-shot upper bound on the decoupling error measured by the quantum relative entropy. Our bound takes an exact exponential form without the need for smoothing and remains valid for arbitrary blocklengths. Informally speaking, for arbitrary decoupling channels \(\mathcal{T}_{\mathsf{A}\to \mathsf{C}}\) with Choi state \(\omega_{\mathsf{A}\mathsf{C}}\) and bipartite states \(\rho_{\mathsf{A}\mathsf{E}}\), we prove that \[\mathbb{E}_{\mathbb{U}(\mathsf{A})} D(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}} (U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*)\|\omega_{\mathsf{C}} \otimes \rho_{\mathsf{E}} ) \leq \inf_{0<s\leq1} \frac{s^s(1-s)^{1-s}}{s} 2^{-s\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho-s\widetilde{H}_{1+s}(\mathsf{A}'|\mathsf{C})_\omega},\] where \(D(\cdot\|\cdot)\) is Umegaki’s relative entropy [15], \(\mathbb{E}_{\mathbb{U}(\mathsf{A})}\) denotes the integration over the Haar measure over the unitary group \(\mathbb{U}(\mathsf{A})\), and \(\widetilde{H}_{\alpha}(\mathsf{A}|\mathsf{E})_\rho\) is the sandwiched conditional Rényi entropy defined later in Eq. 5 . A key structural feature is that the bound is directly expressed in terms of the exponents, is fully non-asymptotic, and involves no smoothing parameters or additive fudge terms. Consequently, in the \(n\)-fold product setting, the induced error exponent is valid for every blocklength \(n\), and is positive whenever the sum-entropy criterion [12], \(\widetilde{H}_{1}(\mathsf{A}|\mathsf{E})_\rho+\widetilde{H}_{1}(\mathsf{A}'|\mathsf{C})_\omega>0\), is satisfied. This stricture is particularly appealing for finite-resource quantum regimes and consequently for applications in physics.

For the standard decoupling scenario, we further establish a lower bound on the one-shot relative entropy decoupling error using the pinching technique combined with an operator inequality. This lower bound is expressed through an information-spectrum quantity, allowing us to obtain an ensemble-tight converse bound that matches the achievable error exponent, provided the subsystem being removed grows at a rate below a certain critical rate. Hence, in the most operationally significant low-cost-rate regime, we determine the exact error exponent of standard quantum decoupling under the relative entropy criterion. While it remains an open question whether this ensemble tightness can be elevated to a fully general converse independent of the Haar ensemble, the appearance of a critical rate is fundamentally analogous to phenomena observed in classical data compression with quantum side information [16], [17], classical-quantum channel coding [18]–[21], privacy amplification [22], [23] and catalytic decoupling [1].

Our general framework applies in particular to decoupling maps implemented by partial isometries. In this setting, we derive an explicit error exponent for quantum state merging [2], [3], [24], when the entanglement cost rate is not too high. Furthermore, a similar approach also provides an achievability bound on the error exponents for entanglement distillation assisted by local operation and classical communication (LOCC) and a broad class of quantum communication tasks, including subspace transmission, entanglement transmission, entanglement generation as well as their one-way LOCC-assisted counterparts. Our bounds offer strong performance guarantees for both unassisted and LOCC-assisted protocols: Specifically, for coding rates below the first-order asymptotic capacity, we show that the error decays exponentially for every blocklength \(n\). This provides a stronger large-deviation characterization than that offered by conventional first-order analyses. Notably, while broader axiomatic classes of operations exist, our focus remains on the physically motivated unassisted and LOCC-assisted scenarios. For certain classes of bipartite states and quantum channels, we establish corresponding exact error exponents. These applications demonstrate both the versatility of our framework and the analytical advantages of a relative entropy based approach. A comprehensive summary of the obtained error exponents is provided in Table ¿tbl:table:main?.

Relation to previous work. The closest related result is due to Cheng et al. [13] who established a one-shot upper bound on the decoupling error under the trace distance for arbitrary decoupling channels \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\) as \[\label{eq:previous} \frac{1}{2}\mathbb{E}_{\mathbb{U}(\mathsf{A})} \|\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}} (U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*)-\omega_{\mathsf{C}} \otimes \rho_{\mathsf{E}} \|_1 \leq\inf_{\alpha \in [1,2]}2^{\frac{1-\alpha}{\alpha}\left(\widetilde{H}_\alpha^{\uparrow}(\mathsf{A}|\mathsf{E})_\rho+\widetilde{H}_\alpha^{\uparrow}(\mathsf{A}'|\mathsf{C})_\omega+\log 3\right)}.\tag{1}\] Such one-shot bounds imply achievability results for the error exponent under the trace-distance. However, the absence of corresponding converse bounds prevents a full characterization of the error exponent. In contrast, focusing on the quantum relative entropy — a more demanding criterion for measuring errors — our work establishes both achievability and ensemble-tight converse bounds for standard decoupling and shows that their coincidence in the low-rate regime. This yields the first exact characterization of the error exponent of quantum information decoupling under the quantum relative entropy. We also remark that by reverse Pinsker’s inequality, the error exponent established in Eq. 1 does not cover ours (see Remark 5).

A recent work [25] also studies the achievability of quantum information decoupling under quantum relative entropy, where the error bound is only obtained for \(\alpha = 2\). However, no tight exponent characterization was derived there, and achieving the general Rényi parameter regime \(\alpha \in (1,2)\) remained open. Another line of work studies catalytic decoupling, where auxiliary systems assist the protocol. Tight one-shot characterizations, second-order asymptotics and large deviation analysis have been obtained under purified distance [1], [26]–[28]. However, these results rely on catalytic assistance and do not address the structureless setting considered here.

While recent work [29] establishes the exact error exponent for entanglement distillation assisted by the more powerful class of non-entangling operations, we focus here on the physically motivated LOCC-assisted setting. Since the non-entangling distillation capacity can strictly exceed that achievable under LOCC [30], our achievability bound may at first sight appear more conservative than the corresponding result in [29]. However, the converse analysis of [29] continues to provide a valid upper bound in the LOCC regime. By combining these two viewpoints, we show that for maximally correlated states the optimal error exponent in the high-distillation-rate regime is already achievable using one-way LOCC.
Outline. The remainder of this paper is organized as follows. In Section 2, we introduce the notations and definitions used throughout the paper. Section 3 formalizes the quantum information decoupling problem and presents our main results. Applications of these results to quantum state merging, entanglement distillation and several quantum communication tasks are discussed in Sections 4, 5 and 6, respectively. Finally, Section 7 concludes the paper with a discussion and several open problems.

2 Preliminaries↩︎

2.1 Notation↩︎

For a finite-dimensional Hilbert space \(\mathcal{H}\), let \(\mathcal{L}(\mathcal{H})\) denote the set of all linear operators acting on \(\mathcal{H}\) and let \(\mathcal{P}(\mathcal{H})\) be the set of the positive semi-definite operators on \(\mathcal{H}\). The set of normalized and sub-normalized quantum states on \(\mathcal{H}\) are defined respectively as \[\begin{align} \mathcal{S}(\mathcal{H})&=\{ \rho \in \mathcal{P}(\mathcal{H})~|~\operatorname{Tr}\rho=1\},\\ \mathcal{S}_{\leq}(\mathcal{H})&=\{ \rho \in \mathcal{P}(\mathcal{H})~|~\operatorname{Tr}\rho \leq 1\}. \end{align}\] We use \(|\mathcal{H}|\) to denote the dimension of \(\mathcal{H}\), and \(I_{\mathcal{H}}\) for the identity operator on \(\mathcal{H}\). When \(\mathcal{H}\) is associated with a quantum system \(\mathsf{A}\), the above notations \(\mathcal{L}(\mathcal{H})\), \(\mathcal{P}(\mathcal{H})\), \(\mathcal{S}(\mathcal{H})\), \(\mathcal{S}_{\leq}(\mathcal{H})\), \(|\mathcal{H}|\) and \(I_{\mathcal{H}}\) also can be written as \(\mathcal{L}(\mathsf{A})\), \(\mathcal{P}(\mathsf{A})\), \(\mathcal{S}(\mathsf{A})\), \(\mathcal{S}_{\leq}(\mathcal{H})\), \(|\mathsf{A}|\) and \(I_{\mathsf{A}}\), respectively.

For an operator \(X \in \mathcal{L}(\mathcal{H})\), \({\operatorname{supp}}(X)\) denotes the support of \(X\). For \(A, B \in \mathcal{P}(\mathcal{H})\), \(\{A \geq B\}\) denotes the projection onto the positive part of \(A-B\), namely the subspace spanned by the eigenvectors corresponding to the non-negative eigenvalues of \(A-B\), \(\{A>B\}\), \(\{A \leq B\}\) and \(\{A<B\}\) are defined analogously. A bipartite state \(\rho_{\mathsf{A}\mathsf{B}} \in \mathcal{S}(\mathsf{A}\mathsf{B})\) is called maximally correlated if there exist orthonormal bases \(\{|a_x\rangle\}_x\) and \(\{|b_x\rangle\}_x\) of \(\mathcal{H}_{\mathsf{A}}\) and \(\mathcal{H}_{\mathsf{B}}\) such that \({\operatorname{supp}}(\rho_{\mathsf{A}\mathsf{B}})\) is contained in the subspace spanned by \(\{|a_x\rangle\otimes|b_x\rangle \}_x\).

The purified distance between two quantum states \(\rho, \sigma \in \mathcal{S}(\mathcal{H})\) is given by \[P(\rho,\sigma):=\sqrt{1-F^2(\rho,\sigma)},\] where \(F(\rho,\sigma):=\|\sqrt{\rho}\sqrt{\sigma}\|_1\) is the fidelity function. The trace distance between \(\rho\) and \(\sigma\) is defined as \[d(\rho,\sigma):=\frac{1}{2}\|\rho-\sigma\|_1.\]

For isomorphic systems \(\mathsf{A}\cong \mathsf{A}'\), \(\Phi_{\mathsf{A}\mathsf{A}'}\) denotes the normalized maximally entangled state between \(\mathsf{A}\) and \(\mathsf{A}'\), given by \[\Phi_{\mathsf{A}\mathsf{A}'}=\frac{1}{|\mathsf{A}|} \sum^{|\mathsf{A}|}_{i,j =1}|i\rangle\langle j|_{\mathsf{A}}\otimes |i\rangle\langle j|_{\mathsf{A}'}.\]

A quantum channel \(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}\) is a completely positive and trace-preserving linear map from \(\mathcal{L}(\mathsf{A})\) to \(\mathcal{L}(\mathsf{B})\). For a quantum channel \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\), its normalized Choi state is defined as \[\omega_{\mathsf{A}'\mathsf{C}}=\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(\Phi_{\mathsf{A}\mathsf{A}'}).\] It follows that for any quantum state \(\rho_{\mathsf{A}\mathsf{E}}\), \[\label{equ:choi} \mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(\rho_{\mathsf{A}\mathsf{E}})=|\mathsf{A}|^2 \langle\Phi_{\mathsf{A}\mathsf{A}'}| \omega_{\mathsf{A}'\mathsf{C}}\otimes \rho_{\mathsf{A}\mathsf{E}} |\Phi_{\mathsf{A}\mathsf{A}'}\rangle.\tag{2}\] Let \(A\) be a self-adjoint operator with spectral projections \(P_1, \ldots, P_r\). The pinching channel \(\mathcal{E}_A\) associated with \(A\) is defined as \[\mathcal{E}_A : X\mapsto\sum_{i=1}^r P_i X P_i.\] The pinching inequality [31] states that for any \(\sigma \in \mathcal{P}(\mathcal{H})\), we have \[\sigma \leq v(A) \mathcal{E}_A(\sigma),\] where \(v(A)\) is the number of different eigenvalues of \(A\).

Let \(G\) be a group with unitary representations \(\{U_g\}_{g \in G}\) on the Hilbert space \(\mathcal{H}_{\mathsf{A}}\) and \(\{V_g\}_{g \in G}\) on \(\mathcal{H}_{\mathsf{B}}\). A quantum channel \(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}\) is said to be covariant with respect to \(G\) if \[V_g \mathscr{N}(\cdot) V_g^*=\mathscr{N}(U_g\cdot U_g^*), \quad \forall g \in G.\]

2.2 Quantum divergences↩︎

The Umegaki relative entropy [15], also known as the quantum relative entropy, is one of the most fundamental information measures in quantum information theory. For \(\rho, \sigma \in \mathcal{P}(\mathcal{H})\), it is defined as \[D(\rho\|\sigma):= \begin{cases} \operatorname{Tr}(\rho(\log\rho-\log\sigma)) & \text{ if }{\operatorname{supp}}(\rho)\subseteq{\operatorname{supp}}(\sigma), \\ +\infty & \text{ otherwise.} \end{cases}\] For a bipartite state \(\rho_{\mathsf{A}\mathsf{B}} \in \mathcal{S}(\mathsf{A}\mathsf{B})\), the conditional entropy and coherent information of \(\rho_{\mathsf{A}\mathsf{B}}\) are given by \[\begin{align} H(\mathsf{A}|\mathsf{B})_\rho:&= -\min_{\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})} D(\rho_{\mathsf{A}\mathsf{B}} \| I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}}) \\ &=-D(\rho_{\mathsf{A}\mathsf{B}} \|I_{\mathsf{A}} \otimes\rho_{\mathsf{B}}), \\ I(\mathsf{A}\rangle \mathsf{B})_\rho:&=\min_{\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})} D(\rho_{\mathsf{A}\mathsf{B}} \| I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}})\\ &=D(\rho_{\mathsf{A}\mathsf{B}} \|I_{\mathsf{A}} \otimes\rho_{\mathsf{B}}). \end{align}\]

Quantum Rényi divergences are parameterized families of divergence measures that extend the quantum relative entropy. Among the various proposals, the Petz Rényi divergence [32] and the sandwiched Rényi divergence [33], [34] play a particularly prominent role due to their favorable mathematical properties and well-established operational interpretations in quantum information theory.

