Time-bin criterion

The time-bin criterion (TBK) is a quality criterion . The knowledge of the values ​​about visibility (German interference contrast) and TBK enable the assessment of the functional quality of an experimental setup in time-bin configuration (TBI) for the purpose of realizing the time-bin coding (TBE).

General

The main motive for using the time-bin criterion is the creation of a quick possibility of qualitative assessment of the measurement results on the experimental setup. The TBK is described in detail in and.

Causes of development

One difficulty in building the experimental setup is the unavailability of expensive single photon sources . Therefore, a heavily damped laser is often used. This has the advantage of being very robust , which is particularly important in the rough operation of development. However, the attenuation should not be too high as long as the experimental setup is still imperfect . In the first phase it is advantageous not to use a pulse laser . A continuous wave laser is better because it simplifies some of the requirements for the detectors . A disadvantage of this pioneering is that can not be distinguished whether the detected surface of the satellite and originates or occurs from other causes. Regardless of what the measured visibility says. The TBK is used to avoid this. ${\displaystyle S_{1}}$ ${\displaystyle S_{2}}$

Real time bin configuration with ${\displaystyle \kappa }$

Development of the leading satellite , the central impulse and the lagging satellite of the time-bin configuration depending on the coupled imperfection and imbalance. Change of line width not taken into account. ${\displaystyle S_{1}}$ ${\displaystyle Z}$ ${\displaystyle S_{2}}$ ${\displaystyle \kappa }$

For an experimental setup in Time-bin configuration without the need for phase manipulation on the leading end satellite are at the detector , the central pulse and the trailing satellite observed. Depending on the particular imperfection and imbalance with their services are defined. ${\displaystyle \varphi }$ ${\displaystyle S_{1}}$ ${\displaystyle Z}$ ${\displaystyle S_{2}}$ ${\displaystyle \kappa }$ ${\displaystyle 0\leq \kappa \leq 2}$

• Satellite S ₁

${\displaystyle P(\psi _{K})(\kappa ,\varphi _{1}=0,\varphi _{2}=0)=\kappa ^{2}\cdot (2-\kappa )^{4}}$

• Central pulse Z

${\displaystyle P(\psi _{M})(\kappa ,\varphi _{1}=0,\varphi _{2}=0)=4\cdot \kappa ^{6}\cdot (2-\kappa )^{6}}$

• Satellite S ₂

${\displaystyle P(\psi _{L})(\kappa ,\varphi _{1}=0,\varphi _{2}=0)=\kappa ^{4}\cdot (2-\kappa )^{2}}$

Minimum and maximum measurement ${\displaystyle MIN}$ ${\displaystyle MAX}$

Development of the minimum and maximum measurement within the time bin configuration depending on the coupled imperfection and imbalance. ${\displaystyle MIN_{SZS}}$ ${\displaystyle MAX_{SZS}}$ ${\displaystyle \kappa }$

There are basically two methods for determining and . Firstly, the measurement of the two satellites and the central pulse together (index SZS ) or just the central pulse (index Z ) alone. ${\displaystyle MIN}$ ${\displaystyle MAX}$

• Method SZS

This method is used in an expansion stage of the measuring equipment, which cannot detect, select and measure the individual central pulse.

${\displaystyle MAX_{SZS}=(2-\kappa )^{2}\cdot \kappa ^{2}\cdot (\kappa ^{2}+4\cdot (2-\kappa )^{4}\cdot \kappa ^{4}+(2-\kappa )^{2})}$

${\displaystyle MIN_{SZS}=(2-\kappa )^{2}\cdot \kappa ^{2}\cdot (\kappa ^{2}+(2-\kappa )^{2})}$

• Method Z

If the evaluation electronics are ultimately fast and precise enough, another method can be used.

${\displaystyle MAX_{Z}=4}$

${\displaystyle MIN_{Z}=4\cdot (2-\kappa )^{6}\cdot \kappa ^{6}}$

proofreader ${\displaystyle m_{\bullet }}$

The value of is defined for a . This is required for the later calculation of the value . ${\displaystyle m_{\bullet }}$ ${\displaystyle \kappa =1}$ ${\displaystyle TBK}$

${\displaystyle m_{\bullet }={\frac {MAX_{\bullet ,\kappa =1}}{MIN_{\bullet ,\kappa =1}}}}$

This is fixed with and . ${\displaystyle m_{\bullet }}$ ${\displaystyle m_{SZS}=3}$ ${\displaystyle m_{Z}=1}$

Derivation

The time-bin criterion can be defined with the help of the intensities on the detectors for different states within the experimental setup.

