Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution
<p>Schematic of the continuous-variable version of quantum key distribution (CV-QKD) protocols with an untrusted source. Both the entangled Gaussian source and the quantum channels are fully controlled by Eve. However, Eve has no access to the apparatuses in Alice’s and Bob’s stations. Alice and Bob can perform either homodyne (Hom) or heterodyne (Het) detection, using either direct or reverse reconciliation.</p> ">
<p>(<b>a</b>) Entanglement-based CV-QKD protocol with an untrusted relay where the displacement operator is placed at Bob’s side. (<b>b</b>) Entanglement-based scheme where the displacement operator is placed at both Alice’s and Bob’s sides.</p> ">
<p>Entanglement-in-the-middle protocol. Equivalent channels and squeezing: an Einstein–Podolsky–Rosen (EPR) state <span class="html-italic">λ</span> sent through two Gaussian channels of transmittance <span class="html-italic">T</span><sub>1</sub>, <span class="html-italic">T</span><sub>2</sub> and excess noise <span class="html-italic">ε</span><sub>1</sub>, <span class="html-italic">ε</span><sub>2</sub>, followed by two successful noiseless linear amplifiers (NLAs), has the same final covariance matrix with a state <span class="html-italic">ς</span> sent through two Gaussian channels of transmittance <span class="html-italic">η</span><sub>1</sub>, <span class="html-italic">η</span><sub>2</sub> and excess noise <math display="inline"> <mrow> <msubsup> <mi>ε</mi> <mn>1</mn> <mi>g</mi></msubsup></mrow></math>, <math display="inline"> <mrow> <msubsup> <mi>ε</mi> <mn>2</mn> <mi>g</mi></msubsup></mrow></math>, without two NLAs.</p> ">
<p>Improvement of the CV-QKD protocols with entanglement-in-the-middle. A comparison among the secret key rates for the protocols where (left panel) Alice uses homodyne detection and Bob uses homodyne detection and DR(equivalent to Alice using homodyne detection and Bob using homodyne detection and RR); and (right panel) Alice uses heterodyne detection and Bob uses homodyne detection and DR (equivalent to Alice using homodyne detection and Bob using heterodyne detection and RR), under the following situations: no NLAs (<span class="html-italic">g</span><sub>1</sub> = 1, <span class="html-italic">g</span><sub>2</sub> = 1), using an NLA at Alice’s side (<span class="html-italic">g</span><sub>2</sub> = 1), using an NLA at Bob’s side (<span class="html-italic">g</span><sub>1</sub> = 1) and using two NLAs at both sides. Here, we use the realistic parameters: <span class="html-italic">V</span> = 1.7, <span class="html-italic">β</span> = 0.948, <span class="html-italic">ε</span> = 0.002 and <math display="inline"> <mrow> <msub> <mi>P</mi> <mrow> <mtext>total</mtext></mrow></msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>g</mi> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>N</mi> <mi>A</mi></msub></mrow></msubsup> <msubsup> <mi>g</mi> <mn>2</mn> <mrow> <mn>2</mn> <msub> <mi>N</mi> <mrow> <mi>B</mi> <mo stretchy="false">|</mo> <mi>A</mi></mrow></msub></mrow></msubsup></mrow> <mo>)</mo></mrow></mrow></math>.</p> ">
<p>Improvement of the CV-QKD protocols with entanglement-in-the-middle. A comparison among the secret key rates for the protocols where (left panel) Alice uses homodyne detection and Bob uses heterodyne detection and DR (equivalent to Alice using heterodyne detection and Bob using homodyne detection and RR); and (right panel) Alice uses heterodyne detection and Bob uses heterodyne detection and DR (equivalent to Alice using heterodyne detection and Bob using heterodyne detection and RR), under the following situations: no NLAs (<span class="html-italic">g</span><sub>1</sub> = 1, <span class="html-italic">g</span><sub>2</sub> = 1), using an NLA at Alice’s side (<span class="html-italic">g</span><sub>2</sub> = 1), using an NLA at Bob’s side (<span class="html-italic">g</span><sub>1</sub> = 1) and using two NLAs at both sides. Here, we use the realistic parameters: <span class="html-italic">V</span> = 1.7, <span class="html-italic">β</span> = 0.948, <span class="html-italic">ε</span> = 0.002 and <math display="inline"> <mrow> <msub> <mi>P</mi> <mrow> <mtext>total</mtext></mrow></msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>g</mi> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>N</mi> <mi>A</mi></msub></mrow></msubsup> <msubsup> <mi>g</mi> <mn>2</mn> <mrow> <mn>2</mn> <msub> <mi>N</mi> <mrow> <mi>B</mi> <mo stretchy="false">|</mo> <mi>A</mi></mrow></msub></mrow></msubsup></mrow> <mo>)</mo></mrow></mrow></math>.</p> ">
<p>Entanglement-based scheme of the modified CV-QKD protocol with an untrusted relay, where a displacement operator <span class="html-italic">D</span> is placed at both Alice’s and Bob’s sides and the two NLAs are placed before the measurement devices.</p> ">