When \(\alpha\) tends to \(1\), \(D_\alpha(\rho\|\sigma)\) and \(D_\alpha^*(\rho\|\sigma)\) converge to the quantum relative entropy and when \(\alpha\) goes to infinity, \(D_\alpha^*(\rho\|\sigma)\) converges to the max-relative entropy [35] \[D_{\rm{max}}(\rho\|\sigma):=\inf\{\lambda~|~\rho \leq 2^\lambda \sigma\}.\]

Based on these Rényi divergences, one can define Rényi generalizations of conditional entropy and coherent information, which are widely used in the analysis of non-asymptotic quantum information theory. For \(\alpha\in (0,\infty)\setminus \{1\}\) and \(\rho_{\mathsf{A}\mathsf{B}} \in \mathcal{S}(\mathsf{A}\mathsf{B})\), the Petz and sandwiched conditional Rényi entropies are defined as \[\begin{align} \tag{3} H^{\uparrow}_\alpha(\mathsf{A}|\mathsf{B})_\rho:&=-\inf_{\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})} D_{\alpha}(\rho_{\mathsf{A}\mathsf{B}}\|I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}}), \\ \tag{4} H_\alpha(\mathsf{A}|\mathsf{B})_\rho:&= D_{\alpha}(\rho_{\mathsf{A}\mathsf{B}}\|I_{\mathsf{A}} \otimes\rho_{\mathsf{B}}), \\ \tag{5} \widetilde{H}^{\uparrow}_\alpha(\mathsf{A}|\mathsf{B})_\rho:&=-\inf_{\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})} \widetilde{D}_{\alpha}(\rho_{\mathsf{A}\mathsf{B}}\|I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}}), \\ \widetilde{H}_\alpha(\mathsf{A}|\mathsf{B})_\rho:&= \widetilde{D}_{\alpha}(\rho_{\mathsf{A}\mathsf{B}}\|I_{\mathsf{A}} \otimes\rho_{\mathsf{B}}). \end{align}\] Similarly, the Petz and sandwiched Rényi coherent information are defined as \[\begin{align} \tag{6} I_\alpha(\mathsf{A}\rangle \mathsf{B})_\rho:&=\inf_{\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})} D_{\alpha}(\rho_{\mathsf{A}\mathsf{B}}\|I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}}), \\ \tag{7} \widetilde{I}_\alpha(\mathsf{A}\rangle \mathsf{B})_\rho:&=\inf_{\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})} \widetilde{D}_{\alpha}(\rho_{\mathsf{A}\mathsf{B}}\|I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}}). \end{align}\]

In the following proposition, we summarize several important properties of these quantum Rényi divergences.

3 Quantum information decoupling↩︎

3.1 Achievability bound for quantum information decoupling↩︎

Let \(\rho_{\mathsf{A}\mathsf{E}} \in \mathcal{S}(\mathsf{A}\mathsf{E})\) be a bipartite state. Quantum information decoupling aims to suppress correlations between systems \(\mathsf{A}\) and \(\mathsf{E}\) by applying a random unitary transformation on \(\mathsf{A}\), followed by a quantum channel \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\).

For a given state \(\rho_{\mathsf{A}\mathsf{E}}\), we quantify the decoupling performance of \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\) by the expected quantum relative entropy between the resulting state and the ideal product state. Specifically, the decoupling error is defined as \[\epsilon_{\rm{dec}}(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}},\rho_{\mathsf{A}\mathsf{E}}) :=\mathbb{E}_{\mathbb{U}(\mathsf{A})} D(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*)\|\omega_{\mathsf{C}} \otimes \rho_{\mathsf{E}}),\] where the expectation is taken with respect to the Haar measure on the unitary group \(\mathbb{U}(\mathsf{A})\), and \(\omega_{\mathsf{C}}:=\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(\frac{I_{\mathsf{A}}}{|\mathsf{A}|})\) denotes the output of the maximally mixed state on \(A\) under \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\).

In this subsection, we study the behavior of \(\epsilon_{\rm{dec}}(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}},\rho_{\mathsf{A}\mathsf{E}})\) for any completely positive and trace non-increasing map \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}\). Our main result establishes a one-shot upper bound on the decoupling error.

Proof. Firstly, by direct calculation, we have \[\label{equ:gendec1} \begin{align} &\epsilon_{\rm{dec}}(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}},\rho_{\mathsf{A}\mathsf{E}}) \\ =&\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \operatorname{Tr}\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \log \mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}} (U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*)-\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \operatorname{Tr}\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \log \omega_{\mathsf{C}}\otimes\rho_{\mathsf{E}} \\ =&\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \operatorname{Tr}\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \log \mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}} (U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*)-\operatorname{Tr}\omega_{\mathsf{C}}\otimes\rho_{\mathsf{E}}\log \omega_{\mathsf{C}}\otimes\rho_{\mathsf{E}}. \end{align}\tag{8}\] Next, we use Eq. (2 ) to evaluate the first term in Eq. (8 ): \[\label{equ:pregiv} \begin{align} &\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \operatorname{Tr}\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \log \mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}} (U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \\ =&\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} |\mathsf{A}|^2\operatorname{Tr}\langle\Phi_{\mathsf{A}\mathsf{A}'}| U_{\mathsf{A}}\rho_{\mathsf{A}\mathsf{E}}U_{\mathsf{A}}^*\otimes\omega_{\mathsf{A}'\mathsf{C}} |\Phi_{\mathsf{A}\mathsf{A}'}\rangle \log |\mathsf{A}|^2\operatorname{Tr}\langle\Phi_{\mathsf{A}\mathsf{A}'}| U_{\mathsf{A}}\rho_{\mathsf{A}\mathsf{E}}U_{\mathsf{A}}^*\otimes\omega_{\mathsf{A}'\mathsf{C}} |\Phi_{\mathsf{A}\mathsf{A}'}\rangle \\ =&\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} |\mathsf{A}|^2\operatorname{Tr}(U_{\mathsf{A}}\rho_{\mathsf{A}\mathsf{E}}U_{\mathsf{A}}^*\otimes\omega_{\mathsf{A}'\mathsf{C}}) ( \Phi_{\mathsf{A}\mathsf{A}'}\otimes\log |\mathsf{A}|^2\operatorname{Tr}\langle\Phi_{\mathsf{A}\mathsf{A}'}| U_{\mathsf{A}}\rho_{\mathsf{A}\mathsf{E}}U_{\mathsf{A}}^*\otimes\omega_{\mathsf{A}'\mathsf{C}} |\Phi_{\mathsf{A}\mathsf{A}'}\rangle ) \end{align}\tag{9}\] Because for any pure state \(\phi_{\mathsf{A}} \in \mathcal{S}(\mathsf{A})\) and \(T_{\mathsf{B}} \in \mathcal{P}(\mathsf{B})\), it holds that \[\label{equ:purel} \phi_{\mathsf{A}} \otimes \log T_{\mathsf{B}}=\log \phi_{\mathsf{A}} \otimes T_{\mathsf{B}}.\tag{10}\] Eq. (9 ) and Eq. (10 ) imply that \[\label{equ:cyc} \begin{align} &\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \operatorname{Tr}\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \log \mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}} (U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \\ =& \mathbb{E}_{\mathbb{U}_{\mathsf{A}}} |\mathsf{A}|^2\operatorname{Tr}(U_{\mathsf{A}}\rho_{\mathsf{A}\mathsf{E}}U_{\mathsf{A}}^*\otimes\omega_{\mathsf{A}'\mathsf{C}}) ( \log \Phi_{\mathsf{A}\mathsf{A}'}\otimes|\mathsf{A}|^2\operatorname{Tr}\langle\Phi_{\mathsf{A}\mathsf{A}'}| U_{\mathsf{A}}\rho_{\mathsf{A}\mathsf{E}}U_{\mathsf{A}}^*\otimes\omega_{\mathsf{A}'\mathsf{C}} |\Phi_{\mathsf{A}\mathsf{A}'}\rangle ) \\ =& \mathbb{E}_{\mathbb{U}_{\mathsf{A}}} |\mathsf{A}|^2\operatorname{Tr}(\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}) ( \log U_{\mathsf{A}}^*\Phi_{\mathsf{A}\mathsf{A}'}U_{\mathsf{A}}\otimes|\mathsf{A}|^2\operatorname{Tr}\langle\Phi_{\mathsf{A}\mathsf{A}'}| U_{\mathsf{A}}\rho_{\mathsf{A}\mathsf{E}}U_{\mathsf{A}}^*\otimes\omega_{\mathsf{A}'\mathsf{C}} |\Phi_{\mathsf{A}\mathsf{A}'}\rangle ) \\ =& \mathbb{E}_{\mathbb{U}_{\mathsf{A}}} |\mathsf{A}|^2\operatorname{Tr}\left(\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}\right) \left( \log|\mathsf{A}|^2 \operatorname{Tr}_{\tilde{\mathsf{A}}\bar{\mathsf{A}}} (U_{\mathsf{A}}^*\Phi_{\mathsf{A}\mathsf{A}'}U_{\mathsf{A}}\otimes\Phi_{\tilde{\mathsf{A}}\bar{\mathsf{A}}}\otimes I_{\mathsf{C}\mathsf{E}})( I_{\mathsf{A}\mathsf{A}'}\otimes U_{\tilde{\mathsf{A}}}\rho_{\tilde{\mathsf{A}}\mathsf{E}}U_{\tilde{\mathsf{A}}}^*\otimes\omega_{\bar{\mathsf{A}}\mathsf{C}}) \right) \\ =& \mathbb{E}_{\mathbb{U}_{\mathsf{A}}} |\mathsf{A}|^2\operatorname{Tr}\left(\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}\right) \left( \log|\mathsf{A}|^2 \operatorname{Tr}_{\tilde{\mathsf{A}}\bar{\mathsf{A}}} (U_{\mathsf{A}}^*\Phi_{\mathsf{A}\mathsf{A}'}U_{\mathsf{A}}\otimes U^*_{\tilde{\mathsf{A}}}\Phi_{\tilde{\mathsf{A}}\bar{\mathsf{A}}}U_{\tilde{\mathsf{A}}}\otimes I_{\mathsf{C}\mathsf{E}})( I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\tilde{\mathsf{A}}\mathsf{E}}\otimes\omega_{\bar{\mathsf{A}}\mathsf{C}}) \right) , \end{align}\tag{11}\] where the first and the last equality come from the cyclicity of the trace. Making use of the operator concavity of the logarithm, we can further upper bound Eq. (11 ) as

\[\label{equ:gendec2} \begin{align} &\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \operatorname{Tr}\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \log \mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}} (U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^*) \\ \leq & |\mathsf{A}|^2\operatorname{Tr}\left(\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}\right) \left( \log|\mathsf{A}|^2 \operatorname{Tr}_{\tilde{\mathsf{A}}\bar{\mathsf{A}}} \big(\mathbb{E}_{\mathbb{U}_{\mathsf{A}}} (U_{\mathsf{A}}^*\Phi_{\mathsf{A}\mathsf{A}'}U_{\mathsf{A}}\otimes U^*_{\tilde{\mathsf{A}}}\Phi_{\tilde{\mathsf{A}}\bar{\mathsf{A}}}U_{\tilde{\mathsf{A}}})\otimes I_{\mathsf{C}\mathsf{E}}\big)\big( I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\tilde{\mathsf{A}}\mathsf{E}}\otimes\omega_{\bar{\mathsf{A}}\mathsf{C}}\big) \right) \\ =& |\mathsf{A}|^2\operatorname{Tr}(\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}) \Bigg( \log\operatorname{Tr}_{\tilde{\mathsf{A}}\bar{\mathsf{A}}} \Big( \big(\frac{1}{|\mathsf{A}|^2-1}I_{\mathsf{A}\mathsf{A}'\tilde{\mathsf{A}}\bar{\mathsf{A}}}+\frac{1}{|\mathsf{A}|^2-1}F_{\mathsf{A}\tilde{\mathsf{A}}}\otimes F_{A'\bar{A}}-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}F_{\mathsf{A}\tilde{\mathsf{A}}}\otimes I_{\mathsf{A}'\bar{\mathsf{A}}} \\ &-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}I_{\mathsf{A}\tilde{\mathsf{A}}}\otimes F_{A'\bar{A}}\big)\otimes I_{\mathsf{C}\mathsf{R}}\Big)\big( I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\tilde{\mathsf{A}}\mathsf{E}}\otimes\omega_{\bar{\mathsf{A}}\mathsf{C}}\big) \Bigg) \\ =& |\mathsf{A}|^2\operatorname{Tr}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}} \log (\frac{1}{|\mathsf{A}|^2-1}I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\mathsf{E}}\otimes\omega_{\mathsf{C}}+\frac{1}{|\mathsf{A}|^2-1}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}\rho_{\mathsf{A}\mathsf{E}}\otimes I_{\mathsf{A}'}\otimes\omega_{\mathsf{C}}\\ &-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}I_{\mathsf{A}}\otimes\rho_{\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}) \\ \leq &\operatorname{Tr}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}} \log (\frac{1}{|\mathsf{A}|^2-1}I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\mathsf{R}}\otimes\omega_{\mathsf{C}}+\frac{1}{|\mathsf{A}|^2-1}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}\rho_{\mathsf{A}\mathsf{E}}\otimes I_{\mathsf{A}'}\otimes\omega_{\mathsf{C}}\\ &-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}I_{\mathsf{A}}\otimes\rho_{\mathsf{R}}\otimes\omega_{\mathsf{A}'\mathsf{C}}) \\ \leq &\operatorname{Tr}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}} \log (I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\mathsf{E}}\otimes\omega_{\mathsf{C}}+\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}) , \end{align}\tag{12}\] where the second inequality is because that \(\frac{1}{|\mathsf{A}|^2-1}I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\mathsf{R}}\otimes\omega_{\mathsf{C}}+\frac{1}{|\mathsf{A}|^2-1}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}\rho_{\mathsf{A}\mathsf{E}}\otimes I_{\mathsf{A}'}\otimes\omega_{\mathsf{C}} -\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}I_{\mathsf{A}}\otimes\rho_{\mathsf{R}}\otimes\omega_{\mathsf{A}'\mathsf{C}}\) is a subnormalized state, \(\operatorname{Tr}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}} \log (\frac{1}{|\mathsf{A}|^2-1}I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\mathsf{R}}\otimes\omega_{\mathsf{C}}+\frac{1}{|\mathsf{A}|^2-1}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}-\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}\rho_{\mathsf{A}\mathsf{E}}\otimes I_{\mathsf{A}'}\otimes\omega_{\mathsf{C}} -\frac{1}{|\mathsf{A}|^3-|\mathsf{A}|}I_{\mathsf{A}}\otimes\rho_{\mathsf{R}}\otimes\omega_{\mathsf{A}'\mathsf{C}}) \leq 0\), and the last inequality is from the operator monotonicity of the logarithm.

