Probabilistically cloning two single-photon states using weak cross-Kerr nonlinearities

The states secretly chosen from the linearly independent set {$|{{\Psi }_{1}}\rangle$, $|{{\Psi }_{2}}\rangle$,⋯, $|{{\Psi }_{n}}\rangle$} can be exactly cloned in a probabilistic way. A Duan-Guo PQC machine consists of a unitary evolution together with a projection measurement. It is written as follows:

$$\begin{eqnarray}&&U{{(|{{\Psi }_{i}}\rangle }_{x}}|\Sigma {{\rangle }_{y}}|{{Z}^{(0)}}{{\rangle }_{z}})=\sqrt{{{\Upsilon }_{i}}}|{{\Psi }_{i}}{{\rangle }_{x}}|{{\Psi }_{i}}{{\rangle }_{y}}|{{Z}^{(i)}}{{\rangle }_{z}}+\sqrt{1-{{\Upsilon }_{i}}}|{{\Phi }^{(i)}}{{\rangle }_{xyz}},\;\;\;(i=1,2,\cdots ,n),\end{eqnarray} \tag{ 1 }$$

where U is the unitary evolution; x, y, and z represent the to-be-cloned system, the cloning system, and the assistant system respectively; $|{{Z}^{(0)}}\rangle$ and $|{{Z}^{(i)}}\rangle$ are normalized states of the assistant system z (not generally orthogonal); $|{{\Phi }^{(1)}}\rangle$, $|{{\Phi }^{(2)}}\rangle$, ⋯, $|{{\Phi }^{(n)}}\rangle$ are n normalized states of the composite system xyz (not generally orthogonal); and $\langle {{Z}^{(i)}}|{{\Phi }^{(j)}}\rangle =0$ ($i,j=1,2,\cdots ,n$). During the cloning process, after an appropriate unitary evolution U, a projection measurement is performed on the assistant system z. With the probability Υ_i, one can get the measurement result (MR) $|{{Z}^{(i)}}\rangle$. Following this projection, the state of the system xy should be $|{{\Psi }_{i}}{{\rangle }_{x}}|{{\Psi }_{i}}{{\rangle }_{y}}$ and the cloning is realized successfully.

The success probabilities of the Duan-Guo PQC machine satisfy theorem 2 in [27]. That is, the matrix ${{X}^{(1)}}-\sqrt{\Gamma }X_{z}^{(2)}\sqrt{{{\Gamma }^{\dagger }}}$ should be positive semidefinite, where ${{X}^{(1)}}=[\langle {{\Psi }_{i}}|{{\Psi }_{j}}\rangle ]$, $X_{z}^{(2)}=[{{\langle {{\Psi }_{i}}|{{\Psi }_{j}}\rangle }^{2}}\langle {{Z}_{i}}|{{Z}_{j}}\rangle ]$ and $\Gamma ={\rm diag}({{\Upsilon }_{1}},{{\Upsilon }_{2}},\cdots ,{{\Upsilon }_{n}})$. From theorem 2, the best success probabilities of PQC can be worked out. Specifically, corresponding to the two-state set ($|{{\varphi }_{1}}\rangle$, $|{{\varphi }_{2}}\rangle$), the optimal success probabilities Υ₁ and Υ₂ satisfy

$$\begin{eqnarray}&&\frac{{{\Upsilon }_{1}}+{{\Upsilon }_{2}}}{2}\leqslant \frac{1}{1+|\langle {{\varphi }_{1}}|{{\varphi }_{2}}\rangle |}.\end{eqnarray} \tag{ 2 }$$

We now present Chen et alʼs PQC machine and network for cloning two nonorthogonal qubit states. [34] mentions that Chen et alʼs network is simpler than those in [50, 51]. The to-be-cloned state is prepared in

$$\begin{eqnarray}&&|{{\varphi }_{\pm }}\rangle ={\rm cos} \xi |0\rangle \pm {\rm sin} \xi |1\rangle ,\;\;\;\;\xi \in [0,\frac{\pi }{4}],\end{eqnarray} \tag{ 3 }$$

where $|0\rangle$ and $|1\rangle$ are two orthogonal bases of a single qubit. Chen et alʼs simple PQC machine is expressed as

$$\begin{eqnarray}&&\mathcal{U}|{{\varphi }_{\pm }}{{\rangle }_{x}}|0{{\rangle }_{y}}|0{{\rangle }_{z}}=\sqrt{\Upsilon }|{{\varphi }_{\pm }}{{\rangle }_{x}}|{{\varphi }_{\pm }}{{\rangle }_{y}}|0{{\rangle }_{z}}+\sqrt{1-\Upsilon }|\Phi {{\rangle }_{xy}}|1{{\rangle }_{z}},\end{eqnarray} \tag{ 4 }$$

where $\mathcal{U}$ is the unitary evolution of qubits x, y, and z; $\Upsilon =1/(1+{\rm cos} 2\xi )$ is the optimal success probability of PQC; and $|\Phi {{\rangle }_{xy}}$ is the normalized state of qubits x and y. After the unitary operation $\mathcal{U}$, a projection measurement is performed on the assistant qubit z under bases $|0\rangle$ and $|1\rangle$. From equation (4) one can see that, if the MR is $|0\rangle$ with the probability Υ, the original state $|{{\varphi }_{\pm }}\rangle$ is successfully cloned. On the other hand, with the probability $1-\Upsilon$ one can obtain the MR $|1\rangle$. In this case, qubits x and y collapse to $|\Phi {{\rangle }_{xy}}$ and the cloning fails.

