Onset of superactivation of quantum capacity

Bjarne Bergh; Marco Parentin; Mark M. Wilde; Nilanjana Datta

arxiv: 2604.27042 · v2 · submitted 2026-04-29 · 🪐 quant-ph

Onset of superactivation of quantum capacity

Marco Parentin , Bjarne Bergh , Nilanjana Datta , Mark M. Wilde This is my paper

Pith reviewed 2026-05-07 10:51 UTC · model grok-4.3

classification 🪐 quant-ph

keywords superactivationquantum capacityfinite-blocklengtherasure channelPPT channelquantum channelsfidelitynumerical certification

0 comments

The pith

As few as 17 uses of the joint channel allow qubit transmission at a fidelity impossible with any number of uses of either channel alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that superactivation of quantum capacity occurs in the finite-blocklength regime rather than only in the asymptotic limit of infinitely many uses. It focuses on the 50% erasure channel and the positive-partial-transpose channel, both of which have zero quantum capacity when used separately, and shows through numerical certification that their combination yields positive capacity with small numbers of uses. This matters because it moves the phenomenon from a purely theoretical curiosity toward something observable with modest experimental resources in quantum communication. The authors also introduce a definition of finite-blocklength superactivation and methods to certify it numerically.

Core claim

The central claim is that the tensor product of the 50% erasure channel and the positive-partial-transpose channel exhibits a strong finite-blocklength form of superactivation. Specifically, 17 uses of the combined channel permit transmission of a qubit with fidelity strictly higher than the fidelity achievable by any finite number of uses of either channel by itself.

What carries the argument

Numerical certification of finite-blocklength quantum capacity by bounding achievable fidelities for the joint channel versus the separate channels.

Load-bearing premise

The optimization procedures used to compute upper and lower bounds on fidelity are sufficiently complete that they neither underestimate the performance of the individual channels nor overestimate the performance of their joint channel.

What would settle it

An improved numerical search or an analytical argument that shows some large but finite number of uses of one individual channel can match or exceed the fidelity achieved by 17 joint uses.

Figures

Figures reproduced from arXiv: 2604.27042 by Bjarne Bergh, Marco Parentin, Mark M. Wilde, Nilanjana Datta.

**Figure 1.** Figure 1: FIG. 1. Comparison between lower bounds on the view at source ↗

**Figure 2.** Figure 2: FIG. 2. Lower bounds on the channel fidelity view at source ↗

read the original abstract

Superactivation of quantum capacity is the phenomenon whereby two quantum channels, each with zero quantum capacity, can exhibit a strictly positive capacity when used in tandem. In this work, we explore superactivation in the previously unexplored non-asymptotic regime of finitely many channel uses. We give a definition of finite-blocklength superactivation and propose numerical methods that can certify it. Then, focusing on the 50% erasure and positive-partial-transpose channels considered in the original work on superactivation, we show that as few as 17 uses of the joint channel already enable qubit transmission with a fidelity unattainable by any number of uses of either channel alone, demonstrating a strong finite-blocklength form of superactivation and opening the door to experimental demonstration.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Finite-blocklength superactivation is numerically certified at 17 uses for the classic erasure-plus-PPT pair, but the optimization lacks explicit error bounds that would make the fidelity advantage airtight.

read the letter

The main advance is a definition of finite-blocklength superactivation together with a numerical example showing that 17 joint uses of the 50% erasure channel and the PPT channel already give a qubit fidelity higher than either channel can achieve with any number of uses. The work takes the original asymptotic superactivation result and makes it concrete by moving to small block lengths and proposing optimization methods to certify the effect. That choice of channels keeps the link to prior literature direct, and the small number 17 is the kind of detail that could matter for thinking about experiments. The paper does a clean job of stating the definition and reporting the numerical outcome. The soft spot is the numerical certification itself. The claim rests on an optimization that produces an encoding-decoding pair whose fidelity exceeds the individual-channel threshold, yet there is no detailed account of how the search is guaranteed to be complete, how numerical precision is controlled in the fidelity calculation, or how large an error margin remains. Without those controls it is possible that the reported advantage at 17 uses is not yet strictly established. This is a paper for quantum information researchers who work on channel capacities outside the asymptotic limit. Someone already thinking about finite-blocklength effects or experimental feasibility would find the definition and the concrete example worth reading. It deserves peer review because the finite-blocklength framing is new and the claim is specific enough to be checked. I would send it out, asking referees to focus on the optimization procedure and any error analysis that is supplied.

