An Improved Correction Term for Dimension Reduction in Quantum Key Distribution Twesh Upadhyaya1Thomas van Himbeeck1 2and Norbert L utkenhaus1 1Institute for Quantum Computing and Department of Physics and Astronomy

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An Improved Correction Term for Dimension Reduction in Quantum Key Distribution

Twesh Upadhyaya,1, ∗Thomas van Himbeeck,1, 2 and Norbert L¨utkenhaus1

1Institute for Quantum Computing and Department of Physics and Astronomy

University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

2Department of Electrical & Computer Engineering,

University of Toronto, Toronto, Ontario, Canada M5S 3G4

(Dated: October 27, 2022)

The dimension reduction method [1] enables security proofs of quantum key distribution (QKD)

protocols that are originally formulated in inﬁnite dimensions via reduction to a tractable ﬁnite-

dimensional optimization. The reduction of dimensions is associated with a correction term in

the secret key rate calculation. The previously derived correction term is loose when the protocol

measurements are nearly block-diagonal with respect to the projection onto the reduced ﬁnite-

dimensional subspace. Here, we provide a tighter correction term. It interpolates between the

two extreme cases where all measurement operators are block-diagonal, and where at least one

has maximally large oﬀ-diagonal blocks. This new correction term can reduce the computational

overhead of applying the dimension reduction method by reducing the required dimension of the

chosen subspace.

I. INTRODUCTION

Quantum key distribution (QKD) is a promising quan-

tum technology, enabling two parties to communicate se-

curely, even if an eavesdropper has unlimited computa-

tional power [2–4].

A security analysis of a QKD protocol derives the rate

at which a secret key can be generated at the speciﬁed

security level. Recently, numerical tools have been intro-

duced to perform these key rate calculations, by reliably

solving an optimization over the joint state of Alice and

Bob [5, 6]. These tools are useful for practical QKD secu-

rity proofs as they enable modelling of imperfect devices,

encapsulate extended side channel models, and take ad-

vantage of speciﬁc details of the observed data to get an

increased key generation rate.

As most QKD protocols are implemented optically,

the underlying Hilbert space is inﬁnite-dimensional. The

relevant optimization is then over inﬁnite-dimensional

states, so the aforementioned numerical tools cannot be

applied directly. For discrete-variable (DV) protocols,

techniques such as squashing maps or the more gen-

eral ﬂag-state squasher can be used to map the prob-

lem to an eﬀective ﬁnite-dimensional optimization [7, 8].

These tools, however, are not straightforwardly applica-

ble to continuous-variable (CV) protocols. This is be-

cause they rely on all the protocol’s measurement op-

erators commuting with a projector on an underlying

low-dimensional subspace, a subspace which essentially

captures the protocol’s behaviour.

Recently, we have extended the numerical framework

to encompass both CV and DV protocols in inﬁnite-

dimensional Hilbert spaces via the dimension reduction

method [1]. This method provides a tight lower bound

on the inﬁnite-dimensional key rate optimization by op-

∗twesh.upadhyaya@uwaterloo.ca

timizing a speciﬁed ﬁnite-dimensional problem, and sub-

tracting a correction term that bounds the diﬀerence be-

tween the original inﬁnite-dimensional problem and the

ﬁnite-dimensional one. The correction term bounds how

much the key rate can increase under projection. In our

previous work, we found an analytic form for the correc-

tion term that was applicable to any QKD protocol, but

loose in certain cases. We also found that the correc-

tion term is zero when all the POVMs commute with the

same projector. We conjectured that a tighter correction

term exists which interpolates between these two cases;

becoming smaller when the measurements are closer to

block-diagonal. In this work, we ﬁnd such a correction

term. Practically, this is relevant for improving the per-

formance of the dimension reduction method, and en-

abling its applications to more computationally demand-

ing scenarios. It may also be of independent interest to

better understand how the key rate changes under pro-

jection.

II. BACKGROUND

In this section we brieﬂy review the formulation of the

asymptotic key rate as a convex minimization and the

dimension reduction method; focusing in particular on

the correction term. For a more detailed discussion, we

refer the reader to Ref. [1].

A. Key Rate Optimization and Dimension

Reduction Method

In each key generation round of a QKD protocol, Alice

and Bob establish a quantum state ρAB ; and Eve holds

its puriﬁcation in her register E. Alice and Bob measure

their respective subsystems and perform classical data

processing, which may involve public announcements, to

generate a raw key, which is the key before error correc-

arXiv:2210.14296v1 [quant-ph] 25 Oct 2022

tion and privacy ampliﬁcation. The raw key is stored in

the register Z, and any public announcements are stored

in the register C. These measurement and postprocess-

ing steps can be represented as a quantum-to-classical

channel Φ : AB →ZC. We use the notation [E] = EC

for the composite register containing all information ac-

cessible to Eve. The asymptotic key rate per signal sent

under collective attacks is given by the Devetak-Winter

formula [9], which can be expressed as a convex optimiza-

tion [6],

R∞= min

ρAB ∈S∞

[H(Z|[E])Φ(ρABE )]−δleak

EC .(1)

The convex feasible set S∞is constrained by the param-

eter estimation Alice and Bob perform, as well as the re-

duced density matrix constraint for prepare-and-measure

protocols [6]. The error-correction cost δleak

EC can be ob-

served directly and does not need to be optimized over.