The quantum logic network of Chen et alʼs PQC machine is demonstrated in figure 1(a). The network is composed of two controlled-unitary gates (controlled-U₁ and controlled-U₂), two CNOT gates, a single-qubit Hadmard gate, and a single-qubit projection measurement. Here the unitary operations U₁, U₂, and H are written as

$$\begin{eqnarray*}\begin{array}{rcl} {{U}_{1}} & = & {\rm sin} \beta |0\rangle \langle 0|+{\rm cos} \beta |1\rangle \langle 0|+{\rm cos} \beta |0\rangle \langle 1|-{\rm sin} \beta |1\rangle \langle 1|, \\ {{U}_{2}} & = & {\rm sin} \delta |0\rangle \langle 0|+{\rm cos} \delta |1\rangle \langle 0|+{\rm cos} \delta |0\rangle \langle 1|-{\rm sin} \delta |1\rangle \langle 1|, \\ H & = & \frac{1}{\sqrt{2}}(|0\rangle \langle 0|+|1\rangle \langle 0|+|0\rangle \langle 1|-|1\rangle \langle 1|), \\ \end{array}\end{eqnarray*}$$

where $\beta ={\rm arcsin} \left( \sqrt{\frac{1+{{{\rm tan} }^{4}}\xi }{2}} \right)$ and $\delta ={\rm arcsin} \left[ \left( \sqrt{\frac{2}{1+{{{\rm tan} }^{4}}\xi }}+\sqrt{\frac{2}{1+{{{\rm tan} }^{-4}}\xi }} \right)/2 \right]$.

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The Hamiltonian of cross-Kerr nonlinearity is expressed as [49, 52–54],

$$\begin{eqnarray*}&&{{H}_{K}}=-\hbar \kappa a_{P}^{\dagger }{{a}_{P}}a_{S}^{\dagger }{{a}_{S}},\end{eqnarray*}$$

where ${{a}^{\dagger }}$ and a are respectively the creation and annihilation operators, the subscripts P and S denote the probe beam and the signal beam respectively, and κ is the coupling strength of the cross-Kerr nonlinearity. Suppose the input state of the signal mode is the Fock state $|\tilde{n}{{\rangle }_{S}}$, whereas the probe state is initially in a coherent state $|\alpha {{\rangle }_{P}}$. The cross-Kerr nonlinearity then causes the combined system of P and S to evolve as follows:

$$\begin{eqnarray}\begin{array}{rcl} |\tilde{n}{{\rangle }_{S}}|\alpha {{\rangle }_{P}} & \to & {{e}^{i{{H}_{K}}t/\hbar }}|\tilde{n}{{\rangle }_{S}}|\alpha {{\rangle }_{P}} \\ {} & = & |\tilde{n}{{\rangle }_{S}}|{{e}^{in\theta }}\alpha {{\rangle }_{P}}, \\ \end{array}\end{eqnarray} \tag{ 5 }$$

where $\theta =\kappa t$, with t being the interaction time. By the way, for the weak cross-Kerr nonlinearity the assumption $\theta \ll 1$ is always satisfied. The coherent beam picks up a phase shift θ directly proportional to the number of the photons in the Fock state $|\tilde{n}{{\rangle }_{S}}$. Specifically, if n = 0 or 1, the evolution is respectively expressed as

$$\begin{eqnarray}&&|\tilde{0}{{\rangle }_{S}}|\alpha {{\rangle }_{P}}\to |\tilde{0}{{\rangle }_{S}}|\alpha {{\rangle }_{P}},\;\;\;{\rm or}\;\;\;|\tilde{1}{{\rangle }_{S}}|\alpha {{\rangle }_{P}}\to |\tilde{1}{{\rangle }_{S}}|{{e}^{i\theta }}\alpha {{\rangle }_{P}}.\end{eqnarray} \tag{ 6 }$$

From equation (6) one can see that as soon as the phase shift is measured, we can infer the number of photons in the signal mode.

Now we present the QND based on weak cross-Kerr nonlinearity. See figure 2 for the demonstration. Suppose the probe beam is initially in the coherent state $|\alpha {{\rangle }_{P}}$ and the two signal beams are in the following Fock state:

$$\begin{eqnarray}&&{{\eta }_{00}}|\tilde{0}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{\eta }_{11}}|\tilde{1}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{\eta }_{01}}|\tilde{0}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{\eta }_{10}}|\tilde{1}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}},\end{eqnarray} \tag{ 7 }$$

where $|{{\eta }_{00}}{{|}^{2}}+|{{\eta }_{01}}{{|}^{2}}+|{{\eta }_{10}}{{|}^{2}}+|{{\eta }_{11}}{{|}^{2}}=1$. First, the probe beam and the signal beam S₁ simultaneously pass through the first cross-Kerr media (CKM), which induces phase shift $+\theta$ of the probe beam if the signal beam is in the state $|\tilde{1}\rangle$. After that, the probe beam and the signal beam S₂ simultaneously interact with the second CKM, which leads to $-\theta$ for the probe beam if the signal beam S₂ has only one photon. After the two interactions, from equation (6) one can see that the entire state of the probe beam and the two signal beams transforms to

