Enhancing the teleportation of quantum Fisher information by weak measurement and environment-assisted measurement

Abstract

The quantum teleportation is reexamined in a Fisher information perspective. Two proposals of enhancing the teleportation of quantum Fisher information (QFI) under amplitude damping channel are proposed by utilizing the weak measurement and environment-assisted measurement (EAM), respectively. Although both schemes enable us to improve the teleportation of QFI with a certain probability, the latter proposal can completely recover the initial QFI, while the former only approaches to it. A detailed comparison confirms that the scheme of EAM outperforms the WM one in the improvements of QFI. The underlying reason is that the EAM is a post-measurement; thus, it extracts more information from the system and environment than the pre-posed WM. Our research offers an active way to enhance the teleportation of QFI under the amplitude damping decoherence, which is very important for quantum metrology and other quantum information tasks.

Appendices

Appendix A. Derivation of $F_{\phi , C}$

According to the optimal quantum cloning machine [56, 57], one qubit can be cloned into M copies

$$\begin{aligned} \alpha | 0\rangle +\beta |1\rangle \rightarrow \alpha |\varphi _{0}\rangle + \beta |\varphi _{1}\rangle , \end{aligned}$$

(A.1)

where

$$\begin{aligned} |\varphi _{0}\rangle= & {} \sum _{j_{1}=0}^{M-1} \sqrt{\frac{2(M-j_{1})}{M(M+1)}}|\{0,M-j_{1}\},\{1, j_{1}\}\rangle _{C} \otimes |\{0,M-1-j_{1}\},\{1, j_{1}\}\rangle _{A}\\ |\varphi _{1}\rangle= & {} \sum _{j_{2}=0}^{M-1}\sqrt{\frac{2(M-j_{2})}{M(M+1)}} |\{0,M-j_{2}\},\{1, j_{2}\}\rangle _{C} \otimes |\{0,j_{2}\},\{1, M-1-j_{2}\}\rangle _{A}. \end{aligned}$$

In the above, the subscripts C and A indicate the copies hold by M qubits and the $M-1$ ancilla qubits, respectively. $|\{0,M-j\},\{1, j\}\rangle$ denotes the symmetric and normalized state of M qubits with $(M-j)$ qubits in state $|0\rangle$ and j qubits in the orthogonal state $|1\rangle$.

After tracing the ancilla qubits, the copies can be given by

$$\begin{aligned} \rho _{C}= & {} \alpha ^{2} \sum _{j=0}^{M-1} \frac{2(M-j)}{M(M+1)} |\{0, M-j\},\{1, j \}\rangle _{C} \langle \{0,M-j \}, \{1,j \}| \nonumber \\&+ 2 \alpha \beta ^{*} \sum _{j=0}^{M-1} \frac{\sqrt{(M-j)(j+1)}}{M(M+1)} |\{0, M-j\},\{1, j \}\rangle _{C} \langle \{0,M-1-j \}, \{1,j+1\}|\nonumber \\&+ 2 \alpha \beta \sum _{j=0}^{M-1} \frac{\sqrt{(M-j)(j+1)}}{M(M+1)} |\{0, M-1-j\},\{1, j+1 \}\rangle _{C} \langle \{0,M-j \}, \{1,j \}|\nonumber \\&+ |\beta |^{2} \sum _{j=0}^{M-1} \frac{2(M-j)}{M(M+1)} |\{0, j\},\{1,M- j \}\rangle _{C} \langle \{0, j \}, \{1, M-j \}|. \end{aligned}$$

(A.2)

One of the copies can be described by

$$\begin{aligned} \rho= & {} \left( \frac{2M+1}{3M} \alpha ^{2}+\frac{M-1}{3M} |\beta |^{2}\right) | 0\rangle \langle 0 | +\frac{M+2}{3M} \alpha \beta ^{*} | 0\rangle \langle 1 | \nonumber \\&+ \left( \frac{2M+1}{3M} |\beta |^{2}+\frac{M-1}{3M} \alpha ^{2}\right) | 1\rangle \langle 1| + \frac{M+2}{3M} \alpha \beta | 1 \rangle \langle 0 |. \end{aligned}$$

(A.3)

Assume $\alpha = \cos \frac{\theta }{2}$, $\beta = e^{i\phi } \sin \frac{\theta }{2}$. The Bloch vector of the state is

$$\begin{aligned} {\mathbf {r}}_{x}= & {} \frac{M+2}{3M} \sin \theta \cos \phi , \end{aligned}$$

(A.4)

$$\begin{aligned} {\mathbf {r}}_{y}= & {} \frac{M+2}{3M} \sin \theta \sin \phi , \end{aligned}$$

(A.5)

$$\begin{aligned} {\mathbf {r}}_{z}= & {} \frac{M+2}{3M} \cos \theta . \end{aligned}$$

(A.6)

The QFI of $\phi$ encoded in the cloned qubit is

$$\begin{aligned} F_{\phi }=\left( \frac{M+2}{3M}\right) ^2 \sin ^{2} \theta . \end{aligned}$$

(A.7)

In classical situations, one can make infinite number of copies of a state, i.e., $M \rightarrow \infty$. Correspondingly, the $F_{\phi }$ approaches to

$$\begin{aligned} F_{\phi ,C}=F_{\phi }^{M \rightarrow \infty }=\frac{1}{9} \sin ^{2} \theta . \end{aligned}$$

(A.8)

When $\theta =\pi /2$, $F_{\phi ,C}=1/9$, which is the largest QFI obtained with classical operations.

