Can entangling codewords across independent uses of a quantum channel improve the rate of classical communication? An old conjecture states that there is no such entanglement advantage for all channels. Although Hastings's result and follow-up works disproved this conjecture, it turns out that enormous dimension is required to construct a channel where entanglement enhances classical communication. This indicates that there might be no entanglement advantage for many physical channels, or for all low dimensional channels. Motivated by recent progress on (generalized) amplitude damping channels, we propose a general Schur-weighted block entropy inequality, and show that there is no entanglement advantage for all phase-covariant channels including multi-level amplitude damping channels, generalized dephasing channels, and damping-dephasing channels. As a by-product, we also recover the recently established result that there is no entanglement advantage for all qubit channels.
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The paper asks whether entangling inputs across independent channel uses can increase classical communication rates above the single-letter Holevo information $\chi(\mathcal A)$. It establishes strong Holevo additivity, $\chi(\mathcal A\otimes\mathcal M)=\chi(\mathcal A)+\chi(\mathcal M)$ for every partner channel $\mathcal M$, for channels covariant under all diagonal unitaries in a fixed basis, in arbitrary finite dimension. Consequently, their classical capacity satisfies $\mathcal C(\mathcal A)=\chi(\mathcal A)$. This class includes multilevel amplitude damping, generalized dephasing, and damping–dephasing channels.
The central tool is a Schur-weighted block entropy inequality: it lower-bounds the entropy of a positive block matrix plus block-diagonal noise using a smaller matrix determined by block traces and weighted within-block entropies. Its proof combines a rank-one argument with concavity of a variational entropy deficit and joint convexity of a scalar comparison function. Applying this inequality with an entropic additivity criterion that relaxes marginal compatibility proves the phase-covariant result.
The paper also recovers strong additivity for every qubit-to-qubit channel. It decomposes any $d$-dimensional-input, qubit-output channel into a commuting-output entanglement-breaking operation and a completely positive remainder with at most $d-1$ Kraus operators, and proves additivity when the remainder has one Kraus operator.
The method does not establish additivity for arbitrary qubit-output channels with input dimension $d>2$. Covariance under a one-parameter Hamiltonian evolution, rather than all diagonal unitaries, and the dimensional boundary for entanglement-enhanced classical communication remain open.
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the author used ChatGPT-6 Astra and Claude Opus 5.5 to study the OpenAI manuscript~\cite{OpenAI2026} and to explore generalizations and proofs.
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See the companion github repository https://github.com/pxwu24/additivity-of-classical-communication for the presentation of the results and a complete Lean verification
Can entangling codewords across independent uses of a quantum channel improve the rate of classical communication? An old conjecture states that there is no such entanglement advantage for all channels. Although Hastings's result and follow-up works disproved this conjecture, it turns out that enormous dimension is required to construct a channel where entanglement enhances classical communication. This indicates that there might be no entanglement advantage for many physical channels, or for all low dimensional channels. Motivated by recent progress on (generalized) amplitude damping channels, we propose a general Schur-weighted block entropy inequality, and show that there is no entanglement advantage for all phase-covariant channels including multi-level amplitude damping channels, generalized dephasing channels, and damping-dephasing channels. As a by-product, we also recover the recently established result that there is no entanglement advantage for all qubit channels.
The paper asks whether entangling inputs across independent channel uses can increase classical communication rates above the single-letter Holevo information χ(A). It establishes strong Holevo additivity, χ(A⊗M)=χ(A)+χ(M) for every partner channel M, for channels covariant under all diagonal unitaries in a fixed basis, in arbitrary finite dimension. Consequently, their classical capacity satisfies C(A)=χ(A). This class includes multilevel amplitude damping, generalized dephasing, and damping–dephasing channels.
The central tool is a Schur-weighted block entropy inequality: it lower-bounds the entropy of a positive block matrix plus block-diagonal noise using a smaller matrix determined by block traces and weighted within-block entropies. Its proof combines a rank-one argument with concavity of a variational entropy deficit and joint convexity of a scalar comparison function. Applying this inequality with an entropic additivity criterion that relaxes marginal compatibility proves the phase-covariant result.
The paper also recovers strong additivity for every qubit-to-qubit channel. It decomposes any d-dimensional-input, qubit-output channel into a commuting-output entanglement-breaking operation and a completely positive remainder with at most d−1 Kraus operators, and proves additivity when the remainder has one Kraus operator.
The method does not establish additivity for arbitrary qubit-output channels with input dimension d>2. Covariance under a one-parameter Hamiltonian evolution, rather than all diagonal unitaries, and the dimensional boundary for entanglement-enhanced classical communication remain open.
