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 $Q\left(V_{\mathrm{IM}}\right)=Q_0+\frac{R_{\mathrm{err}}}{R_{\mathrm{err}}+R_{\mathrm{cor}}}$
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. A method to substantially minimize quantum bit error rate in a quantum key distribution system comprising: i) determining an initial quantum bit error rate (QBER) for a quantum channel; ii) using an existing intensity modulator bias voltage (VIM) to calculate a dynamic voltage step size ?V that, if applied to VIM, will yield a minimum QBER; iii) applying a voltage step of dynamic voltage step size in a first direction to the existing VIM to yield a stepped VIM; iv) redetermining a QBER existing using the stepped VIM; v) applying the voltage step in a second direction opposite the first direction to the existing V.sub.IM if and only if the redetermined QBER is greater than the initial QBER and leaving VIM unchanged if the redetermined QBER in less than the initial QBER.

2. The method of claim 1 wherein the method is initiated during a start up phase of a quantum key distribution system.

3. The method of claim 1 further comprising: tracking the QBER during operation of a quantum communication; detecting a variation in QBER during the quantum communication; and repeating operations ii)-iv) responsive to the detecting of a variation.

4. The method of claim 1 wherein $Q\left(V_{\mathrm{IM}}\right)=Q_0+\frac{R_{\mathrm{err}}}{R_{\mathrm{err}}+R_{\mathrm{cor}}}$ 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.

5. The method of claim 4, wherein $\frac{R_{err}}{R_{err}+R_{cor}}=\frac{1-\mathrm{Cos}\left[?.Math.\frac{V_{\mathrm{IM}}-V_0}{V_{?}}\right]}{\begin{array}{c}2-\mathrm{Cos}\left[?.Math.\frac{V_{IM}-V_0}{V_{?}}\right]+\\ \frac{\mathrm{Sin}\left[?.Math.\frac{V_{\mathrm{IM}}-V_0}{V_{?}}\right]-\mathrm{Sin}\left[?.Math.\left(\frac{V_{\mathrm{IM}}-V_0}{V_{?}}+{?}_{RF}\right)\right]}{?.Math.{?}_{RF}}\end{array}}$ 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.? is PI-voltage that corresponds to half the voltage difference between two QBER minima ?RF is a scaling factor that accounts for the effective amplitude of the fast RF voltage pulses and which is calibrated beforehand.

6. The method of claim 1, wherein ?V is calculated according to the following formula: $?V=?\frac{V_{?}}{?}.Math.\mathrm{ArcCos}\left[\frac{\begin{array}{c}?\left(1-2?Q\right)\left(?\left(1-?Q\right)-?Q.Math.\mathrm{Sin}\left[?\right]\right)?2\sqrt{2}\\ \sqrt{\begin{array}{c}?Q^3{\mathrm{Sin}\left[\frac{?}{2}\right]}^4({?}^2\left(1-\frac{3?Q}{2}\right)+?Q-\\ ?Q.Math.\mathrm{Cos}\left[?\right]-?\left(1-?Q\right)\mathrm{Sin}\left[?\right])\end{array}}\end{array}}{\begin{array}{c}{?}^2.Math.{\left(1-?Q\right)}^2+2.Math.?Q^2-\\ 2.Math.?Q.Math.\left(?Q.Math.\mathrm{Cos}\left[?\right]+?.Math.\left(1-?Q\right).Math.\mathrm{Sin}\left[?\right]\right)\end{array}}\right]$ where ?Q=Q?Q.sub.0 and ?=?.sub.RF.Math.?.

7. A Quantum Key Distribution (QKD) system comprising: an intensity modulator; a quantum bit error rate (QBER) determination module to determine the QBER based on an existing intensity modulator bias voltage V.sub.IM; a dynamic voltage step size calculation module to calculate a dynamic voltage step size ?V to optimize QBER, wherein if ?V applied to VIM in a correct direction it will yield a minimum QBER; wherein a voltage step of the size ?V is applied in a first direction to an existing VIM to yield a stepped VIM and a QBER existing using the stepped VIM is redetermined; and wherein the voltage step is applied in a second direction opposite the first direction to the existing V.sub.IM if and only if the redetermined QBER is greater than the initial QBER and the VIM is left unchanged if the redetermined QBER is less than the initial QBER.

8. The system of claim 7 where in the dynamic voltage step size is calculated according to the formula: $?V=?\frac{V_{?}}{?}.Math.\mathrm{ArcCos}[\frac{\begin{array}{c}?\left(1-2?Q\right)\left(?\left(1-?Q\right)-?Q.Math.\mathrm{Sin}\left[?\right]\right)?\\ 2\sqrt{2}\sqrt{?Q^3{\mathrm{Sin}\left[\frac{?}{2}\right]}^4\left({?}^2\left(1-\frac{3?Q}{2}\right)+?Q-?Q.Math.\mathrm{Cos}\left[?\right]-?\left(1-?Q\right)\mathrm{Sin}\left[?\right]\right)}\end{array}}{{?}^2.Math.{\left(1-?Q\right)}^2+2.Math.?Q^2-2.Math.?Q.Math.\left(?Q.Math.\mathrm{Cos}\left[?\right]+?.Math.\left(1-\mathrm{?Q}\right).Math.\mathrm{Sin}\left[?\right]\right)}$ where ?Q=Q?Q.sub.0 and ?=?.sub.RF.Math.?.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will be described with reference to the drawings, in which the same reference numerals indicate the same feature. In particular,

(2) FIG. 1a is a block diagram of a conventional optimum IM's bias voltage determining method with a fix voltage step size ?V.

