June 04, 2026
We study the robustness of asymptotic entanglement manipulation beyond the exact i.i.d.regime, focusing on Mazzola–Sutter–Renner (MSR) almost i.i.d.sources, which allow a sublinear number of deviations from a tensor-power structure. For pure MSR sources along a bipartite reference state \(\ket{\phi}_{AB}\), we prove that the entanglement concentration rate is robust: every rate below the entropy of entanglement \(S(\phi_A)\) remains achievable. Moreover, this can be done by a single Schur–Weyl concentration protocol that is universal within the MSR class, depending only on the reference state and not on the particular source sequence. For mixed MSR sources along a reference state \(\rho_{AB}\), we prove a source-dependent entanglement-distillation achievability result: every rate below the coherent information \(I(A\rangle B)_\rho\) of the reference state is achievable, although the entanglement distillation protocol may depend on the particular MSR source sequence. For the reverse task of entanglement dilution, we prove a rate-robustness theorem: the asymptotic entanglement cost of any MSR target sequence along \(\rho_{AB}\) is at most \(E_F^\infty(\rho_{AB})\), the regularized entanglement of formation of the reference state. To establish these results, we prove structural and entropic properties of MSR almost i.i.dsequences which may be useful in other information-theoretic settings. Thus, for the achievability statements considered here, MSR almost i.i.d.perturbations exhibit the same asymptotic behaviour as their i.i.d.reference states, despite allowing sublinear deviations from a tensor-power structure.
Entanglement is a fundamental non-local resource in quantum information theory, playing a central role in quantum teleportation, superdense coding, quantum cryptography and quantum error correction. Since transformations by local operations and classical communication (LOCC) do not increase entanglement, as quantified by entanglement monotones, a central problem in entanglement theory is to determine the optimal rates at which bipartite states can be converted to and from maximally entangled states under LOCC. In entanglement distillation, many copies of a shared bipartite state are converted into maximally entangled states; in the special case of pure states, this task is usually called entanglement concentration. The reverse task, entanglement dilution, asks for the minimum rate at which maximally entangled states must be consumed in order to prepare a target state. The corresponding optimal asymptotic rates are the distillable entanglement \(E_D\) and the entanglement cost \(E_C\), respectively.
For i.i.d.pure-state sources, entanglement concentration and dilution are both governed by the entropy of entanglement \(S(\rho_A)\), where \(\rho_A\) is either marginal of the bipartite pure state [1], [2]. For mixed states, the entanglement cost is given by the regularized entanglement of formation \(E_F^\infty\), whereas the distillable entanglement is generally harder to characterize and may be strictly smaller than the entanglement cost [3], [4]. These tasks have also been studied beyond the first-order i.i.d. setting, including information-spectrum formulations for arbitrary sequences of bipartite states [5]–[8], as well as finite-blocklength and second-order refinements of entanglement concentration, dilution and related pure-state conversion problems [9]–[11].
The i.i.d.assumption is one of the central idealizations in quantum Shannon theory: asymptotic rates are usually derived under the premise that the underlying source is exactly a tensor power. This assumption is mathematically powerful, but it can be unstable under structural perturbations. In realistic settings, correlations, memory effects, local imperfections or preparation errors may lead to sources which approximate tensor-power behaviour only in a weaker structural or statistical sense. This raises a basic robustness question: which information-theoretic conclusions survive when the exact i.i.d.hypothesis is replaced by an appropriate notion of almost i.i.d. behaviour? Recent work has shown that the answer depends crucially on the chosen notion of approximation; being “close to i.i.d.” is not a single mathematical condition, and different relaxations may have different operational consequences (see e.g. [12]–[14]).
At present, three natural notions of almost i.i.d.quantum sources have emerged. The weakest is the class of weakly almost i.i.d.sources, in which fixed-size marginals converge, on average, to the corresponding tensor powers of a reference state; this condition allows arbitrary long-range correlations and multipartite entanglement, and is therefore well suited to concentration statements depending only on local statistics [14]. An intermediate notion is given by Wasserstein almost i.i.d.sources, where the deviation from the tensor-power state is controlled in normalized quantum Wasserstein distance, capturing the idea that only a small fraction of local subsystems is significantly affected [12]. The most restrictive of these three notions is the almost i.i.d.class introduced by Mazzola–Sutter–Renner, in which the state admits a permutation-invariant extension supported on a subspace with only a prescribed number of unrestricted tensor positions. These three classes form a strict hierarchy: Mazzola–Sutter–Renner (MSR) almost i.i.d.states are Wasserstein almost i.i.d., Wasserstein almost i.i.d.states are weakly almost i.i.d., and both inclusions are strict [12]. In this paper, we focus on the MSR class, whose additional structure makes it possible to study the robustness of entanglement manipulation protocols with quantitative control over sublinear deviations from tensor-power structure.
The main goal of this paper is to understand the robustness of asymptotic entanglement manipulation protocols under almost i.i.d.perturbations of the underlying resource state. We focus on the class of MSR almost i.i.d.states introduced by Mazzola, Sutter and Renner, which provide a natural and mathematically tractable model of sources containing a sublinear number of defects.
Our first main result concerns entanglement concentration. For an i.i.d.source \(\ket{\phi}_{AB}^{\otimes n}\), the optimal concentration rate is given by the entropy of entanglement \(S(\rho_A)\), where \(\rho_A=\mathop{\mathrm{Tr}}_B\ket{\phi}\!\bra{\phi}_{AB}\). A natural question is whether this rate remains achievable when the source is replaced by an arbitrary pure MSR source along \(\ket{\phi}_{AB}\). More importantly, one may ask whether the concentration protocol itself can be chosen universally, depending only on the reference state \(\ket{\phi}_{AB}\) and not on the particular MSR source.
Theorem 14 answers this question affirmatively. We prove that every rate below \(S(\rho_A)\) can be achieved by a single concentration protocol that depends only on the reference state \(\ket{\phi}_{AB}\), and whose error tends to zero uniformly over the entire class of pure MSR sources along \(\ket{\phi}_{AB}\). Thus, from the perspective of asymptotic entanglement concentration, MSR perturbations do not alter either the optimal rate or the protocol itself. The proof combines structural properties of the MSR class with spectral-entropy rigidity, Schur–Weyl duality and the Hayashi–Matsumoto concentration protocol [15].
Our second main result concerns the reverse task of entanglement dilution. Here we consider arbitrary mixed MSR sources along a bipartite state \(\rho_{AB}\). Theorem 16 shows that the asymptotic dilution cost remains bounded above by the regularized entanglement of formation \(E_F^\infty(\rho_{AB})\) of the reference state. In other words, the standard i.i.d.achievability bound for entanglement dilution continues to hold throughout the MSR class. This shows that the resources required to create the state are asymptotically unaffected by the presence of a sublinear number of defects.
Several auxiliary results obtained in the proof may be useful beyond the specific entanglement-manipulation tasks considered here. We establish structural stability properties of the MSR class, including stability under local tensor-power channels, taking marginals, and blocking operations. We also prove entropy-rigidity estimates showing that the relevant spectral entropy rates of MSR sources coincide with the corresponding entropy quantities of the reference state. Finally, for the entanglement dilution theorem, we develop a cq-lifting argument which relates an MSR target source to the ensemble description entering the information-spectrum formula for entanglement cost.
These results show that MSR almost i.i.d.sources preserve the asymptotic entanglement content of the underlying i.i.d.reference state under sublinear deviations from tensor-power structure. For pure-state entanglement concentration, the robustness is stronger: not only does the optimal rate remain achievable, but a single Schur–Weyl protocol (of Hayashi and Matsumoto [15]) works universally over the whole MSR class along the fixed reference state. For mixed-state dilution, the asymptotic cost remains bounded by the regularized entanglement of formation of the reference state, while mixed state distillation admits a source-dependent coherent-information achievability result.
After introducing the necessary notation and mathematical preliminaries, we define MSR almost i.i.d.states and sources in Section 2.1. We prove several important properties of MSR sources in Section 3: their structural stability properties and the complexity estimates for their defect spaces are treated in Section 3.1, while the rigidity estimates for their entropic quantities are established in Section 3.2. In Section 4 we introduce the entanglement manipulation tasks of distillation, concentration and dilution for general sequences of states, and define the corresponding operational quantities, namely distillable entanglement and entanglement cost. Entanglement concentration for pure MSR almost i.i.d.sources is studied in Section 5; the main result there is a universal concentration theorem for the whole pure MSR class along a fixed reference state, stated as Theorem 14. This section also contains a source-dependent achievability result for entanglement distillation from mixed MSR sources (see Theorem 10). Section 6 deals with entanglement dilution, where the target is a sequence of MSR almost i.i.d.states along a fixed mixed state; see Theorem 16. We end with a conclusion in Section 7 and some open questions in Section 8. Proofs of many of the technical lemmas are deferred to the appendices in order to improve the readability of the paper.
Let \(\mathcal{H}\) be a finite-dimensional Hilbert space. We write \(\mathcal{L}(\mathcal{H})\) for the algebra of linear operators on \(\mathcal{H}\), and \(\mathcal{D}(\mathcal{H})\) for the set of density operators on \(\mathcal{H}\), i.e.positive semidefinite operators of unit trace. A pure state is a rank-one projection \(\ket{\psi}\!\bra{\psi}\), where \(\ket{\psi}\in\mathcal{H}\) is a unit vector. The identity operator on \(\mathcal{H}\) is denoted by \(\mathbb{1}_{\mathcal{H}}\), or simply by \(\mathbb{1}\) when the underlying Hilbert space is clear. We label different quantum systems by capital Roman letters (\(A,B,F, A'\), etc.) and often use these letters interchangeably with the corresponding Hilbert spaces.
An operator \(\Pi\in\mathcal{L}(\mathcal{H})\) is an orthogonal projection if \(\Pi=\Pi^\dagger=\Pi^2\). Its range is denoted by \({\rm{Ran}}\,(\Pi)\). For a positive semidefinite operator \(X\), we write \(\mathop{\mathrm{supp}}X:={\rm{Ran}}\,(\{X>0\})\), where \(\{X>0\}\) denotes the spectral projection onto the strictly positive part of the spectrum of \(X\). More generally, if \(Q=\sum_i\lambda_i\ket{\psi_i}\!\bra{\psi_i}\) is self-adjoint and \(r\in\mathbb{R}\), we use the notation \(\{Q>r\}:=\sum_{\lambda_i>r}\ket{\psi_i}\!\bra{\psi_i}\), and similarly for \(\{Q\le r\}\), \(\{Q\ge r\}\), and \(\{Q<r\}\).
For \(p\in[1,\infty)\), the Schatten \(p\)-norm of \(X\in\mathcal{L}(\mathcal{H})\) is \(\|X\|_p:=(\mathop{\mathrm{Tr}}|X|^p)^{1/p}\), where \(|X|:=\sqrt{X^\dagger X}\). The Schatten \(\infty\)-norm is the operator norm, denoted by \(\|X\|_\infty\). In particular, \(\|X\|_1=\mathop{\mathrm{Tr}}|X|\) is the trace norm and \(\|X\|_2=(\mathop{\mathrm{Tr}}X^\dagger X)^{1/2}\) is the Hilbert–Schmidt norm.
The von Neumann entropy of \(\rho\in\mathcal{D}(\mathcal{H})\) is \(S(\rho):=-\mathop{\mathrm{Tr}}\rho\log\rho\). Throughout the paper, all logarithms are taken to base \(2\). For a bipartite state \(\rho_{AB} \in {\mathcal{D}}({\mathcal{H}}_A \otimes {\mathcal{H}}_B)\), the conditional entropy is given by \(S(A|B)_\rho = S(\rho_{AB}) - S(\rho_B)\), where \(\rho_B:= \mathop{\mathrm{Tr}}_B \rho_{AB}\) is the reduced state of the system \(B\).
A quantum channel from \(\mathcal{H}\) to \(\mathcal{K}\) is a completely positive trace-preserving linear map \(\pazocal E:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{K})\). We shall also use the same symbol for its restriction to states. The identity channel is denoted by \({\rm id}\).
The fidelity between two states \(\rho,\sigma\in\mathcal{D}(\mathcal{H})\) is defined by \[\begin{align} F(\rho,\sigma) := \|\sqrt{\rho}\sqrt{\sigma}\|_1 . \end{align}\] With this convention, \(0\le F(\rho,\sigma)\le1\), and \(F(\rho,\sigma)=1\) if and only if \(\rho=\sigma\). The purified distance is \[\begin{align} P(\rho,\sigma) := \sqrt{1-F(\rho,\sigma)^2}. \end{align}\] We shall use the Fuchs–van de Graaf inequalities in the form \[\begin{align} 1-F(\rho,\sigma) \le \frac{1}{2}\|\rho-\sigma\|_1 \le P(\rho,\sigma). \end{align}\]
We next recall the min-entropy quantities [16], [17] used below. For a state \(\rho_A\in\mathcal{D}(\mathcal{H}_A)\), the min-entropy is \[\begin{align} H_{\min}(A)_\rho := -\log\|\rho_A\|_\infty . \end{align}\] Equivalently, \(H_{\min}(A)_\rho\) is the largest real number \(\lambda\) such that \(\rho_A\le 2^{-\lambda}\mathbb{1}_A\). For a state \(\rho\), we also use the notation \(H_{\min}(\rho):=-\log\|\rho\|_\infty\).
For a bipartite state \(\rho_{AB}\), the conditional min-entropy is \[\begin{align} H_{\min}(A|B)_\rho := -\inf_{\sigma_B\in\mathcal{D}(\mathcal{H}_B)} D_{\max}\!\left( \rho_{AB}\middle\|\mathbb{1}_A\otimes\sigma_B \right), \end{align}\] where \[\begin{align} D_{\max}(\omega\|\tau) := \inf\{\lambda\in\mathbb{R}:\omega\le 2^\lambda\tau\} \end{align}\] whenever \(\mathop{\mathrm{supp}}\omega\subseteq\mathop{\mathrm{supp}}\tau\), and is \(+\infty\) otherwise [16], [18]. Equivalently, \(H_{\min}(A|B)_\rho\) is the largest real number \(\lambda\) for which there exists a state \(\sigma_B\) such that \(\rho_{AB}\le 2^{-\lambda}\mathbb{1}_A\otimes\sigma_B\).
For \(\rho\in\mathcal{D}(\mathcal{H})\) and \(\varepsilon,\delta\ge0\), we write \[\begin{align} B_{\mathrm P}^{\varepsilon}(\rho) := \{\sigma\in\mathcal{D}(\mathcal{H}):P(\sigma,\rho)\le\varepsilon\}, \qquad B_1^\delta(\rho) := \{\sigma\in\mathcal{D}(\mathcal{H}):\|\sigma-\rho\|_1\le\delta\}. \end{align}\] The \(\varepsilon\)-smooth min-entropy with respect to purified distance is \[\begin{align} H_{\min}^{\varepsilon,\mathrm P}(A)_\rho := \sup_{\sigma_A\in B_{\mathrm P}^{\varepsilon}(\rho_A)} H_{\min}(A)_\sigma, \end{align}\] and the trace-norm smoothed version is defined analogously by optimizing over \(B_1^\delta(\rho_A)\)1. Similarly, for a bipartite state \(\rho_{AB}\), the purified-distance smooth conditional min-entropy is \[\begin{align} H_{\min}^{\varepsilon,\mathrm P}(A|B)_\rho := \sup_{\sigma_{AB}\in B_{\mathrm P}^{\varepsilon}(\rho_{AB})} H_{\min}(A|B)_\sigma, \end{align}\] with the trace-norm version obtained by replacing \(B_{\mathrm P}^{\varepsilon}(\rho_{AB})\) by \(B_1^\delta(\rho_{AB})\).
By the Fuchs–van de Graaf inequalities [19], \(\|\rho-\sigma\|_1\le2P(\rho,\sigma)\). Hence \[\begin{align}\label{eq:ball-inclusion} B_{\mathrm P}^{\varepsilon}(\rho) \subseteq B_1^{2\varepsilon}(\rho). \end{align}\tag{1}\] Consequently, \[\begin{align}\label{eq:h-min} H_{\min}^{2\varepsilon,\|\cdot\|_1}(A)_\rho \ge H_{\min}^{\varepsilon,\mathrm P}(A)_\rho, \qquad H_{\min}^{2\varepsilon,\|\cdot\|_1}(A|B)_\rho \ge H_{\min}^{\varepsilon,\mathrm P}(A|B)_\rho. \end{align}\tag{2}\] Thus any lower bound on the min-entropy smoothed with respect to the purified distance immediately yields the corresponding lower bound for trace-norm smoothing, with smoothing parameter enlarged from \(\varepsilon\) to \(2\varepsilon\).
