Quantum Bit Error Rate Minimization Method

Abstract

The invention relates to a IM bias voltage determining method adapted to determine an IM bias voltage corresponding to a desired Quantum Bit Error Rate based on the following formula

[00001]$Q\left(V_{\mathrm{IM}}\right)=Q_0+\frac{R_{\mathrm{err}}}{R_{\mathrm{err}}+R_{\mathrm{cor}}}$

where Q(V.sub.IM) is the QBER dependent of the IM bias voltage V.sub.IM, Q.sub.0 is the optimal minimal QBER, R.sub.err is the number of erroneous detections, R.sub.cor is the number of correct detections and V.sub.IM is the IM bias voltage.

Claims

1. IM bias voltage determining method adapted to determine an IM bias voltage corresponding to a desired Quantum Bit Error Rate based on the following formula: $Q\left(V_{\mathrm{IM}}\right)=Q_0+\frac{R_{\mathrm{err}}}{R_{\mathrm{err}}+R_{\mathrm{cor}}}$ where Q(V.sub.IM) is the QBER dependent of the IM bias voltage V.sub.IM Q.sub.0 is the optimal minimal QBER R.sub.err is the number of erroneous detections R.sub.cor is the number of correct detections V.sub.IM is the IM bias voltage

2. IM bias voltage determining method according to claim 1, wherein the predetermined Quantum Bit Error Rate is a minimum Quantum Bit Error Rate.

3. IM bias voltage determining method according to claim 1, wherein $\frac{R_{\mathrm{err}}}{R_{\mathrm{err}}+R_{\mathrm{cor}}}=\frac{1-\mathrm{Cos}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right]}{\begin{array}{c}2-\mathrm{Cos}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right]+\\ \frac{\mathrm{Sin}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right]-\mathrm{Sin}\left[\pi .Math.\left(\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}+{\alpha }_{\mathrm{RF}}\right)\right]}{\pi .Math.{\alpha }_{\mathrm{RF}}}\end{array}}$ where Q(V.sub.IM) is the QBER dependent of the IM bias voltage V.sub.IM Q.sub.0 is the minimal achievable QBER limited by other error sources R.sub.err is the number of erroneous detections R.sub.cor is the number of correct detections V.sub.IM is the IM bias voltage V.sub.0 is the optimal set voltage producing a minimum QBER V.sub.π is PI-voltage that corresponds to half the voltage difference between two QBER minima α.sub.RF is a scaling factor that accounts for the effective amplitude of the fast RF voltage pulses and which is calibrated beforehand.

4. IM bias voltage determining method according to claim 1 comprising a δV calculating step consisting in determining a voltage value δV corresponding to a correction value to be applied to V.sub.IM to obtain the optimal set voltage producing a minimum QBER.

5. IM bias voltage determining method according to claim 4, wherein δV calculating step is carried out by applying the following formula: $\partial V=\pm \frac{V_{\pi }}{\pi }.Math.\mathrm{ArcCos}\left[\frac{\begin{array}{c}\beta \left(1-2.Math.\Delta .Math.Q\right).Math.\left(\beta \left(1-\Delta .Math.Q\right)-\Delta .Math..Math.Q.Math.\mathrm{Sin}\left[\beta \right]\right)\pm 2.Math.\sqrt{2}\\ \sqrt{\begin{array}{c}\Delta .Math.Q^3.Math.{\mathrm{Sin}\left[\frac{\beta }{2}\right]}^4.Math.({\beta }^2\left(1-\frac{3.Math.\Delta .Math.Q}{2}\right)+\Delta .Math..Math.Q-.Math.\\ \Delta .Math..Math.Q.Math.\mathrm{Cos}\left[\beta \right]-\beta \left(1-\Delta .Math..Math.Q\right).Math.\mathrm{Sin}\left[\beta \right])\end{array}}\end{array}}{\begin{array}{c}{\beta }^2.Math.{\left(1-\Delta .Math..Math.Q\right)}^2+2.Math.\Delta .Math..Math.Q^2-\\ 2.Math.\Delta .Math..Math.Q.Math.\left(\Delta .Math..Math.Q.Math.\mathrm{Cos}\left[\beta \right]+\beta .Math.\left(1-\Delta .Math..Math.Q\right).Math.\mathrm{Sin}\left[\beta \right]\right)\end{array}}\right]$ where ΔQ=Q−Q.sub.0 and β=α.sub.RF.Math.π.

6. IM bias voltage determining method according to claim 1 further comprising adding said correction value δV to V.sub.IM in a first direction, i.e. positive or negative way, to obtain the optimal set voltage producing a minimum QBER, and if the obtained QBER is not minimal, adding said correction value δV to V.sub.IM in a second direction, opposite to the first direction.

