Unified quantum no-go theorems and transforming of quantum pure states in a restricted set

Abstract

The linear superposition principle in quantum mechanics is essential for several no-go theorems such as the no-cloning theorem, the no-deleting theorem and the no-superposing theorem. In this paper, we investigate general quantum transformations forbidden or permitted by the superposition principle for various goals. First, we prove a no-encoding theorem that forbids linearly superposing of an unknown pure state and a fixed pure state in Hilbert space of a finite dimension. The new theorem is further extended for multiple copies of an unknown state as input states. These generalized results of the no-encoding theorem include the no-cloning theorem, the no-deleting theorem and the no-superposing theorem as special cases. Second, we provide a unified scheme for presenting perfect and imperfect quantum tasks (cloning and deleting) in a one-shot manner. This scheme may lead to fruitful results that are completely characterized with the linear independence of the representative vectors of input pure states. The upper bounds of the efficiency are also proved. Third, we generalize a recent superposing scheme of unknown states with a fixed overlap into new schemes when multiple copies of an unknown state are as input states.

Appendices

Appendix A: Completing proof of the Theorem 1

Assume that there exists a unitary map $\mathcal{U}$ with the matrix representative U satisfying Eq. (3) for all unknown states $|{\psi }\rangle$ in Hilbert space ${\mathbb {H}}$. For two basis states $|0\rangle$ and $|1\rangle$, it follows that

$$\begin{aligned} U\times (|0\rangle \otimes |\Sigma \rangle \otimes |P_0\rangle )\propto & {} \sqrt{p_0}|0\rangle \otimes |\Sigma _0\rangle \otimes |P_1\rangle +\sqrt{p_1}|{{\Phi }_0}\rangle , \end{aligned}$$

(47)

$$\begin{aligned} U\times (|1\rangle \otimes |\Sigma \rangle \otimes |P_0\rangle )\propto & {} \sqrt{p'_0}|{\chi '}\rangle \otimes |{\Sigma _1}\rangle \otimes |{P_1}\rangle +\sqrt{p'_1}{{\Phi }_1}\rangle , \end{aligned}$$

(48)

where $|\chi '\rangle =\alpha |1\rangle +\beta |0\rangle$. $|{\Sigma }_0\rangle$ and $|{\Sigma }_1\rangle$ are the output states of the ancillary system and generally depend on the input states.

For any state $|\psi \rangle = a|0\rangle +b|1\rangle$ (a and b satisfy $|a|^2+|b|^2=1$), we obtain

$$\begin{aligned} U\times (|\psi \rangle \otimes |{\Sigma }\rangle \otimes |P_0\rangle ) \propto \sqrt{p''_0}|\varphi _1\rangle \otimes |\Sigma _2\rangle \otimes |P_1\rangle +\sqrt{p''_1}|{\Phi }_2\rangle \end{aligned}$$

(49)

where $|\varphi _1\rangle =\sqrt{r''}({\alpha }|\psi \rangle +{\beta }|0\rangle )$ and $r''$ is a normalization constant, and $|\Sigma _2\rangle$ is the output state of the ancillary system. Moreover, from the linearity of quantum operations and Eqs. (47) and (47), we obtain

$$\begin{aligned} U \times ( |\psi \rangle \otimes |\Sigma \rangle \otimes |P_0\rangle ) \propto |\Psi \rangle \otimes |P_1\rangle +a\sqrt{p_1}|{\Phi }_0\rangle +b\sqrt{p'_1}|{\Phi }_1\rangle \end{aligned}$$

(50)

where $|\Psi \rangle =a\sqrt{p_0}|0\rangle \otimes |{\Sigma }_0\rangle +b\sqrt{p'_0}|\chi '\rangle \otimes |\Sigma _1\rangle$. Let $|{\Sigma }_i\rangle =a_i|0\rangle +b_i|1\rangle$ with $|a_i|^2+|b_i|^2=1$, $i=1, 2, 3$. From Eqs. (49) and (50), we obtain $a=0$, which is a contradiction to $|\psi \rangle$ in Eq. (49) with arbitrary a and b of $|a|^2+|b|^2=1$.

Appendix B: Proof of Theorem 3

Similar to the proof of Theorem 2, we only need to prove the result for $0<|\beta |<1$. In detail, for $0<|\beta |<1$ assume that there exists a unitary map $\mathcal{U}$ with the matrix representative U for all the states $|\psi \rangle$ in Hilbert space ${\mathbb {H}}$ such that

$$\begin{aligned} U\times (|\psi \rangle ^{\otimes k}\otimes |\Sigma \rangle \otimes |P_0\rangle ) \propto \sqrt{p_0}|\varphi \rangle ^{\otimes n}\otimes |G\rangle \otimes |P_1\rangle +\sqrt{p_1}|\Phi \rangle \end{aligned}$$

(51)

where $|\varphi \rangle =\sqrt{r}(\alpha |\psi \rangle +\beta |\phi \rangle )$ and r is a normalization constant. $|\Sigma \rangle$ is an ancillary state in Hilbert space ${\mathbb {H}}_a={\mathbb {H}}^m$ with $m>n$. $|G\rangle$ is an ancillary state in Hilbert space ${\mathbb {H}}^{m+k-n}$, which may generally depend on the input state $|\varphi \rangle$. $|P_0\rangle$ and $|P_1\rangle$ are two states in Hilbert space ${\mathbb {H}}_p$ with $\mathrm{dim}({\mathbb {H}}_p)\gg 2$. Theses two states are independent of the input states. Let $e^{i\theta }|\varphi \rangle$ be a new state. From Eq. (51) it follows that

$$\begin{aligned} U\times (e^{i\theta }|\psi \rangle ^{\otimes k}\otimes |\Sigma \rangle \otimes |P_0\rangle )\propto & {} \sqrt{p_0} |\varphi '\rangle ^{\otimes n}\otimes |G'\rangle \otimes |P_0\rangle +\sqrt{p_1}|\Phi '\rangle \end{aligned}$$

