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Fundamental Limits of Repeaterless Quantum Communications

Stefano Pirandola, Riccardo Laurenza, Carlo Ottaviani, Leonardo Banchi

arXiv:1510.08863v8quant-ph

TL;DR

The paper asks for ultimate point-to-point rates under unrestricted two-way LOCC, including the limits relevant to repeaterless QKD. It combines an REE upper bound with dimension-independent teleportation stretching, establishing exact capacities for several fundamental channels and the rate-loss trade-off for QKD.

Abstract

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Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. By constructing an upperbound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed "teleportation stretching", we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we determine the fundamental rate-loss trade-off affecting any protocol of quantum key distribution. Our findings set the ultimate limits of point-to-point quantum communications and provide the most precise and general benchmarks for quantum repeaters.

Results

The section develops dimension-independent two-way capacity bounds using relative entropy of entanglement and teleportation stretching. It applies these methods across broad bosonic Gaussian and finite-dimensional channel classes, establishing single-letter bounds and identifying cases where capacities equal entanglement flux.

REE bounds

  • Theorem 1 upper-bounds every generic two-way capacity C(E), including D2, Q2, K, and P2, by the relative entropy of entanglement at finite or infinite dimension.The capacities satisfy D2 = Q2 ≤ K, while unlimited two-way classical communication and one-time-pad arguments give P2(E) = K(E).Source: Results · pp. 2–3

Channel simulation

  • The framework extends channel simulation to arbitrary trace-preserving LOCCs and resource states in finite or infinite dimensions, including continuous-variable systems.For bosonic channels, simulations may be asymptotic limits of finite-energy channels and resource states.Source: Results · p. 3

Teleportation covariance

  • Teleportation-covariant channels are Choi-stretchable at any dimension, with bosonic simulations understood asymptotically; the class includes Gaussian, Pauli, and erasure channels.Teleportation covariance allows input teleportation unitaries to be pushed through the channel and corrected at the output.Source: Results · p. 4

Teleportation stretching

  • Teleportation stretching transforms any adaptive protocol using a stretchable channel into a block protocol whose output is a trace-preserving LOCC applied to n copies of the resource state.For Choi-stretchable channels, the resource is the channel’s Choi matrix, with asymptotic limits required for bosonic channels.Source: Results · p. 6

REE converse

  • The adaptive-to-block reduction preserves both the task and output, so quantum communication, entanglement distribution, and key generation remain the corresponding tasks after stretching.Applying REE monotonicity to the stretched output discards the final LOCC and produces a simple single-letter converse bound.Source: Results · p. 6

General bounds

  • For Choi-stretchable distillable channels, all two-way capacities equal the entanglement flux Φ(E) := E_R(ρ_E), including lossy, quantum-limited amplifier, dephasing, and erasure channels.The entanglement flux measures entanglement distributed by a single EPR state; the one-shot REE bound underpins these capacity results.Source: Results · p. 6

Immediate generalizations

  • For forward and backward distillable channels, the two-way capacity equals the larger individual capacity, while distillable multiband channels have capacity equal to the sum across bands.These results extend stretching to bidirectional and independent multiband communication.Source: Results · pp. 6–7

Bosonic communications

  • For a lossy bosonic channel with transmissivity η, Φ(η) ≤ −log2(1 −η), equal to reverse coherent information, and the channel is distillable.This establishes the achievable fundamental rate-loss scaling for repeaterless optical quantum communication and secret-key distribution.Source: Results · p. 7

Qubit communications

  • For finite-dimensional channels, capacities obey the dimensionality bound and a simplified REE-to-distillable-entanglement chain for Choi-stretchable channels.The dimensionality bound is C(E) ≤ min{log2 dA, log2 dB}.Source: Results · p. 8

Bosonic communications

  • For thermal-loss, amplifier, and additive-noise Gaussian channels, the method produces two-way capacity bounds whose lower and upper bounds converge in low-noise or ideal limits.The thermal-loss bounds coincide at sufficiently high transmissivity, while additive-noise bounds rapidly coincide for small added noise; ideal amplifiers are distillable.Source: Results · p. 8, Figure 7 · p. 9