7. IM bias voltage determining method according to claim 1 adapted to be carried out at start-up phase.

8. IM bias voltage determining method according to any one of claim 1 further comprising a QBER tracking step adapted to detect a QBER variation and adapted to be carried out during quantum communication upon detection of a QBER variation.

9. Quantum communication system comprising an IM bias voltage determining module carrying out the IM bias voltage determining an IM bias voltage corresponding to a desired Quantum Bit Error Rate based on the following formula: $Q\left(V_{\mathrm{IM}}\right)=Q_0+\frac{R_{\mathrm{err}}}{R_{\mathrm{err}}+R_{\mathrm{cor}}}$ where Q(V.sub.IM) is the QBER dependent of the IM bias voltage V.sub.IM Q0 is the optimal minimal QBER Rerr is the number of erroneous detections Rcor is the number of correct detections VIM is the IM bias voltage

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The invention will be described with reference to the drawings, in which the same reference numerals indicate the same feature. In particular,

[0039] FIG. 1a is a block diagram of a conventional optimum IM's bias voltage determining method with a fix voltage step size ΔV.

[0040] FIG. 1b—is a QBER vs IM bias voltage plot showing the several steps carried out in the conventional optimum IM's bias voltage determining method with a fix voltage step size ΔV. The green point indicates the arbitrarily selected starting bias voltage and corresponding QBER value, the red point the desired working point at V.sub.0 that is to be determined by the alignment method. The black lines indicate the individual iterations. As can be seen, if the step size ΔV is too large, the optimum working point can never be reached.

[0041] FIG. 2a is a block diagram of the present optimum IM's bias voltage determining method with a dynamic voltage step size δV.

[0042] FIG. 2b—is a QBER vs IM bias voltage plot showing the present optimum IM's bias voltage determining method with a dynamic voltage step size δV. The green point indicates the arbitrarily selected starting bias voltage and corresponding QBER value, the red point the desired working point at V.sub.0 that is to be determined by the alignment method. As can be seen, by applying the calculated step size δV once, the optimum working point can be reached immediately.

DETAILED DESCRIPTION

[0043] The invention will be described, for better understanding, with reference to specific embodiments. It will however be understood that the invention is not limited to the embodiments herein described but is rather defined by the claims and encompasses all embodiments which are within the scope of the claims.

[0044] As mentioned above, the present invention aims at reducing the start-up phase in quantum communication by increasing the speed of initial alignment and optimization of the intensity modulator's bias voltage for minimizing the QBER.

[0045] FIGS. 1a and 1b show the current method which is carried out to find the desired IM bias voltage for minimizing the QBER during QKD operation with a fix voltage step size ΔV.

[0046] According to this method, one first applies an initial bias voltage V.sub.IM to the IM, and then accumulates sufficient detections related to the QBER parameters and calculates the initial QBER value. In the example of FIG. 1b, the initial bias voltage is V.sub.IM=4V.

[0047] Once the initial QBER value has be determined, one adds a fix voltage value ΔV (ΔV=1V in the example of FIG. 1b) to the currently applied bias voltage in one direction (plus or minus, i.e. left or right in FIG. 1b) and one applies this new voltage V.sub.IM+ΔV (+ or − according to the direction chosen beforehand) and measures the new QBER associated with this new voltage.

[0048] If the new QBER is smaller than the previous one then the preceding step is repeated as many times as necessary to find a desired V.sub.IM=V.sub.0 producing a minimum QBER.

[0049] If the QBER is not smaller than the previous one, and is therefore larger, then one changes the chosen direction and repeats the above steps. Therefore, the direction by which ΔV is applied to V.sub.IM (added or subtracted) is changed whenever a higher QBER is measured in this method (also called regulation cycle).

[0050] As stated above, the duration until the optimal working point is found is usually too long. The inventor has therefore developed an analytical model which describes the relationship between QBER and IM bias voltage.

[0051] The IM transfer function between the applied IM bias voltage V.sub.IM and the resulting optical attenuation is commonly described as

[00005]$\frac{1}{2}-\frac{1}{2}.Math.\mathrm{Cos}\left[\frac{\pi .Math.\left(V-V_0\right)}{v_{\pi }}\right],$

where V.sub.π is called PI-voltage and corresponds to half the voltage between two output minimums (indicated by the grey arrow in FIGS. 1b and 2b). The dynamic voltage V is the sum V=V.sub.RF+V.sub.IM between the IM bias voltage V.sub.IM and the high-frequency modulation voltage V.sub.RF. The voltage V.sub.0 is the a-priori unknown offset voltage that minimizes the optical output power after the IM, and is the one that has to be determined by the regulation method. The modulation voltage V.sub.RF is rapidly varied in accordance with the quantum communication protocol to produce the corresponding optical output amplitude pattern.

