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Efficient capacity-achieving entanglement generation with application to pure-loss Bosonic channels
Avijit Mandal, Christophe Piveteau, Henry D. Pfister, Joseph M. Renes
TL;DR
Efficient decoding schemes that achieve noisy-channel communication limits remain scarce. This paper constructs efficient entanglement-generation schemes for pure discrete phase covariant channels using polar codes and specialized decoders. The schemes achieve coherent information, and degradable channels in the class—including truncated pure-loss Bosonic channels—reach quantum capacity.
Problem
Efficient coding schemes that achieve coherent information or quantum communication limits remain scarce, particularly because constructing a suitable decoder is difficult.
Source: §1 · p. 1Method
The paper uses polar codes for entanglement generation, with classical polar decoding and BPQM implementing the two decoding POVMs.
Source: §1 · p. 2Results
The construction achieves coherent information for pure discrete phase covariant channels, and achieves quantum capacity for degradable channels in the class.
Source: §1 · pp. 1–2Takeaways & Limitations
Truncated pure-loss Bosonic channels are within the class, and their achievable entanglement-generation rate can be made close to the channel's coherent information.
Source: §1 · p. 1, §1.1 · p. 3The finite-register approximation analysis is given for binary BPQM; extension to d-ary eigen lists is left to future work.
Source: §5.4 · p. 18
Abstract
from arXiv · showhide
While the ultimate limits on the rate of quantum communication over noisy channels are well-established, very few efficient coding schemes achieving these limits are known. This contrasts sharply with transmitting classical data over classical channels, where polar codes and low-density parity-check codes both efficiently achieve the classical capacity of any noisy channel. In this work we construct an efficient entanglement generation scheme which achieves the coherent information for a broad class of channels. For degradable channels in the class, which includes truncated versions of the pure loss Bosonic channel, the resulting scheme achieves the quantum capacity. The construction makes use of polar codes for two kinds of classical-input, quantum-output (CQ) channels: those with commuting outputs and those whose outputs are heralded mixtures of pure states. The former case is amenable to decoding by classical polar decoding algorithms, the latter by the belief propagation with quantum messages (BPQM) algorithm. To our knowledge, this construction is the first explicit and efficiently decodable protocol that achieves the energy-constrained quantum capacity of the pure-loss Bosonic channel.
1 Introduction
Efficient decoding remains a major open challenge for codes that approach noisy-channel communication limits. The paper gives an efficient entanglement-generation construction for pure discrete phase covariant channels, achieving coherent information and, for degradable channels in the class, quantum capacity.
1 Introduction
- The construction achieves coherent information for pure discrete phase covariant channels, and removes entanglement assistance when the channel is degradable.The class includes amplitude damping and dephasing channels of any dimension, resonant multilevel amplitude damping, and truncated pure-loss and pure-amplification Bosonic channels.Source: §1 · pp. 1–2
1.1 Technical overview
- After syndrome extraction, Bob coherently applies two POVMs to recover standard- and Fourier-basis information, then removes logical-information-dependent phases to generate entanglement.If the two POVMs have average errors ϵZ and ϵX, the output fidelity is at least 1 − (√ϵX + √ϵZ)^2.Source: §1.1 · p. 2
- Polar-code decoding uses classical successive cancellation for one POVM and BPQM for the other, with each decoder requiring O(d^2 m log m) arithmetic operations.The polar-code syndrome extraction circuit uses O(m log m) controlled additions.Source: §1 · p. 2, §1.1 · p. 3
- The net entanglement-generation rate equals the input state's coherent information, as established by the EACSS rate calculation.Polar codes ensure the required stabilizer counts approach the relevant conditional entropies.Source: §1.1 · p. 3
- For pure-loss Bosonic channels, truncating a two-mode squeezed input at fixed average local energy gives a rate close to the channel's coherent information.The truncated input has exponentially small fidelity loss in the truncation dimension, and entropy continuity bounds the resulting rate difference.Source: §1.1 · p. 3
1.2 Outline of the paper
- The paper proceeds from background and lemmas to the distillation procedure, then the polar-code rate calculation and efficient implementation of the decoder.Section 5 particularly discusses the connection to BPQM.Source: §1.2 · p. 3
2 Background
The background introduces entanglement-assisted CSS codes and the channel class used by the construction, then reviews the pure-loss channel and its energy-constrained capacity setting.
2.1 Mathematical preliminaries
- Entropies and logarithms use base 2; for prime d, strings are vectors over F_d.The section also defines [m] as {1,...,m}.Source: §2.1 · p. 4
2.2 Entanglement-assisted CSS codes
- Entanglement assistance permits simultaneous measurement of EACSS stabilizers whose X- and Z-type components do not commute.The code assigns Z-type stabilizers to q_A and A, and X-type stabilizers to p_A and A.Source: §2.2 · p. 4
2.3 Useful lemmas
- A conditional-entropy decomposition of coherent information supplies the identity used to show the coding scheme achieves coherent information.The section derives the decomposition using entropy chain rules and relations among the conditional states.Source: §2.3 · p. 5
2.4 Pure discrete phase covariant channels
- PDPC channels have a single environment state per herald value and include amplitude damping, dephasing, ReMAD, and truncated pure-loss and pure-amplifier channels.For these channels, standard-basis inputs yield commuting outputs.Source: §2.4 · p. 5
2.5 Pure loss Bosonic channel
- Under coding schemes where the encoder never outputs a state with energy per mode above n̄, the pure-loss channel’s quantum capacity is zero for η ≤ 1/2; conditional-entropy error decays exponentially with cutoff d.Source: §2.5 · p. 6
3 Entanglement distillation
The distillation procedure coherently combines two decoding POVMs with a phase correction. Its output fidelity is bounded by the average errors of the two decoders.
