Cryptographic protocols often involve the assistance of public side channels to which all parties have perfectly noiseless access. For instance, in the BB84 quantum key distribution protocol, the side channel is used to share the bases in which Alice and Bob encoded or measured their qubits. In this paper, we find that in the case of continuous variable communication, by slightly altering this model such that Eve's copy of the initial round of feedback is corrupted by an iota of noise while keeping Alice's copies noiseless, the capacity can be increased dramatically. Specifically, it is known that the private capacity with noiseless feedback for a pure-loss bosonic channel is at most -log(1-η) bits per mode, where η is the transmissivity, in the limit of infinite input photon number. This is a very pessimistic result as there is a finite rate limit even with an arbitrarily large number of input photons. We refer to this as a loss limited rate. However, in our altered model we find that we can achieve a rate of (1/2) log(1+4ηNS) bits per mode with weak security, where NS is the input photon number. This rate diverges with NS, in sharp contrast to the result for the original model. This suggests that physical considerations behind the eavesdropping model should be taken more seriously, as they can create strong dependencies of the achievable rates on the model. For by a seemingly inconsequential weakening of Eve, we obtain a loss-unlimited rate. Our protocol also works verbatim for arbitrary i.i d, noise (not even necessarily Gaussian) injected by Eve in every round, and even if Eve is given access to copies of the initial transmission and noise. The error probability of the protocol decays super-exponentially with the blocklength.
Cryptographic protocols often involve the assistance of public side channels to which all parties have perfectly noiseless access. For instance, in the BB84 quantum key distribution protocol, the side channel is used to share the bases in which Alice and Bob encoded or measured their qubits. In this paper, we find that in the case of continuous variable communication, by slightly altering this model such that Eve's copy of the initial round of feedback is corrupted by an iota of noise while keeping Alice's copies noiseless, the capacity can be increased dramatically. Specifically, it is known that the private capacity with noiseless feedback for a pure-loss bosonic channel is at most -log(1-η) bits per mode, where η is the transmissivity, in the limit of infinite input photon number. This is a very pessimistic result as there is a finite rate limit even with an arbitrarily large number of input photons. We refer to this as a loss limited rate. However, in our altered model we find that we can achieve a rate of (1/2) log(1+4ηNS) bits per mode with weak security, where NS is the input photon number. This rate diverges with NS, in sharp contrast to the result for the original model. This suggests that physical considerations behind the eavesdropping model should be taken more seriously, as they can create strong dependencies of the achievable rates on the model. For by a seemingly inconsequential weakening of Eve, we obtain a loss-unlimited rate. Our protocol also works verbatim for arbitrary i.i d, noise (not even necessarily Gaussian) injected by Eve in every round, and even if Eve is given access to copies of the initial transmission and noise. The error probability of the protocol decays super-exponentially with the blocklength.
Sending quantum information over long distances is ultimately limited by photon loss. The rate limit of a lossy bosonic channel of transmissivity η is the two-way assisted capacity -log2(1-η) (the PLOB bound), whose achievability rests on unlimited classical feedback: without feedback the quantum capacity vanishes for all η ≤ 1/2, so classical back-communication acts as an on/off switch for quantum communication. Classical bits are priced for entanglement transformations and, for a fixed bipartite state, the one-way distillation trade-off between classical rate and ebit yield has a regularized entropic characterization; but no framework prices the feedback that assists a channel's quantum capacity — feedback that may steer subsequent channel inputs — and no such trade-off has been evaluated for any bosonic channel. Here we show that quantum communication can be metered in classical bits. Writing C(η,B) for the capacity of the pure-loss channel under a feedback budget of B classical bits per channel use, we construct the first achievable-rate frontier C_{ach}(η,B) ≤ C(η,B) and prove the onset law C_{ach}(η,B) ≥ ηlog2(m_B)/m_B, m_B = ⌈ H_2(η)/B⌉, valid for every B>0 with envelope (η/H_2(η)) Blog2(H_2(η)/B) as B → 0 — an exact identity R = η Blog2(1/B) under fixed-length accounting, so the Blog(1/B) order is accounting-convention invariant. The B=0 switch thereby resolves, at the level of achievable rates, into a continuous but non-analytic onset: even for η ≤ 1/2 any positive budget buys a positive quantum rate, each bit unlocking ~log2(1/B) ebits at small budgets and refuting the naive pricing C ≤ Q + B. At large budgets a metered quantum-state-merging family, consuming backward one-way classical communication only, achieves C_{ach}(η,B) ≥ PLOB - [eη^2/(2ln 2)] 2^{-B}(1+o(1)): each backward bit per use halves the achievable distance to PLOB, which exposes the 73.6% plateau of the purification families as a family artifact. Standard entanglement-measure converses are provably blind to B, so what remains open is only strict saturation at finite B, now confined to an exponentially small window. The frontier is also a design tool: on a network of lossy links, the joint allocation of per-link feedback budgets and photon directions under per-site classical budgets is a convex program, and on a 20-site fibre network its optimum doubles the end-to-end entanglement rate relative to a fixed orientation. An appendix extends the frontier to multimode loss, where Grassmann-quantized beamforming controls the PLOB loss at rate 2^{-B/(M-1)}. Preprint (companion paper: Conservation laws for the classical communication cost of one-way entanglement distillation, with entropic converse bounds for the pure-loss channel). Ancillary files: reproduction scripts and data; code maintained at https://github.com/deeptell-inc/fmplob.