[0052] The integration of the output amplitude over the valid time-bins is proportional to number of detections R.sub.cor that will be correctly detected and gives

[00006]$R_{\mathrm{cor}}\propto \frac{\pi .Math.{\alpha }_{\mathrm{RF}}+\mathrm{Sin}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right]-\mathrm{Sin}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}+\pi .Math.{\alpha }_{\mathrm{RF}}\right]}{\pi .Math.{\alpha }_{\mathrm{RF}}}.$

Here, α.sub.RF is a filling factor that accounts for the effective amplitude of the fast RF voltage pulses and which is calibrated beforehand. The integration of the output amplitude over the invalid time-bins is proportional to number of erroneous detections R.sub.err and gives

[00007]$R_{\mathrm{err}}\propto 1-\mathrm{Cos}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right].$

Therefore, the QBER that results from this model is given by

[00008]$Q\left(V_{\mathrm{IM}}\right)=\frac{R_{\mathrm{err}}}{R_{\mathrm{err}}+R_{\mathrm{cor}}}=Q_0+\frac{1-\mathrm{Cos}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right]}{\begin{array}{c}2-\mathrm{Cos}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right]+\\ \frac{\mathrm{Sin}\left[\pi .Math.\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}\right]-\mathrm{Sin}\left[\pi .Math.\left(\frac{V_{\mathrm{IM}}-V_0}{V_{\pi }}+{\alpha }_{\mathrm{RF}}\right)\right]}{\pi .Math.{\alpha }_{\mathrm{RF}}}\end{array}}$

[0053] where Q(V.sub.IM) is the QBER dependent of the IM bias voltage V.sub.IM [0054] Q.sub.0 is the minimal achievable QBER limited by other error sources [0055] R.sub.err is the number of erroneous detections [0056] R.sub.cor is the number of correct detections [0057] V.sub.IM is the IM bias voltage [0058] V.sub.0 is the optimal set voltage producing a minimum QBER [0059] V.sub.π is PI-voltage that corresponds to half the voltage difference between two QBER minima [0060] α.sub.RF is a scaling factor that accounts for the effective amplitude of the fast RF voltage pulses and which is calibrated beforehand.

[0061] By inverting the above equation, a formula for the required dynamic step size ∂V=V.sub.IM−V.sub.0 can be derived. Since the inversion can result in different realizations of the same formula, here only one is given exemplarily:

[00009]$\partial V=\pm \frac{V_{\pi }}{\pi }.Math.\mathrm{ArcCos}\left[\frac{\begin{array}{c}\beta \left(1-2.Math.\Delta .Math.Q\right).Math.\left(\beta \left(1-\Delta .Math.Q\right)-\Delta .Math..Math.Q.Math.\mathrm{Sin}\left[\beta \right]\right)\pm 2.Math.\sqrt{2}\\ \sqrt{\begin{array}{c}\Delta .Math.Q^3.Math.{\mathrm{Sin}\left[\frac{\beta }{2}\right]}^4.Math.({\beta }^2\left(1-\frac{3.Math.\Delta .Math.Q}{Z}\right)+\Delta .Math..Math.Q-.Math.\\ \Delta .Math..Math.Q.Math.\mathrm{Cos}\left[\beta \right]-\beta \left(1-\Delta .Math..Math.Q\right).Math.\mathrm{Sin}\left[\beta \right])\end{array}}\end{array}}{\begin{array}{c}{\beta }^2.Math.{\left(1-\Delta .Math..Math.Q\right)}^2+2.Math.\Delta .Math..Math.Q^2-\\ 2.Math.\Delta .Math..Math.Q.Math.\left(\Delta .Math..Math.Q.Math.\mathrm{Cos}\left[\beta \right]+\beta .Math.\left(1-\Delta .Math..Math.Q\right).Math.\mathrm{Sin}\left[\beta \right]\right)\end{array}}\right]$

[0062] For simplification, it is ΔQ=Q−Q.sub.0 and β=α.sub.RF.Math.π.

[0063] This analytical result allows calculating the required step value δV and therefore the bias voltage that minimizes the QBER immediately on the basis of the previously measured QBER. Hence, it's sufficient to apply the new IM bias voltage V.sub.IM+δV to find the optimal working point. Since the formula gives no indication about the direction in which the voltage step has to be applied (whether V.sub.IM+∂V or V.sub.IM−∂V), the first iteration may result in an incorrect bias voltage, in which case the direction has to be changed by applying V.sub.IM−∂V.

[0064] It is therefore possible to drastically reduce the time needed for measuring the optimal IM bias voltage needed for minimizing the QBER during quantum communication both, at start-up phase, and during operation in case of QBER raise due to temperature or charge drifts.

[0065] While the embodiments have been described in conjunction with a number of embodiments, it is evident that many alternatives, modifications and variations would be or are apparent to those of ordinary skill in the applicable arts. Accordingly, this disclosure is intended to embrace all such alternatives, modifications, equivalents and variations that are within the scope of this disclosure. This is for example particularly the case regarding the different apparatuses which can be used.