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arXiv:2605.15439

Additivity Results for the Rényi-2 Entanglement of Purification

Shokoufe Faraji, Zahra Baghali Khanian

quant-phcs.ITSubmitted May 14, 2026• Announced May 18, 2026• 0 citations
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Abstract

We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for $p>1$, this formulation can be expressed in terms of a constrained maximal output Schatten $p$-norm. More precisely, for a completely positive map $Ω:L(B')\to L(A)$, we consider the quantity $\upsilon_p(Ω)$ defined by optimizing $\|(Ω\otimes \mathrm{id}_E)(σ^{B'E})\|_p$ over all bipartite states $σ^{B'E}$ whose $B'$-marginal is maximally mixed. We focus on the case $p=2$. First, we compute $\upsilon_2$ for the transpose-depolarizing channel and prove that it is multiplicative under tensor powers. We then establish a general multiplicativity criterion: whenever a completely positive map $N:L(B')\to L(A)$ satisfies $N^{\dagger} \mathbin{\circ} N=a\,\mathrm{id}_A+b\,\mathrm{Tr}[\cdot]\,I_d$ for some constants $a,b\ge 0$, where $N^{\dagger}$ denotes the Hilbert-Schmidt adjoint of $N$, the quantity $\upsilon_2(N)$ is multiplicative under tensor powers. Examples of channels satisfying this criterion include the transpose-depolarizing channel, the depolarizing channel, and their respective complementary channels. Furthermore, we show that, for every completely positive map $Ω$, multiplicativity of $\upsilon_p(Ω)$ implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.