$$\begin{eqnarray}&&{{\eta }_{00}}|\tilde{0}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}|\alpha {{\rangle }_{P}}+{{\eta }_{11}}|\tilde{1}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}|\alpha {{\rangle }_{P}}+{{\eta }_{01}}|\tilde{0}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}|{{e}^{-i\theta }}\alpha {{\rangle }_{P}}+{{\eta }_{10}}|\tilde{1}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}|{{e}^{i\theta }}\alpha {{\rangle }_{P}}.\end{eqnarray} \tag{ 8 }$$

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Using the homodyne detector, the different coherent states can be distinguished. If the homodyne detector is in the coherent state $|\alpha {{\rangle }_{P}}$, from equation (8) it can be seen that the two signal modes will be projected into the state ${{\eta }_{00}}|\tilde{0}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{\eta }_{11}}|\tilde{1}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}$ (without normalization). On the other hand, if the result of the homodyne detector is $|{{e}^{i\theta }}\alpha {{\rangle }_{P}}$ (or $|{{e}^{-i\theta }}\alpha {{\rangle }_{P}}$), the collapse state of the two signal modes is $|\tilde{1}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}$ (or $|\tilde{0}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}$). Here we choose the local oscillator phase $\pi /2$ offset from the probe phase. In this case it is called the X quadrature homodyne measurement. With this choice, the states $|{{e}^{i\theta }}\alpha {{\rangle }_{P}}$ and $|{{e}^{-i\theta }}\alpha {{\rangle }_{P}}$ cannot be distinguished [53]. After the X quadrature homodyne measurement, the state of modes S₁ and S₂ is

$$\begin{eqnarray}&&\begin{array}{l} f(X,\alpha ){{({{\eta }_{00}}|\tilde{0}\tilde{0}\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{\eta }_{11}}|\tilde{1}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}) \\ \quad +f(X,\alpha {\rm cos} \theta ){{[{{e}^{-i\vartheta (X)}}{{\eta }_{01}}|\tilde{0}\tilde{1}\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{e}^{i\vartheta (X)}}{{\eta }_{10}}|\tilde{1}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}], \\ \end{array}\end{eqnarray} \tag{ 9 }$$

where we have defined $f(X,Y)\equiv {\rm exp} [-{{\left( X-2Y \right)}^{2}}/4]/{{\left( 2\pi \right)}^{4}}$ and $\vartheta (X)\equiv \alpha X{\rm sin} \theta -{{\alpha }^{2}}{\rm cos} \theta ({\rm Mod}2\pi )$. In equation (9), $f(X,\alpha )$ and $f(X,\alpha {\rm cos} \theta )$ are two Gaussian curves with the midpoint between the peaks located at ${{X}_{0}}=\alpha (1+{\rm cos} \theta )$, and the peaks are separated by a distance ${{X}_{d}}=2\alpha (1-{\rm cos} \theta )$. If the distance is large enough, the overlap between the two Gaussian curves is small. When $X\gt {{X}_{0}}$ or $X\lt {{X}_{0}}$, we respectively have the following two outputs:

$$\begin{eqnarray}&&\sim {{\eta }_{00}}|\tilde{0}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{\eta }_{11}}|\tilde{1}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}},\;\;\;{\rm or}\;\;\;\sim {{e}^{-i\vartheta (X)}}{{\eta }_{01}}|\tilde{0}\tilde{1}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}+{{e}^{i\vartheta (X)}}{{\eta }_{10}}|\tilde{1}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}},\end{eqnarray} \tag{ 10 }$$

where ∼ means that there is a low probability of error in distinguishing the two states in equation (10) from each other. By the way, these two states are not normalized. Specifically, if there are no photons in the signal beam S₂ [i.e., ${{\eta }_{01}}={{\eta }_{11}}=0$ in equation (7)], from equation (6) one can re-express equation (10) as

$$\begin{eqnarray}&&\sim {{\eta }_{00}}|\tilde{0}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}},\;\;\;{\rm or}\;\;\;\sim {{e}^{i\vartheta (X)}}{{\eta }_{10}}|\tilde{1}\tilde{0}{{\rangle }_{{{S}_{1}}{{S}_{2}}}}.\end{eqnarray} \tag{ 11 }$$

The QND using weak cross-Kerr nonlinearities has now been introduced. It is the kernel of the following experimental scheme for economical PQC.