Referee Report

2 major / 2 minor

Summary. The manuscript explores superactivation of quantum capacity in the non-asymptotic, finite-blocklength regime. It defines finite-blocklength superactivation and proposes numerical methods to certify it. Focusing on the 50% erasure channel and positive-partial-transpose (PPT) channel (each with zero quantum capacity), the central claim is that as few as 17 joint uses already permit qubit transmission with a fidelity strictly exceeding what either channel can achieve with any number of uses, providing a concrete finite-blocklength demonstration.

Significance. If the numerical certification holds with rigorous error bounds, the result is significant because it supplies the first explicit finite-use example of superactivation, moving the phenomenon from purely asymptotic theory toward potential experimental realization. It also introduces a practical definition and certification framework that could be applied to other channel pairs, strengthening the connection between capacity theory and finite-resource quantum communication.

major comments (2)

[Abstract and numerical certification section] Abstract and the section presenting the numerical results: the claim that 17 joint uses achieve a fidelity unattainable by the individual channels rests on an optimization whose procedure, convergence criteria, and error analysis are not detailed. Without an explicit, rigorous lower-bound certificate (e.g., SDP duality gap or precision-controlled fidelity evaluation), it is impossible to confirm that the reported fidelity exceeds the individual-channel threshold by more than numerical tolerance.
[Numerical methods and certification] The section on finite-blocklength certification: the methods are asserted to 'certify' positive capacity, yet no concrete error bound or completeness guarantee is supplied for the optimization (local minima, relaxation tightness, or floating-point effects in fidelity computation). This directly undermines the load-bearing assertion that the 17-use scheme is strictly superior.

minor comments (2)

[Definition section] The formal definition of finite-blocklength superactivation would benefit from an explicit mathematical statement (e.g., an inequality involving achievable fidelity after n uses) rather than a prose description.
[Figures] Figure captions and axis labels for the fidelity plots should explicitly state the numerical tolerance used and whether the plotted values are lower bounds or point estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater transparency in the numerical certification procedure. We address each major comment below and will revise the manuscript to incorporate the requested details on optimization, convergence, and error bounds.

read point-by-point responses

Referee: [Abstract and numerical certification section] Abstract and the section presenting the numerical results: the claim that 17 joint uses achieve a fidelity unattainable by the individual channels rests on an optimization whose procedure, convergence criteria, and error analysis are not detailed. Without an explicit, rigorous lower-bound certificate (e.g., SDP duality gap or precision-controlled fidelity evaluation), it is impossible to confirm that the reported fidelity exceeds the individual-channel threshold by more than numerical tolerance.

Authors: We agree that the current manuscript does not provide sufficient detail on the optimization procedure, convergence criteria, and rigorous error analysis. The numerical results rely on a semidefinite programming relaxation of the fidelity optimization over joint channel uses. In the revised version we will add an explicit subsection describing the SDP formulation, the solver parameters, convergence tolerances, and how duality gaps are used to certify a strict lower bound on the achievable fidelity that exceeds the individual-channel thresholds (which are known exactly to be zero for the erasure channel and bounded for the PPT channel) by a margin larger than the duality gap and floating-point precision. revision: yes
Referee: [Numerical methods and certification] The section on finite-blocklength certification: the methods are asserted to 'certify' positive capacity, yet no concrete error bound or completeness guarantee is supplied for the optimization (local minima, relaxation tightness, or floating-point effects in fidelity computation). This directly undermines the load-bearing assertion that the 17-use scheme is strictly superior.