Qubit communications

  • Pauli, depolarising, dephasing, and generalized dephasing channels receive finite-dimensional capacity formulas, with dephasing channels proven distillable and Q2(Pdeph) = Q(Pdeph).The depolarising result extends to arbitrary dimension, with nonzero capacity only below its stated noise threshold.Source: Results · p. 9
  • Erasure channels are distillable: measuring outputs identifies unerased EPR pairs, yielding at least 1−p ebits per copy and matching the entanglement-flux upper bound.The result generalizes to arbitrary dimension, where the erasure state occupies an additional dimension.Source: Results · p. 10
  • For the amplitude damping channel, the bound in Eq. (47) is simple but only good for strong damping (p > 0.9), whereas a tighter bound uses squashed entanglement.Source: Results · p. 10, Figure 8 · p. 10

Discussion

The work establishes ultimate point-to-point rates across dimensions and derives exact capacity formulas for several fundamental channels. These results provide repeater benchmarks and characterize the ultimate rate-loss scaling for repeaterless QKD.

Discussion

  • The resulting limits are fundamental benchmarks that only quantum repeaters may surpass.The benchmarks apply to point-to-point quantum communication, entanglement distribution, and secret-key generation at any dimension.Source: Discussion · p. 10
  • Exact formulas for Q2, D2, and K establish fundamental capacities for bosonic lossy, quantum-limited amplifier, dephasing, and erasure channels.The formulas follow from bounding two-way capacities with single-letter quantities after teleportation stretching.Source: Discussion · p. 11
  • −log2(1 −η) bits per channel use, approximately 1.44η at high loss, is the ultimate achievable rate for repeaterless QKD.The bound remains tight down to approximately one mean photon and can be reached using one-way classical communication costing about 4.68 bits per use.Source: Discussion · p. 11
  • Teleportation stretching has been extended beyond this work to repeater chains, multihop networks, quantum parameter estimation, and quantum channel discrimination.Network capacities can be characterized through quantum widest-path and max-flow min-cut formulations, depending on routing.Source: Discussion · p. 11

Methods

The methods develop Gaussian-state and relative-entropy tools alongside teleportation constructions to characterize teleportation-covariant channels and reduce adaptive bosonic-channel protocols to LOCCs acting on resource states. They establish asymptotic stretching and REE-based capacity bounds across finite- and infinite-dimensional, energy-constrained, two-way, and multiband settings.

Relative entropy between Gaussian states

  • The relative-entropy formula is extended to states with nonzero mean values by introducing the mean difference δ = u1 − u2 and replacing x̂ with x̂ − u2.When the Gibbs matrix G becomes singular for pure states or mixed states containing vacuum contributions, the Gibbs-exponential form requires a suitable limiting procedure.Source: Methods · p. 12

Basics of quantum teleportation

  • Teleportation is formulated with Bell measurements, classical feed-forward, and correction unitaries that recover the input state and its ancillary correlations.For continuous variables, finite-energy TMSV resources and quasi-projections are used first, followed by the limit μ → +∞ to recover ideal teleportation.Source: Methods · pp. 12–13

Teleportation unitaries

  • Teleportation unitaries form an abelian-group-indexed Weyl–Heisenberg structure: generalized Paulis for finite dimensions and phase-space displacements for continuous variables.Their composition closes up to a phase, enabling a common group representation across finite- and infinite-dimensional systems.Source: Methods · p. 13

Teleportation-covariant channels

  • Teleportation covariance is expressed through generally different unitary representations of the same abelian group, including Weyl-covariant bosonic Gaussian channels and dimension-altering erasure channels.Finite-dimensional Weyl-covariant channels are Pauli channels, whereas amplitude damping is not teleportation-covariant.Source: Methods · pp. 13–14

LOCC-averaging in teleportation stretching

  • LOCC-averaging replaces adaptive measurement branches by a trace-preserving averaged LOCC, preserving the mean output state and averaging conditional target rates.For a fixed channel simulation, conditional simulation probabilities factor across transmissions, allowing the adaptive protocol to be reorganized into a single LOCC acting on repeated resource states.Source: Methods · pp. 14–15

Teleportation stretching with bosonic channels

  • Bosonic-channel stretching is necessarily asymptotic because the Choi matrix and ideal Bell detection are energy-unbounded.The procedure applies finite-energy simulations and resources at parameter μ, then takes the μ-limit of the output state and any associated operation or functional.Source: Methods · p. 15

