Unidimensional Continuous-Variable Quantum Key Distribution with Untrusted Detection under Realistic Conditions
<p>(Color online) The PM scheme of the unidimensional CV-QKD protocol with untrusted detection with the unidimensional modulator in both sides, where both Alice and Bob perform unidimensional modulation. Replacing the unidimensional modulation in Bob’s side with the standard Gaussian modulation corresponds to the case of unidimensional modulation only in Alice’s side, while replacing the unidimensional modulation in Alice’s side with the standard Gaussian modulation corresponds to the case of unidimensional modulation only in Bob’s side. In particular, the quantum channels and Charlie are fully controlled by Eve.</p> "> Figure 2
<p>(Color online) The EB scheme and the equivalent one-way model of the unidimensional CV-QKD protocol, where the EPR states are two-mode vacuum states, with untrusted detection and coherent states (<b>a</b>) The EB scheme of the unidimentional modulation both in Alice’s and Bob’s side where the detectors are all homodyne detector. (<b>b</b>) The equivalent one-way model of the case that are the unidimentional modulation only in Alice’s side. (<b>c</b>) The equivalent one-way model of the case that are the unidimentional modulation only in Bob’s side. (<b>d</b>) The equivalent one-way model of the case that are the unidimentional modulation both in Alice’s and Bob’s side. In particular, two quantum channels and Charlie are fully controlled by Eve, but Eve has no access to the apparatuses in Alice’s and Bob’s stations.</p> "> Figure 3
<p>(Color online) Excess noise versus distance with different <math display="inline"><semantics> <msub> <mi>V</mi> <mi>B</mi> </msub> </semantics></math> in the situation that unidimensional modulation is in both sides. The dotted lines are under the condition that <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, the dashed lines represent the condition that <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics></math>, and the solid lines represent the case that <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>1.001</mn> </mrow> </semantics></math>. <math display="inline"><semantics> <msub> <mi>ε</mi> <mi>A</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ε</mi> <mi>B</mi> </msub> </semantics></math> is set as <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mi>A</mi> </msub> <mo>=</mo> <msub> <mi>ε</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. In particular, the region is divided into two parts by a black dotted-dashed line, where upper part is a reasonable region, indicating that the excess noise is greater than zero, and the lower half is an unreasonable area, indicating that the excess noise is less than zero.</p> "> Figure 4
<p>(Color online) Secret key rate in the symmetric case (<math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </msub> </mrow> </semantics></math>). the dotted lines are under the ideal condition (<math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>A</mi> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> in the situations of unidimensional modulation only in Alice’s side as well as in both sides) and the solid lines represent the practical condition (<math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>A</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math> in the situations of unidimensional modulation only in Alice’s side as well as in both sides). The red lines represent the case that the unidimentional modulation only exists in Alice’s side, the black lines are on behalf of the case that the unidimensional modulation exists in both sides.</p> "> Figure 5
<p>(Color online) Curves of the correlation between <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </msub> </semantics></math>. The transmission distance from Alice and Charlie <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msub> </semantics></math> is considered to be a function of the distance from Bob to Charlie <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </msub> </semantics></math>. the dotted lines are under the ideal condition (<math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>A</mi> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> in the situations of unidimensional modulation only in Alice’s side as well as in both sides) and the solid lines represent the practical condition (<math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>A</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.98</mn> </mrow> </semantics></math> in the situations of unidimensional modulation only in Alice’s side as well as in both sides). The red lines represent the case that the unidimentional modulation only exists in Alice’s side, the black lines are on behalf of the case that the unidimensional modulation exists in both sides of Alice and Bob. The excess noise to be <math display="inline"><semantics> <mrow> <msub> <mi>ε</mi> <mi>A</mi> </msub> <mo>=</mo> <msub> <mi>ε</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>.</p> "> Figure 6
<p>(Color online) Secret key rate versus transmission distance between Alice and Charlie, and the distance <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </msub> </semantics></math> is set to <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mrow> <mi>B</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Identically, the dotted lines are under the ideal condition, and the solid lines represent the practical condition. The red lines represent the case that the unidimentional modulation only exists in Alice’s side, the black lines are on behalf of the case that the unidimensional modulation exists in both sides of Alice and Bob, and the blue lines are the secret key rates of the standard CV-MDI protocols. The parameters are set the same as the symmetric situation.</p> ">
Abstract
:1. Introduction
2. Results
2.1. Unidimensional CV-QKD Protocol with Untrusted Detection
