Qubit state tomography in a superconducting circuit via weak measurements

The cQED system was originally described by the well-known Jaynes–Cummings model [28]. In the dispersive regime [28], i.e., with the detuning between the cavity frequency (${\omega }_{r}$) and qubit energy (${\omega }_{q}$) being much larger than the coupling strength g, the cQED system can be described by the Hamiltonian [28]

$$\begin{eqnarray}&&{H}_{\mathrm{eff}}={{\rm{\Delta }}}_{r}{a}^{\dagger }a+\displaystyle \frac{{\widetilde{\omega }}_{q}}{2}{\sigma }_{z}+\chi {a}^{\dagger }a{\sigma }_{z}+({\epsilon }_{m}^{* }a+{\epsilon }_{m}{a}^{\dagger }),\end{eqnarray} \tag{ 2 }$$

where ${{\rm{\Delta }}}_{r}={\omega }_{r}-{\omega }_{m}$ arises from the fact that this Hamiltonian is expressed in the rotating frame with the microwave drive frequency (${\omega }_{m}$). The transition frequency of the qubit reads ${\widetilde{\omega }}_{q}={\omega }_{q}+\chi$, which is modified by a dispersive shift $\chi ={g}^{2}/{\rm{\Delta }}$, with ${\rm{\Delta }}={\omega }_{r}-{\omega }_{q}$. In equation (2), ${a}^{\dagger }$ (a) and ${\sigma }_{z}$ are respectively the creation (annihilation) operator of the cavity photon and the quasi-spin (Pauli matrix) of the qubit. ${\epsilon }_{m}$ is the microwave drive amplitude applied to the cavity.

The dispersive coupling characterized by the third term in the above Hamiltonian allows for a homodyne measurement with output current given by [44]

$$\begin{eqnarray}&&I(t)=-\sqrt{{{\rm{\Gamma }}}_{{ci}}(t)}\langle {\sigma }_{z}\rangle +\xi (t),\end{eqnarray} \tag{ 3 }$$

where $\xi (t)$ is a Gaussian white noise originating from fundamental quantum jumps during the measurement. This expression for the current was obtained in the absence of qubit rotation and by eliminating the cavity degrees of freedom (the so-called polaron transformation) [44]. In a bit more detail, ${{\rm{\Gamma }}}_{{ci}}(t)$ is the coherent information gain rate of measurement given by [44]

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{ci}}(t)=\kappa | \beta (t){| }^{2}{\cos }^{2}(\varphi -{\theta }_{\beta }),\end{eqnarray} \tag{ 4 }$$

where φ is the LO's phase in the homodyne measurement, κ is the leakage rate of the cavity photons, and $\beta (t)={\alpha }_{2}(t)-{\alpha }_{1}(t)\equiv | \beta (t)| {{\rm{e}}}^{{\rm{i}}{\theta }_{\beta }}$ with ${\alpha }_{1}(t)$ and ${\alpha }_{2}(t)$ being the cavity fields associated with the qubit states $| 1\rangle$ and $| 2\rangle$, respectively.

In addition to the information gain rate ${{\rm{\Gamma }}}_{{ci}}$, there exists as well a no-information back-action rate, which reads [44]

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{ba}}(t)=\kappa | \beta (t){| }^{2}{\sin }^{2}(\varphi -{\theta }_{\beta }).\end{eqnarray} \tag{ 5 }$$

To understand the physical meaning, let us consider the stochastic evolution of qubit state ${c}_{1}(t)| 1\rangle +{c}_{2}(t)| 2\rangle$, conditioned on measurement records in a single realization. The rate ${{\rm{\Gamma }}}_{{ci}}$ appearing in equation (3) is associated with the attempt to distinguish the qubit basis states (information gain), which causes a change in probability between $| 1\rangle$ and $| 2\rangle$ conditioned on the measurement record, together with a relative phase change between them that conserves the quantum purity of the superposed state—this actually corresponds to the term 'coherent information gain'. Unlike the information gain rate, the rate ${{\rm{\Gamma }}}_{{ba}}$ is only related to phase fluctuation between $| 1\rangle$ and $| 2\rangle$ (no change in probability). The sum of ${{\rm{\Gamma }}}_{{ci}}$ and ${{\rm{\Gamma }}}_{{ba}}$, ${{\rm{\Gamma }}}_{m}={{\rm{\Gamma }}}_{{ci}}+{{\rm{\Gamma }}}_{{ba}}$, gives the total measurement rate. Differing somehow from ${{\rm{\Gamma }}}_{m}$, the overall decoherence rate is given by [44]