We finally recall the key entropic quantities of the quantum information-spectrum framework [20], [21] (see also, e.g. [22] and references therein). Let \(\boldsymbol{\rho}=(\rho_n)_{n\ge1}\) and \(\boldsymbol{\sigma}=(\sigma_n)_{n\ge1}\) be sequences of positive semidefinite operators acting on finite-dimensional Hilbert spaces \((\mathcal{H}_n)_{n\ge1}\), with each \(\rho_n\) a state. Following the formulation of [23], the spectral sup-divergence rate and spectral inf-divergence rate are defined by \[\begin{align} \overline{D}(\boldsymbol{\rho}\|\boldsymbol{\sigma}) := \inf\left\{ \gamma\in\mathbb{R}: \lim_{n\to\infty} \mathop{\mathrm{Tr}}\!\left[ \{\rho_n-2^{n\gamma}\sigma_n>0\}\rho_n \right] =0 \right\}, \end{align}\] and \[\begin{align} \underline D(\boldsymbol{\rho}\|\boldsymbol{\sigma}) := \sup\left\{ \gamma\in\mathbb{R}: \lim_{n\to\infty} \mathop{\mathrm{Tr}}\!\left[ \{\rho_n-2^{n\gamma}\sigma_n>0\}\rho_n \right] =1 \right\}. \end{align}\]
The corresponding spectral entropy rates are obtained by comparing the state sequence with the identity sequence, i.e.if \(\boldsymbol{\rho}=(\rho_n)_n\) is a sequence of states on \((\mathcal{H}_n)_n\), and \(\boldsymbol{\mathbb{1}}=(\mathbb{1}_n)_n\), where \(\mathbb{1}_n\) is the identity operator on \(\mathcal{H}_n\), then \[\begin{align}\label{eq:sup-spectral-entropy-rate} \overline{S}(\boldsymbol{\rho}) := -\underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}}), \qquad \underline S(\boldsymbol{\rho}) := -\overline{D}(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}}). \end{align}\tag{3}\] We use overlines for spectral sup-rates and underlines for spectral inf-rates.
Remark 1. Equivalently, \(\overline{S}(\boldsymbol{\rho})\) is the infimum over all real \(R\) for which there exists a sequence of projections \((\Pi_n)_n\) satisfying \(\mathop{\mathrm{Tr}}(\Pi_n\rho_n)\to1\) as \(n\to\infty\), and \(\mathop{\mathrm{Tr}}\Pi_n\le 2^{nR}\) for all sufficiently large \(n\). (We include an explanation of this remark in Appendix 9 for the convenience of readers who are unfamiliar with the Information Spectrum Framework.)
We refer to \(\overline{S}(\boldsymbol{\rho})\) and \(\underline S(\boldsymbol{\rho})\), respectively, as the spectral sup-entropy rate and the spectral inf-entropy rate of the source \(\boldsymbol{\rho}\). Note that \(\underline S(\boldsymbol{\rho})\le\overline{S}(\boldsymbol{\rho})\).
For a sequence of bipartite states \(\boldsymbol{\rho}_{AB}=(\rho_{A^nB^n})_n\), the spectral conditional sup- and inf-entropy rates are, respectively, defined by \[\begin{align}\label{eq:cond-sup-spec} \overline{S}(A|B)_{\boldsymbol{\rho}} := -\underline D\!\left( \boldsymbol{\rho}_{AB} \middle\| (\mathbb{1}_{A^n}\otimes\rho_{B^n})_n \right), \end{align}\tag{4}\] and \[\begin{align} \underline S(A|B)_{\boldsymbol{\rho}} := -\overline{D}\!\left( \boldsymbol{\rho}_{AB} \middle\| (\mathbb{1}_{A^n}\otimes\rho_{B^n})_n \right), \end{align}\] where \(\rho_{B^n}:=\mathop{\mathrm{Tr}}_{A^n}\rho_{A^nB^n}\). These quantities reduce to the usual conditional entropy \(S(A|B)_\rho=S(\rho_{AB})-S(\rho_B)\) in the i.i.d.case \(\rho_{A^nB^n}=\rho_{AB}^{\otimes n}\).
Let \(S_n\) denote the permutation group on \(\{1,\ldots,n\}\), and let \({\mathcal{D}}(\mathcal{H})\) be the set of density operators acting on a finite-dimensional Hilbert space \(\mathcal{H}\). The symmetric subspace of \(\mathcal{H}^{\otimes n}\) is defined by \[\begin{align} \mathrm{Sym}^n(\mathcal{H}) := \operatorname{span} \{ \ket{\phi}^{\otimes n}:\ket{\phi}\in\mathcal{H} \}. \end{align}\]
Given a vector \(\ket{\theta}\in\mathcal{H}\) and an integer \(m\le n\), let \[\begin{align} \mathcal{V}(\mathcal{H}^{\otimes n},\ket{\theta}^{\otimes m}) := \left\{ \pi\!\left( \ket{\theta}^{\otimes m}\otimes\ket{\Omega} \right) : \pi\in S_n,\; \ket{\Omega}\in\mathcal{H}^{\otimes(n-m)} \right\}, \end{align}\] that is, the set of vectors obtained by permuting a tensor product consisting of \(m\) copies of \(\ket{\theta}\) together with an arbitrary state on the remaining \(n-m\) subsystems. We further define \[\begin{align} \mathrm{Sym}^n(\mathcal{H},\ket{\theta}^{\otimes m}) := \mathrm{Sym}^n(\mathcal{H}) \cap \operatorname{span} \mathcal{V}(\mathcal{H}^{\otimes n},\ket{\theta}^{\otimes m}). \end{align}\]
The notion of almost i.i.d.states was originally introduced for pure symmetric states: a pure state \(\ket{\Psi^{(n)}}\in \mathrm{Sym}^n(\mathcal{H},\ket{\theta}^{\otimes(n-r)})\) was called a \(\binom{n}{r}\)-almost i.i.d.state in \(\ket{\theta}\). To accommodate mixed states and more general correlated structures, we use the following broader definition.
Definition 2 ((MSR almost i.i.d.states)). Let \(\mathcal{H}_A\) be a finite-dimensional Hilbert space, let \(\sigma_A\in {\mathcal{D}}(\mathcal{H}_A)\), and let \(n\in\mathbb{N}_+\) and \(0\le r\le n\). A state \(\rho_{A_1^n}\in {\mathcal{D}}(\mathcal{H}_A^{\otimes n})\) is called a \(\binom{n}{r}\)-almost i.i.d.state along \(\sigma_A\) if there exist a purification \(\ket{\theta}_{AE}\) of \(\sigma_A\) and an extension \(\rho_{A_1^nE_1^n}\) of \(\rho_{A_1^n}\) such that:
\(\rho_{A_1^nE_1^n}\) is invariant under simultaneous permutations of the subsystem pairs \((A_i,E_i)\);
\[\begin{align}\label{eq:supp} \operatorname{supp}(\rho_{A_1^nE_1^n}) \subseteq \operatorname{span} \mathcal{V} \bigl( \mathcal{H}_{AE}^{\otimes n}, \ket{\theta}_{AE}^{\otimes(n-r)} \bigr). \end{align}\qquad{(1)}\]
The set of all such states are denoted by2 \({\mathcal{S}}^n(\mathcal{H}_A,\sigma_A^{\otimes(n-r)}).\)
In the following, we shall refer to such an extension \(\rho_{A_1^n E_1^n}\) as an MSR extension* of the state \(\rho_{A_1^n}\) along the purification \(\ket{\theta}_{AE}\) of \(\sigma_A\).*
Note that \(\mathcal{V} \bigl( \mathcal{H}_{AE}^{\otimes n}, \ket{\theta}_{AE}^{\otimes(n-r)} \bigr)\) is a set of vectors rather than a linear subspace; the corresponding linear subspace is obtained by taking its span.
Remark 3 ((Unrestricted tensor positions and defect size)). The parameter \(r\) measures the number of tensor positions that are not constrained to carry the reference purification \(\ket{\theta}_{AE}\). More precisely, each vector in \(\mathcal{V} \bigl( \mathcal{H}_{AE}^{\otimes n}, \ket{\theta}_{AE}^{\otimes(n-r)} \bigr)\) has \(n-r\) tensor positions fixed to be in the state \(\ket{\theta}_{AE}\), while the remaining \(r\) tensor positions are arbitrary and may be jointly correlated. We refer to these remaining positions as unrestricted tensor positions, and to \(r\) as the defect size.
Remark 4 ((Defect spaces)). For a purification \(\ket{\theta}_{AE}\) of \(\sigma_A\) and an integer \(r\le n\), we define the associated defect space by \[\begin{align}\label{eq:defect-space} \mathcal{M}_n(\theta,r) := \operatorname{span} \mathcal{V} \bigl( \mathcal{H}_{AE}^{\otimes n}, \ket{\theta}_{AE}^{\otimes(n-r)} \bigr). \end{align}\tag{5}\] Thus, if \(\rho_{A_1^n}\in \mathcal{S}^n(\mathcal{H}_A,\sigma_A^{\otimes(n-r)})\), then by Definition 2 there exists an MSR extension \(\rho_{A_1^nE_1^n}\) satisfying \[\begin{align} \operatorname{supp}(\rho_{A_1^nE_1^n}) \subseteq \mathcal{M}_n(\theta,r). \end{align}\] We shall frequently refer to \(\mathcal{M}_n(\theta,r)\) as the defect space associated with the reference purification \(\ket{\theta}_{AE}\) and defect parameter \(r\).
Definition 5 ((MSR almost i.i.d.sources)). Let \(\rho_A\in\mathcal{D}(\mathcal{H}_A)\), and let \((r_n)_n\) be a sequence of nonnegative integers satisfying \(r_n\le n\). A sequence of states \(\boldsymbol{\rho}_A:=(\rho_{A^n})_{n\ge1}\), with \(\rho_{A^n}\in\mathcal{D}(\mathcal{H}_A^{\otimes n})\), is called an MSR almost i.i.d.source along \(\rho_A\), with defect sizes \((r_n)_n\), if for every \(n\), \[\begin{align} \rho_{A^n} \in \mathcal{S}^n(\mathcal{H}_A,\rho_A^{\otimes(n-r_n)}). \end{align}\] In the asymptotic regime considered in this paper, we shall always assume \(r_n=o(n)\).
The same terminology will be used for bipartite sources \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_n\) by applying the preceding definition to the single composite system \(AB\), with reference state \(\rho_{AB}\).
Remark 6 ((Pure MSR sources)). A sequence \(\boldsymbol{\Psi}_{AB}:=(\ket{\Psi_n}\!\bra{\Psi_n}_{A^nB^n})_{n}\) is called a pure MSR almost i.i.d.source along a bipartite pure state \(\ket{\phi}_{AB}\), with defect sizes \((r_n)_n\), if \[\begin{align}\label{eq:pure-pi-msr} \ket{\Psi_n}_{A^nB^n} \in {\rm Sym}^n\!\left( \mathcal{H}_A\otimes\mathcal{H}_B, \ket{\phi}_{AB}^{\otimes(n-r_n)} \right) \end{align}\tag{6}\] for every \(n\). In particular, each \(\ket{\Psi_n}\) is invariant under simultaneous permutations of the \(n\) bipartite tensor factors.
Remark 7. (Terminology) For brevity we often refer to an MSR almost i.i.d.sequence of states (or source) along a reference state simply as an MSR sequence (or source) along a reference state.
In this section we establish the basic stability and rigidity properties of MSR almost i.i.d.states which will be used throughout the paper. The first group of results concerns the behaviour of the MSR structure under natural operations, such as local tensor-power channels, and gives subexponential bounds on the size of the associated defect spaces. The second group of results shows that, despite the possible presence of sublinear deviations from the reference tensor-power structure, MSR sources retain the relevant asymptotic spectral entropy properties of their reference states.
Lemma 1 ((MSR stability under local tensor-power channels)). Let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_n\) be MSR almost i.i.d.along \(\rho_{AB}\), with defect sizes \(r_n=o(n)\). Let \(\Lambda_A:\mathcal{D}(\mathcal{H}_A)\to\mathcal{D}(\mathcal{H}_{A'})\) be a quantum channel, and define \[\begin{align} \omega_{A'^nB^n} := (\Lambda_A^{\otimes n}\otimes\mathop{\mathrm{id}}_{B^n})(\rho_{A^nB^n}), \qquad \omega_{A'B} := (\Lambda_A\otimes\mathop{\mathrm{id}}_B)(\rho_{AB}). \end{align}\] Then \(\boldsymbol{\omega}_{A'B}:=(\omega_{A'^nB^n})_n\) is MSR almost i.i.d.along \(\omega_{A'B}\), with the same defect sizes \(r_n=o(n)\).
Proof. See Appendix 10.1. ◻
Corollary 8 ((MSR stability under taking marginals)). Let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_n\) be MSR almost i.i.d.along \(\rho_{AB}\), with defect sizes \(r_n=o(n)\). Then the marginal source \(\boldsymbol{\rho}_B:=(\rho_{B^n})_n\), where \(\rho_{B^n}:=\mathop{\mathrm{Tr}}_{A^n}\rho_{A^nB^n}\), is MSR almost i.i.d.along \(\rho_B:=\mathop{\mathrm{Tr}}_A\rho_{AB}\), with the same defect sizes.
Proof. Apply Lemma 1 with the channel \(\Lambda_A=\mathop{\mathrm{Tr}}_A\), whose output system is one-dimensional. Then \[\begin{align} (\Lambda_A^{\otimes n}\otimes\mathop{\mathrm{id}}_{B^n})(\rho_{A^nB^n}) = \mathop{\mathrm{Tr}}_{A^n}\rho_{A^nB^n} = \rho_{B^n}, \end{align}\] and \[\begin{align} (\Lambda_A\otimes\mathop{\mathrm{id}}_B)(\rho_{AB}) = \mathop{\mathrm{Tr}}_A\rho_{AB} = \rho_B. \end{align}\] Hence \(\boldsymbol{\rho}_B\) is MSR almost i.i.d.along \(\rho_B\), with the same defect sizes \(r_n=o(n)\). ◻
The following lemma quantifies the size of the defect spaces appearing in Definition 1. It shows that when the number of defects is sublinear, the corresponding defect spaces have only subexponential dimension.
Lemma 2 ((Subexponential defect complexity)). Let \(\ket{\theta}_{AE}\in\mathcal{H}_A\otimes\mathcal{H}_E\), and let \((r_n)_n\) be a sequence of integers satisfying \(0\le r_n\le n\) and \(r_n=o(n)\). Then the corresponding MSR defect spaces \(\mathcal{M}_n(\theta,r_n)\), defined in 5 , satisfy \[\begin{align} \dim \mathcal{M}_n(\theta,r_n)=2^{o(n)}. \end{align}\] Consequently, every vector in \(M_n(\theta,r_n)\) can be expanded using at most \(2^{o(n)}\) basis vectors.
Proof. By Remark 2.4(c) of [24], \[\begin{align} \dim \mathcal{M}_n(\theta,r_n) \le \binom{n}{r_n}d^{r_n}, \end{align}\] where \(d=\dim(\mathcal{H}_A\otimes\mathcal{H}_E)\). Taking logarithms and dividing by \(n\), we obtain \[\begin{align} \frac{1}{n}\log \dim \mathcal{M}_n(\theta,r_n) \le \frac{1}{n}\log\binom{n}{r_n} + \frac{r_n}{n}\log d. \end{align}\] Since \(r_n=o(n)\), we have \(r_n/n\to0\) as \(n \to \infty\). Moreover, the standard bound \(\log\binom{n}{r_n}\le n h(r_n/n)\), where \(h\) denotes the binary entropy function (which follows immediately from the binomial theorem), gives \[\begin{align} \frac{1}{n}\log\binom{n}{r_n} \le h\!\left(\frac{r_n}{n}\right) \to 0 \quad \text{as} \quad n \to \infty. \end{align}\] Hence \[\begin{align} \lim_{n\to\infty} \frac{1}{n}\log \dim \mathcal{M}_n(\theta,r_n) =0, \end{align}\] which is equivalent to \(\dim\mathcal{M}_n(\theta,r_n)=2^{o(n)}\).