Case 1: unidimensional modulation only in Alice’s side Step 1. Alice produces coherent states and randomly selects the or quadrature along which the prepared states are displaced according to a random Gaussian variable with displacement variance . At the same time, Bob randomly prepares coherent states , where and are Gaussian distributed with modulation variance . Subsequently, the states are sent to the untrusted party Charlie through two different channels whose length are and , respectively. Step 2. After receiving the mode from Alice and the mode from Bob, Charlie combines them with a 50:50 beamsplitter. The output are mode C and D. Subsequently, Charlie performs measurement on the quadrature of the mode C and the quadrature of the mode D with two homodyne detectors, and then announces the results and publicly through the classical channels. Step 3. According to the information Charlie announces, Bob modifies his data as , , where k is the amplification coefficient. Here Alice keeps her data unchanged. Step 4. Alice and Bob perform post-processing, including information reconciliation, privacy amplification, and so on. |
Case 2: unidimensional modulation only in Bob’s side Step 1. Alice randomly prepares coherent states , where and are Gaussian distributed with modulation variance . Meanwhile, Bob produces coherent states and randomly selects the or along which the prepared states are displaced according to a random Gaussian variable with displacement variance . Subsequently, the states are sent to the untrusted party Charlie through two different channels whose length are and , respectively. The next steps are the same as those in Case 1. |
Case 3: unidimensional modulation in both sides Step 1. Both Alice and Bob produce coherent states and simultaneously select the or quadrature along which the prepared states are displaced according to two random Gaussian variables with displacement variance and , respectively. Subsequently, the states are sent to the untrusted party Charlie through two different channels whose length are and , respectively. The next steps are the same as those in Case 1. |
Case 1: unidimensional modulation only in Alice’s side Step 1. Alice generates Einstein-Podolsky-Rosen (EPR) states with variance . Then she keeps mode and squeezes the other mode on a squeezer. The output is mode , which is sent to the untrusted party Charlie through a channel with length . Meanwhile, Bob generates another Einstein-Podolsky-Rosen (EPR) state with variance . Then he keeps mode and sends the other mode through a channel with length . Step 2. Modes and received by Charlie interfere at a 50:50 beamsplitter with two output modes C and D. Subsequently, Charlie performs measurement on the quadrature of the mode C and the quadrature of the mode D with two homodyne detectors, and then announces the results and publicly through the classical channels. Step 3. According to the information Charlie announces, Bob displaces mode by operation , where , and g represents the gain of displacement. The relationship between k and g is well studied in reference [42]. Then Bob measures mode to get the final data , using heterodyne detection. Alice uses mode to get the final data () using homodyne detection. Step 4. Alice and Bob perform post-processing, including information reconciliation, privacy amplification, and so on. |
Case 2: unidimensional modulation only in Bob’s side Step 1. Alice generates Einstein-Podolsky-Rosen (EPR) states with variance . Then she keeps mode and sends the other mode through a channel with length . Meanwhile, Bob generates another Einstein-Podolsky-Rosen (EPR) state with variance . Then he keeps mode and squeezes the other mode on a squeezer. The output is mode , which is sent to the untrusted party Charlie through a channel with length . Step 2 and Step 4 are the same as those in Case 1. Step 3. According to the information Charlie announces, Bob displaces mode by operation , where . Then Bob measures mode to get the final data () using homodyne detection. Alice uses mode to get the final data , using heterodyne detection. |
Case 3: unidimensional modulation in both sides Step 1. Both Alice and Bob generate Einstein-Podolsky-Rosen (EPR) states with variance and respectively. Alice and Bob keep mode and mode of their own EPR state separately. The other two modes, and , are squeezed on two squeezers, and the output are modes and . Then the modes and are sent to the untrusted party Charlie through two different channels with length and . Step 2 and Step 4 are the same as those in Case 1. Step 3. According to the information Charlie announces, Bob displaces mode by operation , where . Then Alice measures mode , Bob measures mode to get the final data (), () using homodyne detection, respectively. |
2.2. Security Analysis
2.2.1. Using Unidimensional Modulated Coherent States Only in Alice’s Side
2.2.2. Using Unidimensional Modulated Coherent States ONLY in Bob’s Side
2.2.3. Using Unidimensional Modulated Coherent States Both in Alice’s and Bob’s Side
2.3. Numeral Simulation
3. Discussion and Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Using Unidimensional Modulated Coherent States Only in Alice’s Side | Using Unidimensional Modulated Coherent States Only in Bob’s Side | Using Unidimensional Modulated Coherent States Only in Both Sides | |
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Huang, L.; Zhang, Y.; Chen, Z.; Yu, S. Unidimensional Continuous-Variable Quantum Key Distribution with Untrusted Detection under Realistic Conditions. Entropy 2019, 21, 1100. https://doi.org/10.3390/e21111100
Huang L, Zhang Y, Chen Z, Yu S. Unidimensional Continuous-Variable Quantum Key Distribution with Untrusted Detection under Realistic Conditions. Entropy. 2019; 21(11):1100. https://doi.org/10.3390/e21111100
Chicago/Turabian StyleHuang, Luyu, Yichen Zhang, Ziyang Chen, and Song Yu. 2019. "Unidimensional Continuous-Variable Quantum Key Distribution with Untrusted Detection under Realistic Conditions" Entropy 21, no. 11: 1100. https://doi.org/10.3390/e21111100