We develop entanglement-of-formation potentials for feedback-assisted classical communication and one-way randomness distillation. For a finite-dimensional quantum channel without initial shared entanglement, an exact amortization theorem bounds the increase of message information plus entanglement remaining across the communicating laboratories. This proves a classical-feedback capacity bound using a mixed-convex-roof expression previously studied by Winter and Yang in the context of potential capacities. We determine the capacity of flagged mixtures of noiseless and entanglement-breaking channels, improve a published depolarizing-channel converse, and evaluate the new bound exactly for the amplitude damping channel. A state analogue gives a finite-blocklength upper bound on the net shared randomness obtainable with one-way classical communication. Amplitude-damping Choi states provide exact benchmarks, while a full-rank generalized amplitude-damping Choi state demonstrates a useful improvement where we do not evaluate the operational rate exactly. We also distinguish one-way from two-way randomness distillation: a two-round weak-measurement protocol strictly exceeds the exact one-way rate of every nontrivial qubit isotropic state. The protocol preserves quantum side information during its first decoding and extracts additional randomness in the reverse direction. This shows why the one-way state potential is not a general two-way converse. Together, these results establish a common entanglement-based approach to sharper communication and randomness bounds, while revealing how retained quantum correlations enable two-way distillation to surpass one-way limits.
We develop entanglement-of-formation potentials for feedback-assisted classical communication and one-way randomness distillation. For a finite-dimensional quantum channel without initial shared entanglement, an exact amortization theorem bounds the increase of message information plus entanglement remaining across the communicating laboratories. This proves a classical-feedback capacity bound using a mixed-convex-roof expression previously studied by Winter and Yang in the context of potential capacities. We determine the capacity of flagged mixtures of noiseless and entanglement-breaking channels, improve a published depolarizing-channel converse, and evaluate the new bound exactly for the amplitude damping channel. A state analogue gives a finite-blocklength upper bound on the net shared randomness obtainable with one-way classical communication. Amplitude-damping Choi states provide exact benchmarks, while a full-rank generalized amplitude-damping Choi state demonstrates a useful improvement where we do not evaluate the operational rate exactly. We also distinguish one-way from two-way randomness distillation: a two-round weak-measurement protocol strictly exceeds the exact one-way rate of every nontrivial qubit isotropic state. The protocol preserves quantum side information during its first decoding and extracts additional randomness in the reverse direction. This shows why the one-way state potential is not a general two-way converse. Together, these results establish a common entanglement-based approach to sharper communication and randomness bounds, while revealing how retained quantum correlations enable two-way distillation to surpass one-way limits.
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.
We determine the optimal success probability for transmitting a fixed number of classical messages through a finite number of uses of the quantum erasure channel, assisted by noiseless classical feedback and without initial shared entanglement. For an input dimension d, an erasure probability p, a blocklength n, and M equiprobable messages, the optimal success probability is equal to E[min{1,dK/M}], where K is binomial with parameters n and 1−p. The converse allows arbitrary adaptive quantum encoders, quantum memories, and receiver instruments. Its main ingredient is an elementary dimension bound for noiseless quantum communication with classical feedback, proved by fixing the classical controls without conditioning the sender's state on the observed transcript. A classical protocol that transmits the base-d digits of an integer representing the message, repeating each digit until the receiver acknowledges its reception or the prescribed blocklength is reached, attains the bound for every integer M. We also establish a relative-majorization property of erasure-channel outputs and exactly evaluate a hypothesis-testing converse, recovering the same numerical bound without feedback. That converse need not be achievable without feedback: four uses of the qubit erasure channel and four messages give a strict gap. A product-state code employing tetrahedral qubit states nevertheless outperforms every classical binary erasure code with these parameters. We give an explicit message-size formula and show that the bounded-remainder normal approximation and the average-success strong-converse exponent are unchanged without feedback. We also determine the feedback-assisted error exponent below capacity, prove that it agrees with the no-feedback exponent above a critical rate, and give no-feedback bounds at lower rates.
We derive explicit, absolutely convergent series formulas for the quantum, unassisted classical, and entanglement-assisted classical capacities of the qubit amplitude damping channel, eliminating residual optimizations and implicit roots. We also obtain analytical upper and lower bounds on two-way assisted quantum and private communication capacities by evaluating a balanced-squashing bound and optimizing the reverse coherent information. Our results characterize the optimal input populations, establish convergence and truncation properties, and provide analytical benchmarks for quantum and classical communication over dissipative channels.
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.
This paper addresses the problem of generating a common random string with min-entropy k using an unlimited supply of quantum isotropic states, with minimal communication between Alice and Bob. Quantum isotropic states can be seen as maximally entangled states polluted by a depolarizing channel. The paper considers two communication models – one-way classical communication and one-way quantum communication, and derives upper bounds on the optimal common randomness rates for both models. We show that in the case of classical communication, quantum isotropic states have no advantage over noisy classical correlation. In the case of quantum communication, we demonstrate that the common randomness rate can be increased by using superdense coding on quantum isotropic states. We also prove an upper bound on the optimal common randomness rate achievable by using one-way quantum communication. As an application, our result yields an upper bound on the classical capacity of the noiseless quantum channel assisted by quantum isotropic states.
Abstract: We derive an improved upper bound on the two-way-assisted capacities of the qubit amplitude-damping channel with decay probability γ, given by log2[1+(1−γ)γγ/(1−γ)]. It applies to quantum communication, entanglement distribution, private communication, and secret-key agreement, and strictly improves the PLOB relative-entropy bound, the analytic max-relative-entropy bound, and the optimized balanced squashed-entanglement bound. At high damping, a known triple-rail protocol attains more than 99.54% of its leading coefficient. Our method extends sector teleportation simulation to qubits: phase rotations and permutations give exact simulations within fixed-excitation subspaces while preserving the two-level input structure. We determine the exact, additive relative entropy of entanglement of the amplitude-damping resources and the exact regularized entanglement gain at fixed input population. We then derive explicit bounds for generalized amplitude damping, which includes thermal excitation. Combining the amplitude-damping gain with unital and entanglement-breaking channels gives a closed expression that recovers the zero-temperature result and vanishes throughout the entanglement-breaking region. It improves established benchmarks over broad parameter ranges, while the exact entanglement of the generalized resources remains open. Both results admit refinements under an average-excitation constraint. The bounds provide quantitative limits for adaptive communication through relaxing qubits and show how a sector simulation can remain useful beyond channels whose resource entanglement is known exactly.