The final assertion follows by choosing an orthonormal basis of \(\mathcal{M}_n(\theta,r_n)\). ◻
We next establish the entropy estimates which ensure that MSR almost i.i.d.sources with sublinear defect size have the same asymptotic spectral entropy as their reference i.i.d.state. The point is that, although the states in the MSR class need not be close in trace norm to \(\rho^{\otimes n}\), their deviation from the tensor-power structure is controlled by a sublinear defect size. As a consequence, this deviation has no effect on the entropy rate. The following lemma makes this precise in the language of information spectrum quantities, and will be used repeatedly to replace spectral entropy rates of MSR sources by the ordinary von Neumann entropy of the reference state.
Lemma 3 ((Spectral entropy rates of MSR almost i.i.d.sources)). Let \(\rho\in\mathcal{D}(\mathcal{H})\), and let \(\boldsymbol{\rho}:=(\rho_n)_n\) be such that \(\rho_n\in S^n(\mathcal{H},\rho^{\otimes(n-r_n)})\) for every \(n\), with \(r_n=o(n)\). Then \[\begin{align} \underline S(\boldsymbol{\rho})=\overline{S}(\boldsymbol{\rho})=S(\rho). \end{align}\]
Proof. We first prove \(\underline S(\boldsymbol{\rho})\ge S(\rho)\). By Theorem 1 of [25], \[\begin{align} \underline S(\boldsymbol{\rho}) = \lim_{\varepsilon\to0} \liminf_{n\to\infty} \frac{1}{n} H_{\min}^{\varepsilon,\|\cdot\|_1}(\rho_n). \end{align}\] Here the smoothing is with respect to trace norm.
The Strong AEP for almost i.i.d.states, Proposition 4.2 of [24], applied with trivial conditioning system, gives, for every fixed \(\varepsilon\in(0,1)\), \[\begin{align} \frac{1}{n} H_{\min}^{\varepsilon,\mathrm P}(\rho_n) = S(\rho)+\frac{a_n}{n}, \end{align}\] where where \(a_n\in\mathbb{R}\) satisfies \(a_n=o(n)\)3. The smoothing is with respect to purified distance. In particular, since \(a_n/n\to0\) as \(n \to \infty\), we obtain \[\begin{align} \liminf_{n\to\infty} \frac{1}{n} H_{\min}^{\varepsilon,\mathrm P}(\rho_n) \ge S(\rho), \end{align}\] where the smoothing is with respect to purified distance. By (2 ), \(H_{\min}^{2\varepsilon,\|\cdot\|_1}(\rho_n) \ge H_{\min}^{\varepsilon,\mathrm P}(\rho_n).\) Consequently, for every fixed \(\varepsilon\in(0,1)\), \[\begin{align} \liminf_{n\to\infty} \frac{1}{n} H_{\min}^{2\varepsilon,\|\cdot\|_1}(\rho_n) \ge S(\rho). \end{align}\] Taking \(\varepsilon\to 0\) yields \(\underline S(\boldsymbol{\rho})\ge S(\rho)\).
We next prove \(\overline{S}(\boldsymbol{\rho})\le S(\rho)\). Since MSR almost i.i.d.sources with \(r_n=o(n)\) are weakly almost i.i.d.along their reference state, Lemma 3 of [14] applies to \(\boldsymbol{\rho}\). Fix \(R>S(\rho)\). Since by this lemma, \(h_q:=-\mathop{\mathrm{Tr}}\rho\log((1-q)\rho+q\mathbb{1}/d) \to S(\rho)\) as \(q\downarrow0\), we may choose \(q\in(0,1)\) and \(\delta>0\) such that \(h_q+\delta<R\). The universal typical projector used in this quoted lemma, which is defined as \[\begin{align} \Pi_n:=\{-\frac{1}{n}\log\sigma_q^{\otimes n}\le h_q+\delta\}, \end{align}\] satisfies \(\mathop{\mathrm{Tr}}(\Pi_n\rho_n)\to1\) as \(n \to \infty\) and \[\begin{align} \mathop{\mathrm{Tr}}\Pi_n\le e^{n(h_q+\delta)}\le e^{nR}. \end{align}\] By the projector characterization of the sup-spectral entropy rate \(\overline{S}(\boldsymbol{\rho})\) (see 3 and the remark below it), the existence of projectors \(\Pi_n\) satisfying \(\mathop{\mathrm{Tr}}(\Pi_n\rho_n)\to1\) as \(n \to \infty\), and \(\mathop{\mathrm{Tr}}\Pi_n\le 2^{nR}\) implies \(\overline{S}(\boldsymbol{\rho})\le R\). Since \(R>S(\rho)\) was arbitrary, \(\overline{S}(\boldsymbol{\rho})\le S(\rho)\).
Finally, since always \(\underline S(\boldsymbol{\rho})\le\overline{S}(\boldsymbol{\rho})\), the two bounds imply \[\begin{align} \underline S(\boldsymbol{\rho})=\overline{S}(\boldsymbol{\rho})=S(\rho). \end{align}\] ◻
Lemma 4 ((Conditional spectral entropy bound for MSR almost i.i.d.states)). Let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_n\) be MSR almost i.i.d.along \(\rho_{AB}\), with defect sizes \(r_n=o(n)\). Then \[\begin{align} \overline{S}(A|B)_{\boldsymbol{\rho}} \le S(A|B)_\rho . \end{align}\]
Proof. By stability of the MSR property under taking marginals given by Corollary 8, the marginal source \(\boldsymbol{\rho}_B:=(\rho_{B^n})_n\) is MSR almost i.i.d.along \(\rho_B\), with defect sizes \(r_n=o(n)\). Hence, Lemma 3, applied to \(\boldsymbol{\rho}_{AB}\) and to \(\boldsymbol{\rho}_B\), gives \[\begin{align} \overline{S}(\boldsymbol{\rho}_{AB})=S(\rho_{AB}), \qquad \underline S(\boldsymbol{\rho}_B)=S(\rho_B). \end{align}\] Using the information-spectrum chain-rule inequality given by Proposition 9 of [23], \[\begin{align} \overline{S}(A|B)_{\boldsymbol{\rho}} \le \overline{S}(\boldsymbol{\rho}_{AB})-\underline S(\boldsymbol{\rho}_B), \end{align}\] we obtain \[\begin{align} \overline{S}(A|B)_{\boldsymbol{\rho}} \le S(\rho_{AB})-S(\rho_B) = S(A|B)_\rho . \end{align}\] ◻
The entropy-rigidity estimates above show that MSR almost i.i.d.sources retain the relevant asymptotic entropy behaviour of their reference state. In the applications below, however, we need a uniform version of this estimate: the weight assigned to the spectral projection \(\{\rho_n\ge 2^{-n\gamma}\mathbb{1}_n\}\) must vanish uniformly over the whole MSR class with a prescribed sublinear defect bound, whenever \(\gamma<S(\rho)\). Equivalently, no source in this class can asymptotically assign non-negligible weight to eigenvalues larger than \(2^{-n\gamma}\) for any \(\gamma<S(\rho)\). This is the form in which the entropy-rigidity estimate will be used later, and is made precise in the following corollary.
Corollary 9 ((Uniform lower spectral tail bound for MSR sources)). Let \(\rho\in\mathcal{D}(\mathcal{H})\), and let \((r_n)_n\) satisfy \(r_n=o(n)\). Then, for every \(\gamma<S(\rho)\), \[\begin{align} \sup_{\boldsymbol{\rho}=(\rho_n)_n} \mathop{\mathrm{Tr}}\!\left[ \left\{ \rho_n\ge 2^{-n\gamma}\mathbb{1}_n \right\} \rho_n \right] \to 0 \quad \text{as} \quad n \to \infty, \end{align}\] where the supremum is over all states \(\rho_n\in S^n(H,\rho^{\otimes(n-s_n)})\) with \(s_n\le r_n\), or equivalently over all MSR source sequences with defect at most \(r_n\) at blocklength \(n\).
Proof. Fix \(\gamma<S(\rho)\), and choose \(\kappa\) such that \(\gamma<\kappa<S(\rho)\). Let \(\varepsilon\in(0,1)\).
By Proposition 4.2 of [24], the error term in the preceding AEP estimate may be chosen uniformly over all MSR almost i.i.d.sources along \(\rho\) with defect size at most \(r_n\). Thus, for every fixed \(\varepsilon\in(0,1)\), \[\frac{1}{n} H_{\min}^{\varepsilon,P}(\rho_n) = S(\rho)+\frac{a_n}{n},\] where \(a_n\in\mathbb{R}\), \(a_n=o(n)\), and \(a_n\) depends only on \(\rho\), the dimension, \(\varepsilon\), and the defect-size bound \((r_n)_n\), but not on the particular state \(\rho_n\). Since \(\kappa<S(\rho)\), and since \(a_n/n\to0\) as \(n \to \infty\), it follows that, for all sufficiently large \(n\), \[\begin{align} \frac{1}{n} H_{\min}^{\varepsilon,\mathrm P}(\rho_n) \ge \kappa \end{align}\] uniformly over the class.
Thus, for each such \(\rho_n\), there exists a state \(\widetilde{\rho}_n\) such that \(P(\widetilde{\rho}_n,\rho_n)\le \varepsilon\) and \(\widetilde{\rho}_n\le 2^{-n\kappa}\mathbb{1}\). By the Fuchs–van de Graaf inequality, \(\|\widetilde{\rho}_n-\rho_n\|_1\le2\varepsilon\).
Set \(Q_n:=\{\rho_n\ge 2^{-n\gamma}\mathbb{1}\}.\) Since \(Q_n\) is the projector onto eigenvalues of \(\rho_n\) at least \(2^{-n\gamma}\), and since \(\mathop{\mathrm{Tr}}\rho_n=1\), we have \(\operatorname{rank}Q_n\le 2^{n\gamma}\). Therefore \[\begin{align} \mathop{\mathrm{Tr}}[Q_n\widetilde{\rho}_n] \le 2^{-n\kappa}\operatorname{rank}Q_n \le 2^{-n(\kappa-\gamma)}. \end{align}\] Moreover, since \(0\le Q_n\le\mathbb{1}\), \[\begin{align} \begin{aligned} \mathop{\mathrm{Tr}}[Q_n\rho_n] &= \mathop{\mathrm{Tr}}[Q_n\widetilde{\rho}_n] + \mathop{\mathrm{Tr}}[Q_n(\rho_n-\widetilde{\rho}_n)] \\ &\le \mathop{\mathrm{Tr}}[Q_n\widetilde{\rho}_n] + \|\rho_n-\widetilde{\rho}_n\|_1 \\ &\le 2^{-n(\kappa-\gamma)}+2\varepsilon . \end{aligned} \end{align}\] Taking the supremum over the MSR class and then the limit superior gives \[\begin{align} \limsup_{n\to\infty} \sup_{\boldsymbol{\rho}} \mathop{\mathrm{Tr}}[Q_n\rho_n] \le 2\varepsilon . \end{align}\] Since \(\varepsilon>0\) was arbitrary, the claim follows. ◻
We shall also need a simple stability property of conditional spectral entropy rates under the addition of auxiliary systems of subexponential size. This situation arises when an auxiliary description of an MSR source includes finite classical labels and an additional quantum register whose dimension grows only subexponentially with the blocklength. The following lemma shows that such auxiliary registers do not change the normalized sup-conditional spectral entropy rate, except for the expected contribution coming from the logarithm of the dimension of the added quantum register.
Lemma 5 ((Finite-register stability of conditional spectral entropy)). Let \(\boldsymbol{\gamma}_{ABCZ}:=(\gamma_{A_lB_lC_lZ_l})_{l\ge1}\) be a sequence of states, where \(Z_l\) is a classical register, as in the applications below, and let \(d_{C,l}:=\dim\mathcal{H}_{C_l}\). Then \[\begin{align} \overline{S}(AC|BZ)_{\boldsymbol{\gamma}} \le \overline{S}(A|B)_{\boldsymbol{\gamma}} + \limsup_{l\to\infty}\frac{1}{l}\log d_{C,l}. \end{align}\] In particular, if \(\log d_{C,l}=o(l)\), then \(\overline{S}(AC|BZ)_{\boldsymbol{\gamma}}\le \overline{S}(A|B)_{\boldsymbol{\gamma}}\). Thus adjoining to Alice’s side a system whose logarithmic dimension is sublinear, and adjoining a classical register to the conditioning system, does not increase the normalized sup-conditional spectral entropy rate.
Proof. See Appendix 11. ◻
In this section we recall the basic entanglement manipulation tasks that will be considered in the sequel. We first discuss entanglement distillation, and its pure-state specialization, entanglement concentration, and then turn to the reverse task of entanglement dilution. We introduce the corresponding operational quantities for general sequences of bipartite states and recall their standard expressions in the i.i.d. setting.
Let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_{n\ge1}\) be a sequence of bipartite states shared by two distant parties, Alice and Bob. An entanglement-distillation protocol for this sequence consists, for each \(n\), of an LOCC map \(\pazocal D_n\) acting on \(A^nB^n\), whose aim is to extract a maximally entangled state \(\Phi_{M_n}\) of Schmidt rank \(M_n\). A rate \(R\ge0\) is said to be achievable if there exists such a sequence of LOCC maps \((\pazocal D_n)_n\) such that \[\begin{align} \liminf_{n\to\infty}\frac{1}{n}\log M_n\ge R \end{align}\] and \[\begin{align} \lim_{n\to\infty} F\!\left( \pazocal D_n(\rho_{A^nB^n}), \Phi_{M_n} \right) = 1. \end{align}\] The distillable entanglement of the sequence \(\boldsymbol{\rho}_{AB}\) is defined as \[\begin{align} E_D(\boldsymbol{\rho}_{AB}) := \sup\{R:\;R\text{ is achievable for }\boldsymbol{\rho}_{AB}\}. \end{align}\] When the sequence is i.i.d., \(\boldsymbol{\rho}_{AB}=(\rho_{AB}^{\otimes n})_n\), this reduces to the usual distillable entanglement \(E_D(\rho_{AB})\). For general mixed states, no single-letter formula for \(E_D(\rho_{AB})\) is known in general, although the hashing inequality gives the coherent information as an achievable lower bound [26]. Thus, in particular, \[\begin{align} E_D(\rho_{AB})\ge \max\{0,I(A\rangle B)_\rho\}, \end{align}\] where \(I(A\rangle B)_\rho:= - S(A|B)_\rho\) denotes the coherent information. For pure states, however, the problem is completely characterized: if \(\psi_{AB}:=\ket{\psi}\!\bra{\psi}_{AB}\), then \[\begin{align} E_D(\psi_{AB}) = S(\rho_A), \qquad \rho_A:=\mathop{\mathrm{Tr}}_B\psi_{AB}. \end{align}\]
In the special case where each \(\rho_{A^nB^n}\) is pure, say \(\rho_{A^nB^n}=\ket{\Psi_n}\!\bra{\Psi_n}_{A^nB^n}\), the same task is usually called entanglement concentration. Thus, for a pure-state sequence \(\boldsymbol{\Psi}_{AB}:=(\ket{\Psi_n}\!\bra{\Psi_n}_{A^nB^n})_n\), a rate \(R\ge0\) is achievable for entanglement concentration if there exist LOCC maps \(\pazocal C_n\) such that \[\begin{align} \liminf_{n\to\infty}\frac{1}{n}\log M_n\ge R, \qquad \lim_{n\to\infty} F\!\left( \pazocal C_n(\Psi_{A^nB^n}), \Phi_{M_n} \right) = 1. \end{align}\] The optimal rate is denoted by \(E_D(\boldsymbol{\Psi}_{AB})\), or simply referred to as the concentration rate of the sequence. In the i.i.d.pure-state case, \(\boldsymbol{\Psi}_{AB}=(\psi_{AB}^{\otimes n})_n\), this rate is equal to the entropy of entanglement \(S(\rho_A)\), by the entanglement concentration theorem of Bennett, Bernstein, Popescu and Schumacher [27], [28]. For arbitrary pure-state sequences, entanglement concentration admits an information-spectrum characterization [5].