Asymptotic teleportation stretching

  • Teleportation stretching converts adaptive bosonic-channel outputs into LOCC maps acting on tensor powers of simulation resource states, with convergence controlled by an energy-constrained diamond norm.The argument assumes finite register-energy bounds N(n), including energy-decreasing and energy-increasing channels, and takes the simulation limit in trace norm.Source: Methods · pp. 15–16

REE bounds

  • The one-shot REE argument gives a two-way capacity upper bound for bosonic channels, applying to both energy-constrained protocols and limits of energy-unconstrained protocols.For Choi-stretchable channels, the channel’s REE equals the REE of its asymptotic Choi resource state.Source: Methods · p. 16

One-shot REE bound for DV channels

  • For finite-dimensional Choi-stretchable channels, teleportation stretching establishes K(E) = K(ρE), while the regularized REE of ρE upper-bounds this distillable key.This equality cannot be directly extended to bosonic channels because their Choi matrix is energy-unbounded and its distillable key is not well-defined.Source: Methods · p. 17

Two-way quantum communication

  • For adaptive communication using forward and backward channels, the capacity lies between max{C(E), C(E′)} and max{ER(σ), ER(σ′)}, reducing to the corresponding resource bounds for Choi-stretchable channels.When both channels are distillable, the optimal strategy uses the channel with maximum capacity.Source: Methods · pp. 17–18

Multiband quantum channel

  • For multiband channels, the two-way capacity is bounded below by the sum of individual band capacities and, for distillable bands, becomes additive.Teleportation stretching decomposes the multiband output into resource states for the individual bands, enabling the upper-bound construction.Source: Methods · p. 18

Truncation of infinite-dimensional Hilbert spaces

The truncation framework connects energy-constrained bosonic states to finite-dimensional approximations. It uses energy projectors and truncation channels to obtain finite-support representations suitable for continuous-variable analysis.

Truncation of infinite-dimensional Hilbert spaces

  • Energy-constrained multimode bosonic states are essentially supported on finite-dimensional Hilbert spaces.The construction orders total-photon-number eigenvalues and projects onto a finite-dimensional low-energy subspace.Source: Truncation of infinite-dimensional… · p. 21
  • An energy-constrained state can be projected onto a d-dimensional support with a controlled projection probability and trace-distance approximation.The finite-dimensional projector is built from the ordered eigenstates of the total photon-number operator.Source: Truncation of infinite-dimensional… · p. 21
  • The truncation framework also admits an equivalent formulation directly through a completely positive trace-preserving truncation map.This provides a channel-based version of the finite-dimensional reduction.Source: Truncation of infinite-dimensional… · p. 22
  • A truncation channel maps an energy-constrained bosonic state to a d-dimensional state using identity and collapsing maps on the projected and discarded components.The map uses Π0 := Pd and Π1 := I −Pd, with the discarded component sent to a fixed state in the truncated support.Source: Truncation of infinite-dimensional… · p. 22
  • The finite-dimensional approximation can be controlled by trace-norm bounds derived from the projection probability and convexity of the trace norm.The resulting estimates support connecting continuous-variable and discrete-variable states.Source: Truncation of infinite-dimensional… · pp. 21–22

Local CV-DV mappings

The appendix develops local CV-to-DV truncation tools and connects coherent-information bounds to achievable entanglement rates, including in infinite-dimensional bosonic systems.

Local CV-DV mappings

  • An LOCC truncation channel maps an energy-constrained bipartite CV state to a d × d-dimensional support using local projections and two bits of classical communication.If either party projects outside the local support, both apply a damping channel to a fixed in-support state.Source: Local CV-DV mappings · pp. 22–23

Coherent and reverse coherent information of a quantum channel

  • For unital channels, coherent and reverse coherent information coincide because the relevant reduced states are maximally mixed.These quantities are used to compute achievable one-way entanglement-distillation rates, including for discrete-variable channels.Source: Coherent and reverse coherent… · p. 24

Hashing inequality in infinite dimension

  • Continuity under finite-dimensional truncation extends the hashing inequality to energy-bounded bosonic states, making coherent and reverse coherent information achievable distillation rates.The extension applies in the infinite-dimensional limit for any fixed mean-photon bound.Source: Hashing inequality in infinite dimension · p. 25

Extension to energy-unbounded Choi matrices of bosonic Gaussian channels

The appendix treats energy-unbounded bosonic Choi matrices and supplies arbitrary-dimensional weak-converse arguments, including bounds that control private-state shield growth.