Combining Eq. (8 ), Eq. (12 ) with Lemma 2, we conclude that for any \(s\in (0,1]\), \[\begin{align} & \epsilon_{\rm{dec}}(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{C}},\rho_{\mathsf{A}\mathsf{E}}) \\ \leq & \operatorname{Tr}\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}} \log (I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\mathsf{E}}\otimes\omega_{\mathsf{C}}+\rho_{\mathsf{A}\mathsf{E}}\otimes\omega_{\mathsf{A}'\mathsf{C}}) -\operatorname{Tr}\rho_{\mathsf{A}\mathsf{E}} \otimes\omega_{\mathsf{A}'\mathsf{C}} \log I_{\mathsf{A}\mathsf{A}'}\otimes\rho_{\mathsf{E}}\otimes\omega_{\mathsf{C}} \\ \leq & \frac{s^s(1-s)^{1-s}}{s} 2^{s\left(\widetilde{D}_{1+s}(\rho_{\mathsf{A}\mathsf{E}}\|I_{\mathsf{A}} \otimes\rho_{\mathsf{E}})+\widetilde{D}_{1+s}(\omega_{\mathsf{A}'\mathsf{C}}\|I_{\mathsf{A}'}\otimes\omega_{\mathsf{C}})\right)} \\ =& \frac{s^s(1-s)^{1-s}}{s} 2^{-s\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho-s\widetilde{H}_{1+s}(\mathsf{A}'|\mathsf{C})_\omega}. \end{align}\] We complete the proof. \(\sqcup\)

As a direct consequence of Theorem 2, we obtain an achievability bound on the error exponent of quantum information decoupling, defined as the asymptotic exponential rate of decay of the decoupling error \(\epsilon_{\rm{dec}}(\mathcal{T}^{\otimes n}_{\mathsf{A}\rightarrow \mathsf{C}},\rho_{\mathsf{A}\mathsf{E}}^{\otimes n})\) in the i.i.d. limit.

Proof. By Theorem 2, for any \(s \in (0,1)\) and \(n \in \mathbb{N}\), we have \[\label{equ:achi} \epsilon_{\rm{dec}}(\mathcal{T}^{\otimes n}_{\mathsf{A}\rightarrow \mathsf{C}}, \rho_{\mathsf{A}\mathsf{E}}^{\otimes n}) \\ \leq \frac{s^s(1-s)^{1-s}}{s} 2^{-ns\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho-ns\widetilde{H}_{1+s}(\mathsf{A}'|\mathsf{C})_\omega}.\tag{13}\] Taking the logarithm, dividing by \(-n\), and letting \(n\) tend to infinity, we observe that the prefactor does not affect the exponential decay rate. Consequently, \[\label{equ:achi2} \lim_{n \rightarrow \infty} \frac{-1}{n} \log \epsilon_{\rm{dec}}(\mathcal{T}^{\otimes n}_{\mathsf{A}\rightarrow \mathsf{C}}, \rho_{\mathsf{A}\mathsf{E}}^{\otimes n}) \geq s(\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho+\widetilde{H}_{1+s}(\mathsf{A}'|\mathsf{C})_\omega).\tag{14}\] Since Eq. (14 ) holds for \(s \in (0,1)\), taking the supremum over \(s\) completes the proof. \(\sqcup\)

3.2 Converse bound for standard quantum information decoupling↩︎

In subsection 3.1, we derived a one-shot upper bound on the decoupling error for general quantum channels and obtained an achievable error exponent in the i.i.d. limit. In this subsection, we turn to the standard setting of quantum information decoupling, where the decoupling map is given by a partial trace.

Specifically, let \(\mathcal{H}_{\mathsf{A}}=\mathcal{H}_{\mathsf{A}_1} \otimes \mathcal{H}_{\mathsf{A}_2}\) be a Hilbert space decomposition, and consider the map \(\mathcal{T}_{\mathsf{A}\rightarrow \mathsf{A}_1}=\operatorname{Tr}_{\mathsf{A}_2}\). Our first result establishes a one-shot lower bound on the decoupling error, which serves as a converse bound in this setting.

Proof. Let the spectral decomposition of \(\rho_{\mathsf{E}}\) be given by \[\rho_{\mathsf{E}}=\sum_{i \in I} \lambda_iE_i,\] where \(\{E_i\}_{i \in I}\) are the spectral projections.

From Lemma 8, we have \[\label{equ:low} \begin{align} \epsilon_{\rm{dec}}(\operatorname{Tr}_{\mathsf{A}_2},\rho_{\mathsf{A}\mathsf{E}}) &\geq \mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \operatorname{Tr}(\operatorname{Tr}_{\mathsf{A}_2}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^\ast) - 9\frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \rho_{\mathsf{E}} )_+ \\ & =\mathbb{E}_{\mathbb{U}_{\mathsf{A}}}\operatorname{Tr}\left( \operatorname{Tr}_{\mathsf{A}_2}(U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^\ast) \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} - 9\frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} \otimes \rho_{\mathsf{E}} \right)_{+} \\ &= \mathbb{E}_{\mathbb{U}_{\mathsf{A}}}\operatorname{Tr}\left( \frac{1}{|\mathsf{A}_2|^2} \sum_{k=1}^{|\mathsf{A}_2|^2} (U_k U_{\mathsf{A}} \rho_{\mathsf{A}\mathsf{E}} U_{\mathsf{A}}^\ast U_k^\ast) - 9\frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} \otimes \rho_{\mathsf{E}} \right)_{+} \\ &\geq \mathbb{E}_{\mathbb{U}_{\mathsf{A}}}\operatorname{Tr}\left( \frac{1}{|\mathsf{A}_2|^2} \sum_{k=1}^{|\mathsf{A}_2|^2} (U_k U_{\mathsf{A}} \mathcal{E}_{\rho_{\mathsf{E}}}(\rho_{\mathsf{A}\mathsf{E}}) U_{\mathsf{A}}^\ast U_k^\ast) - 9\frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} \otimes \rho_{\mathsf{E}} \right)_{+} \\ &=\mathbb{E}_{\mathbb{U}_{\mathsf{A}}}\operatorname{Tr}\left( \frac{1}{|\mathsf{A}_2|^2} \sum_{i \in I} \sum_{k=1}^{|\mathsf{A}_2|^2} (U_k U_{\mathsf{A}} E_i \rho_{\mathsf{A}\mathsf{E}} E_i U_{\mathsf{A}}^\ast U_k^\ast) - 9\sum_{i \in I} \frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} \otimes \lambda_i E_i \right)_{+}, \end{align}\tag{15}\] where \(\{U_k\}_{k=1}^{|\mathsf{A}_2|^2}\) are the Heisenberg–Weyl operators on \(\mathcal{H}_{\mathsf{A}_2}\). Leveraging Lemma 3 , we can further lower bound Eq. (15 ) as \[\begin{align} \epsilon_{\rm{dec}}(\operatorname{Tr}_{\mathsf{A}_2},\rho_{\mathsf{A}\mathsf{E}}) &\geq \mathbb{E}_{\mathbb{U}_{\mathsf{A}}} \sum_{i \in I}\sum_{k=1}^{|\mathsf{A}_2|^2} \operatorname{Tr}\left( \frac{1}{|\mathsf{A}_2|^2} U_k U_{\mathsf{A}} E_i \rho_{\mathsf{A}\mathsf{E}} E_i U_{\mathsf{A}}^\ast U_k^\ast- 9 \frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} \otimes \lambda_i E_i \right)_{+} \\ &=\mathbb{E}_{\mathbb{U}_{\mathsf{A}}}\sum_{k=1}^{|\mathsf{A}_2|^2} \operatorname{Tr}\left( \frac{1}{|\mathsf{A}_2|^2} U_k U_{\mathsf{A}} \mathcal{E}_{\rho_{\mathsf{E}}}(\rho_{\mathsf{A}\mathsf{E}}) U_{\mathsf{A}}^\ast U_k^\ast - 9\frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} \otimes \rho_{\mathsf{E}} \right)_{+} \\ &=\sum_{k=1}^{|\mathsf{A}_2|^2} \operatorname{Tr}\left( \frac{1}{|\mathsf{A}_2|^2} \mathcal{E}_{\rho_{\mathsf{E}}}(\rho_{\mathsf{A}\mathsf{E}}) - 9\frac{I_{\mathsf{A}_1}}{|\mathsf{A}_1|} \otimes \frac{I_{\mathsf{A}_2}}{|\mathsf{A}_2|} \otimes \rho_{\mathsf{E}} \right)_{+} \\ &=\operatorname{Tr}\left( \mathcal{E}_{\rho_{\mathsf{R}}}(\rho_{\mathsf{A}\mathsf{E}}) - 9\frac{|\mathsf{A}_2|}{|\mathsf{A}_1|} I_{\mathsf{A}} \otimes \rho_{\mathsf{E}} \right)_{+} \\ &\geq \frac{1}{v(\rho_{\mathsf{E}})}\operatorname{Tr}\!\left( \rho_{\mathsf{A}\mathsf{E}} - 9v(\rho_{\mathsf{E}})\frac{|\mathsf{A}_2|}{|\mathsf{A}_1|} I_{\mathsf{A}} \otimes \rho_{\mathsf{E}} \right)_{+}, \end{align}\] where the last equality comes from the pinching inequality. \(\sqcup\)

Applying the one-shot lower bound in Theorem 3 to \(n\)-fold i.i.d. states, we obtain a converse bound on the error exponent of standard quantum information decoupling.

Proof. By Theorem 3, for any \(n \in \mathbb{N}\), \[\label{equ:conexp} \begin{align} &\epsilon_{\rm{dec}}(\operatorname{Tr}_{{\mathsf{A}}_2^n},\rho^{\otimes n}_{\mathsf{A}\mathsf{E}}) \\ \geq &\frac{1}{v(\rho_{\mathsf{E}}^{\otimes n})}\operatorname{Tr}\!\left( \rho^{\otimes n}_{\mathsf{A}\mathsf{E}} - 9v(\rho_{\mathsf{E}}^{\otimes n})\frac{|{\mathsf{A}}_2^n|}{|A^n_1|} I_{\mathsf{A}}^{\otimes n} \otimes \rho_{\mathsf{E}}^{\otimes n} \right)_{+} \\ =&\frac{1}{v(\rho_{\mathsf{E}}^{\otimes n})}\operatorname{Tr}\!\left( \rho^{\otimes n}_{\mathsf{A}\mathsf{E}} - 9v(\rho_{\mathsf{E}}^{\otimes n})\frac{2^{2nr}}{|\mathsf{A}|^n} I_{\mathsf{A}}^{\otimes n} \otimes \rho_{\mathsf{E}}^{\otimes n} \right)_{+}. \end{align}\tag{16}\] For any \(\delta>0\), there exists a \(N_\delta \in \mathbb{N}\) such that for any \(n\geq N_\delta\), it holds that \[\frac{\log v(\rho_{\mathsf{E}}^{\otimes n})}{2n} \leq \delta.\] For \(n \geq N_\delta\), we lower bound Eq. 16 as \[\label{equ:revcon} \begin{align} &\epsilon_{\rm{dec}}(\operatorname{Tr}_{{\mathsf{A}}_2^n},\rho^{\otimes n}_{\mathsf{A}\mathsf{E}}) \\ \geq&\frac{1}{v(\rho_{\mathsf{E}}^{\otimes n})}\operatorname{Tr}\!\left( \rho^{\otimes n}_{\mathsf{A}\mathsf{E}} - 9v(\rho_{\mathsf{E}}^{\otimes n})\frac{2^{2nr}}{|\mathsf{A}|^n} I_{\mathsf{A}}^{\otimes n} \otimes \rho_{\mathsf{E}}^{\otimes n} \right)_{+} \\ =&\frac{1}{v(\rho_{\mathsf{E}}^{\otimes n})}\operatorname{Tr}\!\left( \rho^{\otimes n}_{\mathsf{A}\mathsf{E}} - 9\cdot2^{2n(r+\frac{\log v(\rho_{\mathsf{E}}^{\otimes n})}{2n})}\frac{I_{\mathsf{A}}^{\otimes n}}{|\mathsf{A}|^n} \otimes \rho_{\mathsf{E}}^{\otimes n} \right)_{+} \\ \geq &\frac{1}{v(\rho_{\mathsf{E}}^{\otimes n})}\operatorname{Tr}\!\left( \rho^{\otimes n}_{\mathsf{A}\mathsf{E}} - 9\cdot2^{2n(r+\delta)}\frac{I_{\mathsf{A}}^{\otimes n}}{|\mathsf{A}|^n} \otimes \rho_{\mathsf{E}}^{\otimes n} \right)_{+} \end{align}\tag{17}\]

