## Security of a discretely signaled continuous variable quantum key distribution protocol for high rate systems

Optics Express, Vol. 17, Issue 14, pp. 12090-12108 (2009)

http://dx.doi.org/10.1364/OE.17.012090

Acrobat PDF (1474 KB)

### Abstract

We propose a continuous variable based quantum key distribution protocol that makes use of discretely signaled coherent light and reverse error reconciliation. We present a rigorous security proof against collective attacks with realistic lossy, noisy quantum channels, imperfect detector efficiency, and detector electronic noise. This protocol is promising for convenient, high-speed operation at link distances up to 50 km with the use of post-selection.

© 2009 Optical Society of America

## 1. Introduction

5. F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature **421**, 238–241 (2003).
[CrossRef] [PubMed]

6. N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A **63**, 052311 (2001).
[CrossRef]

8. F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. **88**, 057902 (2002)
[CrossRef] [PubMed]

9. R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A **74**, 032302 (2006).
[CrossRef]

10. F. Grosshans, “CollectiveAttacks and Unconditional Security in Continuous Variable Quantum KeyDistribution,” Phys. Rev. Lett. **94**, 020504 (2005).
[CrossRef] [PubMed]

11. M. Navascués and A. Acín, “SecurityBounds for Continuous Variables Quantum Key Distribution,” Phys. Rev. Lett. **94**, 020505 (2005).
[CrossRef] [PubMed]

12. R. García-Patrón and N. J. Cerf, “Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. **97**, 190503 (2006).
[CrossRef] [PubMed]

13. M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. **97**, 190502 (2006).
[CrossRef] [PubMed]

14. J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A **76**, 042305 (2007).
[CrossRef]

15. M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A **76**, 022313 (2007).
[CrossRef]

17. Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A **79**, 012307 (2009).
[CrossRef]

19. C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A **73**,022316 (2006).
[CrossRef]

20. A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. **95**, 180503 (2005).
[CrossRef] [PubMed]

21. T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A **76**, 030303(R) (2007).
[CrossRef]

19. C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A **73**,022316 (2006).
[CrossRef]

20. A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. **95**, 180503 (2005).
[CrossRef] [PubMed]

21. T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A **76**, 030303(R) (2007).
[CrossRef]

15. M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A **76**, 022313 (2007).
[CrossRef]

*β*

_{0}, approaches 1. This differs from discrete QKD. Second, reconciliation needs to be simple and fast. In order to correct errors between Alice and Bob, one usually seeks continuous variable based error correction codes to be as efficient as possible. However, highly efficient error correction codes are also slow. We note that codes for binary symmetric channel are usually simpler and faster. By turning the continuous variable based error correction problem into a binary based error correction problem, several advantages come. First, it is easier to find corresponding error correction codes working at a rate very close to Shannon limit while keeping a lower decoding complexity. Furthermore, if the required error correction efficiency is lowered for a given distance, then we may be able to find a reconciliation code with corresponding lower efficiency but greater speed. As a result, the distance and throughput of CVQKD systems would be significantly improved.

## 2. The quantized input-quantized output CVQKD protocol

9. R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A **74**, 032302 (2006).
[CrossRef]

**Step 1**: Alice randomly picks up a random variable

*x*∈ {1,2,3,4} and encodes a coherent state

_{k}*r*is a positive real number depending on Bob’s signal-to-noise ratio and k denotes the index of time slot, and sends it through a lossy and noisy quantum channel. Alice’s encoding scheme can be described in Fig. 1.

**Step 2**Bob receives a quantum state from the quantum channel. With probability

*p*, each measurement is assigned to channel characterization, where Bob randomly chooses a local oscillator phase

*ϕ*of 0,

_{k}*π*/4 or

*π*/2, makes a homodynemeasurement and records the real result [23]. With probability 1-

*p*, that measurement is assigned a data collection index

*k*, where Bob randomly chooses a local oscillator phase

*ϕ*of 0 or

_{k}*π*/2 before performing homodyne detection. If his measurement result is greater than T, where T≥0 is Bob’s decision threshold, then he quantizes the result to

*qk*=1. If Bob’s measurement result is less than -

*T*otherwise, he quantizes the data to

*qk*=−1. For other cases where his measurement result is between -

*T*and

*T*, Bob quantizes his data to

*q*=0. When

_{k}*q*=0, the data from the corresponding time slot will not be selected in the post processing. When

_{k}*T*=0, it reduces to the case without post-selection.