(3) FIG. 1bis 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 ?V. 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 ?V is too large, the optimum working point can never be reached.

(4) FIG. 2a is a block diagram of the present optimum IM's bias voltage determining method with a dynamic voltage step size ?V.

(5) FIG. 2bis a QBER vs IM bias voltage plot showing the present optimum IM's bias voltage determining method with a dynamic voltage step size ?V. 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 ?V once, the optimum working point can be reached immediately.

(6) FIG. 3 is a block diagram of a system of one embodiment of the invention.

DETAILED DESCRIPTION

(7) 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.

(8) 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.

(9) 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 ?V.

(10) 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.

(11) Once the initial QBER value has be determined, one adds a fix voltage value ?V (?V=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+?V (+ or ? according to the direction chosen beforehand) and measures the new QBER associated with this new voltage.

(12) 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.

(13) 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 ?V is applied to V.sub.IM (added or subtracted) is changed whenever a higher QBER is measured in this method (also called regulation cycle).

(14) 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.

(15) The IM transfer function between the applied IM bias voltage V.sub.IM and the resulting optical attenuation is commonly described as

(16) $\frac{1}{2}-\frac{1}{2}\mathrm{Cos}\left[\frac{?.Math.\left(V-V_0\right)}{v_{?}}\right],$
where V.sub.? 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.

(17) 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

(18) $R_{\mathrm{cor}}?\frac{?.Math.{?}_{\mathrm{RF}}+\mathrm{Sin}\left[?.Math.\frac{V_{\mathrm{IM}}-V_0}{V_{?}}\right]-\mathrm{Sin}\left[?.Math.\frac{V_{\mathrm{IM}}-V_0}{V_{?}}+?.Math.{?}_{\mathrm{RF}}\right]}{?.Math.{?}_{\mathrm{RF}}}.$
Here, ?.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

(19) $R_{\mathrm{err}}?1-\mathrm{Cos}\left[?.Math.\frac{V_{\mathrm{IM}}-V_0}{V_{?}}\right].$
Therefore, the QBER that results from this model is given by

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

(21) 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.? is PI-voltage that corresponds to half the voltage difference between two QBER minima ?.sub.RF is a scaling factor that accounts for the effective amplitude of the fast RF voltage pulses and which is calibrated beforehand.

(22) 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:

(23) $?V=?\frac{V_{?}}{?}.Math.\mathrm{ArcCos}\left[\frac{\begin{array}{c}?\left(1-2?Q\right)\left(?\left(1-?Q\right)-?Q.Math.\mathrm{Sin}\left[?\right]\right)?2\sqrt{2}\\ \sqrt{\begin{array}{c}?Q^3{\mathrm{Sin}\left[\frac{?}{2}\right]}^4({?}^2\left(1-\frac{3?Q}{Z}\right)+?Q-\\ ?Q.Math.\mathrm{Cos}\left[?\right]-?\left(1-?Q\right)\mathrm{Sin}\left[?\right])\end{array}}\end{array}}{\begin{array}{c}{?}^2.Math.{\left(1-?Q\right)}^2+2.Math.?Q^2-\\ 2.Math.?Q.Math.\left(?Q.Math.\mathrm{Cos}\left[?\right]+?.Math.\left(1-?Q\right).Math.\mathrm{Sin}\left[?\right]\right)\end{array}}\right]$

(24) For simplification, it is ?Q=Q?Q.sub.0 and ?=?.sub.RF.Math.?.

(25) This analytical result allows calculating the required step value ?V 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+?V 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.

(26) 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.

(27) FIG. 3 is a block diagram of a system of one embodiment of the invention. An emitter 300 communicates with a receiver 330 over a quantum communication channel 320. To facilitate quantum key distribution over the quantum channel 320, it is desirable to optimize the bias voltage V.sub.IM applied to the intensity modulator 302 so that QBER is minimized. A voltage source 304 applies an initial V.sub.IM to the intensity modulator 302. A quantum bit error rate (QBER) detection module 306 determines a QBER at an existing bias voltage based on the error rate seen with the V.sub.IM that is being applied. This determination is performed consistent with the method described above. Using the QBER and the known current V.sub.IM, a dynamic voltage step calculation module 308 calculates a dynamic voltage step ?V that will minimize the QBER when the current V.sub.IM is incremented or decremented by the voltage step ?V. ?V is provided to the voltage source 304 that then shifts the voltage applied to bias the intensity modulator 302 such that V.sub.IM is incremented or decremented by the voltage step ?V. The manner in which ?V may be calculated is described in detail above.

(28) The QBER detection module 306 then redetermines the QBER at the new V.sub.IM and if the QBER is reduced from the prior determination no further action is required. If the QBER has increased, the original V.sub.IM is shifted in the opposite direction (i.e. if it was initially incremented, it is decremented and vice versa) by ?V. In this way a maximum of two iterations are required to achieve a V.sub.IM that yield and optimal QBER. As the system operates, as noted above, e.g., temperature changes may require recalibration of V.sub.IM, but each calibration requires no more than two iterations.

(29) 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.