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arXiv:2609.23298

Static classical-quantum-entanglement trade-offs: an entropic converse and single-letter characterizations

Mark M. Wilde

quant-phcs.ITSubmitted Sep 20, 2026• Announced Sep 22, 2026• 0 citations
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Abstract

The direct static capacity region of a bipartite quantum state describes its asymptotic conversion into classical communication, quantum communication, and entanglement, with all communication directed from Alice to Bob. This paper gives a unified entropic converse for this region. The proof treats generated and consumed resources simultaneously and derives all three information inequalities for one local instrument. Its main ingredients are no-signalling, the chain rule, strong subadditivity, and continuity of conditional entropy. A supporting-hyperplane characterization leads to the static capacity formula, which is helpful in determining single-letterization of the capacity region. For a pure state of entanglement entropy $h$ subjected to erasure with probability $p$, the complete region is the convex hull of $0$ and $(0,-ph,(1-p)h)$, plus the unit-resource cone. This statement holds for arbitrary collective instruments by using subset-entropy inequalities related to earlier methods used for the dynamic capacity region of the erasure channel. Complete characterizations also hold for symmetrically extendible states and locally flagged pure-state mixtures. For maximally correlated states, the optimization over local instruments admits an exact matrix formulation and an additive outer bound. An explicit qubit example shows that a nonorthogonal discarded quantum memory can outperform every efficient instrument and every instrument with conditionally commuting discarded states. For states obtained by qubit dephasing of Schmidt-aligned pure states, an exact threshold characterizes when the static capacity formula vanishes, at every blocklength and after regularization. Full single-letter characterizations for general Hadamard states and for the remaining trade-offs for dephased states remain open, with sufficient additivity conditions identified.

