Entangled-coherent-state quantum key distribution with entanglement witnessing

An entanglement-witness approach to quantum coherent-state key distribution and a system for its practical implementation are described. In this approach, eavesdropping can be detected by a change in sign of either of two witness functions: an entanglement witness $\mathcal{S}$ or an eavesdropping witness $\mathcal{W}$. The effects of loss and eavesdropping on system operation are evaluated as a function of distance. Although the eavesdropping witness $\mathcal{W}$ does not directly witness entanglement for the system, its behavior remains related to that of the true entanglement witness $\mathcal{S}$. Furthermore, $\mathcal{W}$ is easier to implement experimentally than $\mathcal{S}$. $\mathcal{W}$ crosses the axis at a finite distance, in a manner reminiscent of entanglement sudden death. The distance at which this occurs changes measurably when an eavesdropper is present. The distance dependence of the two witnesses due to amplitude reduction and due to increased variance resulting from both ordinary propagation losses and possible eavesdropping activity is provided. Finally, the information content and secure key rate of a continuous variable protocol using this witness approach are given.

Figure 1

Schemes for phase-based coherent-state key distribution with single-photon triggers. (a) Scheme of the current paper. A beam splitter splits a laser beam into two beams in identical coherent states (solid black lines); a phase shifter compensates for the phase gained in the reflected state. A single photon also enters a superposition of two path states (dashed red lines). Due to the joint interaction of coherent state and the photon within Kerr media, the beams enter an equal-weight superposition of product states of pairs of oppositely phase-shifted coherent states. The specific form of the detection unit will be different for each of the applications to be discussed in the text. (b) Scheme of [6], with two additional interferometers to test for Bell violations.

Figure 2

A model of a Gaussian cloner [28] applied by Eve to Bob's beam. The cloner can be realized by combining an amplifier with a beam splitter. Besides the input from the source ($a_{\mathrm{in}}$), there are two additional inputs: one to the amplifier ($b_{\mathrm{in}}$) and the other to the first beam splitter ($c_{\mathrm{in}}$). The result is two outputs with quadratures that have means equal to that of the input. One copy ($a_{\mathrm{out}}$) is sent on to Bob. Eve keeps the other ($c_{\mathrm{out}}$) to make measurements on.

Figure 3

(a) Behavior of entanglement witness $\mathcal{S}$ as a function of distance, assuming that the amplitudes have decay constants $K=0.046$ km${}^{-1}$. Here, $\alpha =100$ and $\varphi =0.1$. The red dashed line assumes symmetric decay. The solid blue line assumes that the source is in Alice's laboratory, so that decay occurs only on one side; the values in this latter case were magnified by a factor of 100 before plotting.

Figure 4

The larger distance of disentanglement for the asymmetric case in Fig. 3 is due to the fact that the two coherent states move apart in phase space, whereas in the symmetric case, both decay toward the same vacuum state.

Figure 5

Behavior of entanglement witness $\mathcal{S}$ as a function of distance, with and without eavesdropping, for $\alpha =1000$ and $\varphi =0.1$, assuming symmetric decay. The curves correspond to no eavesdropping (solid red), $\gamma =0$ (dashed green), $\gamma =-1$ (dash-dotted blue), and $\gamma =-2$ (dotted black).

Figure 6

Eavesdropping witness $\mathcal{W}$ value vs Alice-Bob distance $d=d_1+d_2$. From left to right, the curves have parameter values $\left|\alpha \varphi \right|=10$, $\left|\alpha \varphi \right|=20$, $\left|\alpha \varphi \right|=50$, $\left|\alpha \varphi \right|=100$, $\left|\alpha \varphi \right|=500$, $\left|\alpha \varphi \right|=1000$, and $\left|\alpha \varphi \right|=5000$. $K=0.046$ km${}^{-1}$ is used for the 1550 nm telecom window. An expanded view of the region enclosed in the dashed box is shown in Fig. 7.