Appendix B. Derivation of Eq. (21)

The schemes proposed in this manuscript are non-deterministic because of the non-unitary operations of WM and QMR. WM can be regarded as POVM and the probability could be calculated by following the standard postulate of quantum measurement. The probability of obtaining the measurement outcome of $M_{\mathrm{tot}}=M_{\mathrm{WM}}^{(2)} \otimes M_{\mathrm{WM}}^{(3)}$ is

$$\begin{aligned} {\mathcal {P}}_{1}= & {} tr\left( M_{\mathrm{tot}}^{\dagger }M_{\mathrm{tot}} \rho \right) \\= & {} \frac{(1+\eta ) (1+\bar{p_{1}} \bar{p_{2}})+ {\bar{\eta }} ( \bar{p_{1}}+\bar{p_{2}})}{4}.\nonumber \end{aligned}$$

(B.1)

Then, the reduced state will pass through the AD channel, which yields to

$$\begin{aligned} \varrho _{\mathrm{AD}}=\Big [\sum _{i=0}^{3} K_{i}\big (M_{\mathrm{tot}} \rho M^{\dagger }_{\mathrm{tot}}\big )K_{i}^{\dagger }\Big ]. \end{aligned}$$

(B.2)

The probability of obtaining the measurement outcome of ${\mathcal {W}}_{\mathrm{tot}}={\mathcal {M}}_{\mathrm{QMR}}^{(2)} \otimes {\mathcal {M}}_{\mathrm{QMR}}^{(3)}$ is

$$\begin{aligned} {\mathcal {P}}_{2}=tr\left( {\mathcal {W}}_{\mathrm{tot}}^{\dagger }{\mathcal {W}}_{\mathrm{tot}} \varrho _{\mathrm{AD}}\right) =\frac{N}{{\mathcal {P}}_{1}}, \end{aligned}$$

(B.3)

where N is the normalization factor of Eq. (13). The final success probability depends on the WM and QMR are performed successfully

$$\begin{aligned} P_{\mathrm{WM}}={\mathcal {P}}_{1}{\mathcal {P}}_{2}=N. \end{aligned}$$

(B.4)

This means the final success probability just equals to the final normalization factor of the teleported state. Choosing the optimal QMR strength $q^{\mathrm{opt}}_{\mathrm{WM}} = 1- \sqrt{\frac{B}{A}}$, the above equation reduces to Eq. (21).

Appendix C. Derivation of Eq. (22)

After experienced the AD noise, the Werner state will be evolved into

$$\begin{aligned} \varrho '= & {} \sum _{i=0}^{3} K_{i}\rho K_{i}^{\dagger }\nonumber \\= & {} \varrho '_{11} |00\rangle _{23} \langle 00| +\varrho '_{22} |01\rangle _{23} \langle 01| +\varrho '_{33} |10\rangle _{23} \langle 10|\\&+\varrho '_{44} |11\rangle _{23} \langle 11| +\varrho '_{14} |00\rangle _{23} \langle 11| + \varrho '_{41} |11\rangle _{23} \langle 00|, \nonumber \end{aligned}$$

(C.1)

where

$$\begin{aligned} \varrho '_{11}= & {} [1+\eta +2{\bar{\eta }}\gamma +(1+\eta ) \gamma ^{2}]/4, \\ \varrho '_{22}= & {} \varrho '_{33}=[{\bar{\eta }}{\bar{\gamma }}+(1+\eta )\gamma {\bar{\gamma }}]/4, \\ \varrho '_{44}= & {} (1+\eta ){\bar{\gamma }}^{2}/4, \\ \varrho '_{14}= & {} \varrho '_{41} = \eta {\bar{\gamma }}/2, \end{aligned}$$

where we have assumed $\gamma _{1}=\gamma _{2}=\gamma$. By the assistant of EAM, the quantum state corresponding to “no click” is picked out, i.e.,

$$\begin{aligned} \varrho '_{\mathrm{EAM}}= & {} K_{0}\rho K_{0}^{\dagger }\\= & {} \frac{1}{Q}\left( \begin{array}{cccc}\frac{(1+\eta )}{4} &{} 0 &{} 0 &{} \frac{\eta {\bar{\gamma }} }{2} \\ 0 &{} \frac{(1-\eta ){\bar{\gamma }}}{4} &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{(1-\eta ){\bar{\gamma }}}{4} &{} 0 \\ \frac{\eta {\bar{\gamma }} }{2} &{} 0 &{} 0 &{} \frac{(1+\eta ){\bar{\gamma }}^2 }{4}\end{array}\right) , \end{aligned}$$

with $Q=\frac{1+\eta }{4}+ \frac{1-\eta }{2} {\bar{\gamma }}+ \frac{1+\eta }{4} {\bar{\gamma }}^{2}$. Following the operations of QMR described by Eq. (5), one can get Eq. (22).