Elementary quantum gates with Gaussian states

Abstract

We study the question of converting initially Gaussian states into non-Gaussian ones by two- and three-photon subtraction to improve non-classical properties of the conditional optical fields. We show the photon subtraction may effectively generate non-Gaussian states only in case of small values of the mean values of the position and momentum operators. In particular, the photon-subtracted state can be made arbitrary close to Gaussian state in limiting case of large initial amplitude of displacement. Use of initial displacement in input Gaussian states opens wider prospects to manipulate them. In particular, realization of probabilistic Hadamard gate with input Gaussian states is discussed where photon subtraction is motive force able unevenly to increase measure of non-classicality of the output state. Subtraction of larger number of photons enables to increase fidelity and non-classical measure of the conditional states.

Appendices

Appendix A: Calculation of characteristic and Wigner functions

In this appendix, we present additional information concerning calculation of characteristic and Winger functions. Position- and momentum-like operators are given by

$$\begin{aligned} x&= \frac{a+a^{+}}{2},\end{aligned}$$

(57)

$$\begin{aligned} p&= \frac{a-a^{+}}{2i}, \end{aligned}$$

(58)

where $a$ and $a^{+}$ are the bosonic annihilation and creation operators of quantum oscillator. They satisfy commutation relation $\left[ {x,\,p} \right] \,=\,i/{2}$. Elements of the CM are defined by

$$\begin{aligned} \left( {V_1^{\left( 1 \right) } } \right) _{il} =\frac{1}{2}\left\langle {R_i R_j +R_j R_i } \right\rangle -\left\langle {R_i } \right\rangle \left\langle {R_j } \right\rangle , \end{aligned}$$

(59)

where $\langle O\rangle = Tr( \rho O)$ is the expectation value of the arbitrary operator $O$, vector of the operators is $R = [x,\, p]^{T}$ and $T$ means vector transposition. Then, parameters $A$ and $B$ in CM (1) are given by

$$\begin{aligned} A&= \left\langle {x^{2}} \right\rangle -\left\langle x \right\rangle ^{2},\end{aligned}$$

(60)

$$\begin{aligned} B&= \left\langle {p^{2}} \right\rangle -\left\langle p \right\rangle ^{2}, \end{aligned}$$

(61)

while off-diagonal terms are supposed to be $\left( {V_{1} ^{({1})}} \right) _{12} \,=\,\left( {V_{2} ^{({1})}} \right) _{21} \,=\,(\langle xp\rangle \,+\,\langle px\rangle )/\left( {\langle xp\rangle \,+\,\langle px\rangle } \right) {2}.{2}\,-\,\langle x\rangle \langle p\rangle \,=\,0$. The vector column

$$\begin{aligned} \left\langle R \right\rangle =Tr\left( {\rho R} \right) =\left[ {\left\langle q \right\rangle ,\left\langle p \right\rangle } \right] ^{T}=\left[ {R_x R_p } \right] ^{T} \end{aligned}$$

(62)

is usually referred to first-moment vector. For example, consider pure squeezed coherent state defined as

$$\begin{aligned} \left| \varphi \right\rangle =S\left( r \right) \left| \alpha \right\rangle =S\left( r \right) D\left( \alpha \right) \left| 0 \right\rangle , \end{aligned}$$

(63)

where displacement and squeezing operators are given by [42]

$$\begin{aligned} D\left( \alpha \right)&= \exp \left( {\alpha a^{+}-\alpha ^{*}a} \right) , \end{aligned}$$

(64)

$$\begin{aligned} S\left( r \right)&= \exp \left( {r{\left( {a^{+2}-a^{2}} \right) }/2} \right) , \end{aligned}$$

(65)

with $\alpha = \alpha _{R} + i \alpha _{I}$ being displacement parameter of the coherent state $D(\alpha ){\vert }0 \rangle = {\vert } \alpha \rangle$ and real squeezing parameter $r > 0$. Then, by calculating (60), (61), and (62), we obtain

$$\begin{aligned} A&= {\exp \left( {2r} \right) }/4,\end{aligned}$$

(66)

$$\begin{aligned} B&= {\exp \left( {-2r} \right) }/4,\end{aligned}$$

(67)

$$\begin{aligned} R_x&= \alpha _R \exp \left( r \right) , \end{aligned}$$

(68)

$$\begin{aligned} R_p&= \alpha _I \exp \left( {-r} \right) . \end{aligned}$$

(69)

Relations (66), (67) mean that $p-$ quadrature of the state (63) is squeezed $(B < 1/14.4)$, while $x-$ quadrature of the same state is desqueezed $(A > 1/14.4)$. Thus, we can call the state (63) momentum (or $p-$ ) squeezed state.