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{d}(t)=4\chi \mathrm{Im}[{\alpha }_{1}(t){\alpha }_{2}^{* }(t)],\end{eqnarray} \tag{ 6 }$$

as a result of tracing the cavity degrees of freedom from the whole entangled qubit-cavity state. An interesting point is that ${{\rm{\Gamma }}}_{m}$ is not necessarily equal to ${{\rm{\Gamma }}}_{d}$, owing to certain 'information loss'. It is only for ideal (quantum limited) measurement that ${{\rm{\Gamma }}}_{m}={{\rm{\Gamma }}}_{d}$ and the single quantum trajectory is a quantum-mechanically pure state.

Conditioned on the output currents, equation (3), one can faithfully keep track of the stochastic evolution of the qubit state. In order to get analytic expression for the PPS average, rather than the equation for the quantum trajectory, we apply alternatively the quantum Bayesian rule (BR) [45–48]. Using the output currents during $(0,{t}_{m})$, we update first the diagonal elements ${\rho }_{{jj}}$ (j = 1, 2) of the qubit state (density matrix) [47, 48]

$$\begin{eqnarray}&&{\rho }_{{jj}}({t}_{m})={\rho }_{{jj}}(0)\,{P}_{j}({t}_{m})/{ \mathcal N }({t}_{m}),\end{eqnarray} \tag{ 7 }$$

where ${ \mathcal N }({t}_{m})={\sum }_{j=\mathrm{1,2}}{\rho }_{{jj}}(0)\,{P}_{j}({t}_{m})$. This result simply follows the standard Bayes formula, and the functional distribution of current reads [48]

$$\begin{eqnarray}&&{P}_{1(2)}({t}_{m})=\displaystyle \frac{1}{{{ \mathcal N }}_{I}}\exp \{-\langle {[I(t)-{\bar{I}}_{1(2)}(t)]}^{2}{\rangle }_{{t}_{m}}/(2V)\},\end{eqnarray} \tag{ 8 }$$

where ${\bar{I}}_{1(2)}(t)=\mp \sqrt{{{\rm{\Gamma }}}_{{ci}}(t)}$ and $\langle \bullet {\rangle }_{{t}_{m}}={({t}_{m})}^{-1}{\int }_{0}^{{t}_{m}}{dt}(\bullet )$. $V=1/{t}_{m}$ is the distribution variance, and ${{ \mathcal N }}_{I}$ is the normalization factor of the distribution probability (to be canceled from the numerator and denominator of the Bayes formula). Note that equation (8) differs from our usual knowledge. According to the central limit theorem, corresponding to $| 1\rangle$ and $| 2\rangle$, the averaged stochastic current, ${I}_{m}={({t}_{m})}^{-1}{\int }_{0}^{{t}_{m}}{\rm{d}}{t}\,{I}(t)$, should respectively be centred at ${\bar{I}}_{1(2)}=\mp {({t}_{m})}^{-1}{\int }_{0}^{{t}_{m}}{\rm{d}}{t}\,\sqrt{{{\rm{\Gamma }}}_{{ci}}(t)}$ and satisfy the standard Gaussian distribution

$$\begin{eqnarray}&&{P}_{1(2)}({t}_{m})={(2\pi V)}^{-1/2}\exp [-{({I}_{m}-{\bar{I}}_{1(2)})}^{2}/(2V)].\end{eqnarray} \tag{ 9 }$$

In [48], we have demonstrated that this 'standard' result is valid only for a time-independent rate ${{\rm{\Gamma }}}_{{ci}}$, in equation (3).

Secondly, we update the off-diagonal elements as follows:

$$\begin{eqnarray}&&{\rho }_{12}({t}_{m})={\rho }_{12}(0)[\sqrt{{P}_{1}({t}_{m}){P}_{2}({t}_{m})}/{ \mathcal N }({t}_{m})]D({t}_{m})\,\exp \{-{\rm{i}}[{{\rm{\Phi }}}_{1}({t}_{m})+{{\rm{\Phi }}}_{2}({t}_{m})]\}.\end{eqnarray} \tag{ 10 }$$