When a thermal environment damps the two transverse components of a qubit differently, can entangling signals across many channel uses improve the classical communication rate? This article gives a complete coding answer for an explicit phase-sensitive class and determines the optimal exponential cost of transmitting above capacity. Physical model and communication permissions. The channel acts as N(ρ0X∗Xρ1)=(aρ0+bρ1kX∗+ℓXkX+ℓX∗cρ0+dρ1). All coefficients are nonnegative, a+c=b+d=1, and the main class satisfies k≥ℓ≥0,ad−bc=k2−ℓ2,bc≥ℓ(k+ℓ). The class contains actual squeezed thermal reservoirs, including nonunital channels that are not entanglement breaking. The coding model is unassisted and memoryless, without feedback; arbitrary entangled block codewords and collective decoding are allowed. The coefficient condition is a sufficient region, not a classification of every physically allowed squeezed bath. An exact capacity from one-use data. Write h2 for binary entropy in bits, q for the excited-state input population, and define t(q)=c+(d−c)q,sx(q)=h2(21+[1−2t(q)]2+4(k+ℓ)2q(1−q)). With cosx the lower convex envelope on [0,1], the unrestricted capacity is C(N)=χ(N)=max0≤q≤1{h2[t(q)]−(cosx)(q)}. At most two opposite-real-phase signal pairs, hence at most four pure signals, attain the one-use optimum. The convex envelope retains optimization over all ensembles. Holevo information is additive with every finite-dimensional channel partner, so arbitrarily long entangled encodings cannot exceed this rate. The best success exponent above capacity. Let Pn⋆(R) be the optimal average success probability over all n-use codes with at least ⌈2nR⌉ messages. For R>C(N), Esc(R,N)=limn→∞−n1log2Pn⋆(R)=supα>1αα−1[R−χα(N)]. Here χα is the sandwiched Rényi information radius, reduced in the paper to a scalar reference/population optimization. It is strongly additive at every real α>1. Product codewords with collective decoding attain the asymptotic exponent, while the converse covers arbitrary entangled codewords. The finite-block inequality Pn⋆(R)≤2−nEsc is an upper bound, not a finite-block equality. Why the reduction works. Normal and reversed positive block factors retain the same quantum source and admit a common scalar Gram bound. This yields maximum-output norm multiplicativity with arbitrary completely positive partners and unnormalized diagonal filters. Binary phase symmetry then suffices to make the information-radius reference diagonal, even though continuous phase covariance is absent. The same input-filter comparison also determines the formation cost of the specified diagonal-input Stinespring states across the output and the full purified environment. Relation to existing results. We independently obtained the additivity and strong-converse results for the axially symmetric Davies family that are also established by L. Eriksson, Capacities, strong-converse exponents and complementary simulation for qubit Davies channels, version 3, 26 October 2026. These results and their proofs are retained as a reference case. The phase-sensitive class and the full axial Davies class share the generalized-amplitude-damping boundary; neither is presented as a strict enlargement of the other. Fang's triangular norm-compression theorem and the general information-radius and coding theorems retain their attribution. The 16-page integrated article contains the full proofs, two figure groups and references. The figures separate the physical parameter region, the globally evaluated capacity formula and explicit finite-block success upper bounds.
The problem of sending a single classical bit through a generalized amplitude damping channel is considered. When two transmissions through the channel are available as a resource, we find that two entangled transmissions can enhance the capability of receiver's judging information correctly under certain conditions compared with two product-state transmissions. In addition, we find a special case in which the two entangled transmissions can always make a classical bit more effectively disable the noise influence.
We consider properties of states of many qubits, which arise after sending certain entangled states via various noisy channels (white noise, colored noise, local depolarization, dephasing, and amplitude damping). Entanglement of these states and their ability to violate certain classes of Bell inequalities are studied. States which violate them allow a higher than classical efficiency in solving related distributed computational tasks with constrained communication. This is a direct property of such states---not requiring their further modification via stochastic local operations and classical communication such as entanglement purification or distillation procedures. We identify families of multiparticle states which are entangled but nevertheless allow the local realistic description of specific Bell experiments. For some of them, the ``gap'' between the critical values for entanglement and violation of Bell inequality remains finite even in the limit of infinitely many qubits.
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 p, 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) 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<1, and construct a binary Kraus representation attaining the uniform formation bound.
The entanglement-assisted classical capacity of a noisy quantum channel (C/sub E/) is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that the capacity C/sub E/ is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs /spl rho/, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of /spl rho/ after half of it has passed through the channel. We calculate entanglement-assisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglement-assisted communication - communication over a discrete memoryless channel (DMC) between parties who share prior random information - we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMCs of equal capacity can simulate one another with unit asymptotic efficiency.
Current advancements in communication equipment demand the investigation of classical information transfer over quantum channels, by encompassing realistic scenarios in finite dimensions. To address this issue, we develop a framework for analyzing classical capacities of quantum channels where the set of states used for encoding information is restricted based on various physical properties. Specifically, we provide expressions for the classical capacities of noiseless and noisy quantum channels when the average energy of the encoded ensemble or the energy of each of the constituent states in the ensemble is bounded. In the case of qubit energy-preserving dephasing channels, we demonstrate that a nonuniform probability distribution based on the energy constraint maximizes capacity, while we derive the compact form of the capacity for equiprobable messages. We suggest an energy-constrained dense coding (DC) protocol that we prove to be optimal in the two-qubit situation and obtain a closed-form expression for the DC capacity. Additionally, we demonstrate a no-go result, which states that when the dimension of the sender and the receiver is two, no energy-preserving operation can offer any quantum advantage for energy-constrained entanglement-assisted capacity. We exhibit that, in the energy-constrained situation, classical-quantum noisy channels can show improved capabilities under entanglement assistance, a phenomenon that is unattainable in the unrestricted scenario.