Eq. 17 and Lemma 4 give \[\label{equ:newupper} \begin{align} &\lim_{n \rightarrow \infty} \frac{-1}{n} \log \epsilon_{\rm{dec}}(\operatorname{Tr}_{{\mathsf{A}}_2^n},\rho^{\otimes n}_{\mathsf{A}\mathsf{E}}) \\ \leq &\sup_{s>0} s\left\{2r-\widetilde{D}_{1+s}\left(\rho_{\mathsf{A}\mathsf{E}}\big\|\tfrac{I_{\mathsf{A}}}{|\mathsf{A}|}\otimes \rho_{\mathsf{E}}\right) \right\} \\ =&\sup_{s>0} s\left\{2r-\log|\mathsf{A}|+\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho\right\}. \end{align}\tag{18}\] \(\sqcup\)

For standard quantum information decoupling, Theorem 2 yields an \(n\)-shot upper bound on \(\epsilon_{\rm{dec}}(\operatorname{Tr}_{{\mathsf{A}}_2^n},\rho^{\otimes n}_{\mathsf{A}\mathsf{E}})\): \[\begin{align} \epsilon_{\rm{dec}}(\operatorname{Tr}_{{\mathsf{A}}_2^n},\rho^{\otimes n}_{\mathsf{A}\mathsf{E}}) &\leq \frac{s^s(1-s)^{1-s}}{s} 2^{-ns\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho-s\widetilde{H}_{1+s}({A^n_1}'{{\mathsf{A}}_2^n}'|A_1^n)_\omega} \\ &=\frac{s^s(1-s)^{1-s}}{s} 2^{-ns\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho-s(\log|{\mathsf{A}}_2^n|-\log|A_1^n|)} \\ &=\frac{s^s(1-s)^{1-s}}{s} 2^{-ns\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho+n\log|\mathsf{A}|-2nsr}. \end{align}\] This bound implies the following achievability bound on the error exponent of standard quantum information decoupling: \[\label{equ:nuwlower} \lim_{n \rightarrow \infty} \frac{-1}{n} \log \epsilon_{\rm{dec}}(\operatorname{Tr}_{{\mathsf{A}}_2^n},\rho^{\otimes n}_{\mathsf{A}\mathsf{E}}) \geq \sup_{s \in (0,1)} s\left\{2r-\log|\mathsf{A}|+\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho\right\}.\tag{19}\] We next show that this achievability bound matches the converse bound when the rate \(r\) is below a critical threshold. As a consequence, we obtain an exact characterization of the error exponent in the low-rate regime for standard quantum information decoupling.

Proof. Eq. ?? holds follows from the fact that when \(r \leq R_{\rm{critical}}\), the lower bound of Eq. 19 and the upper bound of Eq. 18 coincide. To see this, consider the function \[f(s)=s\left( 2r-\log|\mathsf{A}|+\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{E})_\rho \right)\] for \(s \in [0,\infty)\). The convexity of \(s \rightarrow s\widetilde{D}_{1+s}(\rho_{\mathsf{A}\mathsf{R}} \| I_{\mathsf{A}}\otimes \rho_{\mathsf{R}})\) shown in [36] implies that \(f(s)\) is concave. So \(\sup_{s \in [0,1]} f(s)=\sup_{s\geq 0} f(s)\) holds if and only if \(f'(1) \leq 0\). This condition is equivalent to \(r \leq R_{\rm{critical}}\). \(\sqcup\)

4 Quantum state merging↩︎

Let \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}\) be a tripartite pure state. Alice, Bob and a referee hold system \(\mathsf{A}\), \(\mathsf{B}\) and \(\mathsf{R}\), respectively. Quantum state merging is the task of transmitting the quantum information contained in system \(\mathsf{A}\) from Alice to Bob, such that Bob eventually holds systems \(\mathsf{A}\) and \(\mathsf{B}\), while preserving the global state with the reference system \(\mathsf{R}\).

A quantum state merging protocol consists of a shared maximally entangled state \(\Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}\) between Alice and Bob, together with an LOCC channel \(\Lambda_{\mathsf{A}\bar{\mathsf{A}}:\mathsf{B}\bar{\mathsf{B}} \rightarrow \tilde{\mathsf{A}}:\mathsf{A}\mathsf{B}\tilde{\mathsf{B}}}\), which transforms the joint state \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}} \otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}\) into a state close to \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}} \otimes \Phi_{\tilde{\mathsf{A}}\tilde{\mathsf{B}}}\), where \(\Phi_{\tilde{\mathsf{A}}\tilde{\mathsf{B}}}\) is a maximally entangled state. The performance of the protocol is quantified by the purified distance between the final state and the ideal target state \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}} \otimes \Phi_{\tilde{\mathsf{A}}\tilde{\mathsf{B}}}\).

It is well known that in order to achieve asymptotically perfect quantum state merging, entanglement must be consumed when \(H(\mathsf{A}|\mathsf{R})_\rho<0\), whereas entanglement can be distilled when \(H(\mathsf{A}|\mathsf{R})_\rho>0\). Accordingly, in the former case we are concerned with the entanglement cost \(\log|\bar{\mathsf{A}}|-\log|\tilde{\mathsf{A}}|\), while in the latter case we consider the entanglement distillation \(\log|\tilde{\mathsf{A}}|-\log|\bar{\mathsf{A}}|\).

When \(H(\mathsf{A}|\mathsf{R})_\rho<0\), we define the optimal error among all quantum state merging protocols with a fixed entanglement cost \(k\) as \[P^{\rm{merg},c}_{\mathsf{A}\rightarrow \mathsf{B}}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},k):=\min_{\mathcal{M}} P(\mathcal{M}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}),\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}\otimes \psi_{\tilde{\mathsf{A}}\tilde{\mathsf{B}}}),\] where the minimization is taken over all quantum state merging protocols \(\mathcal{M}\) with entanglement cost \(k\). When \(H(\mathsf{A}|\mathsf{R})_\rho>0\), the optimal error \(P^{\rm{merg},d}_{\mathsf{A}\rightarrow \mathsf{B}}(\rho_{ABR},k)\) for a fixed entanglement distillation \(k\) is defined analogously.

For a fixed entanglement cost (distillation) rate \(r\), he error exponent of quantum state merging is defined as the exponential decay rate of \(P^{\rm{merg},c}_{\mathsf{A}^n \rightarrow \mathsf{B}^n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n},nr)\) (\(P^{\rm{merg},d}_{\mathsf{A}^n \rightarrow \mathsf{B}^n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n},nr)\)) as \(n \rightarrow \infty\), i.e., \[\begin{align} &E^{\rm{merg},c}(\rho_{ABR},r):=\liminf_{n \rightarrow \infty} \frac{-1}{n} \log P^{\rm{merg},c}_{\mathsf{A}^n \rightarrow \mathsf{B}^n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n},nr), \\ &E^{\rm{merg},d}(\rho_{ABR},r):=\liminf_{n \rightarrow \infty} \frac{-1}{n} \log P^{\rm{merg},d}_{\mathsf{A}^n \rightarrow \mathsf{B}^n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n},nr). \end{align}\]

In this subsection, we study the error exponent of quantum state merging. We first address the achievability part. The main technical tool is the following application of Theorem 2 to partial isometries.

Proof. Applying Theorem 2, we obtain \[\begin{align} &\mathbb{E}_{\mathbb{U}(\mathsf{A})} D(VU(\rho_{\mathsf{A}\mathsf{R}})U^*V^* \big\| \tfrac{I_{\tilde{\mathsf{A}}}}{|\mathsf{A}|}\otimes \rho_{\mathsf{R}}) \\ \leq &\frac{s^s(1-s)^{1-s}}{s} 2 ^{-s\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_\rho-s\widetilde{H}_{1+s}(\mathsf{A}'|\tilde{\mathsf{A}})_{V\Phi_{\mathsf{A}'\mathsf{A}}V^*}}\\ =&\frac{s^s(1-s)^{1-s}}{s} \frac{|\tilde{\mathsf{A}}|^{1+s}}{|\mathsf{A}|} 2^{-s\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_\rho}, \end{align}\] where the equality follows from \(-s\widetilde{H}_{1+s}(\mathsf{A}'|\tilde{\mathsf{A}})_{V\Phi_{\mathsf{A}'\mathsf{A}}V^*}=\log\frac{|\tilde{\mathsf{A}}|^{1+s}}{|\mathsf{A}|}\). \(\sqcup\)

With Lemma 1 in hand, we can establish an achievability bound on the error exponent for quantum state merging stated as follows.

Proof. We consider the two cases separately.

Case 1: \(H(\mathsf{A}|\mathsf{R})_\rho>0\);

Let \(A_{n,r}\) denote the smallest integer larger than \(|\mathsf{A}|^n\) that is divisible by \(2^{nr}\), \(\mathcal{H}_{\bar{\mathsf{A}}_n}\), \(\mathcal{H}_{\tilde{\mathsf{A}}_n}\) be the Hilbert space of dimension \(|\bar{\mathsf{A}}_n|=A_{n,r}\), \(|\tilde{\mathsf{A}}_n|=2^{nr}\), respectively and \(V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}\) be an isometry from \(\mathcal{H}_{\mathsf{A}}^{\otimes n}\) to \(\mathcal{H}_{\bar{\mathsf{A}}_n}\).

We decompose \(\mathcal{H}_{\bar{\mathsf{A}}_n}\) into a direct sum of mutually orthogonal subspaces \(\{\mathcal{H}_{\mathsf{A}_i}\}_{i=1}^{\frac{A_{n,r}}{2^{nr}}}\), each of dimension \(2^{nr}\). For each \(i\), let \(V_i\) be a partial isometry from \(\mathcal{H}_{\mathsf{A}_i}\) to \(\mathcal{H}_{\tilde{\mathsf{A}}_n}\).

Applying Lemma 1 to each \(V_i\), we obtain \[\label{equ:upper} \begin{align} &\mathbb{E}_{\mathbb{U}(\bar{\mathsf{A}}_n)} \sum_{i=1}^{\frac{A_{n,r}}{2^{nr}}} D\left(V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^* \big\| \tfrac{I_{\tilde{\mathsf{A}}_n}}{A_{n,r}}\otimes \rho_{\mathsf{R}}^{\otimes n}\right) \\ \leq &\sum_{i=1} ^{\frac{A_{n,r}}{2^{nr}}} \frac{s^s(1-s)^{1-s}}{s} \frac{2^{nr(1+s)}}{A_{n,r}} 2^{-s\widetilde{H}_{1+s}(\bar{\mathsf{A}}_n|R^n)_{V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}\rho^{\otimes n}V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}}} \\ =&\sum_{i=1} ^{\frac{A_{n,r}}{2^{nr}}} \frac{s^s(1-s)^{1-s}}{s} \frac{2^{nr(1+s)}}{A_{n,r}} 2^{-sn\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_{\rho}} \\ =& \frac{s^s(1-s)^{1-s}}{s} 2^{nrs}2^{-sn\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_{\rho}} , \end{align}\tag{20}\] where the first equality comes from Proposition () and ().

For each \(i\), define \[p_i=\operatorname{Tr}V_i UV_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*.\] Using the inequality \(D(\rho\|\sigma) \geq P^2(\rho,\sigma)\), we lower bound the first term in 20 as \[\label{equ:lower1} \begin{align} &\mathbb{E}_{\mathbb{U}(\bar{\mathsf{A}}_n)} \sum_{i=1}^{\frac{A_{n,r}}{2^{nr}}} D(V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^* \big\| \tfrac{I_{\tilde{\mathsf{A}}_n}}{A_{n,r}}\otimes \rho_{\mathsf{R}}^{\otimes n}) \\ =&\mathbb{E}_{\mathbb{U}(\bar{\mathsf{A}}_n)} \Big\{\sum_{i=1}^{\frac{A_{n,r}}{2^{nr}}} p_i D\left(\tfrac{1}{p_i}V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^* \big\| \tfrac{I_{\tilde{\mathsf{A}}_n}}{|\tilde{\mathsf{A}}_n|}\otimes \rho_{\mathsf{R}}^{\otimes n}\right) + D\left(\{p_1,\ldots, p_{\frac{A_{n,r}}{2^{nr}}}\} \big \| \{\tfrac{2^{nr}}{A_{n,r}},\ldots, \tfrac{2^{nr}}{A_{n,r}}\}\right) \Big\} \\ \geq& \mathbb{E}_{\mathbb{U}(\bar{\mathsf{A}}_n)} \sum_{i=1}^{\frac{A_{n,r}}{2^{nr}}} p_i D\left(\tfrac{1}{p_i}V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^* \big\| \tfrac{I_{\tilde{\mathsf{A}}_n}}{|\tilde{\mathsf{A}}_n|}\otimes \rho_{\mathsf{R}}^{\otimes n}\right) \\ \geq & \mathbb{E}_{\mathbb{U}(\bar{\mathsf{A}}_n)} \sum_{i=1}^{\frac{|\mathsf{A}|^n}{2^{nr}}} p_i P^2\left(\tfrac{1}{p_i}{V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*},\tfrac{I_{\tilde{\mathsf{A}}_n}}{|\tilde{\mathsf{A}}_n|}\otimes \rho_{\mathsf{R}}^{\otimes n}\right) \\ = & \mathbb{E}_{\mathbb{U}(\bar{\mathsf{A}}_n)} \left(1-\sum_{i=1}^{\frac{|\mathsf{A}|^n}{2^{nr}}} p_i F^2\left(\tfrac{1}{p_i}V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*,\tfrac{I_{\tilde{\mathsf{A}}_n}}{|\tilde{\mathsf{A}}_n|}\otimes \rho_{\mathsf{R}}^{\otimes n}\right)\right). \end{align}\tag{21}\] By Ulmman’s theorem, for each \(i\), there exists an isometry \(H^i_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}\) such that \[\label{equ:umequi} \begin{align} &F^2\left(H^i_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}(\tfrac{1}{p_i}{V_i UV_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*}){H^{i*}_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}}, \Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n} \otimes \rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n}\right) \\ =&F^2\left(\tfrac{1}{p_i}{V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*},\tfrac{I_{\tilde{\mathsf{A}}_n}}{|\tilde{\mathsf{A}}_n|}\otimes \rho_{\mathsf{R}}^{\otimes n}\right), \end{align}\tag{22}\] where \(\Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n}\) is the maximally entangled state on \(\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n\).