**Step 3:**When all quantumcommunication has been finished, Bob reveals to Alice which time slots that were used for characterization phasemeasurements. Alice reveals to Bob the state that she has sent for those time slots. Then Bob performs conditional quantum tomography for each one of the four particular coherent states that Alice sent. Only three different collection angles are required to achieve a good estimate of the received state [23]. We know that without Eve, the channel can be modeled as a beamsplitter with two inputs, one of which is Alice’s output to the quantum channel and the other one is the excess channel noise mode.

*b*̂ is the output of beamsplitter going to Bob’s detectors and η is quantum efficiency of the quantum channel. For any field quadrature of Bob, we have

*p*(

_{b}*q*),

*p*(

_{a}*q*) and

*p*(

_{εn}*q*) are the possibility distributions of the three quadratures, we have

*p*(

_{ε}*q*) once we know

*p*(

_{a}*q*) and got

*p*(

_{b}*q*) fromtomography. Therefore, we can also reconstruct

*. For the protocol, Bob performs quantum conditional tomography for all four cases. Then Bob can reconstruct*ρ ˜

_{b}**Step 4:**For each data collection time slot, Bob reveals the local oscillator phase that was chosen. If Bob used

*ϕ*=0, then Alice records

_{k}*a*=1 for the case where

_{k}*x*=0 or

_{k}*x*=1 and

_{k}*a*=−1 for the case where

_{k}*x*=2 or

_{k}*x*=3. If Bob used

_{k}*a*=1 for the case where

_{k}*x*=0 or

_{k}*x*=2 and

_{k}*a*=−1 for the case where

_{k}*x*=1 or

_{k}*x*=3.

_{k}**Step 5:**Bob sends checkbits to Alice over a public channel, i.e.

*reverse reconciliation*. The reconciliation is strictly one-way.

**Step 6:**Alice and Bob perform privacy amplification to distill the final secure key.

## 3. Security analysis

*I*(

*A;B*) is the mutual information between Alice and Bob. However, practical reconciliation codes do not reach the Shannon limit. If we define a reconciliation efficiency

*β*, then the practical secrecy capacity in this case becomes

*β*

_{0}where

*I*(

*A;B*) can be completely determined by the signal-to-noise ratio of Bob.

*I*(

*A;B*) can be calculated as Eq. (7) and Eq. (8),

*h*(

*p*)=−

*p*log

_{2}(

*p*)−(1−

*p*) log

_{2}(1−

*p*) is the binary entropy function.

*χ*(

*B;E*) is the Holevo information between Bob and Eve, which is defined to be

*S*(

*ρ*̃

*E*) is the von Neumann entropy of Eve’s mixed state.

*ρ*̃

*E*|

_{q=i}is Eve’s mixed state given Bob’s measurement result and

*p*is the probability that Bob’s measurement is

_{i}*i*. For our binary symmetric case, we can rewrite

*χ*(

*B;E*) to be

### 3.1. Analytical solution for no excess channel noise case

*α*〉 and Eve’s received quantum states are

_{i}*ρ*̃

*E*,

*χ*(

*B;E*) relates directly to the error rate of the binary symmetric channel. Let’s consider the case where Bob chose

*ϕ*=0 as the phase for homodyne detection. Given Bob’s quantized data q=1, the possibility that Alice sent |

*α*

_{1}〉, |

*α*

_{2}〉, |

*α*

_{3}〉 and

*χ*(

*B;E*) is

*X*quadrature when Alice sent

*α*. η

_{i}_{m}is the detection efficiency of the homodyne detector. Combining Eq. (13) and Eq. (14), we have