<p>Improvement of the modified CV-QKD protocol with an untrusted relay in (<b>a</b>) the symmetric case (<span class="html-italic">i.e.</span>, <span class="html-italic">L<sub>AC</sub></span> = <span class="html-italic">L<sub>BC</sub></span>) and (<b>b</b>) the asymmetric case (<span class="html-italic">i.e.</span>, <span class="html-italic">L<sub>AC</sub></span> ≠ <span class="html-italic">L<sub>BC</sub></span>). A comparison among the secret key rates in DR under the following situations: no NLAs (<span class="html-italic">g</span><sub>1</sub> = 1, <span class="html-italic">g</span><sub>2</sub> = 1), using an NLA at Alice’s side (<span class="html-italic">g</span><sub>2</sub> = 1), using an NLA at Bob’s side (<span class="html-italic">g</span><sub>1</sub> = 1) and using two NLAs one at each side. Here, we use the realistic parameters: <span class="html-italic">V<sub>A</sub></span> = <span class="html-italic">V<sub>B</sub></span> = 1.7, <span class="html-italic">β</span> = 0.948, <span class="html-italic">ε</span> = 0.002 and <math display="inline"> <mrow> <msub> <mi>P</mi> <mrow> <mtext>total</mtext></mrow></msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>g</mi> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>N</mi> <mi>A</mi></msub></mrow></msubsup> <msubsup> <mi>g</mi> <mn>2</mn> <mrow> <mn>2</mn> <msub> <mi>N</mi> <mrow> <mi>B</mi> <mo stretchy="false">|</mo> <mi>A</mi></mrow></msub></mrow></msubsup></mrow> <mo>)</mo></mrow></mrow></math>.</p> ">
Abstract
:1. Introduction
2. Entanglement-Based CV-QKD Protocols
2.1. Entanglement Distribution: Entanglement-Based Protocols with an Untrusted Source
- Step 1: The untrusted third party, Charlie, initially prepares an entangled source. He sends one mode A1 to Alice through Channel 1 and sends the other mode B1 to Bob through Channel 2, where Eve may perform her attack.
- Step 2: Alice and Bob perform either a homodyne (switching) (Hom) or a heterodyne (no switching) (Het) measurement on the received modes A2 and B2. Once Alice and Bob have collected a sufficiently large set of correlated data, they proceed with classical data post-processing, namely error reconciliation and privacy amplification. The reconciliation can be performed in one of two ways: either direct reconciliation (DR) [7] or reverse reconciliation (RR) [8].
2.2. Entanglement Swapping: Entanglement-Based Protocol with an Untrusted Relay
- Step 1: Alice and Bob both generate an Einstein–Podolsky–Rosen (EPR), states EPR1 and EPR2, respectively, with variances VA and VB and they keep modes A2 and B2 at their respective sides. Then, they send their other modes A1 and B1 to the untrusted third party (Charlie) through two different quantum channels with lengths LAC and LBC.
- Step 2: Charlie combines the received two modes and onto a beam splitter (50:50), where we label output modes of the beam splitter as C and D. Charlie then measures the x-quadrature of mode C and the p-quadrature of mode D using homodyne detectors and publicly announces the measurement results xC, pD to Alice and Bob through classical channels. After the measurements of modes C and D, the two initially independent modes A2 and B2 get entangled if channel noise is not too strong.
- Step 3: Bob displaces the mode B2 to B3 by the operation and gets , where represents the density matrix of mode B, β = g (XC + iPD), (↠and â are the creation and annihilation operators, respectively), and g represents the gain of the displacement. Then Bob measures the mode B3 to get the final data {xB, pB} using heterodyne detection. Alice also measures the mode A2 to get the final data {xA, pA}, again using heterodyne detection.
- Step 4: Once Alice and Bob have collected a sufficiently large set of correlated data, they use an authenticated public channel to do parameter estimation from a randomly-chosen sample of final data from {xA, pA} and {xB, pB}. Then, Alice and Bob proceed with classical data post-processing to distil a secret key. The reconciliation can also be done in two ways: either DR or RR.
3. Improvement Using Noiseless Linear Amplifiers
3.1. Noiseless Linear Amplifier
3.2. Entanglement-Based Protocol with an Untrusted Source
3.3. Entanglement-Based Protocol with an Untrusted Relay
4. Conclusion
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Zhang, Y.; Li, Z.; Weedbrook, C.; Marshall, K.; Pirandola, S.; Yu, S.; Guo, H. Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution. Entropy 2015, 17, 4547-4562. https://doi.org/10.3390/e17074547
Zhang Y, Li Z, Weedbrook C, Marshall K, Pirandola S, Yu S, Guo H. Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution. Entropy. 2015; 17(7):4547-4562. https://doi.org/10.3390/e17074547
Chicago/Turabian StyleZhang, Yichen, Zhengyu Li, Christian Weedbrook, Kevin Marshall, Stefano Pirandola, Song Yu, and Hong Guo. 2015. "Noiseless Linear Amplifiers in Entanglement-Based Continuous-Variable Quantum Key Distribution" Entropy 17, no. 7: 4547-4562. https://doi.org/10.3390/e17074547