Combining Eq. (21 ) and Eq. (22 ), and using the concavity of fidelity, we have \[\label{equ:lower2} \begin{align} &\mathbb{E}_{\mathbb{U}(\bar{\mathsf{A}}_n)} \sum_{i=1}^{\frac{A_{n,r}}{2^{nr}}} D\left(V_iU V_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^* \big\| \tfrac{I_{\tilde{\mathsf{A}}_n}}{A_{n,r}}\otimes \rho_{\mathsf{R}}^{\otimes n}\right) \\ \geq &\mathbb{E}_{U \sim \rm{Harr}} \left(1-\sum_{i=1}^{\frac{|\mathsf{A}|^n}{2^{nr}}} p_i F^2\left(H^i_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}(\tfrac{1}{p_I}{V_i UV_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*}){H^{i *}_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}}, \Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n} \otimes \rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n}\right)\right) \\ \geq & \mathbb{E}_{U \sim \rm{Harr}} \left(1- F^2(\sum_{i=1}^{\frac{|\mathsf{A}|^n}{2^{nr}}} H^i_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}(V_i UV_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*){H^{i *}_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}}, \Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n} \otimes \rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n})\right), \end{align}\tag{23}\] Eq. (23 ) and Eq. (20 ) imply that there exists a unitary operator \(U\) such that \[\label{equ:newstmer} \begin{align} &P^{\rm{merg},d}_{\mathsf{A}^n \rightarrow \mathsf{B}^n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n},nr) \\ \leq & P\left(\sum_{i=1}^{\frac{|\mathsf{A}|^n}{2^{nr}}} H^i_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}(V_i UV_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n})V^*_{\mathsf{A}^n \rightarrow \bar{\mathsf{A}}_n}U^*V_i^*)H^{i *}_{\mathsf{B}^n \rightarrow \tilde{\mathsf{B}}_n\mathsf{A}^n\mathsf{B}^n}, \Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n} \otimes \rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n}\right) \\ \leq & \sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{nrs}2^{-sn\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_{\rho}} }. \end{align}\tag{24}\]

From Eq. (24 ), we have

\[E^{\rm{merg},d}(\rho_{ABR},r) \geq \frac{1}{2}\sup_{0<s<1} s\left(\tilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_\rho-r\right)=\frac{1}{2}\sup_{0<s<1} s\left(-H^{\uparrow}_{\frac{1}{1+s}}(\mathsf{A}|\mathsf{B})_\rho-r\right),\] where the equality is due to Proposition ().

Case 2: \(H(\mathsf{A}|\mathsf{R})<0\).

We choose \(r'>r\) and let \(\Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}\) be a maximally entangled state with \(|\bar{\mathsf{A}}|=2^{r'}\). Then \(H(\mathsf{A}\bar{\mathsf{A}}|\mathsf{R})_{\rho \otimes \Phi}>0\) and Case 1 applies to \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n}\otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}^{\otimes n}\). Consequently, there exists a LOCC channel \(\Lambda_{\mathsf{A}^n\bar{\mathsf{A}}^n:\mathsf{B}^n\bar{\mathsf{B}}^n \rightarrow \tilde{\mathsf{A}}_n:\mathsf{A}^nB^n\bar{\mathsf{A}}^n\bar{\mathsf{B}}^n\tilde{\mathsf{B}}_n}\) such that \[\label{equ:larmer} \begin{align} &P\left(\Lambda_{\mathsf{A}^n\bar{\mathsf{A}}^n:\mathsf{B}^n\bar{\mathsf{B}}^n \rightarrow \tilde{\mathsf{A}}_n:\mathsf{A}^nB^n\bar{\mathsf{A}}^n\bar{\mathsf{B}}^n\tilde{\mathsf{B}}_n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n} \otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}^{\otimes n}),\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n} \otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}^{\otimes n} \otimes \Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n}\right) \\ \leq &\sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{n(r'-r)s}2^{-sn\widetilde{H}_{1+s}(\mathsf{A}\bar{\mathsf{A}}|\mathsf{R})_{\rho \otimes \Phi}} }, \end{align}\tag{25}\] where \(\tilde{\mathsf{A}}_n=\tilde{\mathsf{B}}_n=2^{nr'-nr}\). Eq. (25 ) implies that if we apply the LOCC channel \[\operatorname{Tr}_{\bar{A}^n\bar{B}^n} \circ\Lambda_{\mathsf{A}^n\bar{\mathsf{A}}^n:\mathsf{B}^n\bar{\mathsf{B}}^n \rightarrow \tilde{\mathsf{A}}_n:\mathsf{A}^nB^n\bar{\mathsf{A}}^n\bar{\mathsf{B}}^n\tilde{\mathsf{B}}_n}\] to \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n}\otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}^{\otimes n}\), we can obtain \[\label{equ:encost} \begin{align} &P^{\rm{merg},c}_{\mathsf{A}^n \rightarrow \mathsf{B}^n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n},nr) \\ \leq & P\left(\operatorname{Tr}_{\bar{A}^n\bar{B}^n}\circ\Lambda_{\mathsf{A}^n\bar{\mathsf{A}}^n:\mathsf{B}^n\bar{\mathsf{B}}^n \rightarrow \tilde{\mathsf{A}}_n:\mathsf{A}^nB^n\bar{\mathsf{A}}^n\bar{\mathsf{B}}^n\tilde{\mathsf{B}}_n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n} \otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}^{\otimes n}),\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n} \otimes \Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n}\right) \\ \leq &P\left(\Lambda_{\mathsf{A}^n\bar{\mathsf{A}}^n:\mathsf{B}^n\bar{\mathsf{B}}^n \rightarrow \tilde{\mathsf{A}}_n:\mathsf{A}^nB^n\bar{\mathsf{A}}^n\bar{\mathsf{B}}^n\tilde{\mathsf{B}}_n}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n} \otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}^{\otimes n}),\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}^{\otimes n} \otimes \Phi_{\bar{\mathsf{A}}\bar{\mathsf{B}}}^{\otimes n} \otimes \Phi_{\tilde{\mathsf{A}}_n\tilde{\mathsf{B}}_n}\right) \\ \leq & \sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{n(r'-r)s}2^{-sn\widetilde{H}_{1+s}(\mathsf{A}\bar{\mathsf{A}}|\mathsf{R})_{\rho \otimes \Phi}} }\\ =&\sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{-nrs}2^{-sn\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_{\rho }} }. \end{align}\tag{26}\] Eq. (26 ) gives \[E^{\rm{merg},c}(\rho_{ABR},r) \geq \frac{1}{2}\sup_{0<s<1} s\left(\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_\rho+r\right)=\frac{1}{2}\sup_{0<s<1} s\left(-H^{\uparrow}_{\frac{1}{1+s}}(\mathsf{A}|\mathsf{B})_\rho+r\right).\] \(\sqcup\)

In the following, we derive a converse bound on the error exponent for quantum state merging. Our approach relies on the one-shot characterization of the minimal entanglement cost \(E(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\epsilon)\) for merging the state \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}\) within error \(\epsilon\).

It has been shown  that \(E(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\epsilon)\) can be lower bounded by the partially smoothed conditional-min entropy [42]: \[\label{equ:con2} E(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\epsilon) \geq -H^\epsilon_{\rm{min}}(\mathsf{A}|\dot{\mathsf{R}})_\rho\geq -H^\epsilon_{\rm{min}}(\mathsf{A}|\mathsf{R})_\rho,\tag{27}\] where \[\begin{align} H^\epsilon_{\rm{min}}(\mathsf{A}|\dot{\mathsf{R}})_\rho:&=-\inf_{\tilde{\rho}_{\mathsf{A}\mathsf{R}}: P(\tilde{\rho}_{\mathsf{A}\mathsf{R}},\rho_{\mathsf{A}\mathsf{R}}) \leq \epsilon, \tilde{\rho}_R \leq \rho_{\mathsf{R}}} D_{\rm{max}}(\tilde{\rho}_{\mathsf{A}\mathsf{R}}\|I_{\mathsf{A}} \otimes\rho_{\mathsf{R}}), \\ H^\epsilon_{\rm{min}}(\mathsf{A}|\dot{\mathsf{R}})_\rho:&=-\inf_{\tilde{\rho}_{\mathsf{A}\mathsf{R}}: P(\tilde{\rho}_{\mathsf{A}\mathsf{R}},\rho_{\mathsf{A}\mathsf{R}}) \leq \epsilon} D_{\rm{max}}(\tilde{\rho}_{\mathsf{A}\mathsf{R}}\|I_{\mathsf{A}} \otimes\rho_{\mathsf{R}}) \end{align}\] are the partially smoothed conditional-min entropy and smooth conditional-min entropy.

Equivalently, the maximal entanglement distillation for merging the state \(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}}\) within error \(\epsilon\), \(D(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\epsilon)=-E(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\epsilon)\), satisfies \[\label{equ:con1} D(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\epsilon) \leq H^\epsilon_{\rm{min}}(\mathsf{A}|\dot{\mathsf{R}})_\rho\leq H^\epsilon_{\rm{min}}(\mathsf{A}|\mathsf{R})_\rho.\tag{28}\] It follows that \(P^{\rm{merg},c}_{\mathsf{A}\rightarrow \mathsf{B}}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\lambda)\) and \(P^{\rm{merg},d}_{\mathsf{A}\rightarrow \mathsf{B}}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\lambda)\) satisfy \[\begin{align} &P^{\rm{merg},c}_{\mathsf{A}\rightarrow \mathsf{B}}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\lambda)\\ =&\min\{\delta~|~ E(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\delta)\leq \lambda \} \\ \geq &\min\{\delta~|~ H^\delta_{\rm{min}}(\mathsf{A}|\mathsf{R})_\rho\geq -\lambda \} \\ =&\min\{P(\tilde{\rho}_{\mathsf{A}\mathsf{R}},\rho_{\mathsf{A}\mathsf{R}})~|~\tilde{\rho}_{\mathsf{A}\mathsf{R}}\leq 2^{\lambda}I_{\mathsf{A}} \otimes\rho_{\mathsf{R}} \}. \end{align}\] and \[\begin{align} &P^{\rm{merg,d}}_{\mathsf{A}\rightarrow \mathsf{B}}(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\lambda)\\ =&\min\{\delta~|~ D(\rho_{\mathsf{A}\mathsf{B}\mathsf{R}},\delta)\geq \lambda \} \\ \geq &\min\{\delta~|~ H^\delta_{\rm{min}}(\mathsf{A}|\mathsf{R})_\rho\geq \lambda \} \\ =&\min\{P(\tilde{\rho}_{\mathsf{A}\mathsf{R}},\rho_{\mathsf{A}\mathsf{R}})~|~\tilde{\rho}_{\mathsf{A}\mathsf{R}}\leq 2^{-\lambda}I_{\mathsf{A}} \otimes\rho_{\mathsf{R}} \}. \end{align}\] Taking the asymptotic limit, the following error exponent of smooth conditional min-entropy [23] provides a converse bound on the error exponent for quantum state merging: \[\lim_{n \rightarrow \infty} \frac{-1}{n}\log \{\delta~|~ H^\delta_{\rm{min}}(A^n|R^n)_{\rho^{\otimes n}}\geq nr \} =\frac{1}{2}\sup_{s\geq 0} s\left(\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_\rho-r\right).\] As a result, we have \[E^{\rm{merg},c}(\rho_{ABR},r) \leq \frac{1}{2}\sup_{s>0} s\left(\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_\rho+r\right)=\frac{1}{2}\sup_{s>0} s\left(-H^{\uparrow}_{\frac{1}{1+s}}(\mathsf{A}|\mathsf{B})_\rho+r\right),\] and \[E^{\rm{merg},d}(\rho_{ABR},r) \leq \frac{1}{2}\sup_{s>0} s\left(\widetilde{H}_{1+s}(\mathsf{A}|\mathsf{R})_\rho-r\right) =\frac{1}{2}\sup_{s>0} s\left(-H^{\uparrow}_{\frac{1}{1+s}}(\mathsf{A}|\mathsf{B})_\rho-r\right).\]

These converse bounds match the achievability bounds when the entanglement cost rate is below a critical value or the entanglement distillation rate is above a critical point, yielding the exact error exponent in the low- and high-rate regimes.

5 Entanglement distillation↩︎

Let \(\rho_{\mathsf{A}\mathsf{B}} \in \mathcal{S}(\mathsf{A}\mathsf{B})\) be a bipartite quantum state shared by Alice and Bob. In the resource theory of entanglement, entanglement distillation aims at transforming \(\rho_{\mathsf{A}\mathsf{B}}\) into a maximally entangled state by means of a prescribed class of free operations.

We consider three classes of free operations: one-way LOCC, two-way LOCC, and non-entangling channels, i.e., quantum channels that preserve the set of separable states. We denote the corresponding sets by \(\mathcal{F}_{\rm{\rightarrow}}\), \(\mathcal{F}_{\leftrightarrow}\) and \(\mathcal{F}_{\rm{NE}}\), respectively, and define \(\mathcal{G}:=\{\rightarrow,\leftrightarrow,\rm{NE}\}\).