Using the shorthand ffor the convex objective function,

our goal is to compute tight lower bounds on the follow-

ing minimization,

min

ρ∈S∞

f(ρ).(2)

Tractable lower bounds on this inﬁnite-dimensional op-

timization can be computed via the dimension reduction

method [1]. There are four steps to apply this method;

we brieﬂy summarize them here and give references to

the relevant sections of Ref. [1]. First choose a ﬁnite-

dimensional subspace, represented by a projector Π (Sec.

IV A). Next, ﬁnd a bound Won the weight of ρoutside

this subspace (Sec. IV B). Third, determine a correction

term ∆ for the objective function f(Sec. IV C). Finally,

construct a ﬁnite-dimensional set SNsatisfying certain

properties (Sec. IV D). The desired lower bound is then

min

˜ρ∈SN

f(˜ρ)−∆(W)≤min

ρ∈S∞

f(ρ),(3)

where the ﬁnite-dimensional optimization can be solved

numerically and the correction term is computed analyt-

ically. Tildes denote operators that are subnormalized.

B. Correction Term

Intuitively, the correction term limits how much the

function fcan increase under projection. Formally, it

satisﬁes the following property,

Trρ¯

Π≤W=⇒f(ΠρΠ) −f(ρ)≤∆(W),∀ρ∈S∞.

(4)

In this case, we say that fis uniformly close to decreasing

under projection (UCDUP) on S∞, with correction term

∆. The correction term we derive will apply on the set

of all density operators, ˜

D(H∞), and for any choice of

projection Π, so it can be applied to any QKD protocol.

C. Postprocessing Map

As we have noted, the postprocessing map Φ is a

quantum-to-classical channel from AB to ZC. It follows

that Φ can be realized by a measurement. That is, Φ has

the form

Φ(ρABE ) = X

z∈SZ

c∈SC

|zihz|Z⊗|cihc|C⊗TrAB [(Pz,c

AB ⊗E)ρABE ],

(5)

where {Pz,c

AB }z∈SZ

c∈SC

is some positive operator-valued mea-

sure (POVM), over the alphabet of key symbols SZand

public announcements SC[10]. For simplicity, we re-

index the POVM by k∈SK≡SZ×SC.

III. RESULTS

We ﬁrst introduce three lemmas which will be needed

to prove our main theorem, which is a tight correction

term. The ﬁrst lemma is a continuity bound for condi-

tional entropy in terms of trace distance. The second

lemma lets us consider the dephased state instead of the

projected one; dephased means the oﬀ-diagonal blocks

with respect to the projector and its complement are ze-

roed out. The third lemma provides a bound on the trace

norm of a speciﬁc form of operator, which arises in the

proof of the main theorem and is related to Eve’s condi-

tional states.

Lemma 1 (From Ref. [1]).Let HAand HBbe two

Hilbert spaces, where the dimension of HAis |A|while

HBcan be inﬁnite-dimensional. Let ˜ρAB ,˜σAB ∈˜

D(HA⊗

HB)be two subnormalized, classical-quantum states with

Tr(˜ρAB )≥Tr(˜σAB ). If 1

2k˜ρAB −˜σAB k1≤, then

H(A|B)˜σAB −H(A|B)˜ρAB ≤log2|A|+(1+)h

1 + ,

(6)

where h(x)is the binary entropy function.

Proof. See Appendix A of Ref. [1].

Deﬁne ΞAB to be a dephasing channel associated with

the projector Π and its complement ¯

Π as

ΞAB (ρ)≡ΠρΠ + ¯

Πρ¯

Π.(7)

Lemma 2. For any state ρAB ,f(ΠρAB Π) ≤

H(Z|[E])Φ(Ξ(ρABE ))

Proof. Expanding deﬁnitions, we have that

f(ΠρAB Π) = H(Z|[E])Φ(ΠρABE Π) (8)

≤H(Z|[E])Φ(ΠρABE Π) +H(Z|[E])Φ( ¯

ΠρABE ¯

Π)

(9)

≤H(Z|[E])Φ(ΠρABE Π)+Φ( ¯

ΠρABE ¯

Π) (10)

=H(Z|[E])Φ(ΠρABE Π+ ¯

ΠρABE ¯

Π) (11)

=H(Z|[E])Φ(Ξ(ρABE )).(12)

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AnImprovedCorrectionTermforDimensionReductioninQuantumKeyDistributionTweshUpadhyaya,1,ThomasvanHimbeeck,1,2andNorbertLutkenhaus11InstituteforQuantumComputingandDepartmentofPhysicsandAstronomyUniversityofWaterloo,Waterloo,Ontario,CanadaN2L3G12DepartmentofElectrical&ComputerEngineering,UniversityofT...

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An Improved Correction Term for Dimension Reduction in Quantum Key Distribution Twesh Upadhyaya1Thomas van Himbeeck1 2and Norbert L utkenhaus1 1Institute for Quantum Computing and Department of Physics and Astronomy

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