3.1 Setup
- For the EACSS setup, a POVM recovers f(x^m) with probability at least 1−ϵX.The setup assumes decoding POVMs associated with the two bases.Source: §3.1 · p. 7
- The setup coherently applies UΛ and UΓ, followed by W, to construct the joint state used for entanglement distillation.UΛ and UΓ implement the two decoding POVMs; W applies the final controlled phase and inverse Fourier transform.Source: §3.1 · p. 7
3.2 Quality of the structured decoder
- The structured decoder produces an output whose fidelity with a maximally entangled state is at least 1−(√ϵX+√ϵZ)^2.The bound depends on the average errors of the two POVMs.Source: §3.2 · p. 8
4 Polar Coding Scheme
The scheme uses polar codes to specify an EACSS code and decode amplitude and phase information. The two resulting CQ channels have different structures that support different decoding approaches.
4 Polar Coding Scheme
- The polar coding scheme specifies H, the register partition, and amplitude and phase decoders whose average block error probabilities vanish.Amplitude information is measured in the standard basis and phase information in the Fourier-conjugate basis.Source: §4 · p. 9
4.1 Polar transform
- For m a power of 2, the transform uses the bit-reversal permutation; phase decoding reverses the gate direction and proceeds from i=m down to 1.For d=2, the construction reduces to the qubit construction.Source: §4.1 · p. 9
4.2 Amplitude and phase channels
- The amplitude CQ channel has commuting outputs, while the phase channel is a heralded mixture of symmetric d-ary pure-state channels; in the pure-loss case, the herald is the number of lost photons.The amplitude input distribution is {p_k}, and the phase input distribution is uniform.Source: §4.2 · p. 9
4.3 Reliability parameter
The reliability parameter links channel output distinguishability to decoding error. Its bounds cover both classical channels and heralded mixtures of symmetric pure-state channels.
4.3 Reliability parameter
- F(W) ranges from 0 for perfectly distinguishable outputs to 1 when all outputs coincide and the input distribution is uniform.The reliability parameter is defined for a CQ channel with an input distribution.Source: §4.3 · pp. 9–10
- For classical channels, the minimum error probability is at most (d−1)Z(W), where Z(W) is the d-ary Bhattacharyya parameter.The minimum error is attained by the maximum a posteriori decision.Source: §4.3 · p. 10
- For heralded mixtures of symmetric d-ary pure-state channels, the average error is bounded using the pretty good measurement bound for each herald value.For each heralded channel, the pure states are geometrically uniform and the pretty good measurement attains the minimum error.Source: §4.3 · p. 10
4.4 Synthesized channels and speed of polarization
The amplitude information is converted into a classical source with side information, while the phase information forms a synthesized CQ channel. Both constructions polarize, and the phase channels retain a heralded-mixture structure under polar-code combinations.
4.4 Synthesized channels and speed of polarization
- Because WA(k) is diagonal in B’s standard basis, measuring B there loses no information about K; for pure loss, b is the output photon number.This measurement turns K into a classical source with side information b.Source: §4.4 · p. 10
- For every 0<β<1/2, Bhattacharyya parameters of reliable synthesized sources decay as 2^-mβ.The polarization limits apply to the amplitude source and the phase CQ channel.Source: §4.4 · p. 11
- The phase synthesized channels follow the standard polar-code construction with reversed index order, and polarization over prime fields is established for the CQ channel.The phase inputs are uniform, and reversing their order leaves W^⊗m unchanged.Source: §4.4 · p. 11
- Heralded mixtures of symmetric d-ary PSCs are closed under the check-node and bit-node combinations, so each phase synthesized channel is unitarily equivalent to such a mixture.The unitary preserves F and Perr, allowing the polarization result to apply.Source: §4.4 · p. 11
4.5 Index sets and EACSS registers
The amplitude and phase reliable-index sets and their complements partition the block positions into the four EACSS registers, which determine the code parameters and frozen syndromes.
4.5 Index sets and EACSS registers
- An index is reliable for amplitude when its transformed symbol is nearly determined by earlier symbols and B^m, and reliable for phase when it is nearly determined by later phase symbols and A′^mB^m.The complementary sets are the bad sets for the two information types.Source: §4.5 · p. 11
- The four registers are Ā=GA∩GP, qA=BA∩GP, pA=GA∩BP, and A=BA∩BP; their sizes set k, sz, sx, and ℓ.The four registers partition [m].Source: §4.5 · pp. 11–12
- The frozen amplitude and phase values are the Z-type syndrome HZkm on BA and the X-type syndrome GXxm on BP, respectively.The frozen sets are BA=qA∪A and BP=pA∪A.Source: §4.5 · p. 12
4.6 Decoding the amplitude information
Amplitude decoding uses the source-coding structure of the transformed amplitude information and a successive-cancellation decoder that incorporates the nonuniform input prior.