For \(t \in \mathcal{G}\), an \(t\)-assisted entanglement distillation protocol for \(\rho_{\mathsf{A}\mathsf{B}}\) consists of \((d,\mathscr{L}_{\mathsf{A}:\mathsf{B}\rightarrow \mathsf{M}:\mathsf{M}'})\), where \(\mathscr{L}_{\mathsf{A}:\mathsf{B}\rightarrow \mathsf{M}:\mathsf{M}'} \in \mathcal{F}_t\), with \(|\mathsf{M}|=|\mathsf{M}'|=d\). Let \(\Phi_{\mathsf{M}\mathsf{M}'}\) denote the maximally entangled state of Schmidt rank \(d\). The error of this protocol is defined as \[p_{\rm{err}}^t(\mathscr{L}\mid\rho_{\mathsf{A}\mathsf{B}}):=P(\mathscr{L}(\rho_{\mathsf{A}\mathsf{B}}),\Phi_{\mathsf{M}\mathsf{M}'}).\] For a fixed \(d=2^\lambda\), the optimal error among all protocols \((d,\mathscr{L}_{\mathsf{A}:\mathsf{B}\rightarrow \mathsf{M}:\mathsf{M}'})\) is defined as \[p_{\rm{err}}^t(\rho_{\mathsf{A}\mathsf{B}},\lambda):=\min_{(d, \mathscr{L})} p_{\rm{err}}^t(\mathscr{L}\mid\rho_{\mathsf{A}\mathsf{B}}).\] Since \(\mathcal{F}_{\rightarrow} \subset \mathcal{F}_{\leftrightarrow} \subset \mathcal{F}_{\rm{NE}}\), it follows that for all \(\rho_{\mathsf{A}\mathsf{B}}\in \mathcal{S}(\mathsf{A}\mathsf{B})\) and \(\lambda \geq 0\), \[\label{equ:relation4} p_{\rm{err}}^{\rm{NE}}(\rho_{\mathsf{A}\mathsf{B}},\lambda) \leq p_{\rm{err}}^{\leftrightarrow}(\rho_{\mathsf{A}\mathsf{B}},\lambda) \leq p_{\rm{err}}^{\rightarrow}(\rho_{\mathsf{A}\mathsf{B}},\lambda).\tag{29}\]

For a rate \(r \geq 0\), the error exponent for \(t\)-assisted entanglement distillation is defined as \[E^{t}(\rho_{\mathsf{A}\mathsf{B}},r):=\liminf_{n \rightarrow \infty} \frac{-1}{n} \log p_{\rm{err}}^{t}(\rho_{\mathsf{A}\mathsf{B}}^{\otimes n},nr).\] In this section, we first establish achievability bounds on these error exponents.

Proof. We first prove this claim for \(t=\rightarrow\). Fix \(m \in \mathbb{N}\) and a channel \(\mathscr{L}_{\mathsf{A}^m:\mathsf{B}^m \rightarrow \mathsf{C}:\mathsf{D}}\in \mathcal{F}_t\), and define \[\phi_{\mathsf{C}\mathsf{D}}:=\mathscr{L}(\rho_{\mathsf{A}\mathsf{B}}^{\otimes m}).\] Let \(\phi_{CDR}\) be a purification of \(\phi_{\mathsf{C}\mathsf{D}}\).

By the proof of Theorem 6, for any \(k \in \mathbb{N}\), there exists a one-way LOCC channel \[\Lambda_{\mathsf{C}^{k}:\mathsf{D}^{k} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}=\sum_i \mathscr{E}^i_{\mathsf{C}^{k} \rightarrow \bar{\mathsf{M}}_k} \otimes\mathscr{D}^i_{\mathsf{D}^{k} \rightarrow \tilde{\mathsf{M}}_k},\] such that for any \(s\in (0,1)\), \[\label{equ:entdisapp} \begin{align} &P(\Lambda_{\mathsf{C}^{k}:\mathsf{D}^{k} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}\circ\mathscr{L}^{\otimes k}(\rho_{\mathsf{A}\mathsf{B}}^{\otimes mk}),\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k})\\ =&P(\Lambda_{\mathsf{C}^{k}:\mathsf{D}^{k} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}(\phi_{\mathsf{C}\mathsf{D}}^{\otimes k}),\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k}) \\ \leq &\sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{mkrs}2^{ks\widetilde{D}_{1+s}(\phi_{\mathsf{C}\mathsf{R}}\|I_{\mathsf{C}} \otimes\phi_{\mathsf{R}})}}, \end{align}\tag{30}\] where \(\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k}\) is the maximally entangled state of Schmidt rank \(|\bar{\mathsf{M}}_k|=2^{mkr}\).

Eq. (30 ) implies that for \(n=mk\), \[\label{equ:locgen} p^{\rightarrow}_{\rm{err}}(\rho_{\mathsf{A}\mathsf{B}}^{\otimes n},nr) \leq \sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{mkrs}2^{ks\widetilde{D}_{1+s}(\phi_{\mathsf{C}\mathsf{R}}\|I_{\mathsf{C}} \otimes\phi_R)}}.\tag{31}\] Taking logarithms, dividing \(n=mk\), and letting \(k \rightarrow \infty\) gives \[E^{\rightarrow}(\rho_{\mathsf{A}\mathsf{B}},r) \geq \frac{1}{2} s\left(\tfrac{1}{m}I_{\frac{1}{1+s}}(\mathsf{C}\rangle \mathsf{D})_{\mathscr{L}(\rho_{\mathsf{A}\mathsf{B}}^{\otimes m})}-r\right).\] Optimizing over \(m\), \(\mathscr{L}\in \mathcal{F}_\rightarrow\), and \(0<s<1\) yields the claimed bound for \(t=\rightarrow\).

For \(t=\leftrightarrow\) and \(t=\rm{NE}\), the same argument applies since \(\mathcal{F}_\rightarrow \subseteq \mathcal{F}_t\). In particular, the initial block operation \(\mathscr{L}\) can be chosen from the larger class, while the one-way LOCC post-processing \(\Lambda\) constructed above also belongs to \(\mathcal{F}_t\). This establishes the bound for all \(t\in\mathcal{G}\). \(\sqcup\)

The error exponent under non-entangling operations was characterized in [29]. Specifically, for every \(\rho_{\mathsf{A}\mathsf{B}}\in \mathcal{S}(\mathsf{A}\mathsf{B})\) and \(r \geq 0\), \[\label{equ::renlim} E^{\rm{NE}}(\rho_{\mathsf{A}\mathsf{B}},r)=\frac{1}{2} \sup_{s>0} s\left(\lim_{n \rightarrow \infty} \tfrac{1}{n}\inf_{\sigma_{\mathsf{A}^n\mathsf{B}^n} \in \rm{SEP}(\mathsf{A}^n:\mathsf{B}^n)} D_{\frac{1}{1+s}}(\rho_{\mathsf{A}\mathsf{B}}^{\otimes n}\|\sigma_{\mathsf{A}^n\mathsf{B}^n})-r\right),\tag{32}\] where \(\rm{SEP}(\mathsf{A}:\mathsf{B})\) denotes the set of separable states on \(AB\).

Combining the above characterization with the monotonicity relation Eq. (29 ) immediately yields the following converse bound.

Next, we show that for maximally correlated states, the converse bound derived above matches the achievability bound when the rate \(r\) is above a critical point.

Proof. We first establish that for every \(\alpha \in [0,2]\), \[\label{equ:redata2} \min_{\sigma_{\mathsf{A}\mathsf{B}} \in \rm{SEP}(\mathsf{A}:\mathsf{B})}D_\alpha(\rho_{\mathsf{A}\mathsf{B}}\|\sigma_{\mathsf{A}\mathsf{B}}) = I_{\alpha}(\mathsf{A}\rangle \mathsf{B})_{\rho}.\tag{33}\] For any \(\sigma_{\mathsf{A}\mathsf{B}} \in \rm{SEP}(\mathsf{A}:\mathsf{B})\), \[\sigma_{\mathsf{A}\mathsf{B}} \leq I_{A} \otimes\sigma_{\mathsf{B}}.\] Hence, \[\label{equ:relnew1} \min_{\sigma_{\mathsf{A}\mathsf{B}} \in \rm{SEP}(\mathsf{A}:\mathsf{B})}D_\alpha(\rho_{\mathsf{A}\mathsf{B}}\|\sigma_{\mathsf{A}\mathsf{B}}) \geq \min_{\sigma_{\mathsf{A}\mathsf{B}} \in \rm{SEP}(\mathsf{A}:\mathsf{B})}D_\alpha(\rho_{\mathsf{A}\mathsf{B}}\|I_A\otimes\sigma_{\mathsf{B}}) \geq I_{\alpha}(\mathsf{A}\rangle \mathsf{B})_{\rho}.\tag{34}\] Since \(\rho_{\mathsf{A}\mathsf{B}}\) is a maximally correlated state, there exist orthonormal bases \(\{|a_x\rangle\}_x\) and \(\{|b_x\rangle\}_x\) of \(\mathcal{H}_{\mathsf{A}}\) and \(\mathcal{H}_{\mathsf{B}}\) such that \({\operatorname{supp}}(\rho_{\mathsf{A}\mathsf{B}}) \subset \text{span} \{|a_x\rangle\otimes|b_x\rangle \}_x\).

Let \(\Pi=\sum_x | a_x\rangle\!\langle a_x |\otimes| b_x\rangle\!\langle b_x |\) be the corresponding projector. Consider the quantum channel \[\mathscr{E}(\cdot)=\Pi(\cdot)\Pi+(I-\Pi)(\cdot)(I-\Pi).\] By data processing inequality for the Petz Rényi divergence, for any \(\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})\), \[\label{equ:reldata2} \begin{align} &D_\alpha(\rho_{\mathsf{A}\mathsf{B}} \|I_{A} \otimes\sigma_{\mathsf{B}}) \\ \geq &D_\alpha(\mathscr{E}(\rho_{\mathsf{A}\mathsf{B}}) \|\mathscr{E}(I_{A} \otimes\sigma_{\mathsf{B}})) \\ =&D_\alpha(\rho_{\mathsf{A}\mathsf{B}} \|\Pi(I_{A} \otimes\sigma_{\mathsf{B}})\Pi) \\ =&D_\alpha(\rho_{\mathsf{A}\mathsf{B}} \|\sum_x \langle b_x|\sigma_{\mathsf{B}}|b_x\rangle | a_x\rangle\!\langle a_x |_{\mathsf{A}} \otimes| b_x\rangle\!\langle b_x |_{\mathsf{B}})\\ \geq & \min_{\sigma_{\mathsf{A}\mathsf{B}} \in \rm{SEP}(\mathsf{A}:\mathsf{B})}D_\alpha(\rho_{\mathsf{A}\mathsf{B}}\|\sigma_{\mathsf{A}\mathsf{B}}), \end{align}\tag{35}\] Eq. (34 ) and Eq. (35 ) give Eq. (33 ).

For maximally correlated states, the Petz Rényi relative entropy with respect to separable states is additive, i.e., \[\min_{\sigma_{\mathsf{A}^n\mathsf{B}^n} \in \rm{SEP}(\mathsf{A}^n:\mathsf{B}^n)}D_\alpha(\rho_{\mathsf{A}\mathsf{B}}^{\otimes n}\|\sigma_{\mathsf{A}^n\mathsf{B}^n}) =n\min_{\sigma_{\mathsf{A}\mathsf{B}} \in \rm{SEP}(\mathsf{A}:\mathsf{B})}D_\alpha(\rho_{\mathsf{A}\mathsf{B}}\|\sigma_{\mathsf{A}\mathsf{B}}).\] Hence the regularized quantity in Corollary 3 reduces to the single-letter expression \(I^{\downarrow}_{\alpha}(\mathsf{A}\rangle \mathsf{B})_{\rho}\), yielding Eq. (?? ).

Noticing that from the achievability bound (Theorem 7), we have \[\label{equ:loccachicov12} E^{t}(\rho_{\mathsf{A}\mathsf{B}},r)\geq \frac{1}{2}\sup_{0<s<1} s\left( I_{\frac{1}{1+s}}(\mathsf{A}\rangle \mathsf{B})_{\rho}-r\right),\tag{36}\] The exact characterization in the high-rate regime \(r\geq \frac{\rm{d}}{\rm{ds}}sI_{\frac{1}{1+s}}(\mathsf{A}\rangle \mathsf{B})_{\rho}|_{s=1}\) follows immediately by combining Eq. (36 ), Eq. (?? ) and a similar argument as in the proof of Theorem 4. \(\sqcup\)

6 Quantum communication↩︎

In this section, we investigate several fundamental quantum communication tasks over a quantum channel \(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}\). Throughout, performance is quantified using the purified distance.

• Subspace transmission: An subspace transmission protocol \((d, \mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}}, \mathscr{D}_{\mathsf{B}\rightarrow \mathsf{B}'})\) consists of an encoding channel \(\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}}\) and a decoding channel \(\mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}}\), with \(|\mathsf{M}|=|\bar{\mathsf{M}}|=d\).

  The overall protocol induces the effective channel \[\mathscr{M}_{\mathsf{M}\rightarrow \bar{\mathsf{M}}}:=\mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}} \circ \mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}} \circ \mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}}.\] The goal is to approximate the identity channel \(id_{\mathsf{M}\rightarrow \bar{\mathsf{M}}}\). The error is defined as \[p_{\rm{err}}^{\rm{QC}}(\mathscr{E},\mathscr{D};\mathscr{N}):=\max_{\phi_{\mathsf{R}\mathsf{M}}}P( \mathscr{M}_{\mathsf{M}\rightarrow \bar{\mathsf{M}}}(\phi_{\mathsf{R}\mathsf{M}}) , \phi_{\mathsf{R}\bar{\mathsf{M}}} ),\] where the maximization is over all pure states \(\phi_{\mathsf{R}\mathsf{M}}\).

