Witnessing effective entanglement in a continuous variable prepare-and-measure setup and application to a quantum key distribution scheme using postselection

We report an experimental demonstration of effective entanglement in a prepare-and-measure type of quantum key distribution protocol. Coherent polarization states and heterodyne measurement to characterize the transmitted quantum states are used, thus enabling us to reconstruct directly their $Q$ function. By evaluating the excess noise of the states, we experimentally demonstrate that they fulfill a nonseparability criterion previously presented by Rigas et al. [J. Rigas, O. Gühne, and N. Lütkenhaus, Phys. Rev. A 73, 012341 (2006)]. For a restricted eavesdropping scenario, we predict key rates using postselection of the heterodyne measurement results.

Figure 1

(Color online) Simplified theoretical setup.

Figure 2

Graphical representation of the entanglement criterion. For excess variances $E$ [in shot noise units (SNU)] below the curves, the correlated data $p(A;B)$ cannot be explained by separable states. Zero excess variance corresponds to the detection of a pure coherent state. Different curves belong to different quantum channel transmissions $T$. The input state overlap depends only on Alice’s choice of the coherent state amplitude $\alpha$, and not on the channel transmission.

Figure 3

(Color online) Simplified experimental setup. Alice prepares coherent polarization states with a cw laser diode and a Faraday modulator. Bob characterizes the incoming light by a heterodyne measurement consisting of two polarization sensitive homodyne setups. Eve is simulated by changing the loss of the quantum channel.

Figure 4

Demonstration of a detected signal in the time domain. Ten identical measurements were superimposed to show the variations in detector voltage due to quantum noise. The pulse duration was set to $50\mu \mathrm{s}$ at a sample rate of $1\times {10}^2\text{samples}∕\mathrm{s}$ to better visualize the pulse shape and the discrete nature of the sampled data. In the further experiments, a pulse duration of $5\mu \mathrm{s}$ was used.

Figure 5

Marginal distribution of the measured dimensionless polarization values. The dotted black curve represents the shot noise reference, recorded with vacuum in the signal mode $a_S$. The solid gray curve is Bob’s measurement value histogram for two alternating states, corresponding to Fig. 9 and the highlighted line in Table .

Figure 6

Graphical representation of the entanglement criterion. For excess variances below the curves, the correlated data $p(A;B)$ cannot be satisfied by separable states. Different curves correspond to different quantum channel losses. The open diamonds show measured averages of $E\left({\mathbf{S}}_{\mathbf{2}}\right)$ and $E\left({\mathbf{S}}_{\mathbf{3}}\right)$; their numerical values can be seen in Table .

Figure 7

Vacuum noise $Q$ function. ${\mathbf{S}}_{\mathbf{2}}$ and ${\mathbf{S}}_{\mathbf{3}}$ are proportional to the $\mathbf{X}$ and $\mathbf{Y}$ quadrature of the signal mode $a$. The peak height is indicated by the gray line. Note that all displayed quantities are dimensionless.

Figure 8

Mixed $Q$ function of the two signal states ${\rho }_1$ and ${\rho }_2$ after leaving Alice’s preparation.

Figure 9

Mixed $Q$ function of the two signal states ${\rho }_1$ and ${\rho }_2$ in balanced mixture after experiencing 54.3% of channel loss. This $Q$ function corresponds to the highlighted line in Table .

Figure 10

Both $Q$ functions of the ${\rho }_1$ and the ${\rho }_2$ state, measured directly after preparation. This figure corresponds to Fig. 8.

Figure 11

Both $Q$ functions of the ${\rho }_1$ and the ${\rho }_2$ state, measured after propagation with 54.3% losses. This figure corresponds to Fig. 9.

Figure 12

Experimental analysis of relative acceptance and average error rate depending on the postselection threshold. Open squares: Fraction of accepted states after applying a postselection threshold of $\tau$. Black circles: remaining average error rate in the postselected pulses. This measurement was produced with a channel loss of 51.7% (cf. Table , last line).