Extension to energy-unbounded Choi matrices of bosonic Gaussian channels

  • Bosonic Gaussian-channel Choi matrices are defined as limits of channel outputs on two-mode squeezed vacuum states, with their information functionals continuously extended to the asymptotic operator.The relevant functionals are finite, continuous, monotonic, and bounded in the squeezing parameter.Source: Extension to energy-unbounded Choi… · p. 25

UPPER BOUND AT ANY DIMENSION

First proof of the weak converse theorem

  • The first proof uses LOCC truncation to reduce continuous-variable outputs to finite-dimensional states while preserving the private state and applying REE continuity.The resulting derivation accounts for the key and shield dimensions in the weak-converse bound.Source: First proof of the weak converse theorem · p. 26

Private states and size of the shield system

  • The private-state shield need only grow exponentially with the number of channel uses, not super-exponentially, for finite- or infinite-dimensional memoryless channels.Repeating finite blocks, truncating and measuring their key systems, and applying one-way key distillation yields the exponential bound.Source: Private states and size of the shield… · pp. 27–29

Second proof of the weak converse theorem

The appendix presents energy-based and shield-independent proofs of the weak converse, extending the REE argument across discrete- and continuous-variable settings.

Second proof of the weak converse theorem

  • The second proof applies to discrete- and continuous-variable systems under the assumption that output energy grows at most exponentially with channel uses.Local energy truncation produces finite-dimensional states, after which REE continuity yields the asymptotic bound.Source: Second proof of the weak converse… · pp. 29–30

Third proof of the weak converse theorem

  • The third proof removes assumptions on private-state shield dimension by mapping the key part through a relative-entropy distance that remains asymptotically continuous.The construction extends from discrete-variable channels to bosonic channels through truncation, while preserving the key-system dimension.Source: Third proof of the weak converse theorem · pp. 30–31

TECHNICAL DERIVATIONS FOR BOSONIC GAUSSIAN CHANNELS

This section summarizes Gaussian-state tools, canonical Gaussian channels, and the covariance-based characterization used for bosonic channel analysis.

Basic tools for continuous variables

  • Gaussian-state entropies and correlations are characterized through covariance matrices, symplectic spectra, thermal photon numbers, and separability boundaries.The asymptotic two-mode squeezed-vacuum state realizes the ideal continuous-variable EPR correlations.Source: Basic tools for continuous variables · pp. 32–33
  • Gaussian channels transform Gaussian states into Gaussian states and are teleportation-covariant, allowing their Choi-stretchable representation through asymptotic Choi matrices.Input displacements are mapped to output displacements according to the channel's linear transformation.Source: Basic tools for continuous variables · p. 33
  • The canonical single-mode families include thermal-loss, pure-loss, amplifier, quantum-limited amplifier, additive-noise, conjugate-amplifier, and pathological Gaussian forms.Pure-loss and quantum-limited amplifier channels are the zero-thermal-photon cases of the loss and amplifier families.Source: Basic tools for continuous variables · p. 33

Coherent and reverse coherent information of a Gaussian channel

The paper computes coherent and reverse coherent information for important Gaussian channels and uses these quantities as achievable rates and lower bounds. It then evaluates Gaussian entanglement fluxes, obtaining exact capacity results for pure-loss and quantum-limited amplifier channels.

Coherent and reverse coherent information of a Gaussian channel

  • For Gaussian channels, coherent and reverse coherent information are obtained as finite asymptotic limits of quantities computed on increasingly energetic two-mode squeezed-vacuum inputs.The relevant quantities are continuous and increasing with input energy for fixed channel parameters.Source: Coherent and reverse coherent… · pp. 33–34
  • Reverse coherent information provides an achievable lower bound for Gaussian-channel key generation through a coherent protocol with heterodyne detection and backward classical communication.The protocol’s Devetak–Winter rate equals the reverse coherent information under the stated purification and rank-1 measurement conditions.Source: Coherent and reverse coherent… · p. 34

How to compute the entanglement flux of a Gaussian channel

  • The Gaussian entanglement-flux calculation uses relative entropy between a quasi-Choi state and a suitably correlated separable Gaussian state, with regular limits evaluated from statistical moments.For entanglement-breaking channels, the flux is zero.Source: How to compute the entanglement flux… · p. 35