### 3.2. Numerical simulation for the general case with excess channel noise

#### 3.2.1. Validity of channel model

*i*denotes Alice’s picked state and |ψ〉

_{E}denotes Eve’s original ancillary states. Then Bob’s incoming density matrices are given by a trace over Eve’s Hilbert space

*b̂*can be expresses as a superposition of Alice’s mode and another excess noise mode, we can decompose

_{i}*M*̂ into three different unitary operators

*Ô, P̂*and

*Q̂. Ô*creates

*Tr*denotes the trace over the rest Eve’s state beside

_{r}*ε*. The role of operator

_{n}*P*̂ is to interact

*α*〉 on a beamsplitter to create

_{i}*P*̂ can be written as

*Q*̂ is to map the final state back to |¦

_{i}〉. We have

*M*̂ and

*Ô*,

*P*̂,

*Q*̂ give the same output, the decomposition is therefore equivalent to the unitary operator

*M*̂. The idea of decomposition can be found in Fig. 3.

*Q*̂ is a post-processing on Eve’s states. According to quantum data processing theorem [24

24. R. Ahlswede and P. Lober, “Quantum data processing,” IEEE Trans. Inf. Theory , **47**, 1 pp 474–478 (2001).
[CrossRef]

*Q*̂ since

*Q*̂ can only decrease Eve’s accessible information. In anther word, considering

#### 3.2.2. Mathematical description of the channel model

*P*̂. Whatever Bob’s state, he can infer Eve’s input mode because he also knows Alice’s sent state. Mathematically,

*ε*. One should note that although the notation here implies that Eve is using a two mode state, Eve is not actually limited to a two mode state. The quantumtomography only guarantees that the inputmode εn to the beamsplitter be a thermal state. Subscript εr denotes Eve’s arbitrary number of modes, that remain besides

_{n}*ε*. Since Eve’s entire quantum state is pure, the condition in Eq. (22) must satisfy,

_{n}*ϕ*(

*n*)|

*ϕ*(

*n*′)〉=

*δ*′.

_{n,n}*a*′, the state is expressed in the Fock basis and in mode

*a*the state is expressed in the coherent basis. Alice then makes a photon number counting measurement onmode

*a*′, which projects the state of mode a into one of the four coherent states, i.e.,

*B*̂

_{b,hom}(η

_{m}) and

_{1}and BS

_{2}.

*b*′. Each measurement results, by the state reduction postulate of quantum mechanics, in the rest of the system is collapsing into a pure quantum state. Suppose Bob’s homodynemeasurement results in a real-valued number

*X*, then the system collapses into the state

*a*′ and the hom mode, one obtains Eve’s density matrix given the measurement result

*X*,

*ρ*̂

*is generated is*

^{X}_{E}*r*=

_{B}*X*+

*N*that is the sum of

_{el}*X*, from the homodyne measurement, and

*N*, which is a Gaussian distributed random variable denoting the electronic noise.

_{el}*Without*post-selection, the protocol requires that Bob quantize

*r*according to its sign. If

_{B}*r*>0 Bob sets

_{B}*q*=1, otherwise, Bob sets

*q*=−1. We are interested in the conditional density matrix of Eve given Bob’s quantization result. Without loss of generality, we only analyze the case in which

*q*=1.

*X*. However, Bob’s quantization result not only depends on

*X*, but also depends on

*N*, which is independent of

_{el}*X*. We can always regard Eve’s conditional density matrix as a superposition of different

*ρ*̂

*with different probability*

^{X}_{E}*p*(

*ρ*̂

*|*

^{X}_{E}*q*=1). Therefore, Eve’s conditional density matrix can be written as Eq. (30),

*p*(

*ρ*̂

*|q=1). According Bayes’ theorem,*

^{X}_{E}*p*(

*ρ*̂

*) from Eq. (28) and because of Alice’s symmetric signaling, we have*

^{X}_{E}*V*, which is the variance of the electronic noise.