In this subsection, we introduce a scheme for achieving $1\to 2$ PQC based on QNDs combined with linear optics elements. The demonstration of our scheme is shown in figure 3. Photon x is the to-be-cloned photon. The state of x is randomly chosen from the set {$|{{\varphi }_{+}}\rangle ,\;|{{\varphi }_{-}}\rangle$ }, where

$$\begin{eqnarray}&&|{{\varphi }_{\pm }}\rangle ={\rm cos} \xi |H\rangle \pm {\rm sin} \xi |V\rangle ,\end{eqnarray} \tag{ 12 }$$

$0\leqslant \xi \leqslant \frac{\pi }{4}$, and $|H\rangle$ and $|V\rangle$ denote the horizontal and vertical polarization modes of the photon respectively. Note that the value of ξ is given to us, but we do not know whether the to-be-cloned state is $|{{\varphi }_{+}}\rangle$ or $|{{\varphi }_{-}}\rangle$. Moreover, $|{{\varphi }_{+}}\rangle$ and $|{{\varphi }_{-}}\rangle$ are nonorthogonal unless $\xi =\pi /4$ is satisfied. This representation is reasonable because any pair of arbitrary qubit states $|{{\varphi }_{1}}\rangle$ and $|{{\varphi }_{2}}\rangle$ can be transformed into the form of equation (12) by a unitary rotation [34]. To probabilistically clone the state with unit fidelity, an assistant photon z is needed. The initial state of photon z is

$$\begin{eqnarray}&&|\psi {{\rangle }_{z}}={\rm cos} \frac{\beta }{2}|H{{\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|V{{\rangle }_{z}},\end{eqnarray} \tag{ 13 }$$

where $\beta ={\rm arcsin} (\sqrt{\frac{1+{{{\rm tan} }^{4}}\xi }{2}})$. First, photon z is split into paths 0 and 1 while x is split into 2 and 3 by using a 50:50 beam splitter (BS) and a polarizing beam splitter (PBS). Note that a reversal operation σ^x is carried out on the photon in path 1, where ${{\sigma }^{x}}=|H\rangle \langle V|+|V\rangle \langle H|$. This operation is realized by a half-wave plate (HWP) whose axis is set at the angle $\frac{\pi }{4}$. By the way, the HWP induces the following unitary transformation [55]:

$$\begin{eqnarray*}&&{{U}_{HWP}}(\varrho )={\rm cos} 2\varrho |H\rangle \langle H|+{\rm sin} 2\varrho |H\rangle \langle V|+{\rm sin} 2\varrho |V\rangle \langle H|-{\rm cos} 2\varrho |V\rangle \langle V|.\end{eqnarray*}$$

Obviously, ${{U}_{HWP}}(\frac{\pi }{4})={{\sigma }^{x}}$. Then the two photons undergo the following transformation:

$$\begin{eqnarray}\begin{array}{rcl} |{{\varphi }_{\pm }}{{\rangle }_{x}}|\psi {{\rangle }_{z}} & \to & \frac{1}{\sqrt{2}}{{[{\rm cos} \xi {\rm cos} \frac{\beta }{2}|{{H}_{3}}\rangle }_{x}}{{(|{{V}_{0}}\rangle }_{z}}+|{{H}_{1}}{{\rangle }_{z}})+{\rm cos} \xi {\rm sin} \frac{\beta }{2}|{{H}_{3}}{{\rangle }_{x}}{{(|{{H}_{0}}\rangle }_{z}}+|{{V}_{1}}{{\rangle }_{z}}) \\ {} & {} & \pm {\rm sin} \xi {\rm sin} \frac{\beta }{2}|{{V}_{2}}{{\rangle }_{x}}{{(|{{V}_{0}}\rangle }_{z}}+|{{H}_{1}}{{\rangle }_{z}})\pm {\rm sin} \xi {\rm cos} \frac{\beta }{2}|{{V}_{2}}{{\rangle }_{x}}{{(|{{H}_{0}}\rangle }_{z}}+|{{V}_{1}}{{\rangle }_{z}})], \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where the subscripts 0, 1, 2, and 3 denote the different paths of the photons. After that, photons in paths 1 and 2 are sent to a QND (QND1 in figure 3). It should be mentioned that the photons in paths 1 and 2 correspond to beams S₁ and S₂ in figure 2. Subsequently, the photon in path 0 is reflected by the one-way mirror (OWM), whereas the photon in path 1 passes through the OWM. The QND has two possible outputs. In subsection 3.1, we demonstrated that the output of a QND corresponds to the X quadrature homodyne measurement result of the probe beam. Corresponding to the MR $X\gt {{X}_{0}}$ or $X\lt {{X}_{0}}$, from equation (10) one can see that photons x and z respectively collapse to the state of

$$\begin{eqnarray}&&\sim {\rm cos} \xi |{{H}_{3}}{{\rangle }_{x}}{{({\rm cos} \frac{\beta }{2}|V\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|H{{\rangle }_{z}})\pm {\rm sin} \xi |{{V}_{2}}{{\rangle }_{x}}{{({\rm cos} \frac{\beta }{2}|H\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|V{{\rangle }_{z}}),\end{eqnarray} \tag{ 15 }$$

or

$$\begin{eqnarray}&&\begin{array}{l} \sim {{e}^{i\vartheta (X)}}{\rm cos} \xi |{{H}_{3}}{{\rangle }_{x}}{{[{\rm cos} \frac{\beta }{2}|H\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|V{{\rangle }_{z}}] \\ \quad \pm {{e}^{-i\vartheta (X)}}{\rm sin} \xi |{{V}_{2}}{{\rangle }_{x}}{{[{\rm cos} \frac{\beta }{2}|V\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|H{{\rangle }_{z}}]. \\ \end{array}\end{eqnarray} \tag{ 16 }$$