Authors: We acknowledge the concern. The optimization is formulated as a convex SDP, which eliminates local-minima issues, and we employ a relaxation whose tightness can be certified via duality. In the revision we will include concrete error bounds: duality-gap certificates for the lower bound on fidelity, an analysis of floating-point effects with controlled precision arithmetic, and a discussion of how the relaxation gap is bounded to ensure the reported fidelity for 17 uses remains strictly above the individual-channel limits even after accounting for all numerical tolerances. revision: yes

Circularity Check

0 steps flagged

No circularity; numerical certification compares against externally known zero capacities

full rationale

The paper defines finite-blocklength superactivation and proposes numerical optimization methods to compute achievable fidelities for the joint channel (50% erasure + PPT) with finite uses, then directly compares those fidelities to the known zero quantum capacities of each channel individually (established in the original superactivation literature). No equation or step redefines a quantity in terms of itself, fits a parameter to the target fidelity and renames the fit as a prediction, or relies on a self-citation chain whose validity is internal to the present work. The central claim reduces to concrete encoding/decoding schemes whose performance exceeds an independent external benchmark, with no load-bearing ansatz or uniqueness theorem imported from the authors' prior work. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are described in the provided text.

pith-pipeline@v0.9.0 · 5421 in / 983 out tokens · 48778 ms · 2026-05-07T10:51:18.162786+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

This operation can be seen as the application of an appending channel: idA′ 1 ⊗RΦds ⊗|0⟩ ⟨0| C→A′′ 1 A′′ 2 A′ 2 (ρA′

Alice prepares the states Φ ds A′′ 1 A′′ 2 and|0⟩⟨0| A′ 2 . This operation can be seen as the application of an appending channel: idA′ 1 ⊗RΦds ⊗|0⟩ ⟨0| C→A′′ 1 A′′ 2 A′ 2 (ρA′

work page
[2]

:=ρ A′ 1 ⊗Φ ds A′′ 1 A′′ 2 ⊗ |0⟩⟨0|A′ 2 .(S3.42)

work page
[3]

31 eE Φds A′′1A′′2 A′′ 1 A′′ 2 |0⟩A′2 A′ 2 A′ 1 P A FIG

Alice’s input stateA ′ 1 is coherently copied intoA ′ 2 by applying a generalized controlled-NOT gateU CNOT A′ 1A′ 2 (·) =U CNOT A′ 1A′ 2 (·)U CNOT A′ 1A′ 2 , defined by the unitary operator U CNOT A′ 1A′ 2 := K−1X ℓ=0 |ℓ⟩⟨ℓ|A′ 1 ⊗X ℓ A′ 2 ,(S3.43) whereX ℓ A′ 2 is the generalized shift operator. 31 eE Φds A′′1A′′2 A′′ 1 A′′ 2 |0⟩A′2 A′ 2 A′ 1 P A FIG. S4...

work page
[4]

Alice sendsA 1 =A ′ 1A′′ 1 over the PPT channel andA 2 =A ′ 2A′′ 2 over the erasure channel. These operations can be seen as the application of the following channel: eEA′ 1→A′ 1A′′ 1 A′ 2A′′ 2 (·) :=U CNOT A′ 1A′ 2 ◦(id A′ 1 ⊗RΦds ⊗|0⟩ ⟨0| C→A′′ 1 A′′ 2 A′ 2 )(·),(S3.44) so that the output state of the encoderζ A1A2 becomes: ζA1A2 ≡ K−1X i,j=0 ρij |i⟩⟨j|...

work page
[5]

Bob performs a globaluntwistingunitary channelV † B′ 1A′ 2B′′ 1 A′′ 2 on the output systems. Recall- ing (S3.12), the corresponding unitary operator has the form V † B′ 1A′ 2B′′ 1 A′′ 2 := K−1X ℓ=0 K−1X i=0 |i⊕ℓ⟩⟨i⊕ℓ| B′ 1 ⊗ |i⟩⟨i|A′′ 2 ⊗ U i,ℓ A′′ 2 B′′ 1 † ,(S3.51) where each U i,ℓ A′′ 2 B′′ 1 † is a unitary on the shield systemsA ′′ 2B′′ 1 that inverts...

work page
[6]

Bob traces over the shield systemsB ′′ 1 A′′

work page
[7]

These first two operations correspond toprivacy squeezing, as defined in [46]

work page
[8]