_{el}*e*. According to Eq. (7), in order to obtain

_{AB}*e*, we have to calculate the signal-to-noise ratio of Bob. When the quantumchannel is introducedwith some excess thermal noise, Bob’s noise is actually made up of three different parts. The first part is the vacuumnoise, whose variance is always

_{AB}*V*. And the third part is the thermal noise. The variance of the thermal noise depends on τ, which is the squeezing factor of Eve’s EPR source, and η, which is the quantum efficiency of the channel. Let the average thermal photon number be 〈

_{el}*n*〉. We have

_{th}#### 3.3. QIQO CVQKD with post-selection

*N*and the code rate is

*R*=(1-

*ε*)

*C*, whereC is the channel capacity, in this case decoding time complexity is function of

*ε*and

*N*. Typically, it grows polynomially with

*N*. It has also been conjectured in [25] that per-bit complexity of message-passing decoding of LDPC code over any ”typical” channel, such as a binary erasure channel or a binary symmetric channel, is

*π*is the decoding error rate. So the closer the code approaches the Shannon limit, the more complex the code. In other word, the requirement of high

*β*

_{0}leads to very complex codes. Even so, the decoding error probability drops only polynomially with code length for LDPC codes, which requires even more time complexity to reduce the block error rate to suitable levels.

*e*to be very high, i.e., around 25%. In order to fit the error rate to the requirements of those good codes, we need to modify

_{AB}*e*while also changing

_{AB}*β*

_{0}little. Post-selection satisfies these requirements.

*r*=

_{B}*X*+

*N*when electronic noise is included. According to the proposed protocol, when

_{el}*r*> 0, Bob quantizes it into

_{B}*q*=1, otherwise Bob quantizes it into

*q*=−1. With post-selection, we set a threshold

*T*>0. Bob’s quantization rule is modified as follows: for the case

*r*>

_{B}*T*, he quantizes

*q*=1, for the case -

*T*≤

*r*≤

_{B}*T*, he sets

*q*=0 and for the case

*r*<-

_{B}*T*, he sets

*q*=−1. Finally, Alice and Bob discard data where

*q*=0 and only make error correction on those data where

*q*≠0. Bob’s decision rule for post-selection can be visualized in Fig. 4:

*ρ*̂

_{E}under post-selection, we need to reevaluate the probability of each

*ρ*̂

*. We denote the new possibility as*

^{X}_{E}*p*(

*ρ*̂

*|*

^{X}_{E}*q*≠0). Using Bayes’ theorem, it is rewritten to be

*u*, the variance of noise

_{i}*V*and the

_{B}*T*, and can be written as:

*ρ*̂

*can be therefore written as*

^{X}_{E}*p*(

*ρ*̂

*|*

^{X}_{E}*q*=1). According to Eq. (31), several terms must be calculated. The first term on the numerator is exactly the same as Eq. (28). The second term of the numerator can be expanded to

*e*for post-selection. The symmetry of the states implies that the error rate is the same. So for simplicity, we only calculate the error rate when Alice encodes |

_{AB}*α*

_{1}〉. We obtain:

*T*=1, almost 10% of the data is selected. Therefore, if the clock rate is high enough, post-selection will not be the factor that limits the system.

## 4. Discussion of results

*β*

_{0}. This is as expected because we assumed that Eve could make use of the excess noise and thus achieve higher mutual information with Bob. Secondly, it is also clear why

*β*

_{0}increases with increasing signal-to-noise ratio. This is because at higher SNRs, the signal amplitude increases, which leads Eve to better discrimination between the four states sent by Eve. In order to take advantage of coding, we require a relatively low

*β*

_{0}and thus we require error correcting codes that work at low SNR. However, at low SNR, the error probability of the binary symmetric channel increases. As discussed in the previous section, very good codes have been found for binary symmetric channels but they are very sensitive to the error probability of the channel. In order to make those codes applicable to our case, we use post-selection on Bob’s received data so that the secrecy capacity (per retained bit) between Alice and Bob goes up dramatically, the error probability (per retained bit) drops dramatically, while the required

*β*

_{0}remains almost constant. For 25km QIQO CVQKD, a threshold of

*T*=1 is set for post-selection. This leads to postselection of 10% of Bob’s data. For 50km QIQO CVQKD, a threshold