(52)

where $|\varphi '\rangle =\sqrt{r'}(\sqrt{\alpha }e^{i\theta }|\varphi \rangle +\sqrt{\beta }|\phi \rangle )$ and $r'$ is a normalization constant. $|G'\rangle$ is another ancillary state in Hilbert space ${\mathbb {H}}^{m+k-n}$. Note that $|\psi \rangle \propto e^{i\theta }|\psi \rangle$, i.e., the pure state $e^{i\theta }|\psi \rangle$ is physically undiscriminating from the pure state $|\psi \rangle$. From Eq. (51), it follows that

$$\begin{aligned} U\times (e^{i\theta }|\psi \rangle ^{\otimes k}\otimes |\Sigma \rangle \otimes |P_0\rangle )\propto U\times (|\psi \rangle ^{\otimes k}\otimes |\Sigma \rangle \otimes |P_0\rangle ). \end{aligned}$$

(53)

From Eqs. (52) and (53), we obtain $|\varphi \rangle ^{\otimes n}\otimes |G\rangle \propto |\varphi '\rangle ^{\otimes n}\otimes |G'\rangle$ which yields to $|\beta |=0$ or $|\beta |=1$. This is a contradiction to $0<|\beta |<1$ in Eq. (51).

Appendix C: The proof of inequality (26)

From Eq. (22) and the triangle inequality, we obtain

$$\begin{aligned} |a_{ij}|^k\le & {} \sum _{s=1}^{M+k}\sqrt{p_{is}p_{js}}(\sqrt{r_{is}r_{js}} |\alpha _{is}\alpha _{js}|\times |a_{ij}|^s +|\alpha _{is}\beta _{js}\lambda _{ij,s}|\nonumber \\&+|\beta _{is}\alpha _{js}\lambda _{ji,s}| +|\beta _{is}\beta _{js}|) +\sum _{\ell =M+k+1}^N\sqrt{q_{i\ell }q_{j\ell }}. \end{aligned}$$

(54)

Note that for all the complex constants $\alpha _{is}, \beta _{js}$ with $|\alpha _{is}|, |\beta _{js}|\le 1$, we have $|\alpha _{is}\beta _{js} \lambda _{ij,s}|+|\beta _{is}\alpha _{js}\lambda _{ji,s}|\le \max \{|\lambda _{ij,s}|,|\lambda _{ji,s}|\} \times (|\alpha _{is}\beta _{js}|+|\beta _{is}\alpha _{js}|)$, and $|\alpha _{is}\beta _{js}|+|\beta _{is}\alpha _{js}| \le \frac{1}{2}( |\alpha _{is}|^2+|\beta _{js}|^2 +|\beta _{is}|^2+|\alpha _{js}|^2)=1$ from the equalities $|\alpha _{is}|^2+|\beta _{is}|^2=|\alpha _{js}|^2+|\beta _{js}|^2=1$. Hence, from the arithmetic-geometric average inequality, Eq. (54) reduces to

$$\begin{aligned} |a_{ij}|^k\le \sum _{s=1}^{M+k}{p}_{ij,s}\left( \max \{|\lambda _{ij,s}|,|\lambda _{ji,s}|\} +\alpha _{ij,s}{r}_{ij,s}|a_{ij}|^s+\beta _{ij,s}-1\right) +1, \end{aligned}$$

(55)

where ${p}_{ij,s}=(p_{is}+p_{js})/2$, $\alpha _{ij,s}=|\alpha _{is}\alpha _{js}|$, $\beta _{ij,s}=|\beta _{is}\beta _{js}|$ and ${r}_{ij,s}=\sqrt{r_{is}r_{js}}$, and we have taken use of the equalities $\sum _{s=1}^{M+k}p_{is}+\sum _{\ell =M+k+1}^N q_{i\ell }=1$, $i=1, 2, \ldots , m$. Denote $D_{ij,t}:=2(1-|a_{ij}|^t)$ with $t=k$ or s. Since $1-\beta _{ij,s} \ge \alpha _{ij,s}$ for $|\alpha _{is}|^2+|\beta _{is}|^2=|\alpha _{js}|^2+|\beta _{js}|^2=1$, Eq. (55) yields to inequality (26).

Appendix D: The proof of inequality (33)

From Eq. (29) we obtain

$$\begin{aligned} |a_{i_1j_1}a_{i_2j_2}|\le & {} \sqrt{p_{i_1i_2}p_{j_1j_2}} \sqrt{r_{i_1i_2}r_{j_1j_2}} \times (|\alpha _{i_1}\alpha _{j_1}a_{i_1j_1}|+|\beta _{i_2}\beta _{j_2}a_{i_2j_2}| \nonumber \\&+|\alpha _{i_1}\beta _{j_2}a_{i_1j_2}|+|\beta _{i_2}\alpha _{j_1}a_{i_2j_1}| ) +\sqrt{q_{i_ii_2}q_{j_1j_2}} \nonumber \\\le & {} \frac{1}{3}\sqrt{p_{i_1i_2}p_{j_1j_2}} \sqrt{r_{i_1i_2}r_{j_1j_2}} \times (2|\alpha _{i_1}|^2 +2|\alpha _{j_1}|^2 +2|\beta _{j_2}|^2 +2|\beta _{i_2}|^2 \nonumber \\&+|a_{i_1j_1}|+|a_{i_2j_2}| +|a_{i_2j_1}|+|a_{i_1j_2}|) +\sqrt{q_{i_ii_2}q_{j_1j_2}} \nonumber \\\le & {} \frac{1}{3}\sqrt{p_{i_1i_2}p_{j_1j_2}} \sqrt{r_{i_1i_2}r_{j_1j_2}} \times (|a_{i_1j_1}|+|a_{i_2j_2}| +|a_{i_2j_1}| \nonumber \\&+|a_{i_1j_2}|+4)+\sqrt{q_{i_ii_2}q_{j_1j_2}}, \end{aligned}$$

(56)

where the second inequality is derived from the inequality $x^2+y^2+z\ge x^3+y^3+z^3\ge 3 xyz$ for $0\le x,y,z\le 1$, and the last equality is derived from the equalities $|\alpha _{i}|^2+|\beta _{i}|^2=1$, $i=1, 2, \ldots , m$. By using the arithmetic inequality for ${p}_{i_ii_2}{p}_{j_1j_2}$ and $q_{i_ii_2}q_{j_1j_2}$, Eq. (56) yields to inequality (33).