Entanglement flux of a thermal-loss channel

  • The thermal entanglement-flux bound is tighter than earlier squashed-entanglement bounds for thermal-loss channels.The comparison is made against the Takeoka-Guha-Wilde thermal bound [S27].Source: Entanglement flux of a thermal-loss… · p. 36
  • For thermal-loss channels, the entanglement flux generally differs from the reverse-coherent-information lower bound, but the two bounds coincide for a pure-loss channel.The resulting pure-loss capacity is C_loss(η) = −log2(1 −η).Source: Entanglement flux of a thermal-loss… · p. 36

Relation with quantum discord

The paper connects repeaterless capacities with entanglement flux, quantum discord, and Choi-state calculations across Gaussian and discrete-variable channels. It establishes exact formulas for several channels and illustrates how composition without a repeater can sharply reduce communication rates.

Relation with quantum discord

  • For a bosonic lossy channel, the repeaterless limit is C_loss(η) = −log2(1 −η), giving high-loss scaling of 1.44η bits per channel use.This capacity also equals the channel’s reverse coherent information and the discord of its asymptotic Gaussian Choi matrix.Source: Relation with quantum discord · p. 36

Entanglement flux of a quantum amplifier

  • For a quantum-limited amplifier, the two-way capacity equals the unassisted quantum capacity, and infinite amplification yields zero communication rate.At gain 2, the maximum rate is 1 qubit per use.Source: Entanglement flux of a quantum amplifier · p. 37

Entanglement flux of an additive-noise Gaussian channel

  • Composing a gain-2 quantum-limited amplifier with a transmissivity-1/2 lossy channel produces an additive-noise channel limited to Q2 ≲0.278 qubits per use, despite 1-qubit component rates.Using the intermediate party as a quantum repeater avoids this composition penalty and achieves at least 1 qubit per use.Source: Entanglement flux of an additive-noise… · p. 38

TECHNICAL DERIVATIONS FOR DISCRETE-VARIABLE CHANNELS

  • For a d-dimensional unital channel with Choi eigenvalue distribution p, coherent and reverse coherent information equal log2 d −H(p).This follows from the Choi-state entropy calculation.Source: TECHNICAL DERIVATIONS FOR… · p. 39

Erasure channel in arbitrary finite dimension

  • For a d-dimensional erasure channel, all two-way capacities and entanglement flux equal (1 −p) log2 d.The result includes C, D2, Q2, K, and Φ, while the secret-key capacity is identified as the novel result.Source: Erasure channel in arbitrary finite… · p. 40

Qubit Pauli channels

The paper exploits teleportation covariance and Bell-diagonal Choi states to bound two-way capacities of Pauli channels. It obtains exact results for qubit dephasing and bounds for depolarising and higher-dimensional Pauli channels.

Qubit Pauli channels

  • Qubit Pauli channels are teleportation-covariant and therefore Choi-stretchable, with Bell-diagonal Choi matrices whose eigenvalues are the channel probabilities.Their coherent information is 1 −H(p), while REE supplies a tighter upper bound based on p_max.Source: Qubit Pauli channels · pp. 40–41

Qubit depolarising channel

  • For qubit depolarising channels, the two-way capacity is bounded by the Pauli-channel expression and becomes zero for p > 2/3.The stated bounds specialize the general Pauli-channel result.Source: Qubit depolarising channel · p. 41

Qubit dephasing channel

  • For qubit dephasing, all listed capacities and entanglement flux equal 1 −H2(p), matching the unassisted quantum capacity.The equality includes C, D2, Q2, K, and Φ.Source: Qubit dephasing channel · p. 42

Pauli channels in arbitrary finite dimension

  • Pauli channels remain teleportation-covariant in arbitrary finite dimension, and their Bell-diagonal Choi matrices yield bounds involving log2 d −H({pab}).The arbitrary-dimensional channels are also called Weyl channels.Source: Pauli channels in arbitrary finite… · p. 42

Depolarising channel in arbitrary finite dimension

The paper derives arbitrary-dimensional depolarising and dephasing results, then analyzes amplitude damping with lower and upper bounds based on coherent information, teleportation stretching, and squashed entanglement. Exact capacity formulas occur for dephasing, while amplitude damping remains bounded by a sandwich.