Basic assumption

The basis for the derivation of the time-bin criterion is the assumption that for a system with a quantitative maximum and minimum : ${\displaystyle MAX}$ ${\displaystyle MIN}$

${\displaystyle MAX\geq MIN>0}$

As long as the values and distinguishable are measurable, follows for one ${\displaystyle MAX}$ ${\displaystyle MIN}$ ${\displaystyle (m\in \Re )\geq 1}$

${\displaystyle MAX-m\cdot MIN=0}$

${\displaystyle \Rightarrow }$

${\displaystyle MAX-MIN=(m-1)\cdot MIN}$

${\displaystyle MAX+MIN=(m+1)\cdot MIN}$

Time-bin criterion

The time-bin criterion is derived from changing the basic assumption . ${\displaystyle TBK}$

${\displaystyle {\frac {1}{m}}\cdot {\frac {MAX}{MIN}}=1}$

${\displaystyle \Rightarrow }$

${\displaystyle TBK={\frac {1}{m}}\cdot {\frac {MAX}{MIN}}}$

Visibility

By using the relations for and known from the basic assumption , the general definition of visibility is given as: ${\displaystyle MAX}$ ${\displaystyle MIN}$ ${\displaystyle V}$

${\displaystyle {\frac {MAX-MIN}{MAX+MIN}}={\frac {m-1}{m+1}}}$

${\displaystyle \Rightarrow }$

${\displaystyle V={\frac {MAX-MIN}{MAX+MIN}}}$

Corrected visibility

From and with the help of the basic assumption there is a further possibility of displaying the visibility. ${\displaystyle V}$

${\displaystyle V={\frac {TBK\cdot m-1}{TBK\cdot m+1}}=TBK\cdot V_{TBK}}$

${\displaystyle \Rightarrow }$

${\displaystyle V_{TBK}={\frac {V}{TBK}}=m\cdot {\frac {MAX-MIN}{MAX+MIN}}\cdot {\frac {MIN}{MAX}}}$

Whereby represents a corrected visibility. ${\displaystyle V_{TBK}}$ ${\displaystyle TBK}$

Decision criteria

The decision as to when the experimental setup is functioning well requires defining intervals as a function of . Three areas can be defined. ${\displaystyle \kappa }$

The power of the central impulse is greater than that of both satellites

The term provides two solutions with and within the definition range of lie. Both values ​​result in: ${\displaystyle P(\psi _{M})=P(\psi _{K})+P(\psi _{L})}$ ${\displaystyle \kappa _{1}=0{,}636\dots }$ ${\displaystyle \kappa _{2}=1{,}364\dots }$ ${\displaystyle \kappa }$

${\displaystyle V_{TBK,SZS}=0{,}5}$

${\displaystyle \Rightarrow }$

${\displaystyle 2\cdot V_{SZS}=TBK_{SZS}}$

The power of the central impulse is greater than that of a satellite

From the requirements and two usable solutions can be calculated with and . This means that the area in which a time bin configuration ( TBI ) is available is known. ${\displaystyle P(\psi _{K})=P(\psi _{M})}$ ${\displaystyle P(\psi _{M})=P(\psi _{L})}$ ${\displaystyle \kappa _{3}=0{,}597\dots }$ ${\displaystyle \kappa _{4}=1{,}403\dots }$

${\displaystyle \kappa _{1}\approx \kappa _{3}\approx 0{,}6<\kappa _{TBI}<1{,}4\approx \kappa _{4}\approx \kappa _{2}}$

The minimum and maximum measurements with the SZS method become indistinguishable

An exact coincidence of and only occurs with and . It is determined: ${\displaystyle MIN}$ ${\displaystyle MAX}$ ${\displaystyle \kappa =0=\kappa _{min}}$ ${\displaystyle \kappa =2=\kappa _{\mathrm {max} }}$

${\displaystyle {\frac {MAX_{SZS}}{MIN_{SZS}}}=1{,}05}$

The above value implies a time-bin criterion of:

${\displaystyle TBK_{SZS}={\frac {1}{m_{SZS}}}\cdot {\frac {MAX_{SZS}}{MIN_{SZS}}}=0{,}35>TBK_{{SZS}{,min}}={\frac {1}{3}}=TBK_{SZS}(\kappa =0)=TBK_{SZS}(\kappa =2)}$

Two values ​​of are defined, which exclude the presence of the time bin configuration ( NOTBI ) with and . So it follows: ${\displaystyle \kappa }$ ${\displaystyle \kappa _{5}=0{,}254\dots }$ ${\displaystyle \kappa _{6}=1{,}746\dots }$

${\displaystyle \kappa _{\mathrm {min} }=0\leq \kappa _{NOTBI}<0{,}25\approx \kappa _{5}\qquad \kappa _{6}\approx 1{,}75<\kappa _{NOTBI}\leq 2=\kappa _{\mathrm {max} }}$

application

Possible structure of an empty form for the application of the time-bin criterion via the SZS method.