Bob performs a generalized parity measurement on systemsB ′ 1A′ 2, i.e. a projective measurement withKoutcomes given by: ΠM = n (1 B′ 1 ⊗X −ℓ A′ 2 ) ΓK B′ 1A′ 2 (1 B′ 1 ⊗X ℓ A′ 2 ) o ℓ∈{0,...,K−1} ,(S3.52) where ΓK B′ 1A′ 2 := K−1X i=0 |ii⟩⟨ii|B′ 1A′ 2 (S3.53) 33 is the projector onto the maximally correlated subspace andX ℓ A′ 2 is the generalized shift ...

work page
[9]

If the outcome of the measurement isℓ, he applies the correction X −ℓ B′ 1 (·) :=X −ℓ B′ 1 (·)X ℓ B′ 1 to the key systemB ′

Bob corrects the shift error. If the outcome of the measurement isℓ, he applies the correction X −ℓ B′ 1 (·) :=X −ℓ B′ 1 (·)X ℓ B′ 1 to the key systemB ′

work page
[10]

Forℓ= 0, this is the identity

work page
[11]

Bob performs a generalized inverse CNOT onB ′ 1A′ 2, defined by the unitary operator (U CNOT B′ 1A′ 2 )−1 := K−1X i=0 |i⟩⟨i|B′ 1 ⊗X −i A′ 2 .(S3.54)

work page
[12]

Bob discards the systemA ′ 2. Similarly to the encoding procedure, all these operations can be seen as a sequential concatenation of channels: D0(·) := TrA′ 2 ◦(U CNOT B′ 1A′ 2 )−1 ◦(X −ℓ B′ 1L→B′ 1 ⊗id A′ 2)◦ M B′ 1A′ 2→B′ 1A′ 2L ◦Tr B′′ 1 A′′ 2 ◦ V † B′ 1A′ 2B′′ 1 A′′ 2 (·),(S3.55) whereMis a quantum instrument [100] implementing the parity measurement ...

work page
[13]

Bob discards the output system of the erasure channelB 2, since the original input state has been lost to the environment

work page
[14]

This decoder corresponds to the trace-out channel: D1(·) := TrB′′ 1 A′ 2A′′ 2 (·).(S3.56) 34 D1 P A B′1 B′′1 B2 ΠB2 A′2 A′′2 Z=|1⟩⟨1| FIG

Bob discards the other shield qubitB ′′ 1. This decoder corresponds to the trace-out channel: D1(·) := TrB′′ 1 A′ 2A′′ 2 (·).(S3.56) 34 D1 P A B′1 B′′1 B2 ΠB2 A′2 A′′2 Z=|1⟩⟨1| FIG. S7. Scheme of decoderD 1 corresponding to the case that an erasure is detected. S3.5. Proof of equivalence Using the notation introduced above, the claim of the proposition ca...

work page
[15]

physically

To see this, observe that eachℓ-term in (S3.62) is supported on key pairs of the form|i⊕ℓ, i⟩⟨j⊕ℓ, j|, so we get: ω(0) B1A2 =: K−1M ℓ=0 ω(0) B1A2(ℓ), (S3.65) where ω(0) B1A2(ℓ)≡ K−1X i,j=0 ρij pℓ |i⊕ℓ⟩⟨j⊕ℓ| B′ 1 ⊗ |i⟩⟨j| A′′ 2 ⊗(U i,ℓ σℓ (U j,ℓ)†)A′′ 2 B′′ 1 .(S3.66) 36 The action of the untwisting unitary channel (S3.51) on (S3.65) preserves the block-di...

work page
[16]

X x |x⟩ ⟨x|X ⊗p(x)ρ x AB !α#! 1 α   (S5.14) = α α−1 log Tr   TrA

=D(ω RB1B2∥1 R ⊗ω B1B2),(S4.53) whereξ RZB ′ 1 = (id R ⊗ D p B1B2→ZB ′ 1 )(ωRB1B2)andξ ZB ′ 1 ≡Tr R[ξRZB ′ 1].This physically means that the local processingeDon the stateω RB1B2 is one that perfectly preserves the quantum correlations with the reference systemR, as measured by the coherent information, and thus it is optimal for entanglement transmission...