Conversely, entanglement dilution is the task of creating a desired bipartite state from maximally entangled states by LOCC. Let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_{n\ge1}\) be a sequence of bipartite states. An entanglement-dilution protocol for this sequence consists, for each \(n\), of an LOCC map \(\pazocal E_n\) whose input is a maximally entangled state \(\Phi_{M_n}\) of Schmidt rank \(M_n\), and whose output is a state on \(A^nB^n\). A rate \(R\ge0\) is said to be achievable if there exists such a sequence of LOCC maps \((\pazocal E_n)_n\) such that \[\begin{align} \limsup_{n\to\infty}\frac{1}{n}\log M_n\le R \end{align}\] and \[\begin{align} \lim_{n\to\infty} F\!\left( \pazocal E_n(\Phi_{M_n}), \rho_{A^nB^n} \right) = 1. \end{align}\] The entanglement cost of the sequence \(\boldsymbol{\rho}_{AB}\) is defined as \[\begin{align} E_C(\boldsymbol{\rho}_{AB}) := \inf\{R:\;R\text{ is achievable for entanglement dilution of } \boldsymbol{\rho}_{AB}\}. \end{align}\]
In the i.i.d.case, \(\boldsymbol{\rho}_{AB}=(\rho_{AB}^{\otimes n})_{n\ge1}\), this recovers the usual entanglement cost \(E_C(\rho_{AB})\). Its value is given by the regularized entanglement of formation, \[\begin{align} E_C(\rho_{AB}) = E_F^\infty(\rho_{AB}) := \lim_{m\to\infty} \frac{1}{m} E_F(\rho_{AB}^{\otimes m}) = \inf_{m\ge1} \frac{1}{m} E_F(\rho_{AB}^{\otimes m}). \end{align}\] Here the entanglement of formation of a bipartite state \(\omega_{AB}\) is defined by \[\begin{align} E_F(\omega_{AB}) := \inf_{\{p_i,\ket{\psi_i}\}} \sum_i p_i\, S(\psi_{i,A}), \end{align}\] where the infimum is over all finite pure-state ensemble decompositions \[\begin{align} \omega_{AB} = \sum_i p_i \ket{\psi_i}\!\bra{\psi_i}_{AB}, \end{align}\] and \(\psi_{i,A}:=\mathop{\mathrm{Tr}}_B\ket{\psi_i}\!\bra{\psi_i}_{AB}\). Thus \(E_F^\infty(\rho_{AB})\) is the asymptotic rate at which maximally entangled states must be consumed, per copy, in order to prepare \(\rho_{AB}^{\otimes n}\) by LOCC.
Before specializing to pure-state entanglement concentration, we first present a slightly more general, source-dependent achievability result for entanglement distillation from mixed MSR sources, showing that every rate below the coherent information of the reference state is achievable.
Theorem 10 ((Robust entanglement-distillation achievability, source-dependent)). Let \(\rho_{AB}\in\mathcal{D}(\mathcal{H}_A\otimes\mathcal{H}_B)\), and let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_n\) be MSR almost i.i.d.along \(\rho_{AB}\), with defect sizes \(r_n=o(n)\). Then \[\begin{align} E_D(\boldsymbol{\rho}_{AB}) \ge I(A\rangle B)_{\rho_{AB}} = -S(A|B)_{\rho_{AB}} . \end{align}\] Equivalently, if \(I(A\rangle B)_{\rho_{AB}}>0\), then every rate \(0\le R<I(A\rangle B)_{\rho_{AB}}\) is achievable for entanglement distillation from the source \(\boldsymbol{\rho}_{AB}\).
Proof. If \(I(A\rangle B)_{\rho_{AB}}\le0\), then the claim follows immediately from the non-negativity of \(E_D(\boldsymbol{\rho}_{AB})\). We may therefore assume that \(I(A\rangle B)_{\rho_{AB}}>0\).
We use the information-spectrum entanglement-distillation achievability theorem for arbitrary sequences of bipartite mixed states, given by Theorem 1 of [8]. Applied to the sequence \(\boldsymbol{\rho}_{AB}\), it states that every rate \[\begin{align} 0\le R<-\overline{S}(A|B)_{\boldsymbol{\rho}} \end{align}\] is achievable for entanglement distillation from \(\boldsymbol{\rho}_{AB}\).
It remains to compare the conditional spectral entropy rate of the MSR source with the ordinary conditional entropy of the reference state \(\rho_{AB}\). By Lemma 4, since \(\boldsymbol{\rho}_{AB}\) is MSR almost i.i.d.along \(\rho_{AB}\) with \(r_n=o(n)\), we have \[\begin{align} \overline{S}(A|B)_{\boldsymbol{\rho}} \le S(A|B)_{\rho_{AB}}. \end{align}\] Hence \[\begin{align} -\overline{S}(A|B)_{\boldsymbol{\rho}} \ge -S(A|B)_{\rho_{AB}} = I(A\rangle B)_{\rho_{AB}} . \end{align}\] Therefore, for every \(0\le R<I(A\rangle B)_{\rho_{AB}}\), we have \[\begin{align} R<-\overline{S}(A|B)_{\boldsymbol{\rho}}. \end{align}\] The information-spectrum distillation theorem then implies that \(R\) is achievable for entanglement distillation from \(\boldsymbol{\rho}_{AB}\). Since this holds for every \(R<I(A\rangle B)_{\rho_{AB}}\), it follows that \[\begin{align} E_D(\boldsymbol{\rho}_{AB}) \ge I(A\rangle B)_{\rho_{AB}} . \end{align}\] ◻
For pure-state sources, entanglement distillation reduces to entanglement concentration. Applying the preceding achievability result to a pure MSR almost i.i.d.source therefore gives the following source-dependent concentration statement.
Corollary 11. Let \(\ket{\psi}_{AB}\in\mathcal{H}_A\otimes\mathcal{H}_B\) be a bipartite pure state, let \(\psi_{AB}:=\ket{\psi}\!\bra{\psi}_{AB}\), and let \(\rho_A:=\mathop{\mathrm{Tr}}_B\psi_{AB}\). Let \(\boldsymbol{\Psi}_{AB}:=(\ket{\Psi_{A^nB^n}}\!\!\bra{\Psi_{A^nB^n}})_n\) be a pure MSR almost i.i.d.source along \(\psi_{AB}\), with defect sizes \(r_n=o(n)\). Then every rate \(0\le R<S(\rho_A)\) is achievable for entanglement concentration from \(\boldsymbol{\Psi}_{AB}\), possibly by a protocol depending on the source sequence \(\boldsymbol{\Psi}_{AB}\).
Proof. Since \(\psi_{AB}\) is pure, we have \(I(A\rangle B)_\psi=S(\rho_A)\). The claim therefore follows immediately from Theorem 10, applied to the pure-state sequence \(\boldsymbol{\Psi}_{AB}\), which gives \[\begin{align} E_D(\boldsymbol{\Psi}_{AB}) \ge S(\rho_A). \end{align}\] ◻
We now turn to a universal version of the preceding pure-state concentration statement given by Corollary 11.
Fix a bipartite pure state \(\ket{\phi}_{AB}\), let \(\phi_{AB}:=\ket{\phi}\!\bra{\phi}_{AB}\), and denote by \(\mathfrak S^{\mathrm{pMSR}}_{\phi}((r_n)_n)\) the class of all pure-state sequences \(\boldsymbol{\Psi}_{AB}:=(\ket{\Psi_{A^nB^n}}\!\!\bra{\Psi_{A^nB^n}})_n\) which are MSR almost i.i.d.along \(\phi_{AB}\), with defect sizes \(r_n\) at blocklength \(n\), with \(r_n=o(n)\). By 6 , \(\ket{\Psi_n}_{A^nB^n}\in {\rm Sym}^n(\mathcal{H}_A\otimes\mathcal{H}_B)\) for each \(n\).
By a universal entanglement-concentration protocol for \(\mathfrak S^{\mathrm{pMSR}}_{\phi}\), we mean a sequence of LOCC protocols which depends only on the reference state \(\phi_{AB}\) and on the desired rate \(R\), but not on the particular source sequence \(\boldsymbol{\Psi}_{AB}\in\mathfrak S^{\mathrm{pMSR}}_{\phi}\). We shall prove that every rate below \(S(\phi_A)\), where \(\phi_A:=\mathop{\mathrm{Tr}}_B\phi_{AB}\), is achievable uniformly over this class.
To prove this, we use the following consequence of Schur–Weyl duality proved by Fawzi and Renner in [29].
Lemma 6 ([Lemma C.1). of [29]] Let \(\mathcal{H}_A\) and \(\mathcal{H}_B\) be Hilbert spaces with \(\dim(\mathcal{H}_A)=\dim(\mathcal{H}_B)=d\), and let \(\Lambda_{n,d}\) denote the set of Young diagrams of size \(n\) with at most \(d\) rows. For each \(\lambda\in\Lambda_{n,d}\), let \(\mathcal{U}_\lambda\) and \(\mathcal{V}_\lambda\) denote the irreducible representations of \(U(d)\) and \(S_n\), respectively, appearing in the Schur–Weyl decompositions \(\mathcal{H}_A^{\otimes n}\simeq \bigoplus_{\lambda}\mathcal{U}_{A,\lambda}\otimes\mathcal{V}_{A,\lambda}\) and \(\mathcal{H}_B^{\otimes n}\simeq \bigoplus_{\lambda}\mathcal{U}_{B,\lambda}\otimes\mathcal{V}_{B,\lambda}\).
Then there exists a family \(\{\ket{\psi_\lambda}_{\mathcal{V}_{A,\lambda}\mathcal{V}_{B,\lambda}}\}_\lambda\) of maximally entangled normalized vectors such that every vector \(\ket{\Omega}\in{\rm Sym}^n(\mathcal{H}_A\otimes\mathcal{H}_B)\) admits the decomposition \[\begin{align} \ket{\Omega} = \sum_{\lambda\in\Lambda_{n,d}} \ket{\phi_\lambda}_{\mathcal{U}_{A,\lambda}\mathcal{U}_{B,\lambda}} \otimes \ket{\psi_\lambda}_{\mathcal{V}_{A,\lambda}\mathcal{V}_{B,\lambda}}, \end{align}\] where the vectors \(\ket{\phi_\lambda}_{\mathcal{U}_{A,\lambda}\mathcal{U}_{B,\lambda}}\) are not necessarily normalized.
Remark 12. For notational simplicity, we suppress the dependence on the blocklength \(n\) in the Schur–Weyl spaces and write \(\mathcal{U}_{A,\lambda}, \mathcal{U}_{B,\lambda}, \mathcal{V}_{A,\lambda}\) and \(\mathcal{V}_{B,\lambda}\). More precisely, for each \(n\) and each Young diagram \(\lambda\in\Lambda_{n,d}\), these spaces should be written as \(\mathcal{U}_{A,\lambda}^{(n)}, \mathcal{U}_{B,\lambda}^{(n)}, \mathcal{V}_{A,\lambda}^{(n)}\) and \(\mathcal{V}_{B,\lambda}^{(n)}\). The corresponding dimensions and maximally entangled vectors therefore also depend on \(n\), although this dependence will usually be suppressed in the notation.
Remark 13. If \(d_A\neq d_B\) (where \(d_A = {\rm dim} {\mathcal{H}}_A\) and \(d_B = {\rm dim} {\mathcal{H}}_B\) ) we embed the smaller Hilbert space isometrically into a Hilbert space of dimension \(d=\max\{d_A,d_B\}\). This does not affect the state or the LOCC protocol, and allows Lemma 6 to be applied.
In the Schur–Weyl decomposition, each block is a tensor product of a representation space and a multiplicity space. The Hayashi–Matsumoto concentration protocol extracts entanglement from these multiplicity spaces, whose dimensions determine the amount of entanglement obtained from each block.
The next lemma provides the key link between the spectral concentration results of the previous section and the Schur–Weyl entanglement concentration protocol of Section 5. It shows that, uniformly over the class of pure MSR sources along a fixed reference state \(\ket{\phi}_{AB}\), the weight of the reduced state \(\rho_{A^n}\) on Schur sectors whose multiplicity spaces have dimension significantly below \(2^{nS(\rho_A)}\) becomes asymptotically negligible. This uniform concentration property will allow us to apply the Hayashi–Matsumoto protocol of [15] universally throughout the MSR class.
Lemma 7 ((Uniform Schur-block concentration for pure MSR states)). Let \(\ket{\phi}_{AB}\in\mathcal{H}_A\otimes\mathcal{H}_B\) be a bipartite pure state, let \(\rho_A:=\mathop{\mathrm{Tr}}_B\ket{\phi}\!\bra{\phi}_{AB}\), and let \((r_n)_n\) satisfy \(r_n=o(n)\). Let \(\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\) denote the class of all pure MSR sources along \(\ket{\phi}_{AB}\), with defect size at most \(r_n\) at blocklength \(n\).
Consider the Schur–Weyl decomposition \[\begin{align} \mathcal{H}_A^{\otimes n} = \bigoplus_{\lambda\in\Lambda_{n,d_A}} \mathcal{U}_{A,\lambda}\otimes\mathcal{V}_{A,\lambda}, \end{align}\] and let \(P_{A,\lambda}\) denote the projector onto \(\mathcal{U}_{A,\lambda}\otimes\mathcal{V}_{A,\lambda}\). Then, for every \(R<S(\rho_A)\), \[\begin{align} \sup_{\boldsymbol{\Psi}\in\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)} \sum_{\lambda:\,\frac{1}{n} \log\dim\mathcal{V}_{A,\lambda}<R} \mathop{\mathrm{Tr}}\!\left[ P_{A,\lambda}\rho_{A^n} \right] \to 0 \quad \text{as} \quad n \to \infty, \end{align}\] where \(\rho_{A^n} := \mathop{\mathrm{Tr}}_{B^n}\ket{\Psi_n}\!\bra{\Psi_n}.\)
Proof. Let \(\boldsymbol{\Psi} = (\ket{\Psi_n}\!\bra{\Psi_n}_{A^nB^n})_{n} \in \mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\) be arbitrary, and let \(\boldsymbol{\rho}_A = (\rho_{A^n})_{n}.\)
By stability of the MSR property under local tensor-power channels, Lemma 1, applied to the local channel \(\mathop{\mathrm{Tr}}_B:\mathcal{D}(\mathcal{H}_B)\to\mathbb{C}\), the reduced source \(\boldsymbol{\rho}_A=(\rho_{A^n})_n\) is MSR almost i.i.d.along \(\rho_A\), with defect size at most \(r_n\).
Hence, by Corollary 9, for every \(\gamma<S(\rho_A)\), \[\begin{align}\label{eq:unif-conv} \sup_{\boldsymbol{\Psi}\in\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)} \mathop{\mathrm{Tr}}\!\left[ \left\{ \rho_{A^n} \ge 2^{-n\gamma}\mathbb{1}_{A^n} \right\} \rho_{A^n} \right] \to 0 \quad \text{as} \quad n \to \infty . \end{align}\tag{7}\] Fix \(R<S(\rho_A)\), and choose \(R'\) and \(\gamma\) such that \(R<R'<\gamma<S(\rho_A).\)
Define \[\begin{align} \Pi_n^{<R} := \sum_{\lambda:\,\frac{1}{n}\log\dim\mathcal{V}_{A,\lambda}<R} P_{A,\lambda}. \end{align}\]
Since \(d_A\) is fixed, the number of Young diagrams \(|\Lambda_{n,d_A}|\) is polynomial in \(n\). Moreover, by the Weyl dimension formula, the dimensions of the \(U(d_A)\)-irreducible representation spaces \(\mathcal{U}_{A,\lambda}\) are polynomially bounded in \(n\), uniformly in \(\lambda\in\Lambda_{n,d_A}\). Hence \[\begin{align} \sum_{\lambda\in\Lambda_{n,d_A}}\dim\mathcal{U}_{A,\lambda} \le \operatorname{poly}(n). \end{align}\] Since every \(\lambda\) appearing in the definition of \(\Pi_n^{<R}\) satisfies \(\dim\mathcal{V}_{A,\lambda}<2^{nR}\), we obtain \[\begin{align} \begin{aligned} \mathop{\mathrm{Tr}}\Pi_n^{<R} &= \sum_{\lambda:\,n^{-1}\log\dim\mathcal{V}_{A,\lambda}<R} \dim\mathcal{U}_{A,\lambda}\, \dim\mathcal{V}_{A,\lambda} \\ &\le 2^{nR} \sum_{\lambda\in\Lambda_{n,d_A}}\dim\mathcal{U}_{A,\lambda} \le \operatorname{poly}(n)2^{nR}. \end{aligned} \end{align}\] Since \(R'>R\), the polynomial prefactor is negligible compared with \(2^{n(R'-R)}\). Hence, for all sufficiently large \(n\), \[\begin{align} \mathop{\mathrm{Tr}}\Pi_n^{<R} \le \operatorname{poly}(n)2^{nR} \le 2^{nR'}. \end{align}\]
Let \(Q_n := \left\{ \rho_{A^n} \ge 2^{-n\gamma}\mathbb{1}_{A^n} \right\}.\) Since \(Q_n\) is a spectral projector of \(\rho_{A^n}\), it commutes with \(\rho_{A^n}\). Therefore \[\begin{align} \rho_{A^n} = Q_n\rho_{A^n}Q_n + (\mathbb{1}_{A^n}-Q_n)\rho_{A^n}(\mathbb{1}_{A^n}-Q_n), \end{align}\] the cross terms vanishing identically. Consequently, \[\begin{align} \mathop{\mathrm{Tr}}[\Pi_n^{<R}\rho_{A^n}] = \mathop{\mathrm{Tr}}[\Pi_n^{<R}Q_n\rho_{A^n}Q_n] + \mathop{\mathrm{Tr}}[\Pi_n^{<R}(\mathbb{1}_{A^n}-Q_n)\rho_{A^n}(\mathbb{1}_{A^n}-Q_n)]. \end{align}\] Since \(0\le\Pi_n^{<R}\le\mathbb{1}_{A^n}\), \[\begin{align} \mathop{\mathrm{Tr}}[\Pi_n^{<R}Q_n\rho_{A^n}Q_n] \le \mathop{\mathrm{Tr}}[Q_n\rho_{A^n}]. \end{align}\] By the uniform spectral-tail estimate in 7 , the term \(\mathop{\mathrm{Tr}}[Q_n\rho_{A^n}]\) converges to \(0\) uniformly over \(\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\).