Appendix E: Proof of Theorem 7

Note that $|\psi _1\rangle ^{\otimes k}\otimes |\psi _1\rangle ^{\otimes k}$, $|\psi _1\rangle ^{\otimes k}\otimes |\psi _2\rangle ^{\otimes k}$, $\ldots , |\psi _m\rangle ^{\otimes k}\otimes |\psi _m\rangle ^{\otimes k}$ are linearly dependent if and only if $|\psi _1\rangle ^{\otimes k}, |\psi _2\rangle ^{\otimes k}, \ldots , |\psi _m\rangle ^{\otimes k}$ are linearly dependent. Similar to the proof of Theorem 5, the necessity is easily followed from the linearity of quantum operations and the superposition principle. Now, we prove that these CP maps will be presented in a unified form with a nontrivial probability in the following. In fact, from Stinespring dilation theorem [56], assume that there exists a unitary map $\mathcal{U}\in \mathcal{SU}({\mathbb {H}}^k\otimes {\mathbb {H}}^k\otimes {\mathbb {H}}_a\otimes {\mathbb {H}}_p)$ with matrix representative U such that

$$\begin{aligned} U\times \left( |\psi _i\rangle ^{\otimes k}\otimes |{\psi _j}\rangle ^{\otimes k}\otimes |{\Sigma }\rangle \otimes |{P_0}\rangle \right)\propto & {} \sum _{s=1}^{M+2k}\sqrt{p_{ij,s}}|{\varphi _{ij,s}}\rangle \otimes |0\rangle \otimes ^{M+2k-s}|P_s\rangle \nonumber \\&+\sum ^N_{\ell =M+2k+1}\sqrt{q_{ij,\ell }}|{\Phi }_{ij,\ell }\rangle \otimes |P_\ell \rangle \end{aligned}$$

(57)

Here, $|\Sigma \rangle$ is an ancillary state in Hilbert space ${\mathbb {H}}_a={\mathbb {H}}^M$ with $M>\max \{k,s\}$. $p_{ij,s}$ is the success probability of producing the desired state $|\varphi _{ij,s}\rangle$. $|P_0\rangle , |P_1\rangle , \ldots , |P_N\rangle$ are orthogonal states in an ancillary Hilbert space ${\mathbb {H}}_p$ with $\mathrm{dim}({\mathbb {H}}_p)>N+1$. $|{\Phi }_{ij,\ell }\rangle$ is a normalized failure state in ${\mathbb {H}}^{k}\otimes {\mathbb {H}}^{M}$ with the probability $q_{ij,\ell }$ which satisfies $\sum _{s=1}^{M+2k}p_{ij,s}+\sum _{\ell =M+2k+1}^Nq_{ij,\ell }=1$, $i,j=1, 2, \ldots , m$. Each CP map $\mathcal{F}_j$ is specified by the unitary mapping $\mathcal{U}$ and the followed projector $|P_j\rangle \langle P_j|$, $j=1, 2, \ldots , M+k$.

For the input states $|\psi _{i_1}\rangle ^{\otimes k}\otimes |\psi _{i_2}\rangle ^{\otimes k}$ and $|\psi _{j_1}\rangle ^{\otimes k}\otimes |\psi _{j_2}\rangle ^{\otimes k}$, taking the inner product of the output states, it follows that

$$\begin{aligned} a_{i_1j_1}^{k}a_{i_2j_2}^{k}= & {} \sum _{s=1}^{M+2k}\sqrt{p_{i_1i_2,s}p_{j_1j_2,s} r_{i_1i_2,s}r_{j_1j_2,s}}h^{(s)}_{i_1i_2,j_2j_2}\nonumber \\&+\sum _{\ell =M+2k+1}^N \sqrt{q_{i_1i_2,\ell }q_{j_1j_2,\ell }}b_{i_1i_2,j_2j_2,\ell }, \end{aligned}$$

(58)

where $a_{ij}=\langle \psi _i|\psi _j\rangle$, $b_{i_1i_2,j_2j_2,\ell }=\langle \Phi _{i_1i_2,\ell }|\Phi _{j_1j_2,\ell }\rangle$, and $h^{(s)}_{i_1i_2,j_2j_2}=\alpha ^*_{i_1i_2}\alpha _{j_1j_2}a^s_{i_1j_1} +\beta ^*_{i_1i_2}\beta _{j_1j_2}a^s_{i_2j_2} +\alpha ^*_{i_1i_2}\beta _{j_1j_2}a^s_{i_1j_2} +\beta ^*_{i_1i_2}\alpha _{j_1j_2}a^s_{i_2j_1}$ for all integers $1\le i, j, i_1, i_2, j_1, j_2\le m$ and $M+2k+1\le \ell \le N$. Eq. (58) may be briefly represented by a matrix equation:

$$\begin{aligned} A^{\circ k}=\sum _{s=1}^{M+2k}\Lambda _s\times B_s \times \Lambda _s^\dag + \sum _{\ell =M+2k+1}^{N}Q_\ell , \end{aligned}$$