For the application of the time-bin criterion with the SZS method, the experimental setup is supplied by a continuous wave laser. If the setup is able to use method Z, a strongly damped pulse laser or a single photon source is connected. The phase rotation is then changed on the electro-optical modulator (EOM) until a minimum of power appears on the detector. The minimum measurement is recorded. Similarly for the measurement of the maximum value , the phase is changed until a maximum can be seen on the detector. Since the time-bin configuration is highly temperature-dependent, several measurements should be carried out until a temperature-stable state can be recognized. The evaluation can be carried out numerically or graphically after the measurement campaign. The numerical method is described in. If a qualitative statement is sufficient, the graphic evaluation using a form is faster. In this case, only the value has to be calculated depending on the selected method as an entry into the characteristic field. An example for the graphic evaluation is also described in. ${\displaystyle MIN}$ ${\displaystyle MAX}$ ${\displaystyle TBK_{\bullet }}$

${\displaystyle TBK_{SZS}={\frac {1}{3}}\cdot {\frac {MAX_{SZS}}{MIN_{SZS}}}\qquad \qquad TBK_{Z}={\frac {MAX_{Z}}{MIN_{Z}}}}$

Extensions

Time-bin criterion and visibility and signal-to-noise ratio

For the SZS method there is a direct connection between and the signal-to-noise ratio . With a continuous coupling of laser light into the time-bin configuration, the satellites become visible on the detector as background, which more or less cover the central pulse, depending on the value of . ${\displaystyle TBK_{SZS}}$ ${\displaystyle SNR}$ ${\displaystyle \kappa }$

${\displaystyle SNR_{SZS}={\frac {P(\psi _{M})}{P(\psi _{K})+P(\psi _{L})}}={\frac {MAX_{SZS}-MIN_{SZS}}{MIN_{SZS}}}=3\cdot TBK_{SZS}-1}$

With the exposed points:

${\displaystyle SNR_{SZS,\mathrm {max} }=SNR_{SZS}(\kappa =1)=2\equiv +3\,\mathrm {dB} }$

${\displaystyle SNR_{SZS}=1\equiv 0\,\mathrm {dB} }$at and known from the decision criteria. ${\displaystyle \kappa _{1}=0{,}636\dots }$ ${\displaystyle \kappa _{2}=1{,}364\dots }$

Farther:

${\displaystyle SNR_{SZS}=(3\cdot TBK_{SZS}+1)\cdot V_{SZS}}$

Time-bin criterion and visibility and quantum bit error rate

Between the visibility and the quantum bit error rate (QBER - Quantum Bit Error Rate ) one exists proportionality .

${\displaystyle QBER\sim 1-V=1-V_{TBK,Z}\cdot TBK_{Z}}$

Central impulse in an extended description

The description of the central impulse for method Z can be expanded using the corrected visibility and the time-bin criterion . ${\displaystyle P(\psi _{M})}$

${\displaystyle P(\psi _{M})(\kappa ,\varphi _{1},\varphi _{2}=0)=4\cdot \kappa ^{6}\cdot (2-\kappa )^{6}\cdot \cos ^{2}{\frac {\varphi _{1}}{2}}=4\cdot V_{TBK,Z}\cdot \cos ^{2}{\frac {\varphi _{1}}{2}}={\frac {4}{TBK_{Z}}}\cdot \cos ^{2}{\frac {\varphi _{1}}{2}}}$

It can be evaluated directly or with the appropriate experimental equipment . ${\displaystyle V_{TBK,Z}}$ ${\displaystyle TBK_{Z}}$

literature

• Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, Hugo Zbinden: Quantum Cryptography . PDF accessed on August 15, 2019
• Matthias Leifgen: Protocols and components for quantum key distribution . doi: 10.18452 / 17473 (PDF)
• Björnstjerne Zindler: Construction of fiber-based interferometers for quantum cryptography . PDF accessed on August 7, 2019 (German) (2.363 MB)
• Björnstjerne Zindler: Method of error analysis for the Time-Bin configuration under laser irradiation . PDF accessed on August 7, 2019 (German)

Individual evidence

1. ↑ ^{a b c} Björnstjerne Zindler: Method of error analysis for the time-bin configuration under laser irradiation. ( nadirpoint.de [PDF]).
2. ↑ Björnstjerne Zindler: Construction of fiber-based interferometers for quantum cryptography. ( nadirpoint.de [PDF]).
3. ↑ Björnstjerne Zindler: Possibility to design a form for the time-bin criterion according to the SZS method. ( nadirpoint.de [PDF]).
4. ↑ Björnstjerne Zindler: design possibility of a form for the time-bin criterion according to the method Z . ( nadirpoint.de [PDF]).
5. ↑ Matthias Leifgen: Protocols and Components for quantum key distribution . 2016, Chapter 7.1.1.1 "Requirements for the interferometer", doi : 10.18452 / 17473 .
6. ↑ Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel and Hugo Zbinden: cryptography Quantum . Chapter IV-A "Quantum Bit Error Rate".