The state in equation (16) can evolve to that in equation (15) by modulating an additional phase shift $2\vartheta (X)$ of the photon in path 2 and a σ^x operation on photon z. The operations are performed by a classical feed-forward. Incidentally, the feed-forward technique plays an important role in quantum information and computation in that future operations or measurements depend on earlier measurement results. Using current technologies and customized fast electro-optical modulators, we were able to achieve high fidelity ($\gt 99\%$) for detected photons [56]. Before the feed-forward, photons x and z in the state of equations (15) or (16) are delayed in optical single-mode fibers with lengths of 30 m (150 ns). After the classical feed-forward, the state of x and z evolves to equation (15). Then we use a PBS, whose optical axis is placed at the angle $\frac{\beta }{2}$ (PBS $\frac{\beta }{2}$ in figure 3), and two single-photon trigger detectors (SPTDs) to complete the measurement for photon z with the orthogonal basis $|u\rangle$ and $|{{u}^{\bot }}\rangle$, where

$$\begin{eqnarray}&&|u\rangle ={\rm cos} \frac{\beta }{2}|H\rangle +{\rm sin} \frac{\beta }{2}|V\rangle ,\;\;\;|{{u}^{\bot }}\rangle =-{\rm sin} \frac{\beta }{2}|H\rangle +{\rm cos} \frac{\beta }{2}|V\rangle .\end{eqnarray} \tag{ 17 }$$

With the orthogonal basis $|u\rangle$ and $|{{u}^{\bot }}\rangle$, equation (15) can be re-expressed as follows:

$$\begin{eqnarray}&&\begin{array}{l} {\rm cos} \xi |{{H}_{3}}{{\rangle }_{x}}{{({\rm cos} \frac{\beta }{2}|V\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|H{{\rangle }_{z}})\pm {\rm sin} \xi |{{V}_{2}}{{\rangle }_{x}}{{({\rm cos} \frac{\beta }{2}|H\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|V{{\rangle }_{z}}) \\ \quad =\sqrt{\Upsilon }{{\left( \sqrt{2}{{{\rm cos} }^{2}}\xi {\rm sin} \beta |{{H}_{3}}\rangle \pm \frac{{\rm sin} 2\xi }{\sqrt{2}}|{{V}_{2}}\rangle \right)}_{x}}|u{{\rangle }_{z}}+\sqrt{1-\Upsilon }|{{H}_{3}}{{\rangle }_{x}}|{{u}^{\bot }}{{\rangle }_{z}}, \\ \end{array}\end{eqnarray} \tag{ 18 }$$

where Υ is the same as in equation (4). From equation (18), one can see that the upper SPTD in figure 3 captures a photon with the probability $1-\Upsilon$. With this capture, one knows that the MR is $|{{u}^{\bot }}{{\rangle }_{z}}$, and consequently, photon x collapses to the state $|{{H}_{3}}{{\rangle }_{x}}$. It is obvious that the cloning process fails and that further operations are unnecessary. So this capture means the cutoff of the scheme. On the other hand, the right SPTD is triggered, which takes place with the probability Υ. It means that the MR is $|u{{\rangle }_{z}}$. It follows that the collapse state of photon x is

$$\begin{eqnarray}&&|\varphi {{^{\prime} }_{\pm }}{{\rangle }_{x}}=\sqrt{2}{{{\rm cos} }^{2}}\xi {\rm sin} \beta |{{H}_{3}}{{\rangle }_{x}}\pm \frac{{\rm sin} 2\xi }{\sqrt{2}}|{{V}_{2}}{{\rangle }_{x}}.\end{eqnarray} \tag{ 19 }$$

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In the following, we show that the $1\to 2$ cloning of the state $|{{\varphi }_{\pm }}\rangle$ can be deterministically realized from the state $|\varphi {{^{\prime} }_{\pm }}\rangle$. In our scheme, the right SPTD is the switch of the components in the dot-line rectangle (see figure 3). Unless the right SPTD is triggered, these components do not work. As soon as it is triggered, a single-photon source emits the cloning photon y and an HWP is used for generating the state

$$\begin{eqnarray}&&|\phi {{\rangle }_{y}}={\rm cos} \frac{\delta }{2}|H{{\rangle }_{y}}+{\rm sin} \frac{\delta }{2}|V{{\rangle }_{y}},\end{eqnarray} \tag{ 20 }$$

where $\delta ={\rm arcsin} [(\sqrt{\frac{2}{1+{{{\rm tan} }^{4}}\xi }}+\sqrt{\frac{2}{1+{{{\rm tan} }^{-4}}\xi }})/2]$. After a BS in path 5 and an HWP $\frac{\pi }{4}$ in path 4, photons x and y encounter the following evolution:

$$\begin{eqnarray}&&\begin{array}{l} |\varphi {{^{\prime} }_{\pm }}{{\rangle }_{x}}|\phi {{\rangle }_{y}}\to {{({{{\rm cos} }^{2}}\xi {\rm sin} \beta |{{H}_{3}}\rangle }_{x}}\pm \frac{{\rm sin} 2\xi }{2}|{{V}_{2}}{{\rangle }_{x}}) \\ \quad {{[{\rm cos} \frac{\delta }{2}{{(|{{V}_{4}}\rangle }_{y}}+|{{H}_{5}}\rangle }_{y}})+{\rm sin} \frac{\delta }{2}{{(|{{H}_{4}}\rangle }_{y}}+|{{V}_{5}}{{\rangle }_{y}})]. \\ \end{array}\end{eqnarray} \tag{ 21 }$$