4.6 Decoding the amplitude information
- The amplitude frozen values are random source-code syndromes, with |FZ|/m tending to H(A|B)ρ/log d, and the state |σ〉 supplies the nonuniform input distribution without input shaping.This is the rate of source coding with quantum side information.Source: §4.6 · p. 12
- The source decoder yields POVMs for the transformed amplitude information on GA, satisfying the theorem’s condition with εZ= P̄e,Z.Invertibility of H makes the map from strings with fixed syndrome to their GA coordinates bijective.Source: §4.6 · p. 12
- The successive-cancellation decoder sets frozen symbols and uses maximum a posteriori estimates on GA from earlier decisions and a standard-basis measurement of B^m.It uses the prior pk because K is not uniform.Source: §4.6 · p. 12
4.7 Decoding the phase information
Phase decoding treats the frozen transformed phase symbols as uniform and uses a polar channel code. Its successive-cancellation procedure reads heralds and applies measurements tailored to the resulting pure-state channels.
4.7 Decoding the phase information
- The phase frozen values are uniform and form the frozen part of a polar channel code, with |FX|/m tending to H(X|AB)τ/log d.Bob’s output for this channel is A′^mB^m.Source: §4.7 · p. 13
- A single measurement decodes the phase information for all frozen values, with the same error probability for each value.The decoder shifts a frozen value to the all-zero case and uses the unitary on Bob’s systems to transfer the decoder to the heralded mixture.Source: §4.7 · p. 13
- The decoder processes symbols in reverse order, reading each herald and applying the pretty good measurement to its corresponding PSC; sequential-measurement bounds control the error.The synthesized outputs and the projectors for different steps need not commute.Source: §4.7 · pp. 13–14
4.8 Entanglement Generation Rate
The polar-code net rate is identified with coherent information, and the construction reaches quantum capacity for degradable channels in the stated class. For the energy-constrained pure-loss channel, every rate below capacity is achievable.
4.8 Entanglement Generation Rate
- The net rate equals −H(A|B)σ, equivalently log d − H(A|B)ρ − H(X|AB)τ.The rate is identified by equation (60).Source: §4.8 · p. 14
- Every rate below the energy-constrained pure-loss capacity is achievable with vanishing error as blocklength grows at fixed d.A prime d greater than n̄ gives a truncated input whose mean photon number is at most n̄.Source: §4.8 · p. 14
- For degradable PDPC channels, Theorem 2 achieves quantum capacity using an input that maximizes coherent information.Phase covariance and concavity yield the maximizing input described in the passage.Source: §4.8 · p. 14
5 Code Construction and Efficient Implementation
The construction combines classical successive-cancellation decoding for amplitude information with coherent BPQM decoding for phase information. Density evolution selects information sets, and the implementation has polynomial operation cost in m and d.
5.1 Amplitude decoder
- The amplitude successive-cancellation decoder uses O(d^2 m log m) arithmetic operations and is implemented reversibly.Each kernel step uses O(d^2) operations, across O(m log m) steps.Source: §5.1 · p. 15, §5.4 · p. 16
5.2 Phase decoder
- The phase decoder uses d-ary BPQM check- and bit-node unitaries on heralded mixtures, with coherent pretty-good-measurement decisions.The decision process is coherently implemented by reversing the successive-cancellation steps after the decisions.Source: §5.2 · p. 15, §5.4 · p. 17
5.3 Code construction using Density evolution
- Density evolution selects amplitude and phase information sets by estimating synthesized-channel errors against target error probabilities.Phase estimates track populations of eigen lists, while amplitude estimates use Monte Carlo simulations; the theorem's fidelity bound uses true decoder errors.Source: §5.3 · pp. 15–16
5 Code Construction and Efficient Implementation
- The full protocol uses O(d^2 m log m) operations in the real-register model.The construction targets polynomial-time implementation in blocklength m and alphabet size d.Source: §5 · pp. 14–15, §5.4 · p. 18
5.4 Decoder implementation and complexity
- In the real-register model, the total cost is O(d^2 m log m), while finite-bit approximation of d-ary eigen-list registers is left for future work.The registers are infinite-dimensional in this model; finite-bit approximation is described for binary BPQM.Source: §5.4 · p. 18
AI Statement
The authors state that the results and proof ideas are their own and disclose later-stage AI assistance with manuscript presentation.
AI Statement
- The authors report independently identifying PDPC channels as the class where BPQM is optimal for entanglement generation.They state that all results and proof ideas are their own.Source: AI Statement · p. 18
- ChatGPT and Claude were used during later preparation to check and refine proof presentation and improve clarity; the authors reviewed the suggested changes and take responsibility.Source: AI Statement · p. 18