Compared to the original simple BR [45], a couple of correction factors appear in this result, specifically given by [47, 48]

$$\begin{eqnarray}&&D({t}_{m})=\exp \left\{-{\int }_{0}^{{t}_{m}}{\rm{d}}{t}[{{\rm{\Gamma }}}_{d}(t)-{{\rm{\Gamma }}}_{m}(t)]/2\right\},\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{1}({t}_{m})={\int }_{0}^{{t}_{m}}{\rm{d}}{t}\,{\widetilde{{\rm{\Omega }}}}_{q}(t),\end{eqnarray} \tag{ 11b }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{2}({t}_{m})=-{\int }_{0}^{{t}_{m}}{\rm{d}}{t}\sqrt{{{\rm{\Gamma }}}_{{ba}}(t)}\,I(t).\end{eqnarray} \tag{ 11c }$$

Here we have introduced ${\widetilde{{\rm{\Omega }}}}_{q}(t)={\omega }_{q}+\chi +{B}(t)$, i.e., the bare qubit energy ${\omega }_{q}$ is renormalized by the dispersive shift χ and the shift induced by the ac Stark effect, ${B}(t)=2\chi \mathrm{Re}[{\alpha }_{1}(t){\alpha }_{2}^{* }(t)]$. Briefly speaking, the purity degradation factor $D({t}_{m})$ is a result of non-ideality (information loss) in the measurement, while the two phase factors ${{\rm{e}}}^{-{\rm{i}}{{\rm{\Phi }}}_{1}({t}_{m})}$ and ${{\rm{e}}}^{-{\rm{i}}{{\rm{\Phi }}}_{2}({t}_{m})}$ result, respectively, from the dynamic ac Stark effect and the no-information back-action.

In experiments the cQED system is usually prepared in the bad-cavity and weak-response limits. In this case, the cavity field evolves to a stationary state on a timescale much shorter than the measurement time. One can thus carry out the ac Stark shift and all the rates using the stationary coherent-state fields of the cavity, ${\bar{\alpha }}_{1}$ and ${\bar{\alpha }}_{2}$, which read [47]

$$\begin{eqnarray}&&{\bar{\alpha }}_{1(2)}=-{\rm{i}}{\epsilon }_{m}/[-{\rm{i}}({{\rm{\Delta }}}_{r}\pm \chi )+\kappa /2],\end{eqnarray} \tag{ 12 }$$

where ${{\rm{\Delta }}}_{r}={\omega }_{m}-{\omega }_{r}$ is the offset of the measurement and cavity frequencies. For instance, in the bad-cavity and weak-response limits, we obtain the stationary B(t) as $B\simeq 2\chi \bar{n}$, where $\bar{n}=| \bar{\alpha }{| }^{2}$ and $\bar{\alpha }=-{\rm{i}}{\epsilon }_{m}/(\kappa /2)$, which recovers the standard ac Stark shift. Also, a resonant drive (${\omega }_{m}={\omega }_{r}$) results in ${\theta }_{\beta }=0$.

Now let us consider the weak value for finite-strength measurement. For simplicity, we denote the measurement result as $x\equiv {I}_{m}={({t}_{m})}^{-1}{\int }_{0}^{{t}_{m}}{\rm{d}}{t}\,{I}(t)$, and ${\bar{x}}_{j}\equiv {\bar{I}}_{j}={(-1)}^{j}\sqrt{{{\rm{\Gamma }}}_{{ci}}}$. In the same spirit as the AAV WV for an infinitesimal strength of measurement, we employ the following PPS average as a definition for the WV associated with finite-strength measurement [20, 38, 39]:

$$\begin{eqnarray}&&{}_{f}\langle x{\rangle }_{i}=\displaystyle \frac{\int {\rm{d}}{x}\,{{xP}}_{{\psi }_{i}}(x){P}_{x}({\psi }_{f})}{\int {\rm{d}}{x}\,{P}_{{\psi }_{i}}(x){P}_{x}({\psi }_{f})}.\end{eqnarray} \tag{ 13 }$$

