El crecimiento económico
GRÁFICO 4.6 PRODUCTIVIDAD TOTAL DE LOS FACTORES (promedios simples, 1960=1)
Hom
AM PM MixerPII
HV
Error Signal MixerPII
Error SignalDigital Control System
PZT
Figure 7.3: Schematic diagram of the homodyne lock utilized in P&M experiment. The input signal was modulated in both quadratures using electro-optic modula- tors. The two photocurrents coming from the two detectors of homodyne detection were subtracted, and was sent to the FPGA card. The error signal was extracted digitally, providing a proper feedback signal after being passed through a digital PII
(proportional-double integral) controller.
an achievable value for CV-QKD [10, 123, 126]. Assuming it to be constant makes the comparison between different protocols possible. The result of these measurements is illustrated in figure 7.4.
Figure 7.5 depicts the results obtained for the DR coherent state protocol for optimum modulation variances. We show that secure key can be generated after applying 0.6±0.047 dB of loss, in good agreement with our theoretical model, which predicts our current setup would be secure up to applying 0.9 dB of loss. With the P&M protocol, we have much more freedom to vary the modulation variance and hence the virtual entanglement in order to optimize the secret key rate for each loss setting. As such, we could achieve a loss tolerance superior to the EB DR protocol, whilst using only the cheapest and most readily available quantum optical resources.
7.5
Error Estimation
In order to estimate the errors in calculating the key rates and other variables all vari- ables (A,B, ...) are considered to be independent and the error propagation formula is applied as follows: ∆Z= r (∂Z ∂A) 2∆A2+ (∂Z ∂B) 2∆B2+· · · (7.4)
loss = 0dB loss ≈ 0.2dB loss≈ 0.6dB loss ≈ 1.3dB loss ≈ 2.8dB 0.0 0.5 1.0 1.5 2.0 0.2 0.0 -0.2 -0.4 -0.6 Squeezing Parameter K ey r at e (b it s/ s) loss = 0 dB (b) (a) 0.0 0.5 1.0 1.5 2.0 0.4 0.2 0.0 -0.2 -0.4
Figure 7.4: (a) variation of key rates versus effective modulation squeezing parameter for 5 different values of the applied loss. A theory line with the average value of the applied loss is fitted on the experimental data points. Data points surrounded by dashed circles correspond to the optimum modulation squeezing parameters which resulted the highest key rate for each loss setting. The key rates resulting from these optimum modulation variances are shown separately in Fig 7.5. (b) demonstrates the gap between the theoretical model and the realistic model which captures the experimental imperfections and matches well with the experimental results for the
§7.5 Error Estimation 97 Theory DR protocol Experimental point 0.0 0.5 1.0 1.5 2.0 2.5 - 0.2 - 0.4 DR Protocol EPR Steering & Positive Key (4) (1) (2) (3) (5) (1) (2) (3) (4) (5) 3.0 0.0 0.2
Loss (dB)
Key
r
at
e (
b
it
s/
s)
(a)
(b)
Figure 7.5: (a) key rates versus loss in dB scale for P&M coherent state DR protocol. Experimental error bars were estimated using error propagation of uncertainties. Panels (b) show the connection between the measured values of EPR steering and
are functions of the variances and covariances of the collected data. Each measure- ment was repeated 5 times for the this scheme, which provided us sufficient data to estimate the error.
7.6
Computer Modeling
To model the P&M experiment, I benefit from the equivalent EB picture and begin withσin =EPR(S)given as follows :
EPR(s) = cosh(2s) 0 sinh(2s) 0 0 cosh(2s) 0 −sinh(2s) sinh(2s) 0 cosh(2s) 0 0 −sinh(2s) 0 cosh(2s) (7.5)
wheresis the squeezing parameter and is related to the modulation variance via cosh(2s) =VS+1.
Recall that one part of the equivalent EPR state was sent to Bob through a lossy channel where he performed a homodyne detection, and on the other part Alice performed a heterodyne detection. To model the heterodyne detection a vacuum state was introduced to the first mode (Alice’s mode) and mixed with it on a 50:50 beam splitter. Although much more flexible, the coherent state setup naturally still suffers from imperfections which in turn determine the optimum modulation. These imperfect correlations arise partly from cross correlation between orthogonal quadra- tures and partly from our limited ability to maximize the correlation between Alice and Bob’s modes using electronic delay. Both phenomena can be thought of as an unknown rotation in the system. Hence, a rotation operator with small angles is ap- plied to both quadratures of the second mode (Bob’s mode) to model the imperfect correlation between Alice and Bob’s modes.