After that, the QND2, an OWM, and the ${{\sigma }^{x}}$ operation and phase shift $2\vartheta (X)$ corresponding to the outcome of the QND2 are utilized. Here the photons in paths 5 and 2 correspond to beams S₁ and S₂, respectively, in figure 2. From figure 3, it can be easily seen that these operations are exactly the same as the preceding part. One can use calculations similar to those in equations (14)–(16) to get the following state:

$$\begin{eqnarray}&&\sqrt{2}{{{\rm cos} }^{2}}\xi {\rm sin} \beta |{{H}_{3}}{{\rangle }_{x}}{{({\rm cos} \frac{\delta }{2}|V\rangle }_{y}}+{\rm sin} \frac{\delta }{2}|H{{\rangle }_{y}})\pm \frac{{\rm sin} 2\xi }{\sqrt{2}}|{{V}_{2}}{{\rangle }_{x}}{{({\rm cos} \frac{\delta }{2}|H\rangle }_{y}}+{\rm sin} \frac{\delta }{2}|V{{\rangle }_{y}}).\end{eqnarray} \tag{ 22 }$$

After that, photon y passes through HWPω. Here we set the direction of HWPω with ${\rm cos} 2\omega =({\rm cos} \frac{\delta }{2}-{\rm sin} \frac{\delta }{2})/\sqrt{2}$ and ${\rm sin} 2\omega =({\rm cos} \frac{\delta }{2}+{\rm sin} \frac{\delta }{2})/\sqrt{2}$. Incidentally, this HWP can realize the unitary transformation of

$$\begin{eqnarray}&&{{\delta }^{-}}|H\rangle \langle H|+{{\delta }^{+}}|V\rangle \langle H|+{{\delta }^{+}}|H\rangle \langle V|-{{\delta }^{-}}|V\rangle \langle V|\end{eqnarray} \tag{ 23 }$$

where ${{\delta }^{\pm }}=({\rm cos} \frac{\delta }{2}\pm {\rm sin} \frac{\delta }{2})/\sqrt{2}$. At the same time, the photon in path 2 is reflected by a PBS, whereas the photon in path 3 transits the PBS. Then, by intensive calculations, one can see that the state in equation (22) transforms to

$$\begin{eqnarray}&&{{{\rm cos} }^{2}}\xi |H{{\rangle }_{x}}|H{{\rangle }_{y}}\pm {\rm cos} \xi {\rm sin} \xi {{(|V\rangle }_{x}}|V{{\rangle }_{y}}+|V{{\rangle }_{x}}|H{{\rangle }_{y}})+{{{\rm sin} }^{2}}\xi |H{{\rangle }_{x}}|V{{\rangle }_{y}}.\end{eqnarray} \tag{ 24 }$$

After that, photon y is split into paths 6 and 7 and x is split into 8 and 9 by using a PBS and a 50:50 BS, respectively. Subsequently, the photon in path 9 passes through an HWP $\frac{\pi }{4}$. Then state of photons x and y transforms to

$$\begin{eqnarray*}&&\begin{array}{l} {{{\rm cos} }^{2}}\xi \frac{|{{H}_{8}}{{\rangle }_{x}}+|{{V}_{9}}{{\rangle }_{x}}}{\sqrt{2}}|{{H}_{7}}{{\rangle }_{y}}\pm {\rm cos} \xi {\rm sin} \xi \frac{|{{V}_{8}}{{\rangle }_{x}}+|{{H}_{9}}{{\rangle }_{x}}}{\sqrt{2}}{{(|{{V}_{6}}\rangle }_{y}}+|{{H}_{7}}{{\rangle }_{y}}) \\ \quad +\;{{{\rm sin} }^{2}}\xi \frac{|{{H}_{8}}{{\rangle }_{x}}+|{{V}_{9}}{{\rangle }_{x}}}{\sqrt{2}}|{{V}_{6}}{{\rangle }_{y}}. \\ \end{array}\end{eqnarray*}$$

Photons in paths 7 and 8 are sent to QND3. Incidentally, the photons in paths 8 and 7 correspond to signal beams S₁ and S₂, respectively, in figure 2. Later photon x passes through the OWM. Corresponding to the two possible outcomes of the QND, photons x and y respectively collapse to