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81 pages, 5 figures

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The dynamic capacity theorem characterizes the reliable communication rates of a quantum channel when combined with the noiseless resources of classical communication, quantum communication, and entanglement. In priorwork,we proved the converse part of this theorem by making contact with many previous results in the quantum Shannon theory literature. In this work, we prove the theorem with an "ab initio" approach, using only the most basic tools in the quantum information theorist's toolkit: the Alicki-Fannes' inequality, the chain rule for quantum mutual information, elementary properties of quantum entropy, and the quantum data processing inequality. The result is a simplified proof of the theorem that should be more accessible to those unfamiliar with the quantum Shannon theory literature. We also demonstrate that the "quantum dynamic capacity formula" characterizes the Pareto optimal tradeoff surface for the full dynamic capacity region. Additivity of this formula reduces the computation of the trade-off surface to a tractable, textbook problem in Pareto trade-off analysis, and we prove that its additivity holds for the quantum Hadamard channels and the quantum erasure channel. We then determine exact expressions for and plot the dynamic capacity region of the quantum dephasing channel, an example from the Hadamard class, and the quantum erasure channel. © Springer Science+Business Media, LLC 2011.
In this paper, we give tradeoffs between classical communication, quantum communication, and entanglement for processing information in the Shannon-theoretic setting. We first prove a “unit-resource” capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation. The optimal strategy mixes the three fundamental protocols of teleportation, superdense coding, and entanglement distribution. We then provide an achievable rate region and a matching multiletter converse for the “direct-static” capacity theorem. This theorem applies to the scenario where a large number of copies of a noisy bipartite state are available (in addition to consumption or generation of the above three noiseless resources). Our coding strategy involves a protocol that we name theclassically assisted state redistribution protocoland the three fundamental protocols. We finally provide an achievable rate region and a matching multiletter converse for the “direct-dynamic” capacity theorem. This theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources. Our coding strategy combines theclassically enhanced father protocolwith the three fundamental unit protocols.
In this paper, we give tradeoffs between classical communication, quantum communication, and entanglement for processing information in the Shannon-theoretic setting. We first prove a unit-resource capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation. The optimal strategy mixes the three fundamental protocols of teleportation, superdense coding, and entanglement distribution. We then provide an achievable rate region and a matching multiletter converse for the direct-static capacity theorem. This theorem applies to the scenario where a large number of copies of a noisy bipartite state are available (in addition to consumption or generation of the above three noiseless resources). Our coding strategy involves a protocol that we name the classically assisted state redistribution protocol and the three fundamental protocols. We finally provide an achievable rate region and a matching multiletter converse for the direct-dynamic capacity theorem. This theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources. Our coding strategy combines the classically enhanced father protocol with the three fundamental unit protocols. © 2010 IEEE.
Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus rank at most two. For the amplitude damping channel with damping probability ppp, the unassisted classical capacity equals the known single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost is h2((1+p)/2)h_2((1+\sqrt p)/2)h2​((1+p​)/2) ebits per use. The common mechanism is a support criterion for strong superadditivity of entanglement of formation: one marginal has no support on the sector in which both local systems are orthogonal to fixed distinguished vectors. We prove this criterion in arbitrary finite dimensions using a triangular block-matrix entropy inequality and decompositions preserving two expectations. For amplitude damping, we also derive an exact finite-block Holevo deficit, identify the unique optimal average input for p<1p<1p<1, and construct a binary Kraus representation attaining the uniform formation bound.
The one-way distillable entanglement is a central operational measure of bipartite entanglement, quantifying the optimal rate at which maximally entangled pairs can be extracted by one-way LOCC. Despite its importance, it is notoriously hard to compute, since it is defined by a regularized optimization over many copies and adaptive one-way protocols. At present, single-letter formulas are only known for (conjugate) degradable and PPT states. More generally, it has remained unclear when one-way distillable entanglement can still be additive beyond degradability and PPT settings, and how such additivity relates to additivity questions of quantum capacity of channels. In this paper, we address this gap by identifying three explicit families of non-degradable and non-PPT states whose one-way distillable entanglement is nevertheless single-letter. First, we introduce two weakened degradability-type conditions--regularized less-noisy and informationally degradable--and prove that each guarantees additivity and hence a single-letter formula. Second, we show a stability result for orthogonally flagged mixtures: when one component has orthogonal support on Alice's system and zero one-way distillable entanglement, the mixture remains single-letter, even though degradability is typically lost under such mixing. Finally, we propose a generalized spin-alignment principle for entropy minimization in tensor-product settings, which we establish in several key cases, including a complete Rényi-2 result. As an application, we obtain additivity results for generalized direct-sum channels and their corresponding Choi states.
Entanglement distillation over the pure-loss bosonic channel is governed by the PLOB bound, -log2(1-η) ebits per use, but this benchmark presumes unlimited two-way classical communication. When the classical feedback budget is metered at B bits per use, the achievable rate approaches PLOB exponentially in B, yet no matching converse is known, and we argue that the standard route — static entanglement monotones — cannot work even in principle. Here we develop a converse program built on the Devetak–Harrow–Winter (DHW) characterization of one-way distillation. Our central result is a conservation law: for every rank-one instrument applied by the sending party, the single-copy communication charge R and entanglement yield E obey the exact identity R+E=S(B), the local entropy of the measured system, independent of the state; general instruments obey R+E=S(B)-S(E'|B'X), with a correction that vanishes for all projective coarse-grainings. Combining the identity with the proven quantum entropy power inequality yields an unconditional lower bound on the Bob-entropy cost of approaching PLOB, and, for the class of sharp instruments satisfying a per-instrument yield condition, a two-sided bound η^2 2^{-B} lesssim PLOB-C lesssim (e/2) η^2 2^{-B}: in this class the feedback-metered capacity provably falls short of PLOB at every finite B, with the gap determined up to a factor e/2. For two-mode-squeezed inputs — and, non-asymptotically, for the whole class of products of two-mode-squeezed states of arbitrary energies dressed by passive linear optics — we prove a complementary converse that needs no yield condition at all: a separable Gaussian witness pins the relative entropy of entanglement of the loss output to within Theta(1/bar N) of PLOB, reproducing the 2^{-B} gap scaling unconditionally for these input classes, with constants matched to within 0.78–0.94 of the achievability side for two-mode-squeezed inputs (smaller for the passive class at η ≥ 0.5). We further identify the obstruction to removing the yield condition in general: measurements can increase the reverse coherent information by up to their heralding information, E ≤ I_R + I(X;E), and we exhibit state families saturating this and show numerically that the excess is generic even for loss-channel outputs. The sharp-class converse is thereby reduced to a single variational problem, which we state precisely and conjecture to stay below PLOB; beyond the sharp class, we show the conservation law acquires a coarse-graining discount that must be bounded separately. Preprint (converse-side companion of: How many classical bits buy how much quantum capacity? Feedback-metered limits of lossy quantum communication). Ancillary files: reproduction scripts and data; code maintained at https://github.com/deeptell-inc/fmplob.