(59)

where

$$\begin{aligned}&A=\left[ a_{i_1j_1}a^*_{i_2j_2} \right] _{m^2\times m^2},\quad \Lambda _s=\mathrm{diag}(p_{11,s}, p_{12,s}, \ldots , p_{mm,s})_{m^2\times m^2}, \\&H_s= \left[ \sqrt{r_{i_1i_2,s}r_{j_1j_2,s}}h^{(s)}_{i_1i_2,j_2j_2} \right] _{m^2\times m^2},\quad Q_\ell =\left[ \sqrt{q_{i_1i_2,\ell }q_{j_1j_2,\ell }}b_{i_1i_2,j_2j_2,\ell }\right] _{m^2\times m^2}, \end{aligned}$$

Here, the rows or columns of these matrices are represented by two-bit series $i_1i_2$ or $j_1j_2$, respectively. It is sufficient to prove Eq. (59) with physical realizable matrices $Q_\ell$, $\ell =M+2k+1, \ldots , L$. In fact, $A^{\circ k}$ is positive definite from Theorem 4 because A is positive definite from the linear independence of $|\psi _1\rangle ^{\otimes k}, |\psi _2\rangle ^{\otimes k}, \ldots , |\psi _m\rangle ^{\otimes k}$. The matrix $\sum _{s=1}^{M+2k}\Lambda _s H_s \Lambda _s^\dag$ is Hermitian. From Lemma 1, the matrix $A^{\circ k}-\sum _{s=1}^{M+2k}\Lambda _s B_s \Lambda _s^\dag$ is positive definite when matrices $\Lambda _1, \Lambda _2, \ldots , \Lambda _{M+2k}$ satisfy the following inequality

$$\begin{aligned} \left\| \sum _{s=1}^{M+2k}\Lambda _s\times B_s \times \Lambda _s^\dag \right\| _2<\left\| (A^{\circ k})^{-1}\right\| _2^{-1} \end{aligned}$$

(60)

Therefore, there exists a unitary matrix V such that

$$\begin{aligned} V\times \left( A^{\circ k}-\sum _{s=1}^{M+2k}\Lambda _s\times B_s \times \Lambda _s^\dag \right) \times V^\dag =\mathrm{diag}(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{m^2}) \end{aligned}$$

(61)

where $\lambda _{1}, \lambda _{2}, \ldots , \lambda _{m^2}$ are all the eigenvalues of $A^{\circ k}-\sum _{s=1}^{k_3}\Lambda _s \times H_s \times \Lambda _s^\dag$ and satisfy $\lambda _j>0, j=1, 2, \ldots , m^2$. Define $Q_\ell =V^\dag \times \mathrm{diag}(\lambda _{1,\ell }$, $\lambda _{2,\ell }, \ldots , \lambda _{m^2,\ell })\times V$, where $\lambda _{1,\ell }$, $\lambda _{2,\ell }, \ldots , \lambda _{m^2,\ell }$ are positive constants and satisfy $\sum _{\ell =M+2k+1}^NQ_\ell =\mathrm{diag}(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{m^2})$. So, $Q_\ell$ is positive definite and then physically realizable, $\ell =M+2k+1, M+2k+2, \ldots , N$. This completes the proof. $\square$

In the following, the bound of the success probability is proved in terms of the state metric [49]. In detail, from Eq. (58) we obtain

$$\begin{aligned} |a_{i_1j_1}a_{i_2j_2}|^k\le & {} \sum _{s=1}^{M+2k}\sqrt{p_{i_1i_2,s}p_{j_1j_2,s}} \sqrt{r_{i_1i_2,s}r_{j_1j_2,s}} (|\alpha _{i_1i_2}\alpha _{j_1j_2}|\times |a_{i_1j_1}|^s\nonumber \\&+|\beta _{i_1i_2}\beta _{j_1j_2}|\times |a_{i_2j_2}|^s+|\beta _{i_1i_2}\alpha _{j_1j_2}|\times |a_{i_2j_1}|^s +|\alpha _{i_1i_2}\beta _{j_1j_2}|\times |a_{i_1j_2}|^s)\nonumber \\&+\sum _{\ell =M+2k+1}^N\sqrt{q_{i_1i_2,\ell }q_{j_1j_2,\ell }} \nonumber \\\le & {} \frac{1}{3}\sum _{s=1}^{M+2k}\sqrt{p_{i_1i_2,s}p_{j_1j_2,s}} \sqrt{r_{i_1i_2,s}r_{j_1j_2,s}} \left[ 2|\alpha _{i_1i_2}|^2+2|\alpha _{j_1j_2}|^2 +2|\beta _{i_1i_2}|^2\right. \nonumber \\&\left. +2|\beta _{j_1j_2}|^2 +|a_{i_1j_1}|^s+|a_{i_1j_2}|^s+|a_{i_2j_1}|^s+|a_{i_2j_2}|^s \right] \nonumber \\&+\sum _{\ell =M+2k+1}^N\sqrt{q_{i_1i_2,\ell }q_{j_1j_2,\ell }} \nonumber \\= & {} \frac{1}{3}\sum _{s=1}^{M+2k}\sqrt{p_{i_1i_2}p_{j_1j_2}} \sqrt{r_{i_1i_2}r_{j_1j_2}} (|a_{i_1j_1}|^s+|a_{i_2j_2}|^s\nonumber \\&+|a_{i_2j_1}|^s+|a_{i_1j_2}|^s+4)+\sum _{\ell =M+2k+1}^N\sqrt{q_{i_1i_2,\ell }q_{j_1j_2,\ell }}, \end{aligned}$$

(62)

where the second inequality is derived from the inequality $x^2+y^2+z\ge x^3+y^3+z^3\ge 3 xyz$ for $0\le x,y,z\le 1$, and the last equality is derived from the equalities $|\alpha _{ij}|^2+|\beta _{ij}|^2=1$ for any integers $1\le i, j\le m$ and $1\le s\le M+2k$.

