Tight scaling of key rate for differential-phase-shift quantum key distribution

The performance of quantum key distribution protocols is evaluated based on the ease of implementation and key generation rate. Among major protocols, the differential-phase-shift (DPS) protocol has the advantage of simple implementation using a train of coherent pulses and a passive detection unit. Unfortunately, however, its key rate is known to be at least proportional to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msup><a:mi>η</a:mi><a:mn>2</a:mn></a:msup></a:math> with respect to channel transmission <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>η</b:mi><b:mo>→</b:mo><b:mn>0</b:mn></b:mrow></b:math>. If one can only prove the rate proportional to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msup><c:mi>η</c:mi><c:mn>2</c:mn></c:msup></c:math> and cannot improve the analysis beyond that, then the DPS protocol will be deemed inferior to other major protocols, such as the decoy BB84 protocol. In this paper, we consider a type of DPS protocol in which the phase of each emitted block comprising <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>n</d:mi></d:math> pulses is randomized and significantly improves the analysis of its key rate. Specifically, we reveal that the key rate is proportional to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msup><e:mi>η</e:mi><e:mrow><e:mn>1</e:mn><e:mo>+</e:mo><e:mfrac><e:mn>1</e:mn><e:mrow><e:mi>n</e:mi><e:mo>−</e:mo><e:mn>2</e:mn></e:mrow></e:mfrac></e:mrow></e:msup></e:math> and this scaling is tight. This implies that the DPS protocol can achieve a key rate proportional to <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>η</f:mi></f:math> for a large number of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>n</g:mi></g:math>, which is the same scaling as the decoy BB84 protocol. Our result suggests that the DPS protocol can achieve a combination of both advantages of ease of implementation and a high key generation rate. Published by the American Physical Society 2024