Single-mode state $\rho$ is a Gaussian one if its characteristic function is given by

$$\begin{aligned}&\chi \left( \Lambda \right) = \exp \left( {{\Lambda ^{T}\Omega V_1^{\left( 1 \right) } \Omega ^{T}\Lambda }/2-i\Lambda ^{T}\Omega \left\langle R \right\rangle } \right) = \nonumber \\&\exp \left( {-{\left( {BU^{2}+AV^{2}} \right) }/2} \right) \exp \left( {-iUR_p +iVR_x } \right) , \end{aligned}$$

(70)

where $\Lambda = [U,\,V]^{T}$ are the real variables of the characteristic function, and

$\Omega =\left| {\begin{array}{ll} 0&{} 1 \\ {-1}&{} 0 \\ \end{array}}\right|$ is the symplectic matrix. By Fourier transforming the characteristic function (70), we obtain Wigner function of $\rho$ [1, 2]

$$\begin{aligned} W\left( X \right) =\frac{1}{\left( {2\pi } \right) ^{2}}\int \limits _{R^{2}} {\chi \left( \Lambda \right) \exp \left( {i\Lambda { }^T\Omega X} \right) d^{2}\Lambda } . \end{aligned}$$

(71)

Now, we are going to make change of variables $U\,\rightarrow \,{2}U$ and $V\,\rightarrow \,{2}V$ in (71) to deal with the following Wigner function

$$\begin{aligned} W\left( X \right) =\frac{1}{\pi ^{2}}\int \limits _{R^{2}} {\chi \left( \Lambda \right) \exp \left( {i2\Lambda { }^T\Omega X} \right) d^{2}\Lambda } , \end{aligned}$$

(72)

whose characteristic function becomes

$$\begin{aligned} \chi \left( \Lambda \right) =\exp \left( {-{\left( {BU^{2}+AV^{2}} \right) }/2} \right) \exp \left( {-i2UR_p +i2VR_x } \right) , \end{aligned}$$

(73)

where now coefficients $A$ and $B$ in (73) are multiplied by 4 on compared with (1–3) and (60), (61), (66), (67). Such form of the characteristic function (73) is used to simplify Fourier integrating. Characteristic and Wigner functions of the multimode states can be calculated by similar fashion.

By Fourier integrating the characteristic function (23) of the conditional state (19), one obtains final Wigner function of the state

$$\begin{aligned} W\left( {x,p} \right) ={\left( {G_1 W_1 \left( {x,p} \right) -G_2 W_2 \left( {x,p} \right) +G_3 W_3 \left( {x,p} \right) } \right) }/{p_\mathrm{on} }, \end{aligned}$$

(74)

where

$$\begin{aligned} W_1 \left( {x,p} \right)&= \frac{2\exp \left( {-2{\left( {x-R_x \cos Q} \right) ^{2}}/{B_1 -2{\left( {p-R_p \cos Q} \right) ^{2}}/{A_1 }}} \right) }{\pi \sqrt{A_1 B_1 }},\end{aligned}$$

(75)

$$\begin{aligned} W_2 \left( {x,p} \right)&= \frac{2\exp \left( {-2{\left( {x-\gamma _2 R_x } \right) ^{2}}/{\lambda _2 -2{\left( {p-\gamma _1 R_p } \right) ^{2}}/{\lambda _1 }}} \right) }{\pi \sqrt{\lambda _1 \lambda _2 }},\end{aligned}$$

(76)

$$\begin{aligned} W_3 \left( {x,p} \right)&= \frac{2\exp \left( {-2{\left( {x-\gamma _4 R_x } \right) ^{2}}/{\lambda _4 -2{\left( {p-\gamma _3 R_p } \right) ^{2}}/{\lambda _3 }}} \right) }{\pi \sqrt{\lambda _3 \lambda _4 }}. \end{aligned}$$

(77)