By using the arithmetic inequality for $p_{i_1i_2,s}p_{j_1j_2,s}$ and $q_{i_1i_2,s}q_{j_1j_2,s}$, and the equality $\sum _{s=1}^{M+2k}p_{ij,s}+\sum _{\ell =M+2k+1}^Nq_{ij,\ell }=1$ for all $i,j=1, 2, \ldots , m$, Eq. (62) leads to

$$\begin{aligned} D_{i_1j_1,k}D_{i_2j_2,k}\le & {} \frac{2}{3}\sum _{s=1}^{M+2k}p_{i_1i_2,j_1j_2,s} \left[ r_{i_1i_2,j_1j_2}\left( 16-D_{i_1j_1,s}\right. \right. \nonumber \\&\left. \left. -D_{i_1j_2,s}-D_{i_2j_2,s} -D_{i_2j_1,s}\right) -6\right] \nonumber \\&+4D_{i_1j_1,k}+4D_{i_2j_2,k}, \end{aligned}$$

(63)

where $r_{i_1i_2,j_1j_2}=\sqrt{r_{i_1i_2,s}r_{j_1j_2,s}}$, $p_{i_1i_2,j_1j_2,s}=(p_{i_1i_2,s}+p_{j_1j_2,s})/2$ and $D_{ij,s}=2(1-|a_{ij}|^s)$. The inequality has generalized the bound in Eq. (33).

Appendix F: Proof of the uniqueness of Theorem 8

In this appendix, we complete the proof of Theorem 8 according to the proof [17]. Let $|v\rangle =\sqrt{c}(\sqrt{c_1}|0\rangle +\sqrt{c_2}|1\rangle )$ be an ancillary vector and $|\psi \rangle$ and $|\phi \rangle$ be states in ${\mathbb {H}}$, $|X\rangle \in {\mathbb {H}}^{3}$ satisfy the conditions

$$\begin{aligned} \mathrm{Tr}(|X\rangle \langle X|\times (|{\psi }\rangle \langle \psi |\otimes |\phi \rangle \langle \phi |\otimes |\psi \rangle \langle \psi |))= & {} c_1>0, \nonumber \\ \mathrm{Tr}(|X\rangle \langle X|\times (|{\phi }\rangle \langle \phi |\otimes |\psi \rangle \langle \psi |\otimes |\phi \rangle \langle \phi |))= & {} c_2>0. \end{aligned}$$

(64)

Let $\mathcal{F}\in \mathcal{CP}({\mathbb {C}}^2\otimes {\mathbb {H}}^{2}\otimes {\mathbb {H}}^{2},{\mathbb {H}})$ be a CP map satisfying

$$\begin{aligned} \mathcal{F}(|\mu \rangle \otimes |{\psi }\rangle ^{\otimes 2}\otimes |{\phi }\rangle ^{\otimes 2})\propto |{\varphi }\rangle , \end{aligned}$$

(65)

where $|\varphi \rangle$ is given by

$$\begin{aligned} |\varphi \rangle = \sqrt{r}(\alpha e^{i\theta _1} |\psi \rangle + \beta e^{i\theta _2}|\phi \rangle ) \end{aligned}$$

and r is a normalization constant dependent of $\alpha , \beta , |\psi \rangle$ and $|\phi \rangle$, $e^{i\theta _1}=\frac{\langle X|\phi \rangle \otimes |\psi \rangle \otimes |\phi \rangle }{|\langle X|\phi \rangle \otimes |\psi \rangle \otimes |\phi \rangle |}$ and $e^{i\theta _2}=\frac{\langle X|\psi \rangle \otimes |\phi \rangle \otimes |\psi \rangle }{|\langle X|\psi \rangle \otimes |\phi \rangle \otimes |\psi \rangle |}$. Let $\{F_i|F_i:{\mathbb {C}}^2\otimes {\mathbb {H}}^{4}\rightarrow {\mathbb {H}}\}_{i\in J}$, form the Kraus decomposition of $\mathcal{F}$. Using the analogous procedure to Theorem 1 [17], we get

$$\begin{aligned} F_i(|\mu \rangle \otimes |{\psi }\rangle ^{\otimes 2}\otimes |{\phi }\rangle ^{\otimes 2}) = |\varphi \rangle ,\quad \text{ for } \text{ all } i\in J \end{aligned}$$

(66)

up to a global normalization factor. Now consider the single Kraus operator $F_i$. It follows that

$$\begin{aligned} F_i|\mu \rangle \otimes |\psi \rangle ^{\otimes 2}|\phi \rangle ^{\otimes 2} =a(\alpha e^{i\theta _1}|\psi \rangle +\beta e^{i\theta _2}|\phi \rangle ) \end{aligned}$$

(67)

for all vectors $|\psi \rangle$, $|\phi \rangle \in {\mathbb {H}}$ satisfying the condition in Eq. (64) and a constant a which is dependent of $\alpha , \beta , |\psi \rangle$ and $|\phi \rangle$. This definition is unique up to a global factor. In detail, from the linearity of the left side of Eq. (67), a is independent of $\alpha$ and $\beta$. Moreover, from the linearity of $F_i$ and Eq. (67), it follows that

$$\begin{aligned}&a\left( |\psi \rangle ^{\otimes 2},|\phi \rangle ^{\otimes 2}\right) =a\left( e^{i\theta '_1}|\psi \rangle ^{\otimes 2},e^{i\theta '_2}|\phi \rangle ^{\otimes 2}\right) \end{aligned}$$

(68)

for arbitrary phases $\theta '_i$. Hence, we can assume the following form

$$\begin{aligned} |\phi \rangle \otimes |\psi \rangle \otimes |\phi \rangle= & {} \sqrt{c_1}|X\rangle + \sqrt{d_1}|\Phi ^{\bot }\rangle , \end{aligned}$$