Figure 7

Expanded view of the region enclosed in the dashed box in Fig. 6. The curves have parameter values $\left|\alpha \varphi \right|=500$ (dashed brown), $\left|\alpha \varphi \right|=1000$ (dotted violet), and $\left|\alpha \varphi \right|=5000$ (solid red). $K=0.046$ km${}^{-1}$ is used for the 1550 nm telecom window.

Figure 8

The distance $d_0$ at which axis crossing occurs, as a function of the parameter $\left|\alpha \varphi \right|\approx |\alpha sin\varphi |$ for $K=0.046$ km${}^{-1}$. The solid blue curve is the Alice-to-Bob distance. This is the same as the source-to-Bob distance for the case of loss in only one arm (source in Alice's laboratory), and is double the source-to-Bob distance for case of equal distances in both branches (dashed red curve).

Figure 9

The solid blue line is the distance $d\left(\gamma \right)$ (in kilometers) between Alice and Bob at which $\mathcal{W}$ crosses the axis, as a function of eavesdropping parameter $\gamma$. The amplitude and phase values assumed are $\alpha ={10}^5$ and $\varphi =0.1$. The dashed red line shows the distance $d_0$ in the absence of eavesdropping.

Figure 10

The ratio of added variances for Bob and Eve, $r=\frac{{\sigma }_E^2}{{\sigma }_B^2}$, is plotted vs $\gamma$, for $\varphi =0.1$. The curve is independent of $\alpha$.

Figure 11

The curves shift horizontally by approximately a constant amount $\Delta d(\gamma ,\Lambda )$ in the presence of eavesdropping. Here the solid red line is in the absence of eavesdropping for $\alpha =1000$ and $\varphi =0.1$. The dashed blue line is in the presence of eavesdropping with $\gamma =-1$. The crossing of the $\mathcal{W}=\Lambda$ line can be used instead of the $\mathcal{W}=0$ crossing; this allows more photons to still be present for measurement, increasing measurement accuracy.

Figure 12

The change $\Delta d$ in the distance at which the curve of $\mathcal{W}$ crosses the value $\mathcal{W}$ is plotted vs the eavesdropping parameter $\gamma$, for the values $\Lambda =-1$ (dashed blue) and $\Lambda =-10$ (solid red). The curves are independent of $\alpha$ and change very little for $\Lambda <-10$. The value $\varphi =0.1$ was used for the plots.

Figure 13

Mutual information between Alice and Bob, assuming both have the same initial amplitude $\alpha$. From the top line downward, the initial amplitudes are $\alpha ={10}^5$ (red), $\alpha ={10}^4$ (violet), $\alpha ={10}^3$ (black), and $\alpha =100$ (blue). $\varphi =0.1$ for all curves.

Figure 14

Mutual information between Alice and Bob in the presence of eavesdropping, assuming they have equal initial amplitudes and equal losses. Solid red curve: no eavesdropper. Dotted black curve: $\gamma =-1$. Dash-dotted green curve: $\gamma =0$. Dashed blue curve: $\gamma =1$. The values $\varphi =0.1$ and $\alpha =1000$ were used for all curves. The same curves give the mutual information between Bob and Eve, but with $\gamma$ and $-\gamma$ interchanged.

Figure 15

Secret-key rate $\kappa$ between Alice and Bob in the presence of eavesdropping, assuming they have equal initial amplitudes and equal losses. Solid red curve: no eavesdropper. Dotted black curve: $\gamma =2$. Dashed blue curve: $\gamma =1$. Dash-dotted green curve: $\gamma =0$. $\kappa$ vanishes identically for all $\gamma \le 0$. The values $\varphi =0.1$ and $\alpha =1000$ were used for all curves.

Figure 16

Mutual information between Alice and Bob, assuming they have different initial amplitudes $\alpha$ and $\beta$. From the right to left, Bob's initial amplitudes are $\beta ={10}^5$ (red), $\beta ={10}^4$ (violet), $\beta ={10}^3$ (black), and $\beta =100$ (blue). $\varphi =0.1$ and $\alpha =100$ for all curves.