work page
[17]

Seesaw Iteration Method The seesaw method, or alternating convex search method, is an iterative numerical procedure for finding lower bounds on bilinear optimization problems like (S1.17) by iteratively optimizing over one variable while keeping the other fixed. This simple method gives no guarantee about the optimality of the solution nor estimates of th...

work page
[18]

Permutation-Invariant Codes and Symmetric Seesaw In order to tackle the exponential scaling withn, this section introduces a variant of Definition 4 that we call thesymmetric seesaw method, which exploits permutation symmetry to reduce the complexity from exponential to polynomial inn, using the representation theory of the symmetric group. This procedure...

work page
[19]

Therefore, the decoder’s SDP (S5.45), whose complexity depends exponentially on dB (see (S5.67)), quickly becomes a bottleneck for largen

The effective channel eNA→B has input and output dimensions, respectively,d A = 2 and dB = 4. Therefore, the decoder’s SDP (S5.45), whose complexity depends exponentially on dB (see (S5.67)), quickly becomes a bottleneck for largen. However, the classical-quantum (CQ) structure of the output allows us to use a simple decoding protocol that reduces the SDP...

work page
[20]

The concatenation steps in the symmetric seesaw method require constructing all orbit basis coefficients for the tensor product channel. Even if polynomial inn, a quick calculation shows that the amount of orbitsm AB = n+d2 Ad2 B−1 n scales asO(n 15) for qubit channels, so it quickly becomes a bottleneck asnincreases. However, exploiting the sparsity of t...

work page
[21]

The number of free parameters in the SDP (S5.77) depends ond B asO(n d2 B) =O(n 16), so it quickly becomes the bottleneck of the symmetric seesaw method for largen

Exploiting the Classical–Quantum Output Structure The effective channel eNof Proposition 1 has input dimensiond A = 2 and output dimension dB =d Z ·d Q = 4, whereZis the flag classical system andQis the quantum output system. The number of free parameters in the SDP (S5.77) depends ond B asO(n d2 B) =O(n 16), so it quickly becomes the bottleneck of the sy...

work page
[22]

Exploiting the Sparsity of Channels The second bottleneck identified above is the serial concatenation step. Even though the number of joint orbitsm AQ = n+d2 Ad2 Q−1 n =O(n 15) (ford A =d Q = 2) is polynomial inn, it grows rapidly and becomes a bottleneck for largen, both in memory and in time. We now show that the sparsity of the specific channelsN k in...

work page

[1] [1]

This operation can be seen as the application of an appending channel: idA′ 1 ⊗RΦds ⊗|0⟩ ⟨0| C→A′′ 1 A′′ 2 A′ 2 (ρA′

Alice prepares the states Φ ds A′′ 1 A′′ 2 and|0⟩⟨0| A′ 2 . This operation can be seen as the application of an appending channel: idA′ 1 ⊗RΦds ⊗|0⟩ ⟨0| C→A′′ 1 A′′ 2 A′ 2 (ρA′

work page

[2] [2]

:=ρ A′ 1 ⊗Φ ds A′′ 1 A′′ 2 ⊗ |0⟩⟨0|A′ 2 .(S3.42)

work page

[3] [3]

31 eE Φds A′′1A′′2 A′′ 1 A′′ 2 |0⟩A′2 A′ 2 A′ 1 P A FIG

Alice’s input stateA ′ 1 is coherently copied intoA ′ 2 by applying a generalized controlled-NOT gateU CNOT A′ 1A′ 2 (·) =U CNOT A′ 1A′ 2 (·)U CNOT A′ 1A′ 2 , defined by the unitary operator U CNOT A′ 1A′ 2 := K−1X ℓ=0 |ℓ⟩⟨ℓ|A′ 1 ⊗X ℓ A′ 2 ,(S3.43) whereX ℓ A′ 2 is the generalized shift operator. 31 eE Φds A′′1A′′2 A′′ 1 A′′ 2 |0⟩A′2 A′ 2 A′ 1 P A FIG. S4...

work page

[4] [4]