$$\begin{eqnarray}&&\sim {{{\rm cos} }^{2}}\xi |H{{\rangle }_{x}}|{{H}_{7}}{{\rangle }_{y}}\pm {\rm cos} \xi {\rm sin} \xi {{(|H\rangle }_{x}}|{{V}_{6}}{{\rangle }_{y}}+|V{{\rangle }_{x}}|{{H}_{7}}{{\rangle }_{y}})+{{{\rm sin} }^{2}}\xi |V{{\rangle }_{x}}|{{V}_{6}}{{\rangle }_{y}},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\begin{array}{l} \sim {{{\rm cos} }^{2}}\xi {{e}^{-i\vartheta (X)}}|V{{\rangle }_{x}}|{{H}_{7}}{{\rangle }_{y}}\pm {\rm cos} \xi {\rm sin} \xi {{[{{e}^{i\vartheta (X)}}|V\rangle }_{x}}|{{V}_{6}}{{\rangle }_{y}} \\ \quad +{{e}^{-i\vartheta (X)}}|H{{\rangle }_{x}}|{{H}_{7}}{{\rangle }_{y}}]+{{{\rm sin} }^{2}}\xi {{e}^{i\vartheta (X)}}|H{{\rangle }_{x}}|{{V}_{6}}{{\rangle }_{y}}. \\ \end{array}\end{eqnarray} \tag{ 26 }$$

For the collapse of equation (25), the classical feed-forward does nothing. On the other hand, if the collapse state is equation (26), the classical feed-forward performs the additional phase shift $2\vartheta (X)$ of the photon in path 7 and carries out the ${{\sigma }^{x}}$ operation on photon x. Obviously, after the classical feed-forward, the state of photons x and y is that in equation (25). Finally, photon y passes over a PBS. Then equation (25) evolves to

$$\begin{eqnarray}&&{{{\rm cos} }^{2}}\xi |H{{\rangle }_{x}}|H{{\rangle }_{y}}\pm {\rm cos} \xi {\rm sin} \xi {{(|H\rangle }_{x}}|V{{\rangle }_{y}}+|V{{\rangle }_{x}}|H{{\rangle }_{y}})+{{{\rm sin} }^{2}}\xi |V{{\rangle }_{x}}|V{{\rangle }_{y}}=|{{\varphi }_{\pm }}{{\rangle }_{x}}|{{\varphi }_{\pm }}{{\rangle }_{y}}.\end{eqnarray} \tag{ 27 }$$

It is obvious that the $1\to 2$ PQC has been successfully achieved.

Note that in this scheme we have set $0\leqslant \xi \leqslant \frac{\pi }{4}$. Of course, ξ can be $\frac{\pi }{4}$ or 0. If $\xi =\frac{\pi }{4}$, the states $|{{\varphi }_{+}}{{\rangle }_{x}}$ and $|{{\varphi }_{-}}{{\rangle }_{x}}$ are orthogonal. Although our scheme works and the success probability is unit in this case, one can first measure the to-be-cloned state by utilizing a PBS and two SPTDs to confirm that state is $|{{\varphi }_{+}}{{\rangle }_{x}}$ or $|{{\varphi }_{-}}{{\rangle }_{x}}$. Then photon y can be directly prepared in the state $|{{\varphi }_{+}}{{\rangle }_{y}}$ or $|{{\varphi }_{-}}{{\rangle }_{y}}$ according to the MR. By this means, the cloning process is obviously simpler than our previous cloning scheme. On the other hand, if $\xi =0$, we know that $|{{\varphi }_{+}}{{\rangle }_{x}}=|{{\varphi }_{-}}{{\rangle }_{x}}=|H{{\rangle }_{x}}$, That is, there will be no photons in path 2. The evolutions caused by QND1 and QND2 in this case are presented in equation (11). It follows that, in accordance with calculations similar to those in equations(12)–(27), our scheme apparently works and the success probability Υ reaches $1/2$. From equations (4) and (18), it can be seen that $1/2\leqslant \Upsilon \leqslant 1$. If $\xi \to 0$, one can obtain $\Upsilon \to 1/2$. Therefore, $1/2$ is the lower limit of our PQC scheme. It should be pointed out that for PQC of a state, randomly choosing from a two-state set {$|{{\varphi }_{+}}\rangle$, $|{{\varphi }_{-}}\rangle$} (here ξ is arbitrary), one can casually prepare $|{{\varphi }_{+}}{{\rangle }_{x}}|{{\varphi }_{+}}{{\rangle }_{y}}$ (or $|{{\varphi }_{-}}{{\rangle }_{x}}|{{\varphi }_{-}}{{\rangle }_{y}}$). In this way, the probability for getting the correct cloning is just $1/2$. Therefore, the lower limit $1/2$ is a useless probability for PQC, and consequently, our scheme is nonsensical in the case $\xi =0$. The reason is that if $|{{\varphi }_{+}}\rangle =|{{\varphi }_{-}}\rangle$, the two states are linearly dependent. Theorem 1 in [27] has shown that linearly dependent states cannot be probabilistically cloned. In fact, in the case $\xi =0$, the to-be-cloned state is unambiguous to us. So there is only one state in the set, and we can directly prepare two photons in $|H{{\rangle }_{x}}|H{{\rangle }_{y}}$ to complete the cloning. In short, our PQC scheme should not contain the case $\xi =0$.