*T*=2 leads to retention of about 1% of Bob’s data. For example, with a broadband source with a symbol rate of 10GS/sec, the retained 1 GS/sec and 100 MS/sec for 25km and 50km QIQO CVQKD respectively. Therefore, the post-selection would not limit the clock rate of the system. For 25km QIQO CVQKD with post-selection, the ideal working region is at a signal-to-noise ratio about 0.25, where the secrecy capacity is 0.2bit/channel use, the error probability is less than 10% and the required

*β*

_{0}is about 60%. For 50km QIQO CVQKD, the ideal working region is at signal-to-noise ratio about 0.15, where the secrecy capacity is 0.15bit/channel use, the error probability is less than 10% and the required

*β*

_{0}is about 75%.

*β*=80%. The secrecy key rate per channel use when we take the inefficiency of the error correction code into account is Δ

*I*=

*βI*(

*A;B*)−

*χ*(

*B;E*)=0.1 at 25 km at signal-to-noise ratio 0.45. This results after privacy amplification in a final key rate of 60 kbps at 25 km with channel loss 70%. This provides a speed-up by a factor of 25 over the existing experiment that uses a protocol secure against collective attacks.

17. Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A **79**, 012307 (2009).
[CrossRef]

21. T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A **76**, 030303(R) (2007).
[CrossRef]

## 5. Conclusion

## 6. Appendix

*conditional*gaussian states instead of global gaussian states. If the global state, such as

*ρ*̂

_{E}, is not Gaussian, it’s difficult to find a analytical solution for von Neumann entropy when there excess noise is introduced into the channel. Fortunately, as long as the amount of excess noise is small, it’s still possible to get numerical solutions.

*τ*is small, the amplitudes for large photon numbers are so small that those terms are negligible. For this simulation, we have the excess noise about 0.005 of one shot noise unit and this leads to

*τ*=0.033 for 25k

*m*and

*τ*=0.0167 for 50k

*m*. Using only the first three terms of the expansion is then justified. In other words, the terms up to 2 photons are preserved. We let

*EMAX*=2 denoting the maximal number of photons.

*ε*is interacted with mode

_{n}*a*. However, since the quantum state in mode

*ε*is represented in Fock space, it needs to be transformed into coherent form so that the operation of BS

_{n}_{1}is performed on two coherent states and the outcome of the operation is also coherent states. Here we use another approximationwhere |

*n*〉 is represented as the superposition of

*n*+1 coherent states [27

27. J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A **51**, 5 (1995).
[CrossRef]

*c*(

*r*) is used to normalize |n, r〉. We have

*c*|

_{n}^{2}is the probability amplitude of |

*n*〉 in |

*n,r*〉. In practice, we set

*r*=0 for |0〉 and

*r*=0.1 for other Fock states.

*a*′ mode to get the density matrix of mode

*b*′,

*ε*′

_{n},

*εr*and hom.

*b*′,

*ε*′

_{n},

*εr*and

*hom*is made up of 4 different pure states |Ω

_{i}〉 dependingAlice’s measurement of the photon counting on mode

*a*′. This exactly corresponds to the case in which Alice prepares one of the four coherent states and sends it to Bob. For each of the |Ω

_{i}〉, Bob then make a homodynemeasurement on his mode

*b*′. As have been discussed, the outcome of the homodyne measurement is a gaussian distributed continuous random variable with average value

*V*. Physically, for each of the |Ω

_{S}_{i}〉, Bob’s measurement outcome has infinite possibilities. However, only those values distributed close to

*u*occur with high possibility. As an approximation, out simulation only takes the value in the range of [

_{i}*u*−6√

_{i}*V*+6√

_{S},u_{i}*V*]. Another approximation is that instead of processing the continuous data in the range [

_{S}*u*−6√

_{i}*V*+6√

_{S},u_{i}*V*], we divided it into

_{S}*XMAX*bins with equal widths and made un approximation that Bob’s measurement only has

*XMAX*different possibilities instead of infinite possibilities. Suppose each bin has left bound