For the second term, the definition of \(Q_n\) implies \[\begin{align} (\mathbb{1}_{A^n}-Q_n)\rho_{A^n}(\mathbb{1}_{A^n}-Q_n) \le 2^{-n\gamma}(\mathbb{1}_{A^n}-Q_n). \end{align}\]
Hence \[\begin{align} \mathop{\mathrm{Tr}}[\Pi_n^{<R}(\mathbb{1}_{A^n}-Q_n)\rho_{A^n}(\mathbb{1}_{A^n}-Q_n)] \le 2^{-n\gamma}\mathop{\mathrm{Tr}}\Pi_n^{<R} \le 2^{-n(\gamma-R')}, \end{align}\] which converges to \(0\) exponentially fast.
Combining the two bounds yields \[\begin{align} \sup_{\boldsymbol{\Psi}\in\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)} \mathop{\mathrm{Tr}}[\Pi_n^{<R}\rho_{A^n}] \to 0 \quad \text{as} \quad n \to \infty . \end{align}\]
Since \[\begin{align} \mathop{\mathrm{Tr}}[\Pi_n^{<R}\rho_{A^n}] = \sum_{\lambda:\,\frac{1}{n}\log\dim\mathcal{V}_{A,\lambda}<R} \mathop{\mathrm{Tr}}[P_{A,\lambda}\rho_{A^n}], \end{align}\] the claim follows. ◻
The preceding lemma shows that, uniformly over the pure MSR class, almost all of the state is supported on Schur blocks whose multiplicity spaces have exponential rate \(S(\rho_A)\). This is precisely the structural input needed for the Hayashi–Matsumoto concentration protocol: after projecting onto the Schur–Weyl decomposition, the protocol extracts entanglement from these multiplicity spaces. Since the relevant block concentration holds uniformly over the MSR class, the same Schur–Weyl protocol works for all sources in the class. We therefore obtain the following universal concentration theorem.
Theorem 14 ((Universal entanglement concentration for pure MSR sources)). Let \(\ket{\phi}_{AB}\in\mathcal{H}_A\otimes\mathcal{H}_B\) be a bipartite pure state, let \(\rho_A:=\mathop{\mathrm{Tr}}_B\ket{\phi}\!\bra{\phi}_{AB}\), and let \((r_n)_n\) satisfy \(r_n=o(n)\). Let \(\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\) denote the class of pure MSR sources along \(\ket{\phi}_{AB}\) with defect size at most \(r_n\) at blocklength \(n\).
Then every rate \(0\le R<S(\rho_A)\) is universally achievable for entanglement concentration over \(\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\). More precisely, for every such \(R\), there exists a sequence of LOCC protocols \((\mathcal{C}_n)_n\), depending only on \(R\) and the local Hilbert spaces, and not on the particular source in the class, such that, with \(M_n:=\lfloor 2^{nR}\rfloor\), \[\begin{align} \sup_{\boldsymbol{\Psi}=(\Psi_n)_n \in\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)} \left[ 1- F\!\left( \mathcal{C}_n(\Psi_n), \Phi_{M_n} \right) \right] \to 0 \quad\text{as } n\to\infty . \end{align}\] The protocol itself depends only on the target rate and the local Hilbert spaces; the reference state enters only through the admissible range \(R<S(\rho_A)\).
Proof. Fix \(0\le R<S(\rho_A)\), and set \(M_n:=\lfloor 2^{nR}\rfloor\). The protocol is the Schur–Weyl concentration protocol of Hayashi and Matsumoto [15]. Alice and Bob perform the local Schur measurements, namely the projective measurements \(\{P_{A,\lambda}\}_{\lambda}\) and \(\{P_{B,\lambda}\}_{\lambda}\), where \(P_{A,\lambda}\) and \(P_{B,\lambda}\) denote the orthogonal projections onto the Schur subspaces \(\mathcal{U}_{A,\lambda}\otimes\mathcal{V}_{A,\lambda}\) and \(\mathcal{U}_{B,\lambda}\otimes\mathcal{V}_{B,\lambda}\), respectively. If \(d_A\neq d_B\), we first embed the smaller Hilbert space isometrically into an auxiliary Hilbert space of dimension \(d:=\max\{d_A,d_B\}\), as described above. We then apply the Schur–Weyl decomposition to the resulting pair of \(d\)-dimensional local systems. Let \(\boldsymbol{\Psi}=(\ket{\Psi_n}\!\bra{\Psi_n}_{A^nB^n})_{n\ge1} \in \mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\). Since \(\ket{\Psi_n} \in {\rm Sym}^n\!\left( \mathcal{H}_A\otimes\mathcal{H}_B, \ket{\phi}_{AB}^{\otimes(n-r_n)} \right)\), and hence is permutation-invariant, Lemma 6 gives \[\begin{align} \ket{\Psi_n} = \sum_{\lambda} \ket{\varphi_{n,\lambda}}_{\mathcal{U}_{A,\lambda}\mathcal{U}_{B,\lambda}} \otimes \ket{\psi_{\lambda}}_{\mathcal{V}_{A,\lambda}\mathcal{V}_{B,\lambda}}, \end{align}\] where each \(\ket{\psi_{\lambda}}\) is a normalized maximally entangled state on \(\mathcal{V}_{A,\lambda}\otimes\mathcal{V}_{B,\lambda}\), and the vectors \(\ket{\varphi_{n,\lambda}}\) are not necessarily normalized.
Thus only matching Young diagrams occur on Alice’s and Bob’s sides. In particular, the two Schur measurements produce the same label \(\lambda\) with probability one. Conditional on outcome \(\lambda\), Alice and Bob discard the \(\mathcal{U}\)-systems and retain an exact maximally entangled state on \(\mathcal{V}_{A,\lambda}\otimes\mathcal{V}_{B,\lambda}\), whose Schmidt rank is \(\dim\mathcal{V}_{A,\lambda}=\dim\mathcal{V}_{B,\lambda}\).
If \(\dim\mathcal{V}_{A,\lambda}\ge M_n\), then, conditional on the measurement outcome \(\lambda\), Alice and Bob convert the resulting maximally entangled state deterministically by LOCC into \(\Phi_{M_n}\). This is possible because its Schmidt rank is at least \(M_n\), and follows, for example, from Nielsen’s majorization theorem [30]. If instead \(\dim\mathcal{V}_{A,\lambda}<M_n\), then the protocol outputs an arbitrary fixed state. This protocol depends only on \(R\) and on the Schur–Weyl decomposition, and not on the particular source \(\boldsymbol{\Psi}\).
Let \(\rho_{A^n}:=\mathop{\mathrm{Tr}}_{B^n}\ket{\Psi_n}\!\bra{\Psi_n}\). Since only matching Young diagrams occur, we have \[\begin{align} (P_{A,\lambda}\otimes\mathbb{1})\ket{\Psi_n} = (\mathbb{1}\otimes P_{B,\lambda})\ket{\Psi_n} = (P_{A,\lambda}\otimes P_{B,\lambda})\ket{\Psi_n}. \end{align}\] Hence the probability of obtaining the common Schur label \(\lambda\) is \(\mathop{\mathrm{Tr}}[P_{A,\lambda}\rho_{A^n}]\). Therefore the failure probability is \[\begin{align} p_{\rm fail}^{(n)}(\Psi_n) = \sum_{\lambda:\,\dim\mathcal{V}_{A,\lambda}<M_n} \mathop{\mathrm{Tr}}\!\left[ P_{A,\lambda}\rho_{A^n} \right]. \end{align}\] Since \(M_n=\lfloor 2^{nR}\rfloor\), the condition \(\dim\mathcal{V}_{A,\lambda}<M_n\) implies \(\frac{1}{n}\log\dim\mathcal{V}_{A,\lambda}<R\). Therefore, by Lemma 7, \[\begin{align} \sup_{\boldsymbol{\Psi}\in\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)} p_{\rm fail}^{(n)}(\Psi_n) \to 0 \quad\text{as } n\to\infty . \end{align}\]
For each input source, the output of the protocol has the form \[\begin{align} (1-p_{\rm fail}^{(n)}(\Psi_n))\Phi_{M_n} + p_{\rm fail}^{(n)}(\Psi_n)\sigma_n, \end{align}\] where \(\sigma_n\) is the state output conditional on failure, i.e.on the event \(\dim\mathcal{V}_{A,\lambda}<M_n\).
Since \(\Phi_{M_n}\) is pure, for any state \(\omega\), \(F(\omega,\Phi_{M_n})=\sqrt{\mathop{\mathrm{Tr}}[\Phi_{M_n}\omega]}\). Thus, for \(p=p_{\rm fail}^{(n)}(\Psi_n)\), \[\begin{align} F\!\left( (1-p)\Phi_{M_n}+p\sigma_n,\Phi_{M_n} \right) = \sqrt{(1-p)+p\,\mathop{\mathrm{Tr}}[\Phi_{M_n}\sigma_n]} \ge \sqrt{1-p}. \end{align}\] It follows that \[\begin{align} F\!\left( \mathcal{C}_n(\Psi_n), \Phi_{M_n} \right) \ge \sqrt{1-p_{\rm fail}^{(n)}(\Psi_n)}. \end{align}\] Since \(p_{\rm fail}^{(n)}(\Psi_n)\to0\) uniformly over \(\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\), the fidelity converges to \(1\) uniformly over this class.
Finally, since \(M_n=\lfloor 2^{nR}\rfloor\), we have \[\begin{align} \liminf_{n\to\infty}\frac{1}{n}\log M_n=R. \end{align}\] Thus the rate \(R\) is universally achievable over \(\mathfrak S^{\rm pMSR}_{\phi}((r_n)_n)\). ◻
We now turn to the reverse task, entanglement dilution. While the proof of the entanglement concentration theorem (Theorem 14) relies on Schur–Weyl duality and the Hayashi–Matsumoto protocol, entanglement dilution is naturally approached through information-spectrum techniques, together with the MSR entropy-rigidity results established earlier.
We begin with a lemma which provides the technical bridge between the MSR structure of the target states \(\rho_{A^nB^n}\) and the ensemble-based description needed for entanglement dilution. In the i.i.d.setting, a cq-extension of \(\rho_{AB}\) with pure conditional states, \(\rho_{RAB} = \sum_i p_i \ket{i}\!\bra{i}_R \otimes \ket{\psi_i}\!\bra{\psi_i}_{AB},\) induces, for each \(n\), the tensor-product ensemble \(\rho_{AB}^{\otimes n} = \sum_{i^n} p_{i^n} \ket{\psi_{i^n}}\!\bra{\psi_{i^n}}_{A^nB^n},\) where \(p_{i^n}:=p_{i_1}\cdots p_{i_n}\) and \(\ket{\psi_{i^n}} := \ket{\psi_{i_1}}\otimes\cdots\otimes\ket{\psi_{i_n}}.\) Equivalently, this gives a natural cq-extension of \(\rho_{AB}^{\otimes n}\), with a classical register \(R^n\) recording the sequence \(i^n\) of ensemble labels.
For an MSR almost i.i.d.target sequence \((\rho_{A^nB^n})_n\), however, the states \(\rho_{A^nB^n}\) need not arise from such an exact tensor-product ensemble. The lemma below shows that this difficulty can be overcome: any fixed cq-extension of the reference state \(\rho_{AB}\) with pure conditional states can be lifted to cq-extensions of the MSR states \(\rho_{A^nB^n}\), at the cost of an additional classical register \(\widetilde{R}_n\) whose logarithmic dimension is \(o(n)\). This overhead is asymptotically negligible at the level of rates, and the resulting marginal sequence on \(R^nA^nB^n\) is MSR almost i.i.d.along the chosen cq-extension of \(\rho_{AB}\). This allows the spectral-entropy rigidity results proved earlier to be applied to the lifted cq-state, which is the form required in the subsequent entanglement-dilution argument. Note that by the term pure-state cq-extension used below, we mean a cq-extension whose conditional states on \(AB\) are pure.
Lemma 8 ((cq-lifting of MSR target sequences up to sublinear classical overhead)). Let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_{n}\) be an MSR almost i.i.d.target sequence along \(\rho_{AB}\), with defect sizes \(r_n=o(n)\). Let \[\begin{align} \varrho_{RAB} = \sum_i p_i \ket{i}\!\bra{i}_R \otimes \ket{\phi_i}\!\bra{\phi_i}_{AB} \end{align}\] be a pure-state cq-extension of \(\rho_{AB}\). Then there exists a sequence of cq-extensions \((\widehat\varrho^{(n)})_n\) where \[\begin{align} \widehat\varrho^{(n)} \equiv \widehat\varrho_{\widetilde{R}_nR^nA^nB^n} = \sum_{\substack{\alpha\in\mathcal{A}_n,\; i^n:\\ q^{(n)}_{\alpha,i^n}>0}} q^{(n)}_{\alpha,i^n} \ket{\alpha}\!\bra{\alpha}_{\widetilde{R}_n} \otimes \ket{i^n}\!\bra{i^n}_{R^n} \otimes \ket{\phi^{(n)}_{\alpha,i^n}}\! \bra{\phi^{(n)}_{\alpha,i^n}}_{A^nB^n} \end{align}\] of \(\rho_{A^nB^n}\) such that \(\log\dim\widetilde{R}_n=o(n),\) and, writing \[\begin{align} \varrho_{R^nA^nB^n} := \mathop{\mathrm{Tr}}_{\widetilde{R}_n} \widehat\varrho_{\widetilde{R}_nR^nA^nB^n}, \end{align}\] the sequence \(\boldsymbol{\varrho}_{RAB} := (\varrho_{R^nA^nB^n})_{n}\) is MSR almost i.i.d.along \(\varrho_{RAB}\), with defect sizes \(r_n=o(n)\).
Proof. See Appendix 12.1. ◻
The following lemma is a simple but useful blocking observation. Fix a block size \(m\), and write each blocklength as \(n=lm+s\), where \(0\le s<m\). We group the first \(lm\) tensor positions into \(l\) blocks of size \(m\), leaving \(s\) remaining tensor positions. Since \(m\) is fixed, this remainder has size uniformly bounded in \(n\). Each unrestricted tensor position in one of the vectors spanning the MSR support space can affect at most one \(m\)-block. Hence, after blocking, such a vector has at most \(\min\{r_n,l\}\) unrestricted \(m\)-blocks, while the final \(s\) tensor positions are treated as an unrestricted remainder. Thus, for each fixed value of \(s\), the MSR structure is preserved after blocking, with block-defect size \(t_{l,s}=o(l)\). This will allow us to apply MSR entropy estimates to fixed-size blocks while keeping the leftover tensor positions asymptotically negligible.