Alice sendsA 1 =A ′ 1A′′ 1 over the PPT channel andA 2 =A ′ 2A′′ 2 over the erasure channel. These operations can be seen as the application of the following channel: eEA′ 1→A′ 1A′′ 1 A′ 2A′′ 2 (·) :=U CNOT A′ 1A′ 2 ◦(id A′ 1 ⊗RΦds ⊗|0⟩ ⟨0| C→A′′ 1 A′′ 2 A′ 2 )(·),(S3.44) so that the output state of the encoderζ A1A2 becomes: ζA1A2 ≡ K−1X i,j=0 ρij |i⟩⟨j|...

work page

[5] [5]

Bob performs a globaluntwistingunitary channelV † B′ 1A′ 2B′′ 1 A′′ 2 on the output systems. Recall- ing (S3.12), the corresponding unitary operator has the form V † B′ 1A′ 2B′′ 1 A′′ 2 := K−1X ℓ=0 K−1X i=0 |i⊕ℓ⟩⟨i⊕ℓ| B′ 1 ⊗ |i⟩⟨i|A′′ 2 ⊗ U i,ℓ A′′ 2 B′′ 1 † ,(S3.51) where each U i,ℓ A′′ 2 B′′ 1 † is a unitary on the shield systemsA ′′ 2B′′ 1 that inverts...

work page

[6] [6]

Bob traces over the shield systemsB ′′ 1 A′′

work page

[7] [7]

These first two operations correspond toprivacy squeezing, as defined in [46]

work page

[8] [8]

Bob performs a generalized parity measurement on systemsB ′ 1A′ 2, i.e. a projective measurement withKoutcomes given by: ΠM = n (1 B′ 1 ⊗X −ℓ A′ 2 ) ΓK B′ 1A′ 2 (1 B′ 1 ⊗X ℓ A′ 2 ) o ℓ∈{0,...,K−1} ,(S3.52) where ΓK B′ 1A′ 2 := K−1X i=0 |ii⟩⟨ii|B′ 1A′ 2 (S3.53) 33 is the projector onto the maximally correlated subspace andX ℓ A′ 2 is the generalized shift ...

work page

[9] [9]

If the outcome of the measurement isℓ, he applies the correction X −ℓ B′ 1 (·) :=X −ℓ B′ 1 (·)X ℓ B′ 1 to the key systemB ′

Bob corrects the shift error. If the outcome of the measurement isℓ, he applies the correction X −ℓ B′ 1 (·) :=X −ℓ B′ 1 (·)X ℓ B′ 1 to the key systemB ′

work page

[10] [10]

Forℓ= 0, this is the identity

work page

[11] [11]

Bob performs a generalized inverse CNOT onB ′ 1A′ 2, defined by the unitary operator (U CNOT B′ 1A′ 2 )−1 := K−1X i=0 |i⟩⟨i|B′ 1 ⊗X −i A′ 2 .(S3.54)

work page

[12] [12]

Bob discards the systemA ′ 2. Similarly to the encoding procedure, all these operations can be seen as a sequential concatenation of channels: D0(·) := TrA′ 2 ◦(U CNOT B′ 1A′ 2 )−1 ◦(X −ℓ B′ 1L→B′ 1 ⊗id A′ 2)◦ M B′ 1A′ 2→B′ 1A′ 2L ◦Tr B′′ 1 A′′ 2 ◦ V † B′ 1A′ 2B′′ 1 A′′ 2 (·),(S3.55) whereMis a quantum instrument [100] implementing the parity measurement ...

work page

[13] [13]

Bob discards the output system of the erasure channelB 2, since the original input state has been lost to the environment

work page

[14] [14]

This decoder corresponds to the trace-out channel: D1(·) := TrB′′ 1 A′ 2A′′ 2 (·).(S3.56) 34 D1 P A B′1 B′′1 B2 ΠB2 A′2 A′′2 Z=|1⟩⟨1| FIG

Bob discards the other shield qubitB ′′ 1. This decoder corresponds to the trace-out channel: D1(·) := TrB′′ 1 A′ 2A′′ 2 (·).(S3.56) 34 D1 P A B′1 B′′1 B2 ΠB2 A′2 A′′2 Z=|1⟩⟨1| FIG. S7. Scheme of decoderD 1 corresponding to the case that an erasure is detected. S3.5. Proof of equivalence Using the notation introduced above, the claim of the proposition ca...