*lb*and right bound

_{i,k}*rb*, with 1≤

_{i,k}*k*≤

*XMAX*. Then the measurement result is

*X*=(

_{i,k}*lb*+

_{i,k}*rb*)/2 with possibility

_{i,k}*ε*′

_{n},

*ε*and

_{r}*hom*can be approximately written as

*ρ*̂

_{E}, we further trace |

*ψ*〉〈

^{i,k}*ψ*| over mode

^{i,k}*hom*. For low signal-to-noise ratio, the average photon number in mode

*hom*is also low. If we use Fock basis to expand mode

*hom*, we can neglect those quantum states with large photon numbers. Suppose we can present mode

*hom*in Fock space and only keep the states up to

*HMAX*photons. Then

*ρ*̂

_{E}can be written as

*p*(

*j*|

*X*)=〈

_{i,k}*ψ*|

_{i,k}*j*〉〈

*j*|

*ψ*〉. Here we ignored the subscript for |

_{i,k}*j*〉

_{hom}. We define a global possibility

*ρ*̂

*E*in the Eq. (51)

*E*used to describe the mode. The problem of calculating the von Neumann entropy

*S*(

*ρ*̂

*E*) is equivalent to solving the eigenvalue of the correspondingGram matrix [28]. Each element of the Gram matrix is

*G*are equivalent to the non-zero eigenvalues of

*ρ*̂

*E*. Suppose the non-zero eigenvalues of G are {

*λ*

_{1},

*λ*

_{2},…,

*λ*}, then

_{n}*p*(|

*ε*〉|

_{i,j,k}*q*=1) for calculating

*S*(

*ρ*̂

_{E|q=1}). We first rewrite

*p*(|

*ε*〉|

_{i,j,k}*q*=1) according to Bayes’ theorem,

*ρ*̂

_{E|q=1}=∑

*(|*

_{i,j,k}p*ε*〉|

_{i,j,k}*q*=1)|

*ε*〉 we can calculate

_{i,j,k}*S*(

*ρ*̂

_{E|q=1}) following the Gram matrix that we have discussed above.

*p*(|

*ε*〉|

_{i,j,k}*q*≠0). As what we’ve done above, we first rewrite it according to the Bayes’ theorem.

*ρ*̂

_{E}=∑

*(|*

_{i,j,k}p*ε*〉|

_{i,j,k}*q*≠0)|

*ε*〉. We can then use the Gram matrix to calculate

_{i,j,k}*S*(

*ρ*̂

_{E}).

*S*(

*ρ*̂

_{E|q=1}) in the case with post-selection, we need to calculate the possibility

*p*(|

*ε*〉|

_{i,j,k}*q*=1). We first rewrite it according to Bayes’ theorem, which can be found in Eq. (54). But now the first term must be calculated differently. We can calculate it as Eq. (58),

*ρ*̂

_{E|q=1}and calculate

*S*(

*ρ*̂

_{E|q=1}) according the above methods.

*τ*=1×10

^{−6}in our numerical simulation. It shows that the difference of Δ

*I*is only 6.5804×10

^{−7}between numerical simulation and analytical results.

*EMAX, XMAX, HMAX*and

*r*. We can see that when

_{k}*EMAX*>2,

*XMAX*>20,

*HMAX*>6 and

*r*<0.1, we can see only tiny difference among different simulations. We believe that the simulation results are accurate enough for those parameters.

_{k}*m*without post-selection with

*EMAX*=2,

*XMAX*=20,

*HMAX*=6 and

*r*=0.1 as a reference and note it as Δ

*I*. We give the difference of Δ

_{ref}*I*

_{*}with the reference. We calculated

*EMAX*=2,

*XMAX*=20,

*HMAX*=6 and

*r*=0.1, we are already very close to the limit because if we further adjust the parameters, we get much less differences. In our final results, we just set

*EMAX*=3,

*XMAX*=30,

*HMAX*=8 and

*r*=0.05. We believe that these parameters give us numerical simulation results that are extremely close to the exact results.