${P}_{{\psi }_{i}}(x)$ is the distribution probability of the measurement outcomes associated with the pre-selected state $| {\psi }_{i}\rangle$, before the post-selection using $| {\psi }_{f}\rangle$ . ${P}_{x}({\psi }_{f})$ is the post-selection probability given by ${P}_{x}({\psi }_{f})=\langle {\psi }_{f}| \tilde{\rho }(x)| {\psi }_{f}\rangle$, by applying the quantum BR to update the state from ${\rho }_{i}$ to $\tilde{\rho }(x)$, based on the measurement outcome x. After some algebra, we obtain [39]

$$\begin{eqnarray}&&{}_{f}\langle x{\rangle }_{i}=-\displaystyle \frac{{\epsilon }_{1}\mathrm{Re}({\sigma }_{w}^{z})+{\epsilon }_{2}\mathrm{Im}({\sigma }_{w}^{z})}{1+{ \mathcal G }\,(| {\sigma }_{w}^{z}{| }^{2}-1)},\end{eqnarray} \tag{ 14 }$$

where ${\epsilon }_{1}=\sqrt{{{\rm{\Gamma }}}_{{ci}}}$, ${\epsilon }_{2}=\sqrt{{{\rm{\Gamma }}}_{{ba}}}\,{{\rm{e}}}^{-{{\rm{\Gamma }}}_{d}{t}_{m}}$ and ${ \mathcal G }=(1-{{\rm{e}}}^{-{{\rm{\Gamma }}}_{d}{t}_{m}})/2$. In this result, the AAV WV is slightly modified as

$$\begin{eqnarray}&&{\sigma }_{w}^{z}=\displaystyle \frac{\langle {\psi }_{f}| {\sigma }_{z}| {\tilde{\psi }}_{i}\rangle }{\langle {\psi }_{f}| {\tilde{\psi }}_{i}\rangle },\end{eqnarray} \tag{ 15 }$$

where $| {\tilde{\psi }}_{i}\rangle$ differs from the initial state $| {\psi }_{i}\rangle ={c}_{1}| 1\rangle +{c}_{2}| 2\rangle$ by a phase factor as $| {\tilde{\psi }}_{i}\rangle ={c}_{1}{{\rm{e}}}^{-{\rm{i}}{\widetilde{{\rm{\Omega }}}}_{q}{t}_{m}}| 1\rangle +{c}_{2}| 2\rangle$.

We see that, by tuning the LO phase φ based on equation (14), one can conveniently obtain the real and imaginary parts of ${\tilde{\sigma }}_{w}^{z}$, from which an efficient technique of state tomography can be developed for finite-strength measurement.

In experiments, the PPS average ${}_{f}\langle x{\rangle }_{i}$, or the meter's shift in other words, is the sub-ensemble average of output currents, post-selected from the records of the first stage of weak (partial collapse) measurement. In practice, one should keep and average only those 'x' data that are followed by successful post-selection of $| \psi {\rangle }_{f}$. To generate this post-selection for each trajectory, one may rotate the qubit state (quite arbitrarily) at the moment t_m, and determine $| \psi {\rangle }_{f}$ as the rotated version of the former basis state $| 1\rangle$. This procedure actually defines a new basis for the subsequent post-selection, i.e., with $| 1\rangle$ corresponding to $| \psi {\rangle }_{f}$. We may mention that this method of obtaining ${}_{f}\langle x{\rangle }_{i}$ is not affected by the measurement strength. After getting ${}_{f}\langle x{\rangle }_{i}$, the AAV WV can be easily extracted from equation (14).

Following the same definition of the PPS average, equation (13), we have

$$\begin{eqnarray}&&{}_{f}\langle x{\rangle }_{i}=\displaystyle \frac{\displaystyle \int { \mathcal D }[I(t)]\,x[I(t)]\,{P}_{{\psi }_{i}}(\{I(t)\})\,{P}_{\{I(t)\}}({\psi }_{f})}{\displaystyle \int { \mathcal D }[I(t)]\,{P}_{{\psi }_{i}}(\{I(t)\})\,{P}_{\{I(t)\}}({\psi }_{f})}\equiv \displaystyle \frac{{M}_{1}}{{M}_{2}},\end{eqnarray} \tag{ 16 }$$

where $x[I{(t)]=({t}_{m})}^{-1}{\int }_{0}^{{t}_{m}}I(t){\rm{d}}{t}$, and the two probability distribution functionals read

$$\begin{eqnarray*}{P}_{{\psi }_{i}}(\{I(t)\}) & = & {\rho }_{11}(0){P}_{1}({t}_{m})+{\rho }_{22}(0){P}_{2}({t}_{m}),\\ {P}_{\{I(t)\}}({\psi }_{f}) & = & {\tilde{\rho }}_{11}{\rho }_{f11}+{\tilde{\rho }}_{22}{\rho }_{f22}+{\tilde{\rho }}_{12}{\rho }_{f21}+{\tilde{\rho }}_{21}{\rho }_{f12}.\end{eqnarray*}$$