Lemma 9 ((Blocking with bounded remainder)). Let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_{n\ge1}\) be MSR almost i.i.d.along \(\rho_{AB}\), with defect sizes \(r_n=o(n)\). Fix \(m\in\mathbb{N}_+\), and write \(n=lm+s\), where \(0\le s<m\). Then there exists an extension \[\begin{align} \omega_{(A^mB^mE^m)^l A^sB^sE^s} \end{align}\] of \(\rho_{A^{lm+s}B^{lm+s}}\) such that, if \(\ket{\theta}_{ABE}\) is a purification of \(\rho_{AB}\) and \(\ket{\theta_m}_{A^mB^mE^m}:=\ket{\theta}_{ABE}^{\otimes m}\), then \[\begin{align} \mathop{\mathrm{supp}}\omega_{(A^mB^mE^m)^l A^sB^sE^s} \subseteq \operatorname{span} \mathcal{V}\!\left( (\mathcal{H}_{ABE}^{\otimes m})^{\otimes l}, \ket{\theta_m}^{\otimes(l-t_{l,s})} \right) \otimes \mathcal{H}_{ABE}^{\otimes s}, \end{align}\] where \(t_{l,s}:=\min\{r_{lm+s},l\}\). In particular, for each fixed residue \(s\), we have \(t_{l,s}=o(l)\) as \(l\to\infty\). Moreover, \(\omega\) is invariant under permutations of the \(l\) blocks of size \(m\).
Proof. See Appendix 12.2. ◻
The following corollary combines the preceding blocking lemma with the cq-lifting construction. Fixing a block size \(m\), we view the source at blocklength \(n=lm+s\) as consisting of \(l\) blocks of size \(m\), together with a remainder of size \(s<m\). The blocking lemma shows that, for each fixed \(s\), the first \(lm\) systems form an MSR source at the block level, with only \(o(l)\) unrestricted blocks, while the remaining \(s\) systems have fixed size and are therefore asymptotically negligible. We then apply the cq-lifting argument to these \(m\)-blocks, relative to a chosen pure-state cq-extension of \(\rho_{AB}^{\otimes m}\). The result is a pure-state cq-extension of the full state \(\rho_{A^{lm+s}B^{lm+s}}\), with only a sublinear additional classical register \(\widetilde{R}_l\), in the sense that \(\log\dim\widetilde{R}_l=o(l)\). After tracing out this additional register and the final \(A^sB^s\)-systems, the marginal on \(R^l(A^mB^m)^l\) remains MSR almost i.i.d.along the chosen block-level cq-extension.
Corollary 15 ((cq-lifting with bounded remainder)). Fix \(m\in\mathbb{N}_+\) and \(s\in\{0,\ldots,m-1\}\). Let \((\rho_{A^{lm+s}B^{lm+s}})_{l\ge1}\) be as in Lemma 9. Let \[\begin{align} \varrho^{(m)}_{RA^mB^m} = \sum_i p_i^{(m)} \ket{i}\!\bra{i}_R \otimes \ket{\phi_i^{(m)}}\!\bra{\phi_i^{(m)}}_{A^mB^m} \end{align}\] be a pure-state cq-extension of \(\rho_{AB}^{\otimes m}\). Then there exist pure-state cq-extensions \[\begin{align} \widehat\varrho^{(l,s)}_{\widetilde{R}_lR^lA^{lm+s}B^{lm+s}} \end{align}\] of \(\rho_{A^{lm+s}B^{lm+s}}\) such that \[\begin{align} \log\dim\widetilde{R}_l=o(l), \end{align}\] and, after tracing out \(\widetilde{R}_l\) and the bounded remainder, the marginal sequence on \(R^l(A^mB^m)^l\) is MSR almost i.i.d.along \(\varrho^{(m)}_{RA^mB^m}\), with defect sizes \(o(l)\).
Proof. See Appendix 12.3. ◻
Theorem 16 ((Robust achievability of entanglement dilution for MSR sources)). Let \(\rho_{AB}\in\mathcal{D}(\mathcal{H}_A\otimes\mathcal{H}_B)\), and let \(\boldsymbol{\rho}_{AB}:=(\rho_{A^nB^n})_{n\ge1}\) be an MSR almost i.i.d.target sequence along \(\rho_{AB}\), with defect sizes \(r_n=o(n)\). Then \[\begin{align} E_C(\boldsymbol{\rho}_{AB}) \le E_F^\infty(\rho_{AB}). \end{align}\] Equivalently, every rate \(R>E_F^\infty(\rho_{AB})\) is achievable for entanglement dilution of \(\boldsymbol{\rho}_{AB}\). Moreover, for every fixed defect-size sequence \(r_n=o(n)\), \[\begin{align} \sup_{\boldsymbol{\rho}_{AB}} E_C(\boldsymbol{\rho}_{AB}) \le E_F^\infty(\rho_{AB}), \end{align}\] where the supremum is over all MSR almost i.i.d.sources along \(\rho_{AB}\) with defect size at most \(r_n\) at blocklength \(n\).
Proof. We use the information-spectrum formula for the entanglement cost of a general sequence of bipartite states, which states that \[\begin{align} E_C(\boldsymbol{\rho}_{AB}) = \inf_{\boldsymbol{\gamma}\in\mathcal{D}_{\rm cq}(\boldsymbol{\rho}_{AB})} \overline{S}(A|R)_{\boldsymbol{\gamma}}, \end{align}\] where the infimum is over pure-state cq-extensions of the sequence, and
In the blocked argument below, spectral rates denoted by \(\overline{S}^{\rm block}\) are normalized by the number \(l\) of \(m\)-blocks; the rates entering \(E_C\) are normalized by the original blocklength \(n=lm+s\).
We use the information-spectrum characterization of entanglement cost given by Theorem 1 of [7]. For any sequence \(\boldsymbol{\omega}_{AB}:=(\omega_{A^nB^n})_{n\ge1}\), \[\begin{align} E_C(\boldsymbol{\omega}_{AB}) = \inf_{\boldsymbol{\gamma}\in D_{\rm cq}(\boldsymbol{\omega}_{AB})} \overline{S}(A|R)_{\boldsymbol{\gamma}}, \end{align}\]
\(\overline{S}(A|R)\) denotes the spectral sup-conditional entropy rate.
Fix \(m\in\mathbb{N}_+\) and \(\eta>0\). Choose a pure-state ensemble decomposition \[\begin{align} \rho_{AB}^{\otimes m} = \sum_i p_i^{(m)} \ket{\phi_i^{(m)}}\!\bra{\phi_i^{(m)}}_{A^mB^m} \end{align}\] such that \[\begin{align} \sum_i p_i^{(m)} S(\phi^{(m)}_{i,A^m}) \le E_F(\rho_{AB}^{\otimes m})+\eta . \end{align}\] Let \[\begin{align} \varrho^{(m)}_{RA^mB^m} = \sum_i p_i^{(m)} \ket{i}\!\bra{i}_R \otimes \ket{\phi_i^{(m)}}\!\bra{\phi_i^{(m)}}_{A^mB^m}. \end{align}\] Then \(\varrho^{(m)}_{RA^mB^m}\) is a pure-state cq-extension of \(\rho_{AB}^{\otimes m}\), and \[\begin{align} S(A^m|R)_{\varrho^{(m)}} = \sum_i p_i^{(m)} S(\phi^{(m)}_{i,A^m}) \le E_F(\rho_{AB}^{\otimes m})+\eta . \end{align}\]
Write \(n=lm+s\), where \(0\le s<m\). We first fix a value \(s\in\{0,\ldots,m-1\}\) of the remainder in this decomposition. By Corollary 15, applied to the block-level cq-extension \(\varrho^{(m)}_{RA^mB^m}\), there exist pure-state cq-extensions \[\begin{align} \widehat\varrho^{(l,s)}_{\widetilde{R}_lR^lA^{lm+s}B^{lm+s}} \end{align}\] of \(\rho_{A^{lm+s}B^{lm+s}}\) such that \[\begin{align} \log\dim\widetilde{R}_l=o(l). \end{align}\] Moreover, after tracing out \(\widetilde{R}_l\) and the final \(A^sB^s\)-systems, the marginal sequence on \(R^l(A^mB^m)^l\) is MSR almost i.i.d.along \(\varrho^{(m)}_{RA^mB^m}\), with defect sizes \(o(l)\).
By the conditional spectral entropy bound for MSR sources, Lemma 4, applied to this blocked marginal sequence, we have \[\begin{align} \overline{S}_{\rm block}(A^m|R) \le S(A^m|R)_{\varrho^{(m)}} , \end{align}\] where \(\overline{S}_{\rm block}\) denotes the spectral sup-conditional entropy rate normalized by the number \(l\) of \(m\)-blocks.
We now compare this blocked marginal sequence with the full cq-extension sequence for the fixed value of \(s\), \[\begin{align} \boldsymbol{\widehat\varrho}^{(s)} := \bigl( \widehat\varrho^{(l,s)}_{\widetilde{R}_lR^lA^{lm+s}B^{lm+s}} \bigr)_{l\ge1}. \end{align}\] Passing from the blocked marginal on \(R^l(A^mB^m)^l\) to \(\boldsymbol{\widehat\varrho}^{(s)}\) adds the classical register \(\widetilde{R}_l\) to the conditioning system and adds the final \(A^s\)-system to Alice’s side. Since \(\log\dim\widetilde{R}_l=o(l)\) and \(\log\dim\mathcal{H}_A^{\otimes s}=s\log d_A=O(1)\) for fixed \(s<m\), Lemma 5 implies that these additions do not increase the \(l\)-normalized spectral sup-conditional entropy rate. Hence, for each fixed value of \(s\), \[\begin{align} \overline{S}_{\rm block} (A^mA^s|\widetilde{R} R)_{\boldsymbol{\widehat\varrho}^{(s)}} \le S(A^m|R)_{\varrho^{(m)}} . \end{align}\] Equivalently, when rates are normalized per original tensor factor rather than per \(m\)-block, \[\begin{align} \overline{S} (A|\widetilde{R} R)_{\boldsymbol{\widehat\varrho}^{(s)}} \le \frac{1}{m} S(A^m|R)_{\varrho^{(m)}} . \end{align}\]
Since there are only finitely many possible remainders \(s\in\{0,\ldots,m-1\}\), we obtain a pure-state cq-extension sequence of the full source \(\boldsymbol{\rho}_{AB}\) by using, at blocklength \(n\), the construction corresponding to the decomposition \(n=lm+s\). The spectral sup-rate of a sequence obtained by interleaving finitely many subsequences is bounded above by the maximum of the spectral sup-rates of those subsequences. Since the preceding bound is independent of \(s\), the resulting cq-extension sequence \(\boldsymbol{\widehat\varrho}\) satisfies \[\begin{align} \overline{S}(A|\widetilde{R} R)_{\boldsymbol{\widehat\varrho}} \le \frac{1}{m} S(A^m|R)_{\varrho^{(m)}} . \end{align}\] The information-spectrum formula for entanglement cost therefore gives \[\begin{align} E_C(\boldsymbol{\rho}_{AB}) \le \frac{1}{m} S(A^m|R)_{\varrho^{(m)}} . \end{align}\] By the choice of the ensemble, \[\begin{align} E_C(\boldsymbol{\rho}_{AB}) \le \frac{1}{m} E_F(\rho_{AB}^{\otimes m}) + \frac{\eta}{m}. \end{align}\] Since \(\eta>0\) was arbitrary, we obtain \[\begin{align} E_C(\boldsymbol{\rho}_{AB}) \le \frac{1}{m} E_F(\rho_{AB}^{\otimes m}) . \end{align}\] Finally, taking the infimum over \(m\ge1\) yields \[\begin{align} E_C(\boldsymbol{\rho}_{AB}) \le \inf_{m\ge1} \frac{1}{m} E_F(\rho_{AB}^{\otimes m}) = E_F^\infty(\rho_{AB}). \end{align}\] Thus every rate \(R>E_F^\infty(\rho_{AB})\) is achievable for entanglement dilution of \(\boldsymbol{\rho}_{AB}\).
It remains only to justify the claimed uniformity of the bound. If the defect-size sequence \(r_n=o(n)\) is fixed in advance, then the block defect sizes \(t_{l,s}\), the sublinear classical registers \(\widetilde{R}_l\) in the cq-lifting step, and the bounded-remainder contributions depend only on \(r_n\), \(m\), \(s\), and the reference state \(\rho_{AB}\), and not on the particular MSR source. Consequently, the above argument gives \[\begin{align} \sup_{\boldsymbol{\rho}_{AB}} E_C(\boldsymbol{\rho}_{AB}) \le E_F^\infty(\rho_{AB}), \end{align}\] where the supremum is over all MSR almost i.i.d.target sequences along \(\rho_{AB}\) with defect size at most \(r_n\) at blocklength \(n\). ◻
The results of this paper show that the MSR model provides a robust framework for entanglement manipulation beyond the idealized tensor-power setting. The structural properties of MSR sources, in particular their stability under local tensor-power channels, marginals and blocking operations, make it possible to transfer entanglement-manipulation arguments from the i.i.d.setting to sources with sublinear deviations from tensor-power structure. At the same time, the associated entropy-rigidity results show that the relevant spectral quantities retain the entropy values of the underlying reference state. Combining these structural and entropic ingredients, we proved robust achievability results for the two fundamental operational tasks considered here: universal entanglement concentration for pure MSR sources, and entanglement dilution for mixed MSR target sequences at rates bounded by the regularized entanglement of formation of the reference state. Thus, within the MSR framework, sublinear deviations from a tensor-power structure does not affect the standard asymptotic achievability rates for the tasks considered here; in the pure-state concentration setting, a single protocol can be chosen that works for the whole MSR class along the fixed reference state.
Several natural questions remain open. First, while we proved a universal entanglement-concentration theorem for pure MSR sources, the corresponding universal distillation problem for mixed MSR sources remains open. In particular, it would be interesting to determine whether, for a fixed reference state \(\rho_{AB}\), there exists a single sequence of LOCC distillation protocols, depending only on \(\rho_{AB}\) and the target rate, which achieves the coherent-information lower bound uniformly over the whole MSR class along \(\rho_{AB}\). Such a result would require additional ideas beyond the source-dependent information-spectrum achievability argument used here.
A second direction is to extend the present robustness results beyond the MSR model. The MSR class has strong structural features, in particular the existence of permutation-invariant extensions supported on subexponential defect spaces, which play an essential role in our arguments. It would be important to understand whether analogous entanglement concentration, distillation and dilution results continue to hold for broader notions of almost i.i.d.states. A natural first step is to consider generalizations of the MSR framework, before moving to still weaker models such as Wasserstein almost i.i.d.or weakly almost i.i.d. sources. Such an extension would clarify the extent to which entanglement manipulation is robust under approximate tensor-power structure, and would separate the features that are genuinely asymptotic from those that depend on the stronger symmetry and support properties of the MSR class.
ND is grateful to her friend and former collaborator Garry Bowen, who introduced her to the information-spectrum framework of quantum information theory. She fondly remembers their many interesting collaborations, which started her “beyond i.i.d.” journey. She would also like to thank Bjarne Bergh and Liuhang Ye for their comments and help. She is supported by the Engineering and Physical Sciences Research Council [Grant Ref: EP/Y028732/1].
Note that if \(R>\overline{S}(\boldsymbol{\rho})\), then \(-R<\underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}})\), since \(\overline{S}(\boldsymbol{\rho})=-\underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}})\). Hence we may choose \(\gamma\in\mathbb{R}\) such that \(-R<\gamma<\underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}})\). By the definition of \(\underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}})\), \[\begin{align} \mathop{\mathrm{Tr}}\!\left[ \{\rho_n-2^{n\gamma}\mathbb{1}_n>0\}\rho_n \right]\to 1 \quad \text{as} \quad n \to \infty. \end{align}\] Let \(\Pi_n:=\{\rho_n-2^{n\gamma}\mathbb{1}_n>0\}\). Then \(\mathop{\mathrm{Tr}}(\Pi_n\rho_n)\to1\) as \(n \to \infty\). Moreover, on the range of \(\Pi_n\), all eigenvalues of \(\rho_n\) are strictly larger than \(2^{n\gamma}\). Since \(\mathop{\mathrm{Tr}}\rho_n=1\), this gives \(2^{n\gamma}\mathop{\mathrm{Tr}}\Pi_n\le1\), and therefore \(\mathop{\mathrm{Tr}}\Pi_n\le2^{-n\gamma}\). Since \(\gamma>-R\), we have \(\mathop{\mathrm{Tr}}\Pi_n\le2^{nR}\) for all sufficiently large \(n\). Thus every rate strictly larger than \(\overline{S}(\boldsymbol{\rho})\) admits such a sequence of high-probability projections.