This paper strengthens the interpretation and understanding of the classical capacity of the pure-loss bosonic channel, first established in [1]. In particular, we first prove that there exists a trade-off between communication rate and error probability if one imposes only a mean photon number constraint on the channel inputs. That is, if we demand that the mean number of photons at the channel input cannot be any larger than some positive number NS , then it is possible to respect this constraint with a code that operates at a rate g ( ηNS /(1- p )) where p is the code error probability, η is the channel transmissivity, and g ( x ) is the entropy of a bosonic thermal state with mean photon number x . Then we prove that a strong converse theorem holds for the classical capacity of this channel (that such a rate-error trade-off cannot occur) if one instead demands for a maximum photon number constraint, in such a way that mostly all of the “shadow” of the average density operator for a given code is required to be on a subspace with photon number no larger than nNS , so that the shadow outside this subspace vanishes as the number n of channel uses becomes large. Finally, we prove that a small modification of the well-known coherent-state coding scheme meets this more demanding constraint.
Security analyses of quantum cryptographic protocols typically rely on certain conditions; one such condition is that the sender (Alice) and receiver (Bob) have isolated devices inaccessible to third parties. If an eavesdropper (Eve) has a side channel in one of the devices, then the key rate may be sensibly reduced. In this paper we consider an attack on a coherent-state protocol, where Eve not only taps the main communication channel but also hacks Alice's device. This is done by introducing a Trojan horse mode with low mean number of photons n, which is then modulated in a similar way to the signal state. First we show that this strategy can be reduced to an attack without side channels but with higher loss and noise in the main channel. Then we show how the key rate rapidly deteriorates for increasing photons n, being halved at long distances each time n+1 doubles. Our work suggests that Alice's device should also be equipped with sensing systems that are able to detect and estimate the total number of incoming and outgoing photons.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
Quantum cryptography provides absolute security against an all-powerful eavesdropper (Eve). However, in practice Eve's resources may be restricted to a limited aperture size so that she cannot collect all paraxial light without alerting the communicating parties (Alice and Bob). In this paper we study a quantum wiretap channel in which the connection from Alice to Eve is lossy, so that some of the transmitted quantum information is inaccessible to both Bob and Eve. For a pureloss channel under such restricted eavesdropping, we show that the key rates achievable with a two-mode squeezed vacuum state, heterodyne detection, and public classical communication assistance-given by the Hashing inequality-can exceed the secret key distillation capacity of the channel against an omnipotent eavesdropper. We report upper bounds on the key rates under the restricted eavesdropping model based on the relative entropy of entanglement, which closely match the achievable rates. For the pure-loss channel under restricted eavesdropping, we compare the secret-key rates of continuous-variable (CV) quantum key distribution (QKD) based on Gaussian-modulated coherent states and heterodyne detection with the discrete variable (DV) decoystate BB84 QKD protocol based on polarization qubits encoded in weak coherent laser pulses.
We determine the finite-blocklength asymptotics of classical communication over the pure-loss bosonic channel under a maximum-photon-number occupation constraint. Wilde, Renes and Guha previously derived a second-order achievable rate governed by thermal photon-number fluctuations and left its optimality open. We show that fixed-total-photon-number encoding removes input-number fluctuations while preserving the first-order capacity, leaving binomial photon-survival fluctuations as the operational second-order contribution. For fixed transmissivity 0 < η < 1, signal photon number NS > 0, and error probability 0 < ε < 1, we determine the optimal fixed-error expansion through third order. The operational dispersion is Vsh = η(1 − η)NS [log2(1 + 1/(ηNS))]^2, and the optimal code size contains the third-order correction −(1/2)log2 n up to a bounded O(1) remainder. The achievability construction uses fixed-photon-number shell codes, while the converse applies to arbitrary multimode quantum encodings and arbitrary joint decoding measurements within the stated occupation constraint. This Zenodo record contains the preprint, LaTeX source, vector figures, source data, and reproducibility code used for the numerical checks and figure generation.