work page

[15] [15]

physically

To see this, observe that eachℓ-term in (S3.62) is supported on key pairs of the form|i⊕ℓ, i⟩⟨j⊕ℓ, j|, so we get: ω(0) B1A2 =: K−1M ℓ=0 ω(0) B1A2(ℓ), (S3.65) where ω(0) B1A2(ℓ)≡ K−1X i,j=0 ρij pℓ |i⊕ℓ⟩⟨j⊕ℓ| B′ 1 ⊗ |i⟩⟨j| A′′ 2 ⊗(U i,ℓ σℓ (U j,ℓ)†)A′′ 2 B′′ 1 .(S3.66) 36 The action of the untwisting unitary channel (S3.51) on (S3.65) preserves the block-di...

work page

[16] [16]

X x |x⟩ ⟨x|X ⊗p(x)ρ x AB !α#! 1 α   (S5.14) = α α−1 log Tr   TrA

=D(ω RB1B2∥1 R ⊗ω B1B2),(S4.53) whereξ RZB ′ 1 = (id R ⊗ D p B1B2→ZB ′ 1 )(ωRB1B2)andξ ZB ′ 1 ≡Tr R[ξRZB ′ 1].This physically means that the local processingeDon the stateω RB1B2 is one that perfectly preserves the quantum correlations with the reference systemR, as measured by the coherent information, and thus it is optimal for entanglement transmission...

work page

[17] [17]

Seesaw Iteration Method The seesaw method, or alternating convex search method, is an iterative numerical procedure for finding lower bounds on bilinear optimization problems like (S1.17) by iteratively optimizing over one variable while keeping the other fixed. This simple method gives no guarantee about the optimality of the solution nor estimates of th...

work page

[18] [18]

Permutation-Invariant Codes and Symmetric Seesaw In order to tackle the exponential scaling withn, this section introduces a variant of Definition 4 that we call thesymmetric seesaw method, which exploits permutation symmetry to reduce the complexity from exponential to polynomial inn, using the representation theory of the symmetric group. This procedure...

work page

[19] [19]

Therefore, the decoder’s SDP (S5.45), whose complexity depends exponentially on dB (see (S5.67)), quickly becomes a bottleneck for largen

The effective channel eNA→B has input and output dimensions, respectively,d A = 2 and dB = 4. Therefore, the decoder’s SDP (S5.45), whose complexity depends exponentially on dB (see (S5.67)), quickly becomes a bottleneck for largen. However, the classical-quantum (CQ) structure of the output allows us to use a simple decoding protocol that reduces the SDP...

work page

[20] [20]

The concatenation steps in the symmetric seesaw method require constructing all orbit basis coefficients for the tensor product channel. Even if polynomial inn, a quick calculation shows that the amount of orbitsm AB = n+d2 Ad2 B−1 n scales asO(n 15) for qubit channels, so it quickly becomes a bottleneck asnincreases. However, exploiting the sparsity of t...

work page

[21] [21]

The number of free parameters in the SDP (S5.77) depends ond B asO(n d2 B) =O(n 16), so it quickly becomes the bottleneck of the symmetric seesaw method for largen

Exploiting the Classical–Quantum Output Structure The effective channel eNof Proposition 1 has input dimensiond A = 2 and output dimension dB =d Z ·d Q = 4, whereZis the flag classical system andQis the quantum output system. The number of free parameters in the SDP (S5.77) depends ond B asO(n d2 B) =O(n 16), so it quickly becomes the bottleneck of the sy...

work page

[22] [22]

Exploiting the Sparsity of Channels The second bottleneck identified above is the serial concatenation step. Even though the number of joint orbitsm AQ = n+d2 Ad2 Q−1 n =O(n 15) (ford A =d Q = 2) is polynomial inn, it grows rapidly and becomes a bottleneck for largen, both in memory and in time. We now show that the sparsity of the specific channelsN k in...

work page