Consider one of the possible group of non-Gaussian states. Displaced squeezed superpositions of coherent states (DSSCSs) are produced by successive one by one application of displacement and squeezing operators to the following superposition of the coherent states (SCSs)

$$\begin{aligned} \left| {\hbox {DSSCS}_\pm \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle =D\left( {\alpha _\pm } \right) S\left( {r_{1\pm } } \right) \left| {\hbox {SCS}_\pm \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle , \end{aligned}$$

(78)

where SCSs are given by

$$\begin{aligned} \left| {\hbox {SCS}_\pm \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle =N_\pm \left( {\alpha _\mathrm{SCS} } \right) \left( {\left| {0,\alpha _\mathrm{SCS} } \right\rangle \pm \left| {0,-\alpha _\mathrm{SCS} } \right\rangle } \right) , \end{aligned}$$

(79)

where $N_\pm \left( {\alpha _\mathrm{SCS} } \right) =1/{\sqrt{2\left( {1\pm \exp \left( {-2\left| {\alpha _\mathrm{SCS} } \right| ^{2}} \right) } \right) }}$ is a normalization factor and $\alpha _\mathrm{SCS}$ is an amplitude of the SCSs. Here, magnitudes $\alpha _{\pm }$ and $r_{\pm }$ are the displacement and squeezing parameters of the DSSCSs. Subscripts +  and -  are used in definitions of (78, 79) to discriminate even an odd states from each other. The even and odd DSSCSs become orthogonal to each other in the case of $\alpha _+ \,=\,\alpha _- \,=\,\alpha$ and $r_{{1}+} \,=\,r_{{1}-} \,=\,r_{1}$.

$$\begin{aligned}&\left\langle {\left| {\hbox {DSSCS}_- \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle } | {\hbox {DSSCS}_+ \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle \nonumber \\&\quad =\left\langle {\hbox {SCS}_- \left( {\alpha _\mathrm{SCS} } \right) } \right| S^{+}\left( {r_1 } \right) D^{+}\left( \alpha \right) D\left( \alpha \right) S\left( {r_1 } \right) \left| {\hbox {SCS}_+ \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle \nonumber \\&\left\langle {\hbox {SCS}_- \left( {\alpha _\mathrm{SCS} } \right) } | {\hbox {SCS}_+ \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle =0. \end{aligned}$$

(80)

Respectively, a density operator of the DSSCSs is given by

$$\begin{aligned} \rho _{\mathrm{DSSCS}\pm } =\left| {\hbox {DSSCS}_\pm \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle \left\langle {\hbox {DSSCS}_\pm \left( {\alpha _\mathrm{SCS} } \right) } \right| \end{aligned}$$

(81)

Characteristic function of the DSSCSs is given by

$$\begin{aligned} \chi _{\mathrm{DSSCS}\pm } \left( {U,V} \right) \!=\!N_\pm \left( {\alpha _\mathrm{SCS} } \right) \left( {\chi _1 \left( {U,V} \right) \!+\!\chi _2 \left( {U,V} \right) \!\pm \! \chi _3 \left( {U,V} \right) \!\pm \! \chi _4 \left( {U,V} \right) } \right) ,\quad \quad \end{aligned}$$

(82)

where its four terms can be written as

$$\begin{aligned} \chi _1 \left( {U,V} \right)&= \exp \left( {{\left( {\frac{U^{2}}{\exp \left( {2r_1 } \right) }+\frac{V^{2}}{\exp \left( {-2r_1 } \right) }} \right) }\bigg /2} \right) \nonumber \\&\quad \exp \left( {2iV\left( {x_\alpha +{x_\mathrm{SCS} }/{\exp \left( {-r_1 } \right) }} \right) -2iU\left( {p_\alpha +{p_\mathrm{SCS} }/{\exp \left( {r_1 } \right) }} \right) } \right) , \end{aligned}$$

(83)

$$\begin{aligned} \chi _2 \left( {U,V} \right)&= \exp \left( {{\left( {\frac{U^{2}}{\exp \left( {2r_1 } \right) }+\frac{V^{2}}{\exp \left( {-2r_1 } \right) }} \right) }\bigg /2} \right) \nonumber \\&\quad \exp \left( {2iV\left( {x_\alpha -{x_\mathrm{SCS} }/{\exp \left( {-r_1 } \right) }} \right) -2iU\left( {p_\alpha -{p_\mathrm{SCS} }/{\exp \left( {r_1 } \right) }} \right) } \right) , \end{aligned}$$