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We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for p>1p>1p>1, this formulation can be expressed in terms of a constrained maximal output Schatten ppp-norm. More precisely, for a completely positive map Ω:L(B′)→L(A)Ω:L(B')\to L(A)Ω:L(B′)→L(A), we consider the quantity υp(Ω)\upsilon_p(Ω)υp​(Ω) defined by optimizing ∥(Ω⊗idE)(σB′E)∥p\|(Ω\otimes \mathrm{id}_E)(σ^{B'E})\|_p∥(Ω⊗idE​)(σB′E)∥p​ over all bipartite states σB′Eσ^{B'E}σB′E whose B′B'B′-marginal is maximally mixed. We focus on the case p=2p=2p=2. First, we compute υ2\upsilon_2υ2​ for the transpose-depolarizing channel and prove that it is multiplicative under tensor powers. We then establish a general multiplicativity criterion: whenever a completely positive map N:L(B′)→L(A)N:L(B')\to L(A)N:L(B′)→L(A) satisfies N†∘N=a idA+b Tr[⋅] IdN^{\dagger} \mathbin{\circ} N=a\,\mathrm{id}_A+b\,\mathrm{Tr}[\cdot]\,I_dN†∘N=aidA​+bTr[⋅]Id​ for some constants a,b≥0a,b\ge 0a,b≥0, where N†N^{\dagger}N† denotes the Hilbert-Schmidt adjoint of NNN, the quantity υ2(N)\upsilon_2(N)υ2​(N) is multiplicative under tensor powers. Examples of channels satisfying this criterion include the transpose-depolarizing channel, the depolarizing channel, and their respective complementary channels. Furthermore, we show that, for every completely positive map ΩΩΩ, multiplicativity of υp(Ω)\upsilon_p(Ω)υp​(Ω) implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.
The primary entropic measures for quantum states are additive under the tensor product. In the analysis of quantum information processing tasks, the minimum entropy of a set of states, e.g., the minimum output entropy of a channel, often plays a crucial role. A fundamental question in quantum information and cryptography is whether the minimum output entropy remains additive under the tensor product of channels. Here, we establish a general additivity statement for the optimized sandwiched Rényi entropy of quantum channels. For that, we generalize the results of Devetak et al. (Commun Math Phys 266(1):37-63, 2006) to multi-index Schatten norms. As an application, we strengthen the additivity statement of Van Himbeeck and Brown (A tight and general finite-size security proof for quantum key distribution, 2025) thus allowing the analysis of time-adaptive quantum cryptographic protocols. In addition, we establish chain rules for Rényi conditional entropies that are similar to the ones used for the generalized entropy accumulation theorem of Metger et al. (Commun Math Phys 405(11):261, 2024).
We obtain two new additivity results of quantum channels. The first one is the additivity of the channel Rényi information associated with the sandwiched Rényi divergence of order α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1). To prove this, we introduce the completely bounded 1→α1\toα1→α quasi-norms for completely positive maps, with α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1), and show that it is multiplicative. The additivity/multiplicativity derived here extends and complements the results of Devetak {\it et al} (Commun Math Phys 266:37-63, 2006) and Gupta and Wilde (Commun Math Phys 334:867-887, 2015), which deal with the case α>1α>1α>1. The second one is the additivity of the channel dispersion, which is a quantity related to the second-order behavior of quantum information tasks.
We introduce and study multi-indexed Schatten spaces which lift the commutative ℓq1 ⁣[ℓq2 ⁣[ℓq3 ⁣[… ]]]\ell_{q_1}\![\ell_{q_2}\![\ell_{q_3}\![\dots]]]ℓq1​​[ℓq2​​[ℓq3​​[…]]] -spaces to the quantum toolbox. They extend Pisier's operator-valued Schatten norms and the recently introduced two-indexed Schatten quasi-norms. For every ordered tuple Q=(q1,…,qn)\mathcal{Q}=(q_1,\dots,q_n)Q=(q1​,…,qn​) of non-zero indices we define a quasi-norm (for positive indices) or an anti-norm (for negative indices) on operators on an nnn-fold tensor product of Hilbert spaces. We do this for indices of the same sign satisfying the mutual compatibility condition ∣1qi−1qj∣≤1|\frac{1}{q_i}-\frac{1}{q_j}|\leq1∣qi​1​−qj​1​∣≤1. Beyond establishing desirable properties of these quasi- and anti-norms via factorization formulas that combine operator space and matrix-analytic techniques, we lift several central tools and results of the seminal [Devetak, Junge, King, Ruskai, CMP 2006] to the multi-indexed and quasi-normed setting. These include a general non-commutative Minkowski inequality relating swapped systems and two identity-removal equalities. As a consequence, complete hypercontractivity and complete reverse hypercontractivity tensorize for all compatible indices, including negative ones. Concerning quantum information theory we prove that the completely bounded minimal and the maximal output conditional entropies are additive under tensor products of quantum channels for all α≥12α\geq\frac12α≥21​. This extends the case α>1α>1α>1 of [Fawzi, Kochanowski, Rouzé, Van Himbeeck, CMP 2026] and the unconditional results of [Devetak, Junge, King, Ruskai, CMP 2006]. Beyond tensor products, we establish chain rules for H~α↑\tilde H^\uparrow_αH~α↑​, α∈[12,1)α\in[\frac12,1)α∈[21​,1), for compositions of possibly signaling channels, extending upon the smooth max-entropy chain rule used in device-independent entanglement certification to the whole interval α∈[1/2,1)α\in[1/2,1)α∈[1/2,1).
The entanglement of purification EPE_PEP​, introduced by Terhal, Horodecki, Leung, and DiVincenzo in 2002, measures the total correlations of a bipartite state by the entanglement needed to purify it; whether it is additive on tensor powers has been open since its introduction. In 2012 Chen and Winter argued "beyond reasonable doubt", from numerical evidence, that the two-qubit Werner state at singlet fraction f=1/200f = 1/200f=1/200 is a counterexample, and explicitly left a completely rigorous proof to future work. To our knowledge we give the first rigorous proof: for this state WWW, EP∞(W)≤0.9663…<97/100<EP(W)E_P^\infty(W) \le 0.9663\ldots < 97/100 < E_P(W)EP∞​(W)≤0.9663…<97/100<EP​(W), hence the entanglement of purification is strictly nonadditive at some finite tensor power. The lower bound is a finite certificate: analytic reductions send the Schmidt spectrum of every purification in the complete complex Stiefel domain to a point satisfying explicit necessary inequalities in a three-dimensional box, and an exact-rational subdivision of that box has 25,383 leaves, each of which either excludes nonphysical points or proves the entropy bound. The upper bound combines an exact symmetric-support endpoint, an explicit feasible isometry, and a convexity theorem for the regularized quantity, closed by two directed rational logarithm certificates. Every computer-assisted certificate reduces to integer comparisons; a dependency-free program verifies all of them from the supplementary artifact.
We define 2-indexed (q,p)(q,p)(q,p)-Schatten quasi-norms for any q,p>0q,p > 0q,p>0 on operators on a tensor product of Hilbert spaces, naturally extending the norms defined by Pisier's theory of operator-valued Schatten spaces. We establish several desirable properties of these quasi-norms, such as relational consistency and the behavior on block diagonal operators, assuming that ∣1q−1p∣≤1|\frac{1}{q} - \frac{1}{p}| \leq 1∣q1​−p1​∣≤1. In fact, we show that this condition is essentially necessary for natural properties to hold. Furthermore, for linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms. We prove that the q→pq \to pq→p completely bounded co-quasi-norm is super-multiplicative for tensor products of quantum channels for q≥p>0q \geq p>0q≥p>0, extending an influential result of [Devetak, Junge, King, Ruskai, 2006]. Our proofs rely on elementary matrix analysis and operator convexity tools and do not require operator space theory. On the applications side, we demonstrate that these quasi-norms can be used to express relevant quantum information measures such as Rényi conditional entropies for α≥12α\geq \frac{1}{2}α≥21​ or the Sandwiched Rényi Umlaut information for α<1α< 1α<1. Our multiplicativity results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-ααα-entropy for α≥12α\geq\frac{1}{2}α≥21​ extending another important result of [Devetak, Junge, King, Ruskai, 2006], and additivity of the maximum output Rényi-ααα entropy for α≥12α\geq \frac{1}{2}α≥21​.
Entanglement of purification is a fundamental measure of total correlations whose additivity remains unresolved. We study its additivity for classical states on two qubits at different Rényi orders. For every α∈[0,1)α\in[0,1)α∈[0,1), we prove nonadditivity within this family, witnessed by two copies of a single state. We first solve the one-copy optimization exactly for the entire family at every Rényi order. We then restrict the two-copy optimization to a natural finite set of purifications and exhibit one whose entropy is strictly below the product value. In contrast, for α∈[2,∞]α\in[2,\infty]α∈[2,∞] we prove additivity under tensor products within this family. The interval α∈[1,2)α\in[1,2)α∈[1,2), including the von Neumann case α=1α=1α=1, remains open, and we conjecture additivity there throughout the same family.