• One-way LOCC-assisted subspace transmission: A one-way LOCC-assisted subspace transmission protocol \((d,\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}\bar{\mathsf{A}}},\mathscr{D}_{\bar{\mathsf{A}}:\mathsf{B}\rightarrow \varnothing : \bar{\mathsf{M}}})\) consists of an encoding channel \(\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}\bar{\mathsf{A}}}\) and a one-way LOCC decoding channel \(\mathscr{D}_{\bar{\mathsf{A}}:\mathsf{B}\rightarrow \varnothing : \bar{\mathsf{M}}}\) from Alice to Bob, with \(|\mathsf{M}|=|\bar{\mathsf{M}}|=d\).

  The protocol induces an effective channel \[\mathscr{M}_{\mathsf{M}\rightarrow \bar{\mathsf{M}}}:=\mathscr{D}_{\bar{\mathsf{A}}:\mathsf{B}\rightarrow \varnothing : \bar{\mathsf{M}}} \circ \mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}} \circ\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}\bar{A} }.\] and the error \(p_{\rm{err}}^{(\rm{QC},\rightarrow)}(\mathscr{E},\mathscr{D};\mathscr{N})\) is defined analogously to the subspace transmission setting, namely as the maximum purified distance between the output state and the target state over all pure inputs.

• Entanglement transmission: An entanglement transmission protocol \((d,\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}},\mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}})\) is defined by three elements, in which \(\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}}\) is an encoding channel and \(\mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}}\) is a decoding channel, with \(|\mathsf{M}|=|\bar{\mathsf{M}}|=d\).

  The goal is to transmit one half of a maximally entangled state \(\Phi_{\mathsf{R}\mathsf{M}}\) from Alice to Bob. The output state \[\omega_{\mathsf{R}\bar{\mathsf{M}}}:=(\mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}} \circ \mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}} \circ \mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}}) (\Phi_{\mathsf{R}\mathsf{M}})\] is required to approximate \(\Phi_{\mathsf{R}\bar{\mathsf{M}}}\). The entanglement transmission error of the protocol is \[p_{\rm{err}}^{\rm{ET}}(\mathscr{E},\mathscr{D};\mathscr{N}):=P(\omega_{\mathsf{R}\bar{\mathsf{M}}},\Phi_{\mathsf{R}\bar{\mathsf{M}}}).\]

• One-way LOCC-assisted entanglement transmission: An one-way LOCC-assisted entanglement transmission protocol \((d,\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}\bar{\mathsf{A}}},\mathscr{D}_{\bar{\mathsf{A}}:\mathsf{B}\rightarrow \varnothing: \bar{\mathsf{M}}})\) consists of an encoding channel \(\mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}}\) and a one-way LOCC decoding channel \(\mathscr{D}_{\bar{\mathsf{A}}:\mathsf{B}\rightarrow \varnothing: \bar{\mathsf{M}}}\) from Alice to Bob, with \(|\mathsf{M}|=|\bar{\mathsf{M}}|=d\). The goal of the protocol is to transmit the \(\mathsf{M}\) system of a maximally entangled state \(\Phi_{\mathsf{R}\mathsf{M}}\) such that the final state \[\omega_{\mathsf{R}\bar{\mathsf{M}}}:=(\mathscr{D}_{\bar{A}:B \rightarrow \varnothing:\bar{\mathsf{M}}} \circ \mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}} \circ \mathscr{E}_{\mathsf{M}\rightarrow \mathsf{A}\bar{A}}) (\Phi_{\mathsf{R}\mathsf{M}})\] is close to the initial maximally entangled state. The error of the protocol is defined as \[p_{\rm{err}}^{(\rm{ET}, \rightarrow)}(\mathscr{E},\mathscr{D};\mathscr{N}):=P(\omega_{\mathsf{R}\bar{\mathsf{M}}},\Phi_{\mathsf{R}\bar{\mathsf{M}}}).\]

• Entanglement generation: An entanglement generation protocol is defined by three elements \((d, \Psi_{\mathsf{M}'\mathsf{A}}, \mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}})\), where \(\Psi_{\mathsf{M}'\mathsf{A}}\) is a pure state and \(\mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}}\) is a decoding channel, with \(|\bar{\mathsf{M}}|=|\mathsf{M}'|=d\).

  The output state \[\sigma_{\mathsf{M}'\bar{\mathsf{M}}}:=(\mathscr{D}_{\mathsf{B}\rightarrow \bar{\mathsf{M}}}\circ \mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}})(\Psi_{\mathsf{M}'\mathsf{A}})\] is required to approximate a maximally entangled state \(\Phi_{\mathsf{M}'\bar{\mathsf{M}}}\). The entanglement generation error of the protocol is given by \[p_{\rm{err}}^{\rm{EG}}(\Psi_{\mathsf{M}'\mathsf{A}}, \mathscr{D};\mathscr{N}):=P(\sigma_{\mathsf{M}'\bar{\mathsf{M}}},\Phi_{\mathsf{M}'\bar{\mathsf{M}}}).\]

• One-way LOCC-assisted entanglement generation: An one-way LOCC-assisted entanglement generation protocol consists of a triplet \((d,\Psi_{\mathsf{A}'\mathsf{A}},\{\mathscr{E}^x_{\mathsf{A}'\mathsf{A}\rightarrow \mathsf{M}'\mathsf{A}}\}_x,\{\mathscr{D}^x_{\mathsf{B}\rightarrow \bar{\mathsf{M}}}\}_x)\), where \(\Psi_{\mathsf{A}'\mathsf{A}}\) is a pure state with \(|M'|=d\), \(\{\mathscr{E}^x_{\mathsf{A}'\mathsf{A}\rightarrow \mathsf{M}'\mathsf{A}}\}_x\) is a set of completely positive maps indexed by a finite alphabet \(\mathcal{X}\) such that \(\sum_{x} \mathscr{E}^x_{\mathsf{A}'\mathsf{A}\rightarrow \mathsf{M}'\mathsf{A}}\) is trace preserving, \(\{\mathscr{D}^x_{\mathsf{B}\rightarrow \bar{\mathsf{M}}}\}_x\) is a set of quantum channels indexed by \(\mathcal{X}\), with \(|\bar{\mathsf{M}}|=d\). The goal of the protocol is to transmit the system \(A\) such that the final state \[\sigma_{\mathsf{M}'\bar{\mathsf{M}}}^\rightarrow:=\sum_x (\mathscr{D}^x_{\mathsf{B}\rightarrow \bar{\mathsf{M}}} \circ \mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}} \circ \mathscr{E}^x_{\mathsf{A}'\mathsf{A}\rightarrow \mathsf{M}'\mathsf{A}})(\Psi_{\mathsf{A}'\mathsf{A}})\] is close to a maximally entangled state of Schmidt rank \(d\). The error of the protocol is defined as \[p_{\rm{err}}^{\rm{EG, \rightarrow}}(\Psi_{\mathsf{A}'\mathsf{A}},\{\mathscr{E}_x \}_x, \{\mathscr{D}_X\}_x;\mathscr{N}):=P(\sigma_{\mathsf{M}'\bar{\mathsf{M}}}^\rightarrow,\Phi_{\mathsf{M}'\bar{\mathsf{M}}}).\]

Let \[\mathcal{T}:=\{\rm{ST},\rm{ET},\rm{EG},(\rm{ST}, \rightarrow), (\rm{ET},\rightarrow), (\rm{EG},\rightarrow)\}\] denote the set of this collection of tasks.

For \(\epsilon \in [0,1]\), a protocol is called \((d,\epsilon)\)-achievable for task \(t \in \mathcal{T}\) if its error does not exceed \(\epsilon\). For \(t \in \mathcal{T}\), the minimal achievable error among all protocols with fixed \(d=2^\lambda\) is defined as \[p_{\rm{err}}^{t}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda):=\inf\{\epsilon~|~\exists~(d,\epsilon)-\text{achievable protocol for task}~t\}.\] From the above definitions, we can directly obtain that for any quantum channel \(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}\) and \(\lambda \geq 0\), \[\begin{align} \tag{37} &p_{\rm{err}}^{\rm{ET},\rightarrow}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda) \leq p_{\rm{err}}^{\rm{ST},\rightarrow}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda) \leq p_{\rm{err}}^{\rm{ST}} (\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda),\\ \tag{38} &p_{\rm{err}}^{\rm{ET},\rightarrow}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda) \leq p_{\rm{err}}^{\rm{ET}}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda) \leq p_{\rm{err}}^{\rm{ST}}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda). \end{align}\] Moreover, by Lemma 6 and Lemma 7, \[\begin{align} \label{equ:relation3} p_{\rm{err}}^{\rm{EG},\rightarrow} (\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda)= p_{\rm{err}}^{\rm{EG}}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda) \leq p_{\rm{err}}^{\rm{ET}}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda) \leq 2 p_{\rm{err}}^{\rm{EG}}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\lambda). \end{align}\tag{39}\] For a fixed rate \(r\), the error exponent of task \(t \in \mathcal{T}\) is defined as \[E^{t}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},r):=\liminf_{n \rightarrow \infty} \frac{-1}{n} \log p_{\rm{err}}^{t}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}^{\otimes n}, nr).\] In the following, we study these error exponents and first establish an achievability bound.

Proof. Fix a integer \(m\) and a pure state \(\phi_{\bar{\mathsf{A}}^m\mathsf{A}^m}\). Let \[\phi_{\bar{\mathsf{A}}^m\mathsf{B}^m\mathsf{E}^m}=V^{\otimes m}_{\mathsf{A}\rightarrow \mathsf{B}\mathsf{E}}\phi_{\bar{\mathsf{A}}^m\mathsf{A}^m}V^{* \otimes m}_{\mathsf{A}\rightarrow \mathsf{B}\mathsf{E}}\] be the purification of the channel output, where \(V_{\mathsf{A}\rightarrow \mathsf{B}\mathsf{E}}\) is the Stinespring isometry of \(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}\).

By the proof of Theorem 6, for any \(k \in \mathbb{N}\), there exists an one-way LOCC channel \[\Lambda_{\bar{\mathsf{A}}^{mk}:\mathsf{B}^{mk} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}=\sum_i \mathscr{E}^i_{\bar{\mathsf{A}}^{mk} \rightarrow \bar{\mathsf{M}}_k} \otimes\mathscr{D}^i_{\mathsf{B}^{mk} \rightarrow \tilde{\mathsf{M}}_k},\] such that \[\label{equ:entdisapp1} P(\Lambda_{\bar{\mathsf{A}}^{mk}:\mathsf{B}^{mk} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}(\phi_{\bar{\mathsf{A}}^m\mathsf{B}^m}^{\otimes k}),\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k}) \leq \sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{mkrs}2^{ks\widetilde{D}_{1+s}(\phi_{\bar{\mathsf{A}}^m\mathsf{E}^m}\|I_{\bar{\mathsf{A}}}^{\otimes m} \otimes\phi_{\mathsf{E}^m})}},\tag{40}\] where \(\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k}\) is the maximally entangled state of Schmidt rank \(|\bar{\mathsf{M}}_k|=2^{mkr}\).

Eq. (40 ) implies that for \(n=mk\), \[\label{equ:locgen1} p_{\rm{err}}^{\rm{EG},\rightarrow}(\mathscr{N}^{\otimes n}_{\mathsf{A}\rightarrow \mathsf{B}},nr) \leq \sqrt{\frac{s^s(1-s)^{1-s}}{s} 2^{mkrs}2^{ks\widetilde{D}_{1+s}(\phi_{\bar{\mathsf{A}}^m\mathsf{E}^m}\|I_{\bar{\mathsf{A}}}^{\otimes m} \otimes\phi_{\mathsf{E}^m})}}.\tag{41}\] Taking logarithms and letting \(n \rightarrow \infty\), optimizing over \(m\), pure state \(\phi_{\bar{\mathsf{A}}^m\mathsf{A}^m}\) and \(0<s<1\) yields the bound for for \(t=(\rm{EG},\rightarrow)\). By Eq. (39 ), this extends to \(t\in \{\rm{EG}, \rm{ET}\}\).

Next, we prove Eq. (?? ) for \(t \in \{(\rm{ST},\rightarrow), (\rm{ET},\rightarrow) \}\). Using the standard teleportation protocol, the maximally entangled state \(\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k}\) in Eq. (40 ) can be converted via LOCC into a noiseless quantum channel. Specifically, there exists an one-way LOCC channel from Alice to Bob \[\Pi_{\bar{\mathsf{M}}_k\mathsf{M}_k:\tilde{\mathsf{M}}_k \rightarrow \varnothing: \mathsf{M}'_k}\] such that \[\Pi_{\bar{\mathsf{M}}_k\mathsf{M}_k:\tilde{\mathsf{M}}_k \rightarrow \varnothing :\mathsf{M}'_k}((\cdot)\otimes\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k})={\operatorname{id}}_{\mathsf{M}_k \rightarrow \mathsf{M}'_k}(\cdot).\] Now for \(n=mk\), we construct an one-way LOCC-assisted quantum communication protocol for \(\mathscr{N}^{\otimes n}_{\mathsf{A}\rightarrow \mathsf{B}}\) as follows. The encoder prepares the fixed state \(\phi^{\otimes k}_{\bar{\mathsf{A}}^m\mathsf{A}^m}\), while the decoder consists of the concatenation of \(\Lambda_{\bar{\mathsf{A}}^{mk}:\mathsf{B}^{mk} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}\) and the teleportation channel \(\Pi\).