Depolarising channel in arbitrary finite dimension

  • For arbitrary-dimensional depolarising channels, the coherent information is κ(d,p) −f log2(d + 1), while the REE flux supplies the corresponding upper-bound framework.The capacity is zero beyond the stated threshold p > d/(d + 1).Source: Depolarising channel in arbitrary… · p. 43

Dephasing channel in arbitrary finite dimension

  • For arbitrary-dimensional dephasing channels, lower and upper bounds coincide, giving C = D2 = Q2 = K = Φ = log2 d −H(P).The channel is therefore distillable within the paper’s analysis.Source: Dephasing channel in arbitrary finite… · p. 44

Amplitude damping channel

  • Amplitude damping is not teleportation-covariant, but it decomposes into local qubit-to-single-rail mappings and a bosonic loss channel, enabling asymptotic teleportation stretching.The middle lossy channel has transmissivity η(p) = 1 −p.Source: Amplitude damping channel · p. 44

Amplitude damping channel: Upper bound based on the squashed entanglement

MAXIMUM RATES ACHIEVABLE BY CURRENT QKD PROTOCOLS

This section benchmarks ideal asymptotic continuous-variable QKD protocols under loss, comparing their maximum secret-key rates and high-loss behavior. It also notes a trade-off: two-way protocols have lower lossy-channel rates but greater robustness to excess noise.

MAXIMUM RATES ACHIEVABLE BY CURRENT QKD PROTOCOLS

  • Ideal asymptotic CV protocols are compared under unit efficiencies, zero excess noise, large modulation, and unit reconciliation efficiency.Finite-size effects are neglected, and the analysis covers current practical continuous-variable protocols.Source: MAXIMUM RATES ACHIEVABLE BY CURRENT… · p. 48

Continuous-variable protocols

  • The no-switching protocol uses coherent states with heterodyne detection and is identified as the practical CV protocol with the highest secret-key rate.Its reverse-reconciliation rate is analyzed over a lossy channel with transmissivity η.Source: Continuous-variable protocols · p. 48
  • For high loss (η ≃ 0), the protocol scales as ≃η/(2 ln 2), which is 1/2 of the secret-key capacity.Source: Continuous-variable protocols · p. 48
  • CV-MDI-QKD uses coherent states sent to an untrusted relay, with total transmissivity η = ηAηB and rates depending on relay symmetry.When the relay coincides with Alice, ηA = 1, the protocol recovers the one-way rate.Source: Continuous-variable protocols · p. 48
  • Two-way CV protocols have lower key rates than one-way protocols in lossy channels but higher security thresholds under excess noise.The two variants differ in Bob’s heterodyne versus homodyne detection, with the latter switching between quadratures.Source: Continuous-variable protocols · p. 48

Discrete-variable protocols

This section covers discrete-variable QKD, energy-constrained repeaterless bounds, classical-communication costs, and the paper’s general channel-simulation framework. Together, these passages compare ideal DV rates with fundamental limits and explain how channel simulation supports non-adaptive capacity-achieving protocols across dimensions.

Discrete-variable protocols

  • Under ideal conditions, BB84 with perfect single-photon sources achieves the maximum current DV rate RBB84-1ph = η/2.The assumptions include unit detector efficiency, zero dark counts, perfect visibility, and perfect classical error correction.Source: Discrete-variable protocols · p. 49
  • With weak coherent pulses and decoy states, optimizing the intensity gives µ = 1 and RBB84-decoy = η/(2e).The ideal intensity-dependent rate is maximized over µ.Source: Discrete-variable protocols · p. 49
  • For DV-MDI-QKD, the maximum occurs at µA = µB = 1, yielding RDV-MDI = η/(2e^2).The result depends on the two channel transmissivities through η = ηAηB.Source: Discrete-variable protocols · p. 49

Input energy constraints

  • For bosonic Gaussian channels, energy-constrained two-way capacities increase with the photon-number cap but remain bounded by the channel’s entanglement flux; Q2, D2, and K have finite unconstrained limits.The teleportation-stretching procedure continues to apply under an input energy constraint.Source: Input energy constraints · pp. 49–50
  • The unconstrained lossy-channel bound Φ(η) is rapidly approached from below by reverse coherent information with only approximately 1–5 input photons.For input energy constrained to a few photons, the bound remains tight.Source: Input energy constraints · p. 50
  • Φ(η) is tighter than squashed-entanglement bounds even at one mean photon, especially beyond 25 km; at shorter distances, highly modulated CV-QKD can approach infinite-energy behavior.At intermediate-to-long distances, ideal DV-QKD operates at one mean photon per channel use, while highly modulated CV-QKD is not limited by input energy.Source: Input energy constraints · pp. 50–51, Figure 2 · p. 50, Figure 3 · p. 51