(84)

$$\begin{aligned} \chi _3 \left( {U,V} \right)&= \exp \left( {{\left( {\frac{U}{\exp \left( {r_1 } \right) }+2x_\mathrm{SCS} } \right) ^{2}}\bigg /2} \right) \exp \left( {{\left( {\frac{V}{\exp \left( {-r_1 } \right) }+2p_\mathrm{SCS} } \right) ^{2}}\bigg /2} \right) \nonumber \\&\quad \exp \left( {2i\left( {Vx_\alpha -Up_\alpha } \right) } \right) , \end{aligned}$$

(85)

$$\begin{aligned} \chi _4 \left( {U,V} \right)&= \exp \left( {{\left( {\frac{U}{\exp \left( {r_1 } \right) }-2x_\mathrm{SCS} } \right) ^{2}}\bigg /2} \right) \exp \left( {{\left( {\frac{V}{\exp \left( {-r_1 } \right) }-2p_\mathrm{SCS} } \right) ^{2}}\bigg /2} \right) \nonumber \\&\quad \exp \left( {2i\left( {Vx_\alpha -Up_\alpha } \right) } \right) , \end{aligned}$$

(86)

where $\alpha _\mathrm{SCS} \,=\,x_{scs} \,+\,ip_\mathrm{SCS}$ and $\alpha \,=\,x_\alpha \,+\,ip_\alpha$.

Appendix B: Beam splitter operator and simplified model of analysis

Both UBS and BBS are described by the following operator

$$\begin{aligned} B_{ij} \left( Q \right) =\exp \left( {QX_{ij} } \right) , \end{aligned}$$

(87)

where $X_{ij} \,=\,a_i ^{+}a_j \,-\,a_j ^{+}a_i ,a_i$ and $a_{i}^{+}$ ($a_{j}$ and $a_{j}^{+})$ are the bosonic annihilation and creation operators of the modes $i$ and $j$ ($i,\,j = 1,\,2,\,3$) of the beam splitter and the beam splitter’s parameter $Q$ defines both transmittance $T = \cos ^{2}Q$ and reflectivity $R = \sin ^{2}Q$. Value $Q = \pi /4$ of the beam splitter means $T = R=1/2$, otherwise operator (87) is associated with the UBS with $Q \ne \pi /4$ and $T \ne R$. The following relation between input and output operators takes place [42]

$$\begin{aligned} \left| {\begin{array}{l} {a_1 } \\ {a_2 } \\ \end{array}}\right| _\mathrm{Out} =\left| {\begin{array}{ll} {\cos Q}&{} {-\sin Q} \\ {\sin Q}&{} {\cos Q} \\ \end{array}}\right| \left| {\begin{array}{l} {a_1 } \\ {a_2 } \\ \end{array}}\right| _\mathrm{In} . \end{aligned}$$

(88)

Simplified model of analysis is based on decomposition of the beam splitter operator (87)

$$\begin{aligned} B_{12} \left( Q \right) =\exp \left( {QX_{12} } \right) =1+QX_{12} +{Q^{2}X_{12}^2 }/{2!}+{Q^{3}X_{12}^3 }/{3!}+\ldots . \end{aligned}$$

(89)

Simplified model assumes substitution $B_{12} \left( Q \right) \,\rightarrow \,a_{1} ^{2}a_{2} ^{+{2}}$ and $B_{12} \left( Q \right) \,\rightarrow \,a_{1} ^{{3}}a_{2} ^{+{3}}$ followed by the projective measurement onto either the state ${\vert }2\rangle$ or ${\vert }3\rangle$ in auxiliary mode 2. The simplified model can be valid when UBS parameter $Q < < 1$ is chosen or the same when a small fraction of input beam is tapped off via a beam splitter and projective measurement onto either the states ${\vert }2 \rangle$ or ${\vert }3 \rangle$ are successfully done in mode 2. Then, the output density operators $\rho _{\pm }^{(2S)},\,\rho _{\pm }^{(3S)}$ of mode 1 conditioned on two- or three-photon click of the photon number resolving detector measuring auxiliary mode 2 become

$$\begin{aligned} \rho _\pm ^{\left( {2S} \right) }&= \frac{tr_2 \left( {\rho _{12\pm }^{\left( {2S} \right) } \Pi _2 } \right) }{tr_{12} \left( {\rho _{12\pm }^{\left( S \right) } \Pi _2 } \right) }, \end{aligned}$$