The resulting error satisfies \[\label{equ:evaqu} \begin{align} &p_{\rm{err}}^{\rm{ST}, \rightarrow}(\mathscr{N}^{\otimes n}_{\mathsf{A}\rightarrow \mathsf{B}},nr)\\ \leq& \max_{\phi_{\mathsf{M}_k\mathsf{R}_k}}P(\Pi_{\bar{\mathsf{M}}_k\mathsf{M}_k:\tilde{\mathsf{M}}_k \rightarrow \varnothing: \mathsf{M}'_k}(\Lambda_{\bar{\mathsf{A}}^{mk}:\mathsf{B}^{mk} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}(\phi_{\bar{\mathsf{A}}^m\mathsf{B}^m}^{\otimes k})\otimes\phi_{\mathsf{M}_k\mathsf{R}_k}),id_{\mathsf{M}_k \rightarrow \mathsf{M}'_k}(\phi_{\mathsf{M}_k\mathsf{R}_k})) \\ =& \max_{\phi_{\mathsf{M}_k\mathsf{R}_k}}P(\Pi_{\bar{\mathsf{M}}_k\mathsf{M}_k:\tilde{\mathsf{M}}_k \rightarrow \varnothing :\mathsf{M}'_k}(\Lambda_{\bar{\mathsf{A}}^{mk}:\mathsf{B}^{mk} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}(\phi_{\bar{\mathsf{A}}^m\mathsf{B}^m}^{\otimes k})\otimes\phi_{\mathsf{M}_k\mathsf{R}_k}),\Pi_{\bar{\mathsf{M}}_k\mathsf{M}_k:\tilde{\mathsf{M}}_k \rightarrow \varnothing :\mathsf{M}'_k}(\phi_{\mathsf{M}_k\mathsf{R}_k}\otimes\phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k})) \\ \leq & \max_{\phi_{\mathsf{M}_k\mathsf{R}_k}} P(\Lambda_{\bar{\mathsf{A}}^{mk}:\mathsf{B}^{mk} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}(\phi_{\bar{\mathsf{A}}^m\mathsf{B}^m}^{\otimes k})\otimes\phi_{\mathsf{M}_k\mathsf{R}_k},\Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k}\otimes\phi_{\mathsf{M}_k\mathsf{R}_k} ) \\ =&P\left(\Lambda_{\bar{\mathsf{A}}^{mk}:\mathsf{B}^{mk} \rightarrow \bar{\mathsf{M}}_k:\tilde{\mathsf{M}}_k}(\phi_{\bar{\mathsf{A}}^m\mathsf{B}^m}^{\otimes k}), \Phi_{\bar{\mathsf{M}}_k\tilde{\mathsf{M}}_k}\right)\\ \leq & \sqrt{\left(\tfrac{|\mathsf{A}|^{mk}2^{mkr}-1}{|\mathsf{A}|^{mk}-1}\right)^s2^{ks\widetilde{D}_{1+s}(\phi_{\bar{\mathsf{A}}^m\mathsf{E}^m}\|I_{\bar{\mathsf{A}}}^{\otimes m} \otimes\phi_{\mathsf{E}^m})}}. \end{align}\tag{42}\] so the same bound as in Eq. (41 ) applies. Optimizing as above yields the bound for \(t=(\rm{ST},\rightarrow)\), and Eq. (37 ) gives the result for \(t=(\rm{ET},\rightarrow)\). \(\sqcup\)

Since the Petz Rényi coherent information is not additive in general, the regularization appearing in Theorem 9 is in general unavoidable. However, for covariant quantum channels with respect to the full unitary group, we obtain a single-letter converse bound on the error exponent.

We begin with a one-shot upper bound on the quantity \(R^{\rm{ET},\rightarrow}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},\epsilon)\), which denotes the maximal number of qubits that can be transmitted with error at most \(\epsilon \in (0,1)\) under one-way LOCC assistance. For any covariant quantum channel \(\mathscr{N}_{A\to \mathsf{B}}\) with respect to the full unitary group and any \(n\in\mathbb{N}\), it was shown in [43] that \[\label{equ:locchy} R^{\rm{ET},\rightarrow}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}^{\otimes n},\epsilon)\leq \min_{\sigma_{\mathsf{A}\mathsf{B}} \in \rm{PPT}'(\mathsf{A}:\mathsf{B})} D_{\rm{H}}^{\epsilon^2}(\mathscr{N}_{\mathsf{A}' \rightarrow \mathsf{B}}(\Phi_{\mathsf{A}\mathsf{A}'})^{\otimes n}\|\sigma_{\mathsf{A}\mathsf{B}}^{\otimes n}),\tag{43}\] where \[{\rm{PPT}}'(\mathsf{A}:\mathsf{B}):=\left\{\tau_{\mathsf{A}\mathsf{B}}~\big|~\left\|\tau_{\mathsf{A}\mathsf{B}}^{\top_{\mathsf{B}}}\right\|_1 \leq 1 \right\}\] denotes the Rains set and \[D_{\rm{H}}^\epsilon(\rho\|\sigma):=-\log \min \{\operatorname{Tr}\sigma T ~|~\operatorname{Tr}\rho T\geq 1-\epsilon~\land~0\leq T \leq I \}\] is the hypothesis testing relative entropy.

Eq. (43 ) implies that \[\label{equ:horbas} \begin{align} &p_{\rm{err}}^{\rm{ET},\rightarrow}(\mathscr{N}^{\otimes n}_{\mathsf{A}\rightarrow \mathsf{B}}, nr) \\ =&\inf \left\{\epsilon~\middle|~ R^{\rm{ET},\rightarrow}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}^{\otimes n},\epsilon) \geq nr \right\} \\ \geq&\inf \left\{\epsilon~\Big|~ \min_{\sigma_{\mathsf{A}\mathsf{B}} \in \rm{PPT}'(\mathsf{A}:\mathsf{B})}D_{\rm{H}}^{\epsilon^2}(\mathscr{N}_{\mathsf{A}' \rightarrow \mathsf{B}}(\Phi_{\mathsf{A}\mathsf{A}'})^{\otimes n}\|\sigma_{\mathsf{A}\mathsf{B}}^{\otimes n}) \geq nr \right\} \\ =&\max_{\sigma_{\mathsf{A}\mathsf{B}} \in {\rm{PPT}}'(\mathsf{A}:\mathsf{B})}2^{-\frac{1}{2}D_{\rm{H}}^{2^{-nr}}(\sigma_{\mathsf{A}\mathsf{B}}^{\otimes n}\|\mathscr{N}_{\mathsf{A}' \rightarrow \mathsf{B}}(\Phi_{\mathsf{A}\mathsf{A}'})^{\otimes n})}. \end{align}\tag{44}\] Applying the quantum Hoeffding exponent of quantum hypothesis testing to Eq. (44 ) and combining Eq. (37 ), Eq. (38 ) and Eq. (39 ), we obtain the following conclusion.

Compared to the achievability bound, the converse bound involves only a single-letter quantity, making its expression more concise. The trade-off is that the converse bound is valid only for covariant quantum channels and is generally not tight. Nevertheless, for covariant generalized dephasing channels, we can show that the converse bound coincides with the achievability bound for coding rates above a critical value.

A covariant generalized dephasing channel \(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}\) is a covariant channel with respect to the full unitary group of the form: \[\label{equ:cgdcform} \mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}(\rho)=\sum_{x,y=0}^{|\mathsf{A}|-1}\langle x|_{\mathsf{A}} \rho |y\rangle_{\mathsf{A}} \langle \psi_y|\psi_x \rangle|x\rangle\langle y|_{\mathsf{B}}.\tag{45}\]

For such channels, we obtain the following result.

Proof. We first establish that, for any \(\alpha \in [0,2]\), \[\label{equ:redata} \min_{\sigma_{\mathsf{A}\mathsf{B}} \in {\rm{PPT}}'(\mathsf{A}:\mathsf{B})}D_\alpha(\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'})\|\sigma_{\mathsf{A}\mathsf{B}}) \leq I_{\alpha}(\mathsf{A}\rangle \mathsf{B})_{\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'})}.\tag{46}\] Since \(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}}\) has the form in Eq. (45 ), the support of \(\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'})\) lies in the projector \[\Pi=\sum_{x=0}^{|\mathsf{A}|-1} | x\rangle\!\langle x |_{\mathsf{A}} \otimes| x\rangle\!\langle x |_{\mathsf{B}}.\] Applying the data-processing inequality for the quantum channel \(\mathscr{E}(\cdot)=\Pi(\cdot)\Pi+(I-\Pi)(\cdot)(I-\Pi)\), we obtain, for any \(\sigma_{\mathsf{B}} \in \mathcal{S}(\mathsf{B})\), \[\label{equ:reldata} \begin{align} &D_\alpha(\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'}) \|I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}}) \\ \geq &D_\alpha(\mathscr{E}(\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'})) \|\mathscr{E}(I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}})) \\ =&D_\alpha(\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'}) \|\Pi(I_{\mathsf{A}} \otimes\sigma_{\mathsf{B}})\Pi) \\ =&D_\alpha(\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'}) \big\|\sum\nolimits_{x=0}^{|\mathsf{A}|-1} \langle x|\sigma_{\mathsf{B}}|x\rangle | x\rangle\!\langle x |_A \otimes| x\rangle\!\langle x |_B)\\ \geq &\min_{\sigma_{\mathsf{A}\mathsf{B}} \in {\rm{PPT}}'(\mathsf{A}:\mathsf{B})}D_\alpha(\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'})\|\sigma_{\mathsf{A}\mathsf{B}}), \end{align}\tag{47}\] which proves Eq. (46 ) and Eq. (?? ) follows from Eq. (46 ) and Corollary 4.

Noticing that the achievability bound (Theorem 9) gives \[\label{equ:loccachicov} E^{t}(\mathscr{N}_{\mathsf{A}\rightarrow \mathsf{B}},r)\geq \frac{1}{2}\sup_{0<s<1} s\left( I_{\frac{1}{1+s}}(\mathsf{A}\rangle \mathsf{B})_{\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'})}-r\right),\tag{48}\] where \(t \in \mathcal{T} \setminus \{\rm{ST}\}\). The exact characterization in the high-rate regime \(r\geq \frac{\rm{d}}{\rm{ds}}sI_{\frac{1}{1+s}}(\mathsf{A}\rangle \mathsf{B})_{\mathscr{N}(\Phi_{\mathsf{A}\mathsf{A}'})}|_{s=1}\) follows immediately by combining Eq. (48 ), Eq. (?? ) and a similar argument as in the proof of Theorem 4. \(\sqcup\)

7 Conclusion and Discussion↩︎

We derived fundamental one-shot upper bounds on the error of quantum information decoupling, together with a complementary one-shot lower bound in the standard decoupling setting. These bounds induce corresponding achievability and ensemble-tight converse bounds on the decoupling error exponent. By showing that the two bounds coincide whenever the decoupling rate \(r\) satisfies \(r \leq R_{\text{critical}}\), we obtained an exact characterization of the error exponent of standard quantum information decoupling in the low-rate regime.

As applications, we characterized the exact error exponent for quantum state merging and established achievability bounds on the error exponents for entanglement distillation and a broad class of quantum communication tasks, including subspace transmission, entanglement transmission, entanglement generation, as well as their one-way LOCC-assisted counterparts. We expect that the techniques and results developed here will find further applications in other quantum information processing tasks.

Our analysis also reveals the existence of a critical rate for the error exponents of standard decoupling and quantum state merging, beyond which the exact error exponents remain unknown. The appearance of such a critical rate, separating regimes of complete and partial characterization, is a common phenomenon in the study of error exponents [18], [23], [44]. In classical channel coding, combinatorial effects are known to play a decisive role in this regime [18], [44], and it remains an open question whether analogous mechanisms govern quantum information decoupling and quantum state merging.

Throughout this work, we quantified the decoupling error using quantum relative entropy. A natural direction for future research is to determine the corresponding error exponents under alternative distance measures, such as the trace distance. While prior work has established achievability bounds in trace distance, and related techniques — such as complex interpolation — have successfully been applied to other quantum covering problems, including quantum soft covering [45], convex splitting [46], and privacy amplification [47], the development of matching converse bounds in fully quantum settings remains open.

Finally, our achievability bounds on the error exponents for entanglement distillation, subspace transmission, entanglement transmission, entanglement generation, and their one-way LOCC-assisted counterparts are expressed in terms of regularized Petz–Rényi coherent informations. Establishing concise single-letter characterizations or matching converse bounds for these tasks constitutes an important open problem in quantum information theory.

MB and YY acknowledge support by the European Research Council (ERC Grant Agreement No. 948139) and from the Excellence Cluster Matter and Light for Quantum Computing (ML4Q-2). HC is supported under grants No. NSTC 114-2628-E-002 -006, NSTC 114-2119-M-001-002, and NSTC 115-2124-M-002-014, NTU-114V2016-1, NTU-CC115L893705, and NTU-115L900702.

8 Auxiliary Lemmas↩︎

The following lemmas are used in our analysis, with notation as in the main text.

Proof. Set \(Q := \{\rho > 9\sigma\}\). Denote \(p = \operatorname{Tr}(\rho Q)\) and \(q = \operatorname{Tr}(\sigma Q)\), which are the probabilities of obtaining the outcome associated with \(Q\) when a projective measurement \(\{Q, I-Q\}\) is applied to \(\rho\) and \(\sigma\), respectively. Then, it is easy to see that \[Q \rho Q \geq 9 Q \sigma Q,\] which gives \[p \geq 9q. \label{eq:pgeq9q}\tag{49}\]

Now by the monotonicity of the quantum relative entropy under quantum measurements, we have \[\begin{align} D(\rho\|\sigma) &\geq D\big( (p, 1-p)\|(q, 1-q) \big) \\ &\geq p\log\frac{p}{q}+(1-p)\log \frac{1-p}{1-q} \\ &\geq 3p + (1-p)\log(1-p), \end{align}\] where the last line is from Eq. 49 . Noticing that \(2x+(1-x)\log(1-x) \geq 0\) for \(x \in [0,1]\), we obtain \[\begin{align} D(\rho\| \sigma) \geq p \geq \operatorname{Tr}(\rho - 9\sigma)_{+}. \end{align}\] \(\sqcup\)

References↩︎