Now we present the quantum network for our economical $1\to 2$ PQC in subsection 3.2 (see figure 4). In this network, qubit x is the to-be-cloned qubit and the state is randomly chosen from the set $\{|{{\varphi }_{+}}\rangle ,|{{\varphi }_{-}}\rangle \}$ (see equation (3)). Qubit z is the assistant qubit that indicates whether the cloning progress is successful. The initial state of qubit z is set as $|\psi \rangle ={\rm cos} \frac{\beta }{2}|0\rangle +{\rm sin} \frac{\beta }{2}|1\rangle$. To clone the initial state $|{{\varphi }_{\pm }}\rangle$ economically, a CNOT gate is first performed on qubits x and z. Here the CNOT gate is represented as

$$\begin{eqnarray}&&C=|01\rangle \langle 00|+|00\rangle \langle 01|+|10\rangle \langle 10|+|11\rangle \langle 11|.\end{eqnarray} \tag{ 28 }$$

Therefore, qubits x and z undergo the following transformation:

$$\begin{eqnarray}\begin{array}{rcl} {{C}_{xz}}|{{\varphi }_{\pm }}{{\rangle }_{x}}|\psi {{\rangle }_{z}} & = & {\rm cos} \xi |0{{\rangle }_{x}}{{({\rm cos} \frac{\beta }{2}|1\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|0{{\rangle }_{z}})\pm {\rm sin} \xi |1{{\rangle }_{x}}{{({\rm cos} \frac{\beta }{2}|0\rangle }_{z}}+{\rm sin} \frac{\beta }{2}|1{{\rangle }_{z}}) \\ {} & = & \sqrt{\Upsilon }{{\left( \sqrt{2}{{{\rm cos} }^{2}}\xi {\rm sin} \beta |0\rangle \pm \frac{{\rm sin} 2\xi }{\sqrt{2}}|1\rangle \right)}_{x}}|u{{\rangle }_{z}}+\sqrt{1-\Upsilon }|0{{\rangle }_{x}}|{{u}^{\bot }}{{\rangle }_{z}}, \\ \end{array}\end{eqnarray} \tag{ 29 }$$

where $|u\rangle ={\rm cos} \frac{\beta }{2}|0\rangle +{\rm sin} \frac{\beta }{2}|1\rangle$ and $|{{u}^{\bot }}\rangle =-{\rm sin} \frac{\beta }{2}|0\rangle +{\rm cos} \frac{\beta }{2}|1\rangle$. Incidentally, it can be easily verified that $\langle u|{{u}^{\bot }}\rangle =0$. After the CNOT gate, a projection measurement is performed on qubit z with the measurement bases $|u\rangle$ and $|{{u}^{\bot }}\rangle$. From equation (29) one can see that, if the MR is $|{{u}^{\bot }}\rangle$, qubit x collapses to $|0\rangle$. Obviously, this state does not contain any information about the original to-be-cloned qubit. In this case, the cloning process is discontinued. Otherwise, with the probability Υ, the MR is $|u\rangle$ and qubit x collapses to

$$\begin{eqnarray}&&|\varphi {{^{\prime} }_{\pm }}\rangle =\sqrt{2}{{{\rm cos} }^{2}}\xi {\rm sin} \beta |0\rangle \pm \frac{{\rm sin} 2\xi }{\sqrt{2}}|1\rangle .\end{eqnarray} \tag{ 30 }$$

Note that the probability Υ is just the upper limit of the PQC of two linearly independent states. In this case, one can introduce cloning qubit y and achieve the cloning process deterministically by two CNOT gates and a single-qubit unitary operation. The original state of cloning qubit y is chosen as

$$\begin{eqnarray}&&|\phi \rangle ={\rm cos} \frac{\delta }{2}|0\rangle +{\rm sin} \frac{\delta }{2}|1\rangle .\end{eqnarray} \tag{ 31 }$$

The second CNOT gate in figure 4 is simply C in equation (28). The single-qubit unitary operation and the third CNOT gate are respectively represented as

$$\begin{eqnarray}\begin{array}{rcl} U & = & \frac{1}{\sqrt{2}}[({\rm cos} \frac{\delta }{2}-{\rm sin} \frac{\delta }{2})|0\rangle \langle 0|+({\rm cos} \frac{\delta }{2}+{\rm sin} \frac{\delta }{2})|1\rangle \langle 0| \\ {} & {} & +({\rm sin} \frac{\delta }{2}+{\rm cos} \frac{\delta }{2})|0\rangle \langle 1|-({\rm cos} \frac{\delta }{2}-{\rm sin} \frac{\delta }{2})|1\rangle \langle 1|], \\ \end{array}\end{eqnarray} \tag{ 32 }$$

and

$$\begin{eqnarray}&&C^{\prime} =|00\rangle \langle 00|+|01\rangle \langle 01|+|11\rangle \langle 10|+|10\rangle \langle 11|.\end{eqnarray} \tag{ 33 }$$

After these operations, one can easily obtain the following transformation:

$$\begin{eqnarray}&&C{{^{\prime} }_{yx}}{{U}_{y}}{{C}_{xy}}|\varphi {{^{\prime} }_{\pm }}{{\rangle }_{x}}|\phi {{\rangle }_{y}}=|{{\varphi }_{\pm }}{{\rangle }_{x}}|{{\varphi }_{\pm }}{{\rangle }_{y}}.\end{eqnarray} \tag{ 34 }$$

This transformation shows that the state of qubit x is cloned to qubit y. The quantum network for $1\to 2$ PQC has now been presented in detail.

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