Conversely, suppose that, for some \(R\in\mathbb{R}\), there exists a sequence of projections \((\Pi_n)_n\) such that \(\mathop{\mathrm{Tr}}(\Pi_n\rho_n)\to1\) and \(\mathop{\mathrm{Tr}}\Pi_n\le2^{nR}\) for all sufficiently large \(n\). Fix \(\delta>0\), and define \(Q_n:=\{\rho_n-2^{-n(R+\delta)}\mathbb{1}_n>0\}\). On the complement of \(Q_n\), we have \(\rho_n\le2^{-n(R+\delta)}\mathbb{1}_n\). Hence \[\begin{align} \mathop{\mathrm{Tr}}\!\left[\Pi_n(\mathbb{1}_n-Q_n)\rho_n\right] \le 2^{-n(R+\delta)}\mathop{\mathrm{Tr}}\Pi_n \le 2^{-n\delta} \end{align}\] for all sufficiently large \(n\). It follows that \(\mathop{\mathrm{Tr}}(Q_n\rho_n)\ge \mathop{\mathrm{Tr}}(\Pi_n\rho_n)-2^{-n\delta}\to1\), or equivalently, \[\begin{align} \mathop{\mathrm{Tr}}\!\left[ \{\rho_n-2^{-n(R+\delta)}\mathbb{1}_n>0\}\rho_n \right]\to 1 \quad \text{as} \quad n \to \infty. \end{align}\] By the definition of \(\underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}})\), this implies \(-R-\delta\le \underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}})\). Since \(\delta>0\) is arbitrary, we obtain \(\overline{S}(\boldsymbol{\rho})=-\underline D(\boldsymbol{\rho}\|\boldsymbol{\mathbb{1}})\le R\). Taking the infimum over all such \(R\) gives the claimed characterization of \(\overline{S}(\boldsymbol{\rho})\).
Proof. Let \(V:A\to A'F\) be a Stinespring isometry for \(\Lambda_A\), so that \(\Lambda_A(\cdot)=\mathop{\mathrm{Tr}}_F[V(\cdot)V^\dagger]\). Choose a purification \(\ket{\theta}_{ABE}\) of \(\rho_{AB}\). By Remark 2.4(a) of [24], for this choice of purification there exists an MSR extension \(\rho_{A^nB^nE^n}\) of \(\rho_{A^nB^n}\) satisfying the MSR permutation-invariance and support conditions. Define \[\begin{align} \ket{\theta'}_{A'BFE} := (V\otimes\mathbb{1}_{BE})\ket{\theta}_{ABE}. \end{align}\] Then \(\ket{\theta'}_{A'BFE}\) is a purification of \(\omega_{A'B}\).
Since \(\rho_{A^nB^n}\) is MSR almost i.i.d.along \(\rho_{AB}\), there exists an extension \(\rho_{A^nB^nE^n}\) of \(\rho_{A^nB^n}\) such that:
\(\rho_{A^nB^nE^n}\) is permutation invariant under simultaneous permutations of the triples \((A_i,B_i,E_i)\);
\[\begin{align} \operatorname{supp}(\rho_{A^nB^nE^n}) \subseteq \operatorname{span} \mathcal{V} \bigl( \mathcal{H}_{ABE}^{\otimes n}, \ket{\theta}_{ABE}^{\otimes(n-r_n)} \bigr). \end{align}\]
Define \[\begin{align} \widetilde{\omega}_{A'^nB^nF^nE^n} := (V^{\otimes n}\otimes\mathbb{1}_{B^nE^n}) \rho_{A^nB^nE^n} (V^{\otimes n}\otimes\mathbb{1}_{B^nE^n})^\dagger . \end{align}\] Tracing out \(F^nE^n\) yields \[\begin{align} \mathop{\mathrm{Tr}}_{F^nE^n} \widetilde{\omega}_{A'^nB^nF^nE^n} = \omega_{A'^nB^n}, \end{align}\] so \(\widetilde{\omega}_{A'^nB^nF^nE^n}\) is an extension of \(\omega_{A'^nB^n}\).
Moreover, \(V^{\otimes n}\otimes\mathbb{1}_{B^nE^n}\) commutes with simultaneous permutations of the tensor factors. Hence \(\widetilde{\omega}_{A'^nB^nF^nE^n}\) is permutation invariant under simultaneous permutations of the tuples \((A'_i,B_i,F_i,E_i)\).
It remains to verify the support condition (i.e.the condition ?? of Definition 2). Since \(\widetilde{\omega}_{A'^nB^nF^nE^n}\) is obtained from \(\rho_{A^nB^nE^n}\) by applying the isometry \(V^{\otimes n}\otimes\mathbb{1}_{B^nE^n}\), it suffices to check how this isometry acts on the spanning vectors of the original MSR defect space.
Let \[\begin{align} \ket{\Psi} = U_\pi^{ABE} \bigl( \ket{\theta}_{ABE}^{\otimes(n-r_n)} \otimes \ket{\Omega}_{ABE}^{(r_n)} \bigr) \end{align}\] be one such spanning vector, where \(U_\pi^{ABE}\) denotes the simultaneous permutation of the \(n\) triples \((A_i,B_i,E_i)\), and \(\ket{\Omega}_{ABE}^{(r_n)}\) is an arbitrary vector on the remaining \(r_n\) triples. Since \(V^{\otimes n}\) acts identically on each \(A_i\), we have the intertwining relation \[\begin{align} (V^{\otimes n}\otimes\mathbb{1}_{B^nE^n})U_\pi^{ABE} = U_\pi^{A'BFE} (V^{\otimes n}\otimes\mathbb{1}_{B^nE^n}), \end{align}\] where \(U_\pi^{A'BFE}\) denotes the simultaneous permutation of the \(n\) quadruples \((A'_i,B_i,F_i,E_i)\). Therefore \[\begin{align} \begin{aligned} (V^{\otimes n}\otimes\mathbb{1}_{B^nE^n})\ket{\Psi} &= U_\pi^{A'BFE} (V^{\otimes n}\otimes\mathbb{1}_{B^nE^n}) \bigl( \ket{\theta}_{ABE}^{\otimes(n-r_n)} \otimes \ket{\Omega}_{ABE}^{(r_n)} \bigr) \\ &= U_\pi^{A'BFE} \bigl( \ket{\theta'}_{A'BFE}^{\otimes(n-r_n)} \otimes \ket{\Omega'}_{A'BFE}^{(r_n)} \bigr), \end{aligned} \end{align}\] where \(\ket{\theta'}_{A'BFE}:=(V\otimes\mathbb{1}_{BE})\ket{\theta}_{ABE}\), and \[\ket{\Omega'}_{A'BFE}^{(r_n)} := (V^{\otimes r_n}\otimes\mathbb{1}_{B^{r_n}E^{r_n}}) \ket{\Omega}_{ABE}^{(r_n)} .\] By linearity, the image of every spanning vector of \(\operatorname{span} \mathcal{V} \bigl( \mathcal{H}_{ABE}^{\otimes n}, \ket{\theta}_{ABE}^{\otimes(n-r_n)} \bigr)\) lies in \[\operatorname{span} \mathcal{V} \bigl( \mathcal{H}_{A'BFE}^{\otimes n}, \ket{\theta'}_{A'BFE}^{\otimes(n-r_n)} \bigr).\] Since \(\rho_{A^nB^nE^n}\) is supported on the original defect space, its isometric image satisfies \[\begin{align} \operatorname{supp} (\widetilde{\omega}_{A'^nB^nF^nE^n}) \subseteq \operatorname{span} \mathcal{V} \bigl( \mathcal{H}_{A'BFE}^{\otimes n}, \ket{\theta'}_{A'BFE}^{\otimes(n-r_n)} \bigr). \end{align}\] Therefore \(\widetilde{\omega}_{A'^nB^nF^nE^n}\) is an MSR extension of \(\omega_{A'^nB^n}\) along the purification \(\ket{\theta'}_{A'BFE}\), with the same defect size \(r_n\). Since \(r_n=o(n)\), the sequence \(\boldsymbol{\omega}_{A'B}\) is MSR almost i.i.d. along \(\omega_{A'B}\). ◻
Proof of Lemma 5
Proof. First, note that adding \(Z_l\) to the conditioning system cannot increase the sup-conditional spectral entropy rate. This follows from the definition 4 and the data processing for the inf-spectral divergence rate (see e.g. Proposition 4 of [23]) under the partial trace over \(Z_l\), \[\begin{align} \underline D\!\left( \boldsymbol{\gamma}_{ABZ} \middle\| (\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l})_l \right) \ge \underline D\!\left( \boldsymbol{\gamma}_{AB} \middle\| (\mathbb{1}_{A_l}\otimes\gamma_{B_l})_l \right). \end{align}\] Therefore \[\begin{align} \overline{S}(A|BZ)_{\boldsymbol{\gamma}} \le \overline{S}(A|B)_{\boldsymbol{\gamma}}. \end{align}\]
We next bound the effect of adjoining \(C_l\) to Alice’s side. Applying data processing for the inf-spectral divergence rate under the partial trace over \(C_l\), we obtain \[\begin{align} \underline D\!\left( \boldsymbol{\gamma}_{ABCZ} \middle\| (\mathbb{1}_{A_lC_l}\otimes\gamma_{B_lZ_l})_l \right) \ge \underline D\!\left( \boldsymbol{\gamma}_{ABZ} \middle\| (d_{C,l}\,\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l})_l \right), \end{align}\] because \(\mathop{\mathrm{Tr}}_{C_l}(\mathbb{1}_{A_lC_l}\otimes\gamma_{B_lZ_l}) = d_{C,l}\,\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l}\).
Let \[\begin{align} L:=\limsup_{l\to\infty}\frac{1}{l}\log d_{C,l} \end{align}\] and set \(a_l:=\frac{1}{l}\log d_{C,l}\), so that \(d_{C,l}=2^{l a_l}\). We claim that \[\begin{align} \underline D\!\left( \boldsymbol{\gamma}_{ABZ} \middle\| (d_{C,l}\,\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l})_l \right) \ge \underline D\!\left( \boldsymbol{\gamma}_{ABZ} \middle\| (\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l})_l \right) - L . \end{align}\] Indeed, this follows from the elementary scaling behaviour of the spectral divergence rate under multiplication of the second argument by a scalar factor. More explicitly, multiplying the second argument at blocklength \(l\) by \(d_{C,l}=2^{l a_l}\) changes the spectral projection appearing in the definition with parameter \(\alpha\) into the spectral projection for the unscaled second argument with parameter \(\alpha+a_l\), since \[\begin{align} 2^{l\alpha} d_{C,l} (\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l}) = 2^{l(\alpha+a_l)} (\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l}) . \end{align}\] Thus the scaling by \(d_{C,l}\) can decrease the inf-spectral divergence rate by at most \(\limsup_{l\to\infty}a_l=L\). This proves the claimed bound.
Combining this with the preceding data-processing estimate gives \[\begin{align} \begin{aligned} \overline{S}(AC|BZ)_{\boldsymbol{\gamma}} &= -\underline D\!\left( \boldsymbol{\gamma}_{ABCZ} \middle\| (\mathbb{1}_{A_lC_l}\otimes\gamma_{B_lZ_l})_l \right) \\ &\le -\underline D\!\left( \boldsymbol{\gamma}_{ABZ} \middle\| (d_{C,l}\,\mathbb{1}_{A_l}\otimes\gamma_{B_lZ_l})_l \right) \\ &\le \overline{S}(A|BZ)_{\boldsymbol{\gamma}} + L . \end{aligned} \end{align}\] Finally, since adding \(Z_l\) to the conditioning system cannot increase the sup-conditional spectral entropy rate, we have \(\overline{S}(A|BZ)_{\boldsymbol{\gamma}}\le \overline{S}(A|B)_{\boldsymbol{\gamma}}\). Hence \[\begin{align} \overline{S}(AC|BZ)_{\boldsymbol{\gamma}} \le \overline{S}(A|B)_{\boldsymbol{\gamma}} + \limsup_{l\to\infty}\frac{1}{l}\log d_{C,l}, \end{align}\] as claimed. ◻
Proof. Let \(\ket{\theta}_{ABE}\) be a purification of \(\rho_{AB}\). Since \(\varrho_{RAB}\) is a cq-extension of \(\rho_{AB}\) with pure conditional states, the vector \[\begin{align} \ket{\eta}_{ABR} := \sum_i \sqrt{p_i}\,\ket{\phi_i}_{AB}\ket{i}_R \end{align}\] is also a purification of \(\rho_{AB}\). Enlarging \(R\), if necessary, by adding zero-probability classical labels, we may assume that there exists an isometry \(W:E\to R\) such that \[\begin{align} (\mathbb{1}_{AB}\otimes W)\ket{\theta}_{ABE} = \ket{\eta}_{ABR}. \end{align}\] Let \(\Delta_R\) denote the dephasing channel in the basis \(\{\ket{i}_R\}_i\). Then \[\begin{align} (\Delta_R\otimes \mathbb{1}_{AB}) \bigl(\ket{\eta}\!\bra{\eta}_{ABR}\bigr) = \varrho_{RAB}. \end{align}\]
By Remark 2.4(a) of [24], for the chosen purification \(\ket{\theta}_{ABE}\) there exists an MSR extension \(\omega_{A^nB^nE^n}\) of \(\rho_{A^nB^n}\) which is invariant under simultaneous permutations of the triples \((A_j,B_j,E_j)\) and satisfies \[\begin{align} \operatorname{supp}\omega_{A^nB^nE^n} \subseteq \operatorname{span}{\mathcal{V}} \bigl( \mathcal{H}_{ABE}^{\otimes n}, \ket{\theta}_{ABE}^{\otimes(n-r_n)} \bigr). \end{align}\] Define \[\begin{align} \tau^{(n)} \equiv \tau_{A^nB^nR^n} := (\mathbb{1}_{A^nB^n}\otimes W^{\otimes n}) \omega_{A^nB^nE^n} (\mathbb{1}_{A^nB^n}\otimes W^{\otimes n})^\dagger . \end{align}\] Then \(\tau^{(n)}\) is invariant under simultaneous permutations of the triples \((A_j,B_j,R_j)\), has marginal \(\rho_{A^nB^n}\) on \(A^nB^n\), and its support is contained in \[\begin{align} \operatorname{span}{\mathcal{V}} \bigl( \mathcal{H}_{ABR}^{\otimes n}, \ket{\eta}_{ABR}^{\otimes(n-r_n)} \bigr). \end{align}\] Thus \((\tau^{(n)})_n\) is MSR almost i.i.d.along the pure reference state \(\ket{\eta}\!\bra{\eta}_{ABR}\), with defect sizes \(r_n\).
Applying Lemma 1 once more, now to the local tensor-power dephasing channel \(\Delta_R^{\otimes n}\) on the \(R\) systems, we obtain that the sequence \((\varrho_{R^nA^nB^n})_n\), where \[\begin{align} \varrho_{R^nA^nB^n} := (\Delta_R^{\otimes n}\otimes\mathop{\mathrm{id}}_{A^nB^n}) (\tau_{A^nB^nR^n}), \end{align}\] is MSR almost i.i.d.along \(\varrho_{RAB}\), with defect sizes at most \(r_n\). Moreover, \(\mathop{\mathrm{Tr}}_{R^n}\varrho_{R^nA^nB^n}=\rho_{A^nB^n}\): since \(\Delta_R^{\otimes n}\) is trace-preserving, dephasing the \(R^n\)-register does not change the marginal on \(A^nB^n\). Hence \[\begin{align} \mathop{\mathrm{Tr}}_{R^n}\varrho_{R^nA^nB^n} = \mathop{\mathrm{Tr}}_{R^n}\tau_{R^nA^nB^n} = \mathop{\mathrm{Tr}}_{E^n}\omega_{A^nB^nE^n} = \rho_{A^nB^n}. \end{align}\]
It remains to refine \(\varrho^{(n)}\) into a cq-extension with pure conditional states on \(A^nB^n\), at the cost of a subexponential classical register. Choose a spectral decomposition \[\begin{align} \tau^{(n)} = \sum_{\alpha\in\mathcal{A}_n} \lambda_\alpha^{(n)} \ket{\Psi_\alpha^{(n)}}\!\bra{\Psi_\alpha^{(n)}}, \end{align}\] where \(\lambda_\alpha^{(n)}>0\) and the vectors \(\{\ket{\Psi_\alpha^{(n)}}\}_{\alpha\in\mathcal{A}_n}\) form an orthonormal basis of \(\operatorname{supp}\tau^{(n)}\). Since \(\tau^{(n)}\) is supported on the defect space associated with \(\ket{\eta}_{ABR}\), Lemma 2 gives \[\begin{align} |\mathcal{A}_n| = \operatorname{rank}\tau^{(n)} \le \dim M_n(\eta,r_n) \le \binom{n}{r_n}d^{r_n}, \qquad d:=\dim(\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_R). \end{align}\] Hence \[\begin{align} \log |\mathcal{A}_n| \le \log\binom{n}{r_n}+r_n\log d = o(n), \end{align}\] because \(r_n=o(n)\).