(90)

$$\begin{aligned} \rho _\pm ^{\left( {3S} \right) }&= \frac{tr_2 \left( {\rho _{12\pm }^{\left( {3S} \right) } \Pi _3 } \right) }{tr_{12} \left( {\rho _{12\pm }^{\left( 3 \right) } \Pi _3 } \right) }, \end{aligned}$$

(91)

where $\rho _{12\pm } ^{\left( {{2}S} \right) }\,=\,a_{1}^{2}a_{2} ^{+{2}}(\rho _{{1}\pm } \,\otimes \,|0\rangle \langle 0|_{2} )\left( {a_{1} ^{2}a_{2} ^{+{2}}} \right) ^{+},\rho _{12\pm } ^{({3}S)}\,=\,a_{1} ^{{3}}a_{2} ^{+{3}}(\rho _{{1}\pm } \,\otimes \,|0\rangle \langle 0|_{2} )\left( {a_{1} ^{{3}}a_{2} ^{+{3}}} \right) ^{+},tr_{2}$ is the trace over mode 2, $tr_{12}$ is the trace over modes 1,2and $\Pi _{2} \,=\,|{2}\rangle \langle {2}|,\Pi _{3} \,=\,|{3}\rangle \langle {3}|$ are the operators of the projective measurements onto the state ${\vert }2 \rangle$ and ${\vert }3 \rangle$, respectively, in the second mode. Here, symbol $S$ in superscripts is used to show that they are computed by means of use of simplified model.

Appendix C: Measure of non-classicality and degree of mixedness

Now, we have two criteria of non-classicality. One is based on the negativity of the Wigner function, and the other is based on the non-existence of the well-behaved positive $P$ function. These two criteria do not necessarily coincide. In fact, the positivity of the $P$ function guarantees the positivity of the Wigner function but the converse is not necessarily true. We are going to discuss only second criterion. A Gaussian state can be classical (coherent and thermal states) or non-classical (squeezed states, Fock number states). In many respects, the Wigner representation appears as the best compromise between a classical phase space density and correct quantum-mechanical behavior. Yet there is a way to define other quasiprobability distributions by introducing some parameter in characteristic function

$$\begin{aligned} \chi _s \left( {U,V} \right) =tr\left( {\rho D\left( {U,V} \right) } \right) \exp \left( {s{\left( {\left| U \right| ^{2}+\left| V \right| ^{2}} \right) }/2} \right) . \end{aligned}$$

(92)

So, values $s = - 1,\,s= 0$, and $s = 1$ correspond to $Q$, Wigner, and $P$ functions, respectively. Definition of degree of non-classicality is based on the following procedure. We can smooth the Glauber-Sudarshan quasiprobability distribution $P(\alpha ) = P(x,\,p)$ corresponding to $s = 1$ by convoluting it with Gaussian distribution having the same width as vacuum, by taking averages of $P$ around each phase space point $(x,\,p)$ with a width that corresponds to the vacuum noise

$$\begin{aligned} R\left( {\alpha ,\tau } \right) ={\int \limits _{R^{2}} P \left( \beta \right) \exp \left( {{-\left| {\alpha -\beta } \right| ^{2}}/\tau } \right) }/{\left( {\pi \tau } \right) }. \end{aligned}$$

(93)

For a given $P$ function, there exists a certain value of $\tau ^{(1)}$ such that the $R$ function becomes positive-definite $R \ge 0$ for $\tau > \tau ^{(1)}$. This threshold value $\tau ^{(1)}$ takes values in the range [0, 1] and value $\tau ^{(1)} > 0$ is regarded as a measure of non-classicality of the state [43, 44]. Sole value $\tau ^{(1)} = 0$ corresponds to classical states and can be named as measure of classicality. It is possible to show the measure of non-classicality of momentum squeezed Gaussian states, for example, state (63) is equal to

$$\begin{aligned} \tau ^{\left( 1 \right) }={\left( {1-A} \right) }/2={\left( {1-\exp \left( {-2r} \right) } \right) }/2. \end{aligned}$$

(94)

Squeezed Gaussian state is non-classical according to $\tau ^{(1)}$ criterion since its non-classical measure (94) is more of zero. Nevertheless, the state can also be considered classical according to criterion based on the positivity of the Wigner function. The positivity of the Wigner function does not guarantee the positivity of the $P$ function. In the case, the measure of classicality can be also determined by minimal positive value of the Wigner function.