Here we have denoted the Bayesian updated state by $\tilde{\rho }=\tilde{\rho }(x)=\tilde{\rho }(\{I(t)\})$. The probabilities ${P}_{\mathrm{1,2}}({t}_{m})$ follow equation (8), being functionals of the current record $\{I(t)\,| \,t\in [0,{t}_{m}]\}$. In equation (16), involved in both the numerator and denominator is the functional (or 'path') integral, $\int { \mathcal D }[I(t)](\cdots )$, which means summing all the possible currents of the measurement. By means of the Gaussian path-integral method, calculation of equation (16) is straightforward. We obtain

$$\begin{eqnarray}{M}_{1} & = & -\left({\displaystyle \int }_{0}^{{t}_{m}}\sqrt{{{\rm{\Gamma }}}_{{ci}}(t)}{\rm{d}}{t}\right)({\rho }_{11}{\rho }_{f11}-{\rho }_{22}{\rho }_{f22})+\left({\displaystyle \int }_{0}^{{t}_{m}}\sqrt{{{\rm{\Gamma }}}_{{ba}}(t)}{\rm{d}}{t}\ \ {{\rm{e}}}^{-{\displaystyle \int }_{0}^{{t}_{m}}{{\rm{\Gamma }}}_{d}(t){\rm{d}}{t}}\right)\\ & & \times \,2\,\mathrm{Im}({\rho }_{21}{\rho }_{f12}{{\rm{e}}}^{{\rm{i}}{\displaystyle \int }_{0}^{{t}_{m}}\tilde{{\rm{\Omega }}}(t){\rm{d}}{t}}),\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{M}_{2}={\rho }_{11}{\rho }_{f11}+{\rho }_{22}{\rho }_{f22}+\left({{\rm{e}}}^{-{\displaystyle \int }_{0}^{{t}_{m}}{{\rm{\Gamma }}}_{d}(t){\rm{d}}{t}}\right)\times \,2\,\mathrm{Re}({\rho }_{21}{\rho }_{f12}{{\rm{e}}}^{{\rm{i}}{\displaystyle \int }_{0}^{{t}_{m}}\tilde{{\rm{\Omega }}}(t){\rm{d}}{t}}).\end{eqnarray} \tag{ 18 }$$

Reorganizing this result further in terms of the AAV WV form, we find that the same expression as equation (14) can be obtained, with only several parameters modified as

$$\begin{eqnarray}&&{\epsilon }_{1}={\displaystyle \int }_{0}^{{t}_{m}}\sqrt{{{\rm{\Gamma }}}_{{ci}}(t)}{\rm{d}}{t},{\epsilon }_{2}={\displaystyle \int }_{0}^{{t}_{m}}\sqrt{{{\rm{\Gamma }}}_{{ba}}(t)}{\rm{d}}{t}\,{{\rm{e}}}^{-{\displaystyle \int }_{0}^{{t}_{m}}{{\rm{\Gamma }}}_{d}(t){\rm{d}}{t}},{ \mathcal G }=(1-{{\rm{e}}}^{-{\displaystyle \int }_{0}^{{t}_{m}}{{\rm{\Gamma }}}_{d}(t){\rm{d}}{t}})/2.\end{eqnarray} \tag{ 19 }$$

The AAV WV of equation (15) is now modified by replacing the initial state $| {\psi }_{i}\rangle$ with $| {\tilde{\psi }}_{i}\rangle ={c}_{1}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Phi }}}_{1}({t}_{m})}| 1\rangle +{c}_{2}| 2\rangle$. ${{\rm{\Phi }}}_{1}({t}_{m})$ is given by equation (11b).

Actually, the above weak-value formalism that results in (17) and (18) can be related to the past quantum state (PQS) formalism proposed recently [33–37], if we replace the density matrix ${\rho }_{{fij}}=\langle i| {\rho }_{f}| j\rangle$ by the PQS Effect Matrix ${E}_{{ij}}=\langle i| \hat{E}| j\rangle$. The PQS formalism generalized the post-selection in the WV problem to successive continuous measurements characterized by the positive-operator valued measure (POVM) operator $\hat{E}$, which in general does not collapse the superposed state of a qubit. However, for the purpose of state tomography, the post-continuous measurement in PQS is harder to repeat than the post-selection in the weak value because of its highly stochastic nature. Therefore the PQS should be hard to apply to the problem of state tomography.