Let \(\widetilde{R}_n\) be a classical register with orthonormal basis \(\{\ket{\alpha}_{\widetilde{R}_n}:\alpha\in\mathcal{A}_n\}\). Thus \[\begin{align} \log\dim\widetilde{R}_n = \log|\mathcal{A}_n| = o(n). \end{align}\] For each \(\alpha\in\mathcal{A}_n\), write \[\begin{align} \ket{\Psi_\alpha^{(n)}}_{R^nA^nB^n} = \sum_{i^n} \ket{i^n}_{R^n}\otimes \ket{\psi_{\alpha,i^n}^{(n)}}_{A^nB^n}. \end{align}\] Define \[\begin{align} q_{\alpha,i^n}^{(n)} := \lambda_\alpha^{(n)} \bigl\|\psi_{\alpha,i^n}^{(n)}\bigr\|^2, \end{align}\] and, whenever \(q_{\alpha,i^n}^{(n)}>0\), \[\begin{align} \ket{\phi_{\alpha,i^n}^{(n)}}_{A^nB^n} := \bigl\|\psi_{\alpha,i^n}^{(n)}\bigr\|^{-1} \ket{\psi_{\alpha,i^n}^{(n)}}_{A^nB^n}. \end{align}\] Now define \[\begin{align} \widehat\varrho^{(n)}_{\widetilde{R}_nR^nA^nB^n} := \sum_{\substack{\alpha\in\mathcal{A}_n,\;i^n:\\ q_{\alpha,i^n}^{(n)}>0}} q_{\alpha,i^n}^{(n)} \ket{\alpha}\!\bra{\alpha}_{\widetilde{R}_n} \otimes \ket{i^n}\!\bra{i^n}_{R^n} \otimes \ket{\phi_{\alpha,i^n}^{(n)}}\! \bra{\phi_{\alpha,i^n}^{(n)}}_{A^nB^n}. \end{align}\] By construction, \[\begin{align} \mathop{\mathrm{Tr}}_{\widetilde{R}_n}\widehat\varrho^{(n)}_{\widetilde{R}_nR^nA^nB^n} = \varrho^{(n)}_{R^nA^nB^n}, \qquad \mathop{\mathrm{Tr}}_{\widetilde{R}_nR^n}\widehat\varrho^{(n)}_{\widetilde{R}_nR^nA^nB^n} = \rho_{A^nB^n}. \end{align}\] Therefore \(\widehat\varrho^{(n)}\) is a pure-state cq-extension of \(\rho_{A^nB^n}\), the additional classical register satisfies \(\log\dim\widetilde{R}_n=o(n)\), and the marginal sequence \((\varrho^{(n)}_{R^nA^nB^n})_n\) is MSR almost i.i.d.along \(\varrho_{RAB}\). This proves the claim. ◻
Proof. Let \(n=lm+s\), with \(0\le s<m\), and fix a purification \(\ket{\theta}_{ABE}\) of \(\rho_{AB}\). By Remark 2.4(a) of [24], for this purification there exists an extension \(\omega_{A^nB^nE^n}\) of \(\rho_{A^nB^n}\), invariant under permutations of the \(n\) triples \((A_j,B_j,E_j)\), such that \[\begin{align} \mathop{\mathrm{supp}}\omega_{A^nB^nE^n} \subseteq \operatorname{span} \mathcal{V}\!\left( \mathcal{H}_{ABE}^{\otimes n}, \ket{\theta}_{ABE}^{\otimes(n-r_n)} \right). \end{align}\] We view the first \(lm\) triples as \(l\) consecutive blocks of size \(m\), and the last \(s\) triples as a remainder.
We claim that the same extension, viewed with this tensor-product decomposition, satisfies the desired block support condition. It suffices to check this on the generating vectors of the MSR defect space. Such a generating vector is obtained from \(\ket{\theta}_{ABE}^{\otimes(n-r_n)}\) by placing an arbitrary vector on the remaining \(r_n\) one-site tensor factors and then applying a permutation of the \(n\) sites.
Consider one such generating vector. Among the first \(lm\) tensor factors, call an \(m\)-block bad if it contains at least one unrestricted one-site factor. If the number of bad blocks is \(b\), then \(b\le r_n\) and \(b\le l\). Hence \[\begin{align} b\le t_{l,s}:=\min\{r_n,l\}=\min\{r_{lm+s},l\}. \end{align}\] Every good \(m\)-block is exactly equal to \(\ket{\theta}_{ABE}^{\otimes m}=\ket{\theta_m}_{A^mB^mE^m}\). The final \(s\) tensor factors are placed in the unrestricted remainder system \(\mathcal{H}_{ABE}^{\otimes s}\). Since \(b\le t_{l,s}\), the vector has at least \(l-t_{l,s}\) blocks fixed to \(\ket{\theta_m}\). Equivalently, by allowing additional reference blocks to be counted among the unrestricted blocks, it lies in \[\begin{align} \operatorname{span} \mathcal{V}\!\left( (\mathcal{H}_{ABE}^{\otimes m})^{\otimes l}, \ket{\theta_m}^{\otimes(l-t_{l,s})} \right) \otimes \mathcal{H}_{ABE}^{\otimes s}. \end{align}\] Taking the span over all generating vectors gives the desired support inclusion for \(\omega_{A^nB^nE^n}\), viewed as a state on \((A^mB^mE^m)^lA^sB^sE^s\).
Since \(n=lm+s\), with \(m\) and \(s\) fixed, we have \[\begin{align} \frac{t_{l,s}}{l} \le \frac{r_{lm+s}}{l} = \frac{r_{lm+s}}{lm+s}\,(m+s/l) \longrightarrow 0 \quad\text{as }l\to\infty. \end{align}\] Hence \(t_{l,s}=o(l)\).
Finally, \(\omega_{A^nB^nE^n}\) is invariant under all permutations of the \(n\) one-site triples. In particular, it is invariant under those permutations which exchange the \(l\) consecutive blocks of size \(m\) and leave the remainder fixed. Therefore, viewed as a state on \((A^mB^mE^m)^lA^sB^sE^s\), it is invariant under permutations of the \(l\) blocks of size \(m\). ◻
Proof. Fix \(m\in\mathbb{N}_+\) and \(s\in\{0,\ldots,m-1\}\), and write \(n=lm+s\). Let \(\ket{\theta}_{ABE}\) be a purification of \(\rho_{AB}\), and set \[\begin{align} \ket{\theta_m}_{A^mB^mE^m} := \ket{\theta}_{ABE}^{\otimes m}. \end{align}\] By Lemma 9, for each \(l\) there exists an extension of \(\rho_{A^{lm+s}B^{lm+s}}\) such that \[\begin{align} \operatorname{supp}\omega \subseteq \operatorname{span} {\mathcal{V}}\!\left((\mathcal{H}_{ABE}^{\otimes m})^{\otimes l}, |\theta_m\rangle^{\otimes(l-t_{l,s})}\right) \otimes \mathcal{H}_{ABE}^{\otimes s}, \end{align}\] where \(t_{l,s}=o(l)\), and such that \(\omega\) is invariant under permutations of the \(l\) blocks of size \(m\).
Let \[\begin{align} \ket{\eta_m}_{A^mB^mR} := \sum_i \sqrt{p_i^{(m)}}\, \ket{\phi_i^{(m)}}_{A^mB^m}\ket{i}_R . \end{align}\] This is a purification of \(\rho_{AB}^{\otimes m}\), while \(\ket{\theta_m}\) is another purification of the same state. Enlarging \(R\), if necessary, by adding zero-probability classical labels, we may assume that there is an isometry \(W_m:E^m\to R\) such that \[\begin{align} (\mathbb{1}_{A^mB^m}\otimes W_m)\ket{\theta_m} = \ket{\eta_m}. \end{align}\] Let \(\Delta_R\) denote the dephasing channel in the basis \(\{\ket{i}_R\}_i\). Then \[\begin{align} (\Delta_R\otimes\mathop{\mathrm{id}}_{A^mB^m}) \ket{\eta_m}\!\bra{\eta_m} = \varrho^{(m)}_{RA^mB^m}. \end{align}\] Let \(U_l\) denote the isometry which applies \(W_m\) to the \(E^m\)-part of each of the \(l\) blocks and acts as the identity on the remaining systems \(A^sB^sE^s\). Define \[\begin{align} \tau^{(l,s)} := U_l\omega U_l^\dagger . \end{align}\] By the support inclusion obtained from Lemma 9, together with the identity \[(\mathbb{1}_{A^mB^m}\otimes W_m)\ket{\theta_m}=\ket{\eta_m},\] the state \(\tau^{(l,s)}\) is supported on \[\begin{align} \operatorname{span} \mathcal{V}\!\left( (\mathcal{H}_{A^mB^mR})^{\otimes l}, \ket{\eta_m}^{\otimes(l-t_{l,s})} \right) \otimes \mathcal{H}_{ABE}^{\otimes s}. \end{align}\] Moreover, it is invariant under permutations of the \(l\) blocks. Applying \(\Delta_R^{\otimes l}\) to the \(R^l\)-system gives \[\begin{align} \zeta^{(l,s)}_{R^l(A^mB^m)^lA^sB^sE^s} := (\Delta_R^{\otimes l}\otimes\mathbb{1}_{(A^mB^m)^lA^sB^sE^s}) (\tau^{(l,s)}). \end{align}\] Since \(\Delta_R\) is a local channel on the \(R\)-system, the same local-channel stability argument as in Lemma 8, now applied to the \(m\)-blocks, shows that the marginal sequence on \(R^l(A^mB^m)^l\), obtained by tracing out the bounded remainder \(A^sB^sE^s\), is MSR almost i.i.d.along \(\varrho^{(m)}_{RA^mB^m}\), with defect sizes \(t_{l,s}=o(l)\).
It remains to produce a pure-state cq-extension of the full state \(\rho_{A^{lm+s}B^{lm+s}}\) and to bound the additional classical overhead. Choose a spectral decomposition \[\begin{align} \tau^{(l,s)} = \sum_{\alpha\in\mathcal{A}_{l,s}} \lambda_\alpha^{(l,s)} \ket{\Psi_\alpha^{(l,s)}}\! \bra{\Psi_\alpha^{(l,s)}}, \end{align}\] with \(\lambda_\alpha^{(l,s)}>0\). Since \(\tau^{(l,s)}\) is supported on the block-defect space above, tensored with the bounded remainder system, its rank is at most \[\begin{align} \binom{l}{t_{l,s}} d_m^{\,t_{l,s}} d_{\rm rem}, \end{align}\] where \[d_m:=\dim(\mathcal{H}_{A^mB^mR}) \qquad\text{and}\qquad d_{\rm rem}:=\dim(\mathcal{H}_{ABE}^{\otimes s}).\] Since \(m\) and \(s\) are fixed and \(t_{l,s}=o(l)\), we have \[\begin{align} \log\operatorname{rank}\tau^{(l,s)} \le \log\binom{l}{t_{l,s}} + t_{l,s}\log d_m + \log d_{\rm rem} = o(l). \end{align}\]
For each \(\alpha\), expand \[\begin{align} \ket{\Psi_\alpha^{(l,s)}} = \sum_{i^l} \ket{i^l}_{R^l} \otimes \ket{\psi_{\alpha,i^l}^{(l,s)}}_{A^{lm+s}B^{lm+s}E^s}, \end{align}\] where the \(E^s\)-system is included in the second tensor factor. Define \[\begin{align} q_{\alpha,i^l}^{(l,s)} := \lambda_\alpha^{(l,s)} \bigl\|\psi_{\alpha,i^l}^{(l,s)}\bigr\|^2 . \end{align}\] Whenever \(q_{\alpha,i^l}^{(l,s)}>0\), set \[\begin{align} \ket{\chi_{\alpha,i^l}^{(l,s)}}_{A^{lm+s}B^{lm+s}E^s} := \bigl\|\psi_{\alpha,i^l}^{(l,s)}\bigr\|^{-1} \ket{\psi_{\alpha,i^l}^{(l,s)}} . \end{align}\] Tracing out \(E^s\), we obtain a state \[\begin{align} \chi_{\alpha,i^l,A^{lm+s}B^{lm+s}}^{(l,s)} := \mathop{\mathrm{Tr}}_{E^s} \ket{\chi_{\alpha,i^l}^{(l,s)}}\! \bra{\chi_{\alpha,i^l}^{(l,s)}} . \end{align}\] Choose a pure-state decomposition \[\begin{align} \chi_{\alpha,i^l,A^{lm+s}B^{lm+s}}^{(l,s)} = \sum_j p_{j|\alpha,i^l}^{(l,s)} \ket{\xi_{\alpha,i^l,j}^{(l,s)}}\! \bra{\xi_{\alpha,i^l,j}^{(l,s)}}_{A^{lm+s}B^{lm+s}} . \end{align}\] Since the rank of \(\chi_{\alpha,i^l,A^{lm+s}B^{lm+s}}^{(l,s)}\) is at most \(\dim\mathcal{H}_E^{\otimes s}\), and this dimension is independent of \(l\), the decomposition may be chosen with a bounded number of nonzero terms. Absorbing this bounded factor into the classical overhead, define \(\widetilde{R}_l\) to be a classical register indexing the pairs \((\alpha,j)\). Then \[\begin{align} \log\dim\widetilde{R}_l \le \log\operatorname{rank}\tau^{(l,s)} + \log \dim\mathcal{H}_E^{\otimes s} = o(l). \end{align}\]
Now define \[\begin{align} \widehat\varrho^{(l,s)}_{\widetilde{R}_lR^lA^{lm+s}B^{lm+s}} := \sum_{\substack{\alpha,i^l,j:\\ q_{\alpha,i^l}^{(l,s)} p_{j|\alpha,i^l}^{(l,s)}>0}} q_{\alpha,i^l}^{(l,s)} p_{j|\alpha,i^l}^{(l,s)} \ket{\alpha,j}\!\bra{\alpha,j}_{\widetilde{R}_l} \otimes \ket{i^l}\!\bra{i^l}_{R^l} \otimes \ket{\xi_{\alpha,i^l,j}^{(l,s)}}\! \bra{\xi_{\alpha,i^l,j}^{(l,s)}}_{A^{lm+s}B^{lm+s}} . \end{align}\] This is a pure-state cq-extension of \(\rho_{A^{lm+s}B^{lm+s}}\). Indeed, tracing over \(\widetilde{R}_lR^l\) gives \[\begin{align} \mathop{\mathrm{Tr}}_{\widetilde{R}_lR^l} \widehat\varrho^{(l,s)} = \mathop{\mathrm{Tr}}_{R^lE^s}\zeta^{(l,s)} = \rho_{A^{lm+s}B^{lm+s}}, \end{align}\] where the last equality follows because dephasing the \(R^l\)-system does not change the marginal on \(A^{lm+s}B^{lm+s}E^s\), and \(\omega\) is an extension of \(\rho_{A^{lm+s}B^{lm+s}}\).
Furthermore, tracing out \(\widetilde{R}_l\) and the final \(A^sB^s\)-systems leaves the same marginal on \(R^l(A^mB^m)^l\) as the one obtained from \(\zeta^{(l,s)}\). Hence this marginal sequence is MSR almost i.i.d.along \(\varrho^{(m)}_{RA^mB^m}\), with defect sizes \(t_{l,s}=o(l)\). This proves the corollary. ◻
That is, \(H_{\min}^{\delta,\|\cdot\|_1}(A)_\rho := \sup_{\sigma_A\in B_1^\delta(\rho_A)} H_{\min}(A)_\sigma\).↩︎
We use the notation of Mazzola–Sutter–Renner [24] here. For a generalization of the MSR almost i.i.d.source see [12].↩︎
Here the error term \(a_n\) depends only on the reference state \(\rho\), the dimension of the underlying Hilbert space, the parameter \(\varepsilon\), and the defect-size sequence \((r_n)_n\) where \(r_n=o(n)\), but not on the particular state \(\rho_n\) in the MSR class.↩︎