(69)

$$\begin{aligned} |\psi \rangle \otimes |\phi \rangle \otimes |\psi \rangle= & {} \sqrt{c_2}|X\rangle + \sqrt{d_2}|\Psi ^{\bot }\rangle , \end{aligned}$$

(70)

where $|X\rangle$ is the vector representative of the state $\rho _X$ and $c_i+d_i=1$, $|\Phi ^{\bot }\rangle$ and $|\Psi ^{\bot }\rangle$ are normalized orthogonal complements of $|X\rangle$. From Eq.(67), we obtain

$$\begin{aligned}&F_i\otimes I\otimes I\left[ |\mu \rangle \otimes (\sqrt{c_1}|\tilde{X}\rangle +d_1|\tilde{\Phi }^{\bot }\rangle ) (\sqrt{c_2}|\tilde{X}\rangle +d_2|\tilde{\Psi }^{\bot }\rangle )\right] \nonumber \\&=\tilde{a}((\alpha \sqrt{c_1}+\beta \sqrt{c_2})|\tilde{X}\rangle +\alpha \sqrt{d_1}|\Psi ^{\bot }\rangle +\beta \sqrt{d_2}|\Phi ^{\bot }\rangle )\otimes |\phi \rangle \otimes |\psi \rangle , \end{aligned}$$

(71)

where the vectors $|\tilde{X}\rangle =(S_{1,2}\otimes I)|X\rangle$, $|\tilde{\Phi }^{\bot }\rangle =(S_{1,2}\otimes I)|{\Phi }^{\bot }\rangle$ and $|\tilde{\Psi }^{\bot }\rangle =(S_{1,2}\otimes I)|{\Psi }^{\bot }\rangle$. For normalized vectors $|\tilde{\Psi }^{\bot }\rangle$ and $|\tilde{\Phi }^{\bot }\rangle$, the function

$$\begin{aligned} (\theta _1, \theta _2)\mapsto \tilde{a}\left( e^{i\theta _1}|\tilde{\Psi }^{\bot }\rangle , e^{i\theta _2}|\tilde{\Phi }^{\bot }\rangle \right) \end{aligned}$$

(72)

is a smooth function on ${\mathbb {S}}_1\times {\mathbb {S}}_1$, where ${\mathbb {S}}_1$ denotes the complex circle on ${\mathbb {C}}^2$. By inserting $e^{i\theta _1}|\tilde{\Psi }^{\bot }\rangle$ and $e^{i\theta _2}|\tilde{\Phi }^{\bot }\rangle$ in Eq. (72), from the Fourier transformation and the linearity of $F_i$, it follows that

$$\begin{aligned} \tilde{a}\left( e^{i\theta _1}|\tilde{\Psi }^{\bot }\rangle , e^{i\theta _2}|\tilde{\Phi }^{\bot }\rangle \right) =\tilde{a}\left( |\tilde{\Psi }^{\bot }\rangle ,|\tilde{\Phi }^{\bot }\rangle \right) . \end{aligned}$$

(73)

Moreover, we obtain

$$\begin{aligned} (F\otimes I\otimes I)\times (|\mu \rangle \otimes |\tilde{X}\rangle \otimes |\tilde{X}\rangle )= & {} \tilde{a}\tilde{c}|\tilde{X}\rangle \otimes |\phi \rangle \otimes |\psi \rangle , \end{aligned}$$

(74)

$$\begin{aligned} (F\otimes I\otimes I)\times (|\mu \rangle \otimes |\tilde{\Phi }^\bot \rangle \otimes |\tilde{\Psi }^{\bot }\rangle )= & {} 0, \end{aligned}$$

(75)

$$\begin{aligned} (F\otimes I\otimes I)\times (|\mu \rangle \otimes |\tilde{\Phi }^\bot \rangle \otimes |\tilde{X}\rangle )= & {} \frac{\tilde{a}\alpha }{\sqrt{c_2}}|\tilde{\Phi }^\bot \rangle \otimes |\phi \rangle \otimes |\psi \rangle , \end{aligned}$$

(76)

$$\begin{aligned} (F\otimes I\otimes I)\times (|\mu \rangle \otimes |\tilde{X}\rangle \otimes |\tilde{\Psi }^\bot \rangle )= & {} \frac{\tilde{a}\beta }{\sqrt{c_2}}|\tilde{\Psi }^\bot \rangle \otimes |\phi \rangle \otimes |\psi \rangle \end{aligned}$$

(77)

where $\tilde{c}=(\frac{\alpha }{\sqrt{c_2}}+\frac{\beta }{\sqrt{c_2}})$. From Eqs. (76) and (77), $\tilde{a}$ should be independent of vectors $|\tilde{\Phi }^\bot \rangle$ and $|\tilde{\Psi }^\bot \rangle$. It means that $\tilde{a}$ is a constant. Similar proof may be followed for the conditions in Eq. (64). This completes the proof. ${} \Box$

The presented proof also holds for $|\psi \rangle$ and $|\phi \rangle$ satisfying $\mathrm{Tr}(|X\rangle \langle X|\times (|{\phi }\rangle \langle \phi |\otimes |\psi \rangle \langle \psi |\otimes |\phi \rangle \langle \phi |))=\lambda c_1$ and $\mathrm{Tr}(|X\rangle \langle X|\times (|{\psi }\rangle \langle \psi |\otimes |\phi \rangle \langle \phi |\otimes |\psi \rangle \langle \psi |))=\lambda c_2$, where $\lambda \in (0, \frac{1}{\max \{c_1,c_2\}}]$. The superpositions are then generated with a probability $p'=\lambda p$. From the uniqueness result, it is impossible to generate superpositions for all input states with a nonzero overlap with $|{X}\rangle$.