## 7. Acknowledgments

## References and links

1. | C. H. Bennett and G. Grassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in |

2. | A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Phys. Rev. Lett. |

3. | M. A. Nielsen and I. L. Chuang, ‘Quantum Computation and Quantum Information, (Cambridge University Press, UK, 2000). |

4. | S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. |

5. | F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature |

6. | N. J. Cerf, M. Lévy, and G. Van Assche, “Quantum distribution of Gaussian keys using squeezed states,” Phys. Rev. A |

7. | F. Grosshans and P. Grangier, “Reverse reconciliation protocols for quantum cryptography with continuous variables,”quant-ph/0204127 (2002). |

8. | F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. |

9. | R. Namiki and T. Hirano, “Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection,” Phys. Rev. A |

10. | F. Grosshans, “CollectiveAttacks and Unconditional Security in Continuous Variable Quantum KeyDistribution,” Phys. Rev. Lett. |

11. | M. Navascués and A. Acín, “SecurityBounds for Continuous Variables Quantum Key Distribution,” Phys. Rev. Lett. |

12. | R. García-Patrón and N. J. Cerf, “Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution,” Phys. Rev. Lett. |

13. | M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. |

14. | J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N.J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A |

15. | M. Heid and N. Lütkenhaus, “Security of coherent-state quantum cryptography in the presence of Gaussian noise” Phys. Rev. A |

16. | V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “Title: A Framework for Practical Quantum Cryptography”arXiv:0802.4155(2008). |

17. | Y. Zhao, M. Heid, J. Rigas, and N. Lütkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A |

18. | Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit” Phys. Rev. A |

19. | C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Coherent-state quantum key distribution without random basis switching” Phys. Rev. A |

20. | A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, and P. K. Lam, “No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light,” Phys. Rev. Lett. |

21. | T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, “Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise,” Phys. Rev. A |

22. | J. Singh, O. Dabeer, and U. Madhow, “Capacity of the Discrete-Time AWGN Channel Under Output Quantization,” arXiv:0801.1185v1 (2008). |

23. | V. Buẑek and G. Drobný, “Quantum tomography via the MaxEnt principle,” J. Mod. Opt. |

24. | R. Ahlswede and P. Lober, “Quantum data processing,” IEEE Trans. Inf. Theory , |

25. | K. Khandekar and R. J. McEliece, “On the complexity of reliable communication on the erasure channel,” in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Jun. 2001, p. 1. |

26. | Z. Zhang and P. L. Voss, “A path towards 10 Gb/s continuous variable QKD,” LPHYS08, Trondheim, Norway. July 2008. |

27. | J. Janszky, P. Domokos, S. Szabó, and P. Adam, “Quantum-state engineering via discrete coherent-state superpositions,” Phys. Rev. A |

28. | R. Jozsa and J. Schlienz, “Distinguishability of States and von Neumann Entropy,” arXiv:quant-ph/9911009v1, 3 (1999). |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 1, 2009

Revised Manuscript: May 28, 2009

Manuscript Accepted: May 30, 2009

Published: July 2, 2009

**Citation**

Zheshen Zhang and Paul L. Voss, "Security of a discretely signaled continuous variable quantum key distribution protocol for high rate systems," Opt. Express **17**, 12090-12108 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-12090