From equation (14) we see that, in the weak limit of measurement (the linear response regime), we may approximate the denominator by unity (neglecting the second term). In this case one can obtain $\mathrm{Re}({\sigma }_{w}^{z})$ and $\mathrm{Im}({\sigma }_{w}^{z})$ from the PPS average of currents by choosing, respectively, the LO's phase $\varphi =0$ and $\pi /2$. In the more general case (the nonlinear response regime), the full denominator of equation (14) should be taken into account. In this case one can extract $\mathrm{Re}({\sigma }_{w}^{z})$ and $\mathrm{Im}({\sigma }_{w}^{z})$ by applying an iterative algorithm. That is, first, set trial values for the real and imaginary parts of the AAV WV through

$$\begin{eqnarray*}&&\mathrm{Re}({\sigma }_{w}^{z})\Leftarrow -({}_{f}\langle x{\rangle }_{i}/{\epsilon }_{1}){| }_{\varphi =0},\mathrm{Im}({\sigma }_{w}^{z})\Leftarrow -({}_{f}\langle x{\rangle }_{i}/{\epsilon }_{2}){| }_{\varphi =\pi /2}.\end{eqnarray*}$$

Then, iteratively evaluate equation (14) several times until convergence is reached. In our numerical example illustrated in the next section, we found that this iterative approach is always efficient. The reason for applying this procedure is that there is a '± sign' problem when solving the AAV WV from the quadratic equation (14). An alternative method to overcome this problem is not solving for the AAV WV from equation (14), but solving for the unknown density matrix elements ${\rho }_{{ij}}$ directly from the linear equations (16)–(18).

Regarding the accuracy of the AAV WV extracted, we find that, by simulating 10⁶ trajectories, an accuracy of 0.5% can be achieved for $\varphi =0$, which decreases to 3% for $\varphi =\pi /2$. The reason is that, in the latter case, the component related to information gain (the first term) in equation (3) vanishes, thus resulting in stronger fluctuations of the output currents. In practice, one may choose $\varphi =\pi /4$ rather than $\pi /2$. Using equation (14), $\mathrm{Re}({\sigma }_{w}^{z})$ and $\mathrm{Im}({\sigma }_{w}^{z})$ can be easily extracted as well. For this choice, the same accuracy as for $\varphi =0$ can be achieved.

With the knowledge of $\mathrm{Re}({\sigma }_{w}^{z})$ and $\mathrm{Im}({\sigma }_{w}^{z})$, based on equation (15), one can directly determine the unknown state, $| {\psi }_{i}\rangle ={c}_{1}| 1\rangle +{c}_{2}| 2\rangle$, as follows. Note that, owing to the dynamic ac Stark effect, the wavefunction involved in the AAV WV is actually 'modified' as $| {\tilde{\psi }}_{i}\rangle ={c}_{1}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Phi }}}_{1}({t}_{m})}| 1\rangle +{c}_{2}| 2\rangle$. Up to a normalization factor, we rewrite this unknown state as

$$\begin{eqnarray}&&| {\tilde{\psi }}_{i}\rangle =| 1\rangle +\tilde{c}| 2\rangle ,\end{eqnarray} \tag{ 20 }$$

where $\tilde{c}=({c}_{2}/{c}_{1}){{\rm{e}}}^{{\rm{i}}{{\rm{\Phi }}}_{1}}$. For a given post-selection state $| {\psi }_{f}\rangle ={b}_{1}| 1\rangle +{b}_{2}| 2\rangle$, the AAV WV can be expressed as

$$\begin{eqnarray}&&{\sigma }_{w}^{z}=\displaystyle \frac{{b}_{1}^{* }-{b}_{2}^{* }\tilde{c}}{{b}_{1}^{* }+{b}_{2}^{* }\tilde{c}}.\end{eqnarray} \tag{ 21 }$$

From this result, we obtain

$$\begin{eqnarray}&&\tilde{c}=\left(\displaystyle \frac{1-{\sigma }_{w}^{z}}{1+{\sigma }_{w}^{z}}\right){\left(\displaystyle \frac{{b}_{1}}{{b}_{2}}\right)}^{* }\equiv {{r}{\rm{e}}}^{{\rm{i}}\tilde{\theta }},\end{eqnarray} \tag{ 22 }$$

which fully characterizes the unknown state $| {\psi }_{i}\rangle$ by noting that ${c}_{2}/{c}_{1}={{r}{\rm{e}}}^{{\rm{i}}(\tilde{\theta }-{{\rm{\Phi }}}_{1})}$.