We now present an explicit protocol to generate the superposition with a higher success probability [17]. Let $\mathcal{G}(|\Phi \rangle \langle \Phi |)=G|\Phi \rangle \langle \Phi | G^\dag$, for a linear mapping $\mathcal{G}:{\mathbb {H}}^{4}\rightarrow {\mathbb {H}}$ defined by $G=G_2G_1$, where

$$\begin{aligned} G_1=\frac{\alpha }{\sqrt{c_1}}I\otimes I\otimes I\otimes I +\frac{\beta }{\sqrt{c_2}}S_{1,3}\otimes S_{2,4}, G_2=I\otimes \langle X|, \end{aligned}$$

$S_{i,j}$ denotes the swapping operation of the ith and jth state in ${\mathbb {H}}$. The action of $G_2$ on the tensor of $|x_1\rangle \otimes |x_2\rangle \otimes |x_3\rangle \otimes |x_4\rangle$ is given

$$\begin{aligned} I\otimes \langle X|x_1\rangle \otimes |x_2\rangle \otimes |x_3\rangle \otimes |x_4\rangle =|x_1\rangle (\langle X|x_2\rangle \otimes |x_3\rangle \otimes |x_4\rangle ) \end{aligned}$$

(78)

for all $\rho _{x_i}\in {\mathbb {H}}$. With forward evaluations, we can obtain

$$\begin{aligned} G\times \left( |\psi \rangle ^{\otimes 2}\otimes |\phi \rangle ^{\otimes 2}\right) =\alpha e^{i\theta _1}|\psi \rangle + \beta e^{i\theta _2}|\phi \rangle \end{aligned}$$

(79)

which shows that $G\times (|\psi \rangle \otimes |\psi \rangle \otimes |\phi \rangle \otimes |\phi \rangle )= |\varphi \rangle$ up a global factor. $\mathcal{G}$ is trace non-increasing if and only if $GG^\dag \le I\otimes I$, i.e,

$$\begin{aligned} G^\dag \times G= & {} \frac{|\alpha |^2}{c_1}I\otimes |X\rangle \langle X| +\frac{|\beta |^2}{c_2} |X\rangle \langle X|\otimes I +\frac{\alpha \beta ^*}{\sqrt{c_1c_2}}I\otimes |X\rangle \langle X| (S_{1,3}\otimes S_{2,4}) \nonumber \\&+\frac{\alpha ^*\beta }{\sqrt{c_1c_2}}(S_{1,3}\otimes S_{2,4}) (I\otimes |X\rangle \langle X|) \end{aligned}$$

(80)

The maximal eigenvalue of $G^\dag \times G$ is given by

$$\begin{aligned} \lambda _{max} =\max \left\{ \frac{|\beta |^2}{c_2}+\frac{|\alpha |^2}{2c_1} \left( \sqrt{\frac{4|\beta |^2}{c_2}+1}+1\right) , \left| \frac{\alpha }{\sqrt{c_1}}+\frac{\beta }{\sqrt{c_2}}\right| ^2\right\} . \end{aligned}$$

(81)

The largest $x\in {\mathbb {R}}^+$ satisfying $\tilde{\mathcal{F}}=x\cdot \mathcal{F}$ being non-increasing is $1/\lambda _\mathrm{max}$. The success probability is

$$\begin{aligned} p_{s}=\mathrm{Tr}\left( \tilde{\mathcal{F}} (|\psi \rangle \langle \psi |^{\otimes 2}\times |{\phi }\rangle \langle \phi |^{\otimes 2})\right) = \frac{N^2_{\varphi }}{\lambda _\mathrm{max}}. \end{aligned}$$

(82)

Hence, $\tilde{p}_{s}\ge p_{s}$ if and only if

$$\begin{aligned} \frac{1}{c_1}+\frac{1}{c_2}\ge \lambda _\mathrm{max} \end{aligned}$$

(83)

for $c_{i}\in (0,1]$ and $|\alpha |^2+|\beta |^2=1$.

Appendix G: Proof of Theorem 9

Define $|v\rangle =\sqrt{c}(\sqrt{c_1}|0\rangle +\sqrt{c_2}|1\rangle )$, where c is a normalization constant. Given two permutations $\tau _1, \tau _2\in {\mathbb {P}}_{2k-1}$, for all states $|\Psi _{\tau _1(1,2,\cdots , 2k-1)}\rangle$ satisfying $\mathrm{Tr}(|X\rangle \langle X|\times |\Psi _{\tau _1(1,2,\ldots , 2k-1)}\rangle \langle \Psi _{\tau _1(1,2,\ldots , 2k-1)}|)=c_1>0$ and states $|\Phi _{{\tau }_2(1,2,\ldots , 2k-1)}\rangle \langle \Phi _{{\tau }_2(1,2,\ldots , 2k-1)}|$ satisfying $\mathrm{Tr}(|X\rangle \langle X|\times |\Phi _{{\tau }_2(1,2,\ldots , 2k-1)}\rangle \langle \Phi _{{\tau }_2(1,2,\ldots ,} {2k-1)}|)=c_2>0$, we can define a CP map $\mathcal{F}_{\tau _1,\tau _2}$ such that

$$\begin{aligned} \mathcal{F}_{\tau _1,\tau _2}\left( |\mu \rangle \otimes |{\psi }\rangle ^{\otimes k} \otimes |{\phi }\rangle ^{\otimes k}\right) \propto |{\varphi }\rangle , \end{aligned}$$