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### References

- C. H. Bennett and G. Grassard, "Quantum Cryptography: Public Key Distribution and Coin Tossing," in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, New York, 1984), pp. 175-179.
- A. K. Ekert, "Quantum Cryptography Based on Bell’s Theorem," Phys. Rev. Lett. 67, 661-663 (1991). [CrossRef] [PubMed]
- M. A. Nielsen and I. L. Chuang, ‘Quantum Computation and Quantum Information, (Cambridge University Press, UK, 2000).
- S. L. Braunstein and P. Van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005). [CrossRef]
- F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, "Quantum key distribution using gaussian-modulated coherent states," Nature 421, 238-241 (2003). [CrossRef] [PubMed]
- N. J. Cerf, M. Levy, and G. Van Assche, "Quantum distribution of Gaussian keys using squeezed states," Phys. Rev. A 63, 052311 (2001). [CrossRef]
- F. Grosshans and P. Grangier, "Reverse reconciliation protocols for quantum cryptography with continuous variables,"quant-ph/0204127 (2002).
- F. Grosshans and P. Grangier, "Continuous Variable Quantum Cryptography Using Coherent States," Phys. Rev. Lett. 88, 057902 (2002) [CrossRef] [PubMed]
- R. Namiki and T. Hirano, "Efficient-phase-encoding protocols for continuous-variable quantum key distribution using coherent states and postselection," Phys. Rev. A 74, 032302 (2006). [CrossRef]
- F. Grosshans, "Collective Attacks and Unconditional Security in Continuous Variable Quantum Key Distribution," Phys. Rev. Lett. 94, 020504 (2005). [CrossRef] [PubMed]
- M. Navascues and A. Acın, "SecurityBounds for Continuous Variables Quantum Key Distribution," Phys. Rev. Lett. 94, 020505 (2005). [CrossRef] [PubMed]
- R. Garcıa-Patron and N. J. Cerf, "Unconditional Optimality of Gaussian Attacks against Continuous-Variable Quantum Key Distribution," Phys. Rev. Lett. 97, 190503 (2006). [CrossRef] [PubMed]
- M. Navascues, F. Grosshans, and A. Acın, "Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography," Phys. Rev. Lett. 97, 190502 (2006). [CrossRef] [PubMed]
- J. Lodewyck, M. Bloch, R. Garcia-Patron, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, "Quantum key distribution over 25 km with an all-fiber continuous-variable system," Phys. Rev. A 76, 042305 (2007). [CrossRef]
- M. Heid and N. Lutkenhaus, "Security of coherent-state quantum cryptography in the presence of Gaussian noise" Phys. Rev. A 76, 022313 (2007). [CrossRef]
- V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, "Title: A Framework for Practical Quantum Cryptography"arXiv:0802.4155(2008).
- Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, "Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks," Phys. Rev. A 79, 012307 (2009). [CrossRef]
- Ch. Silberhorn, T. C. Ralph, N. Lutkenhaus, and G. Leuchs, "Continuous Variable Quantum Cryptography: Beating the 3 dB Loss Limit" Phys. Rev. A 89, 167901 (2002).
- C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, "Coherent-state quantum key distribution without random basis switching" Phys. Rev. A 73,022316 (2006). [CrossRef]
- A. M. Lance, T. Symul, V. Sharma, C. Weedbrook, T. C. Ralph, P. K. Lam, "No-Switching Quantum Key Distribution Using Broadband Modulated Coherent Light," Phys. Rev. Lett. 95, 180503 (2005). [CrossRef] [PubMed]
- T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, "Experimental demonstration of post-selection-based continuous-variable quantum key distribution in the presence of Gaussian noise," Phys. Rev. A 76, 030303(R) (2007). [CrossRef]
- J. Singh, O. Dabeer and U. Madhow, "Capacity of the Discrete-Time AWGN Channel Under Output Quantization," arXiv:0801.1185v1 (2008).
- V. Buzek and G. Drobny, "Quantum tomography via the MaxEnt principle," J. Mod. Opt. 47, 2823-2839 (2000).
- R. Ahlswede and P. Lober, "Quantum data processing," IEEE Trans. Inf. Theory, 47, 474-478 (2001). [CrossRef]
- K. Khandekar and R. J. McEliece, "On the complexity of reliable communication on the erasure channel," in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Jun. 2001, p. 1.
- Z. Zhang and P. L. Voss, "A path towards 10 Gb/s continuous variable QKD," LPHYS08, Trondheim, Norway. July 2008.
- J. Janszky, P. Domokos, S. Szabo, and P. Adam, "Quantum-state engineering via discrete coherent-state superpositions," Phys. Rev. A 51, 5 (1995). [CrossRef]
- R. Jozsa and J. Schlienz, "Distinguishability of States and von Neumann Entropy," arXiv:quant-ph/9911009v1, 3 (1999).

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