Cost of classical communication

  • An adaptive protocol may have zero target-bits-per-second throughput if it requires infinitely many classical feed-forward and feedback rounds, a problem addressed by the paper’s bound.For distillable channels, C(E) = D1(ρE), so an optimal protocol is non-adaptive and requires only one forward or backward classical-communication round.Source: Cost of classical communication · p. 51
  • For a bosonic lossy channel, backward classical communication achieves K(η) = −log2(1 −η), with asymptotic cost γCC(η) and no more than log2(3πe) bits per channel use.A large block with n = 10^9 uses requires at most 4.68 × 10^9 classical bits, under the stated practical assumptions.Source: Cost of classical communication · pp. 51–52

ADVANCES IN PROTOCOL REDUCTION

This section develops teleportation stretching as a task-preserving reduction from adaptive protocols to block protocols, combining it with relative entropy of entanglement to bound two-way capacities across arbitrary channels and dimensions. It also addresses bosonic-channel finite-energy approximations, capacity extensions, and limitations for thermal-loss channels.

ADVANCES IN PROTOCOL REDUCTION

  • Teleportation stretching reduces any adaptive protocol to ρ_b(E⊗n) = Λ̄(σ⊗n) through a trace-preserving LOCC, preserving the original task for any channel and finite or infinite dimension.Unlike the restricted BDSW argument, the method is finite-size, task-preserving, and applies to quantum communication, key generation, parameter estimation, channel discrimination, and more complex networks.Source: ADVANCES IN PROTOCOL REDUCTION · pp. 54–55

Schematic summary of our key findings

  • Combining channel REE with LOCC simulation and teleportation stretching yields the single-letter bound C(E) ≤ E_R(σ) for any σ-stretchable channel at any dimension.The derivation uses REE monotonicity under trace-preserving LOCCs and subadditivity over tensor products; for Choi-stretchable channels, σ is the Choi matrix.Source: Schematic summary of our key findings · p. 56
  • The entanglement flux is tight for many channels, establishing Q2, D2, and K for bosonic lossy, quantum-limited amplifier, and dephasing channels, plus K for finite-dimensional erasure channels.The section also states that the single-letter REE bound is tight for dephasing and erasure channels.Source: ADVANCES IN BOUNDING TWO-WAY CAPACITIES · p. 55, Schematic summary of our key findings · p. 56
  • 1.44η bits per channel use is the long-distance repeaterless-QKD rate-loss limit, achievable with one-way classical communication and equal to the maximum point-to-point throughput.This characterizes the fundamental rate-loss tradeoff for quantum optical communications.Source: Schematic summary of our key findings · p. 56

Repeater-assisted capacities and multi-hop networks

  • Ref. [S69] extends REE-based capacity bounds to repeater-assisted quantum networks, where entanglement cuts give converses and widest-path or maximum-flow algorithms establish achievable routing rates.The framework covers arbitrary network topologies, multiple paths, lossy channels, quantum-limited amplifiers, dephasing, and erasure channels.Source: Repeater-assisted capacities and… · p. 57

Single-hop networks (broadcast, multiple-access and interference channels)

  • REE+teleportation stretching also yields upper bounds for arbitrary-dimension broadcast, multiple-access, and interference channels, including the lossy broadcast capacity region in ref. [S75].Ref. [S74] studies unlimited-two-way-communication multipoint networks, while ref. [S75] applies the method to the bosonic lossy broadcast channel.Source: Single-hop networks (broadcast,… · p. 57

Improving the lower bound for the thermal-loss channel

  • The two-way capacities of the thermal-loss channel remain open: existing lower and upper bounds do not close the gap, despite an improved trusted-noise-detection lower bound from ref. [S76].The improvement applies in a region where reverse coherent information is beaten but remains far from the upper bound.Source: Improving the lower bound for the… · p. 57

Strong converse rates for private communication

  • Ref. [S79] uses REE+teleportation stretching to derive strong converse rates for adaptive private communication over dephasing, erasure, and Gaussian channels, while the present method is more general than its finite-dimensional teleportation-simulable framework.The section distinguishes this from BDSW and ref., emphasizing the first demonstrated finite-n adaptive key-generation reduction for continuous-variable channels.Source: Strong converse rates for private… · p. 58

Simulation and stretching of bosonic channels

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