To model the lossy channel, a thermal state with the variance of 1+χwas intro- duced to the system and mixed with the second mode on a beamsplitter of transmis- sion T. Here χis the excess noise entering to the system through the transmission channel. The excess noiseχwas very small compared to decoherence effects caused by cross correlation between orthogonal quadratures. Hence, it has negligible impact on the key rate calculations. The transmission can again be determined directly by taking the ratio of the correlation at a particular setting with the correlation at full transmission. Here again all the states and operators were assumed to be Gaussian. Therefore, the final covariance matrix can be described by symplectic transformation (see subsection 2.6.1 and equation 6.2) as follows:
σout =S[σin
M Vχ(B)
M
diag(1, 1)]ST, (7.6) where Vχ = diag(1+χx, 1+χp) , and χx(p) is the excess noise introduced to ˆx(p)ˆ
§7.6 Computer Modeling 99
quadratures. The operatorSis given by :
S= RT(θx,θp)BS1,4(1/2)BS2,3(T). (7.7)
where RT is the rotation operator described by the matrix 8.17 and BSis the beam- splitter operator defined by matrix 8.18.
The variation of key rates versus the equivalent modulation squeezing parameter for 5 different transmission was previously shown in Fig 7.4. As is clear from Fig 7.4, using coherent states provides a much greater range over which to tune the equivalent squeezing. When using actual EPR states the maximum achievable value forsis around 0.8, well short of the optimum.
As the modulation was increased, so too was the detrimental effect on the correla- tions, leading to a smaller value for the optimal modulation parameter which for an ideal experiment would depend only uponβ. In inset(b) of Fig 7.4 the gap between the ideal case without cross correlation and the realistic case is shown for the case of zero applied loss.
The key rate resulting from optimum modulation variances for each loss setting is chosen and plotted versus the applied loss in Fig 7.5. (a). In addition, our model predicts that if the cross correlation between Alice and Bob’s modes was zero, the loss tolerance of the system would extend from 0.9 dB to 1.3 dB as depicted in Fig 7.6. This would result the extension of the range of secure communication for this protocol.
Loss (dB)
0.5 0.0 1.0 1.5 2.0 2.5 3.0 0.0 - 0.2 - 0.4 0.2 0.4K
ey Ra
te
D
R P
ro
toc
ol
s
(bi
ts
/s
)
Secure communication range for asystem without imperfections
Currently achieved range of secure communication
Figure 7.6: Comparison of the key rates resulted from optimum modulation variances versus the applied loss in dB scale for our experimental system and an ideal system with out any imperfections. The reconciliation efficiency is taken as 0.95 for both cases. These theoretical lines are produced using the model described in the text.
7.7
Summary
In this chapter I detailed our experimental implementation of one-sided device- independent quantum key distribution using coherent states. I started by giving the overall view of our experiment then I described the way I calibrated the function generator outputs. I explained our control and data acquisition system and presented our experimental results. I detailed the computer modelling I perform to understand the setup better and the way to improve it.
Chapter8
Bell-like Correlations for
Continuous-Variables
8.1
Introduction
In the early 1930s, Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) in their seminal paper[3] pointed out the quantum entanglement to demonstrate that Quan- tum Mechanics was incomplete. They hoped that a more comprehensive and less troubling theory could replace it one day. Their argument was based on two assump- tions of realism (that physical objects have real properties determining the outcome of a measurement) and locality (that the physical reality in one location is not in- fluenced instantaneously by measurements conducted at a distant location), together called "local realism" or "local hidden variables". However, EPR did not provide any test to prove local realistic theories. These concepts were first quantified in 1964 by John Bell [5] through his famous inequalities. Bell assumes that a pair of particles have interacted and separated, where two distant observers perform measurement on them. If local realistic theories are correct, the correlations between different out- comes of measurements should obey certain constraints defined by Bell’s inequalities. While the violation of the Bell’s inequalities disprove all local realistic theories.
In addition, it has particular implication in quantum key distribution where the violation of the Bell’s inequality will rule out any tampering of the quantum source leading to the development of device-independent QKD as described earlier in chap- ter 4.
In this chapter I review the Bell’s inequality and show how it can be extended to the continuous-variables scheme. Then I will discuss the computer simulation I per- formed to model two experiments proposed in ref [1, 2] that can violate the Bell’s inequality in continuous varaibles. The feasibility of these experiments open the door for real-life implementation of device-independent CV-QKD.