(84)

where $|\varphi \rangle =\sqrt{r}(\alpha e^{i\theta _1}|\psi \rangle +\beta e^{i\theta _2}|\phi \rangle )$ with $e^{i\theta _1}=\frac{\langle X|\Phi _{{\tau }_2(1,2,\cdots , 2k-1)}\rangle }{|\langle X|\Phi _{{\tau }_2(1,2,\cdots , 2k-1)}\rangle |}$ and $e^{i\theta _2}=\frac{\langle X|\Psi _{\tau _1(1,2,\cdots , 2k-1)}\rangle }{|\langle X|\Psi _{\tau _1(1,2,\cdots , 2k-1)}\rangle |}$ and r is a normalized constant, and the vector $|\mu \rangle$ is given by $|\mu \rangle =\alpha |0\rangle +\beta |1\rangle$. Let

$$\begin{aligned} \mathcal{F}_{\tau _1,\tau _2}=\mathcal{F}_6\circ \mathcal{F}_5\circ \mathcal{F}_4\circ \mathcal{F}_3\circ \mathcal{F}_2\circ \mathcal{F}_1, \end{aligned}$$

(85)

where $\mathcal{F}_j(|\Phi \rangle \langle \Phi |)=F_j|\Phi \rangle \langle \Phi |F^\dag _j, j=1,2,\ldots , 5$, $\mathcal{F}_6(|\Phi \rangle \langle \Phi |)=\mathrm{Tr}_{1,3,4,\ldots , nk+1}(|\Phi \rangle \langle \Phi |)$, and

$$\begin{aligned} F_1= & {} |0\rangle \langle 0|\otimes I^{2k}+|1\rangle \langle 1|\otimes S_{2,k+2}\otimes I^{k-1}, \\ F_2= & {} |0\rangle \langle 0|\otimes I \otimes S_{\tau _1}+|1\rangle \langle 1| \otimes I^{2k}, \\ F_3= & {} |0\rangle \langle 0|\otimes I^{2k}+|1\rangle \langle 1|\otimes I \otimes S_{\tau _2} \\ F_4= & {} I_2\otimes I\otimes |X\rangle \langle X| , \\ F_5= & {} |v\rangle \langle v|\otimes I^{nk}, \end{aligned}$$

where $I^j$ denotes the identity operator on ${\mathbb {H}}^j (j=1,2, \ldots , nk)$, $I_2$ denotes the identity operator on ${\mathbb {C}}^2$, $S_{\tau _i}$ is a swapping operator induced by the permutation $\tau _i$ in ${\mathbb {P}}_{2k-1}$ and performed on the last $2k-1$ subsystems, $i=1, 2$. It is easy to get the result by forward evaluations from the input states $|{v}\rangle \otimes |{\psi }\rangle ^{\otimes k}\otimes |{\phi }\rangle ^{\otimes k}$.

Appendix H: Proof of Corollary 4

Define an ancillary vector

$$\begin{aligned} |v\rangle =\sqrt{c}\left( \sum _{j=0}^{n-1}\frac{\sqrt{c_j}}{\prod _{j=0}^{n-1}\sqrt{c_j}}|j\rangle \right) , \end{aligned}$$

(86)

where c is a normalization constant. Let

$$\begin{aligned} \mathcal{F}=\mathcal{F}_5\circ \mathcal{F}_4\circ \mathcal{F}_3\circ \mathcal{F}_2\circ \mathcal{F}_1, \end{aligned}$$

(87)

where $\mathcal{F}_j(|\Phi \rangle \langle \Phi |)\!=F_j|\Phi \rangle \langle \Phi |F^\dag _j, j\!=1,2,\cdots , 4$, $\mathcal{F}_5(|\Phi \rangle \langle \Phi |)=\mathrm{Tr}_{1,3,4,\cdots , nk,nk+1}(|\Phi \rangle \langle \Phi |)$, and

$$\begin{aligned} F_1= & {} |0\rangle \langle 0|\otimes I^{nk}+\sum _{j=1}^{n-1}|j\rangle \langle j|\otimes S_{2,jk+2}, \\ F_2= & {} \sum _{j=0}^{n-1}|j\rangle \langle j|\otimes S_{\tau _j}, \\ F_3= & {} I_n\otimes I\otimes |X\rangle \langle X|, \\ F_4= & {} |v\rangle \langle v|\otimes I^{nk}, \end{aligned}$$

where $I^j$ denotes the identity operator on Hilbert space ${\mathbb {H}}^j (j=1,2, \ldots , nk)$, $I_n$ denotes the identity operator on ${\mathbb {C}}^2$, $S_{\tau _j}$ is a swapping operator induced by the permutation $\tau _j$ in ${\mathbb {P}}_{nk-1}$ and performed on the last $nk-1$ subsystems, $j=0, 1, \ldots , n-1$. $F_1$ is used to change the vector $|v\rangle \otimes (\otimes _{i=1}^n |\psi _i\rangle ^{\otimes k})$ into $\sqrt{c}\sum _{j=0}^{n-1}\sqrt{c_j}|j\rangle \otimes |\psi _j\rangle \otimes |\Psi ^{j}_{1,2,\ldots , nk-1}\rangle$. $F_2$ is used to change the vector $\sqrt{c}\sum _{j=0}^{n-1}\sqrt{c_j}|j\rangle \otimes |\psi _j\rangle \otimes |\Psi ^{j}_{1,2,\ldots , nk-1}\rangle$ into $\sum _{j=0}^{n-1}\sqrt{c_j}|j\rangle \otimes |\psi _j\rangle \otimes |\Psi ^{j}_{\tau _j(1,2,\ldots , nk-1)}\rangle$, where $|\Psi ^{j}_{1,2,\ldots , nk-1}\rangle =\otimes _{t=1}^n|\psi _t\rangle ^{\otimes k_{jt}}$ with integers $k_{jt}$ satisfying $k_{jt}=k$ for $j\not =t$ and $k_{jt}=k-1$ for $j=t$. From forward evaluations, we can prove the results using the followed measurements induced by the operators $|X\rangle \langle X|$ and $|v\rangle \langle v|$ [17].