The degree of mixedness in a prepared quantum state can be characterized by its purity $\mu = tr( \rho ^{2})$. The parameter is equal to 1 for pure state. For a Gaussian state with CM (1), the purity becomes

$$\begin{aligned} \mu =\frac{1}{2^{2}\sqrt{\det V_1^{\left( 1 \right) } }}. \end{aligned}$$

(95)

Consider infinite Hilbert space of displaced Fock number states

$$\begin{aligned} \left| {n,\alpha } \right\rangle =D\left( \alpha \right) \left| n \right\rangle , \end{aligned}$$

(96)

where $\alpha$ is an amplitude of displacement and $n$ is the number of photons. The designation (96) was earlier used in [37]. Choose two sets of orthogonal displaced number states

$$\begin{aligned}&\left\{ {\left| {n,\alpha } \right\rangle ,n=0,1,2,...,\infty } \right\} ,\end{aligned}$$

(97)

$$\begin{aligned}&\left\{ {\left| {n,\alpha ^{{\prime }}} \right\rangle ,n=0,1,2,...,\infty } \right\} , \end{aligned}$$

(98)

with different values of amplitude of displacement $\alpha$ and $\alpha ^{{\prime }}$, where in general case $\alpha \ne \alpha ^{{\prime }}$. As far as each of the sets is complete, then every member of one set can be expressed through states from another set. Such decomposition or the same $\alpha -$ representation is not trivial and is presented in [37–41]. So, for example, $\alpha -$ representation of the SCSs (79) is given by

$$\begin{aligned}&\left| {\hbox {SCS}_\pm \left( {\alpha _\mathrm{SCS} } \right) } \right\rangle =N_\pm \left( {\alpha _\mathrm{SCS} } \right) \exp \left( {-{\left( {\alpha _\mathrm{SCS}^2 +\left| \alpha \right| ^{2}} \right) }/2} \right) \nonumber \\&\sum _{l=0}^\infty {\frac{\left( {\exp \left( {\alpha _\mathrm{SCS} \alpha ^{*}} \right) \left( {-\delta } \right) ^{l}\pm \exp \left( {-\alpha _\mathrm{SCS} \alpha ^{*}} \right) \left( {-\delta ^{{\prime }}} \right) ^{l}} \right) }{\sqrt{l!}}\left| {l,\alpha } \right\rangle }. \end{aligned}$$

(99)

It is known that if a single-mode pure state $\rho$ is orthogonal to a certain coherent state $\langle \alpha \left| \rho \right| \alpha \rangle$ for some $\alpha$, then the state is maximally non-classical, its measure of non-classicality is equal $\tau ^{\left( {1} \right) }\,=\,{1}$ [43, 44]. It means that if $c_{0}$ wave amplitude in $\alpha -$ representation of arbitrary pure state $|\Psi \rangle \,=\,\sum \,_{k=0} c_k |k,\,\alpha \rangle$ equals 0, then the state is maximally non-classical. It follows from decomposition (99) the wave amplitudes $c_{0\pm }$ are proportional to

$$\begin{aligned} c_{0\pm } \,\sim \,\hbox {exp}\,\left( {\alpha _\mathrm{SCS} \alpha ^{*}} \right) \,\pm \,\hbox {exp}\,\left( {\,-\,\alpha _\mathrm{SCS} \alpha ^{*}} \right) . \end{aligned}$$

(100)

It is evident that for any value of $\alpha _\mathrm{SCS}$, it is possible to choose at least one value $\alpha$ for which $c_{0\pm }$ (100) become zero. So for example, $c_{0+}$ and $c_{0+}$ are equal to zero for $\alpha _+ \,=\,i\pi /i\pi \left( {{2}\alpha _\mathrm{SCS} } \right) .\left( {{2}\alpha _\mathrm{SCS} } \right)$ and $\alpha _- \,=\,0$, respectively. Moreover, $\alpha -$ representation can be applied to arbitrary coherent qubit $|\Psi \rangle \,=\,N(a|\varepsilon _{1} \rangle \,+\,b|\varepsilon _{2} \rangle )$, where $a,b,\varepsilon _{1} ,\varepsilon _{2}$ are arbitrary values and $N$ is normalization factor, to show that $c_{0} = 0$ under certain values of $\alpha$ and consequently $\tau ^{(1)} = 1$ for such state.