Reversible quantum measurement with arbitrary spins

We propose a physically reversible quantum measurement of an arbitrary spin-$s$ system using a spin-$j$ probe via an Ising interaction. In the case of a spin-$1∕2$ system $(s=1∕2)$, we explicitly construct a reversing measurement and evaluate the degree of reversibility in terms of fidelity. The recovery of the measured state is pronounced when the probe has a high spin $(j>1∕2)$, because the fidelity changes drastically during the reversible measurement and the reversing measurement. We also show that the reversing measurement scheme for a spin-$1∕2$ system can serve as an experimentally feasible approximate reversing measurement for a high-spin system $(s>1∕2)$. If the interaction is sufficiently weak, the reversing measurement can recover a cat state almost deterministically in spite of there being a large fidelity change.

Figure 1

$\mid a_{m^{\prime }\sigma }^{\left(j\right)}(\theta ,\varphi ){\mid }^2$ as a function of $m^{\prime }$ ($\sigma =\pm 1∕2$, $j=10$, $g=0.25$, $\theta =\pi ∕6$, $\varphi =\pi ∕6$).

Figure 2

Transitions of the measured state by successive measurements $\left\{{\widehat{T}}_m\right(\theta ,\varphi \left)\right\}$ and $\left\{{\widehat{T}}_m\right(\pi -\theta ,\pi -\varphi \left)\right\}$. The first measurement on the state $\mid \psi {⟩}_q$ yields an outcome $m$ $(=j,j-1,\dots ,-j+1,-j)$ with probability $p_m$, causing a state reduction to $\mid {\psi }_m{⟩}_q$. The second measurement on $\mid {\psi }_m{⟩}_q$ then yields an outcome $m^{\prime }$ with conditional probability $p_{m{m}^{\prime }}∕p_m$, causing a state reduction to $\mid {\psi }_{m{m}^{\prime }}{⟩}_q$.

Figure 3

Probability $p_m$ and fidelity $F_m$ of the first measurement as functions of the outcome $m$ ($\mid c_{1∕2}{\mid }^2=\mid c_{-1∕2}{\mid }^2=1∕2$, $j=10$, $g=0.25$, $\theta =\pi ∕6$, $\varphi =\pi ∕6$).

Figure 4

Joint probability $p_{m{m}^{\prime }}$ of the first and second measurements as a function of the outcomes $m$ and $m^{\prime }$ ($\mid c_{1∕2}{\mid }^2=\mid c_{-1∕2}{\mid }^2=1∕2$, $j=10$, $g=0.25$, $\theta =\pi ∕6$, $\varphi =\pi ∕6$).

Figure 5

Fidelity $F_{m{m}^{\prime }}$ after the second measurement as a function of the outcomes $m$ and $m^{\prime }$ ($\mid c_{1∕2}{\mid }^2=\mid c_{-1∕2}{\mid }^2=1∕2$, $j=10$, $g=0.25$, $\theta =\pi ∕6$, $\varphi =\pi ∕6$). $F_{m{m}^{\prime }}$ depends only on $m^{\prime }+m$ with $F_{m,-m}=1$.

Figure 6

Probability $p\left(m\right)$ of obtaining outcome $m$ for the first measurement and the corresponding information gain $I\left(m\right)$ and fidelity $F\left(m\right)$, with $j=10$, $g=0.25$, $\theta =\pi ∕6$, $\varphi =\pi ∕6$, and $\gamma =\pi ∕6$. Also shown are the expected information gain $I^{\prime }\left(m\right)$ and expected fidelity $F^{\prime }\left(m\right)$ after the reversing measurement, given that the outcome of the first measurement is $m$.

Figure 7

Average squared fidelity after the first measurement ${\sum }_m{p}_m{F}_m^2$ and total probability of recovery $q$ as functions of $j$ ($\mid c_{1∕2}{\mid }^2=\mid c_{-1∕2}{\mid }^2=1∕2$, $j=10$, $g=0.25$, $\theta =\pi ∕6$, $\varphi =\pi ∕6$). Although the strength of the interaction $g$ is much smaller $(\sim {10}^{-8})$ in the real situation discussed in Sec. 5, it can be enhanced by a cavity-assisted interaction or by collective enhancement via large $j$ and $s$.

Figure 8

Initial spin distribution $\mid c_{\sigma }^{\prime }{\mid }^2$ and final spin distribution ${\rho }_m\left(\sigma \right)$ as functions of $\sigma$ ($j=s=10$, $g=0.25$, $m=5$). ${\rho }_m\left(\sigma \right)$ has a pair of highest peaks at $\sigma =\pm 4$ (the other peaks are too small to be seen on the scale of this figure). The probability $p_m^{\prime }$ in Eq. (88) is calculated to be $0.016$.

Figure 9

Probability $p_m$ and fidelity $F_m$ of the first measurement on the state $\mid S_x=s\hslash {⟩}_q$ as functions of the outcome $m$ ($j=50$, $s=10$, $g=0.01$, $\theta =\pi ∕12$, $\varphi =\pi ∕4$).

Figure 10

Joint probability $p_{m{m}^{\prime }}$ of the first and second measurements on the state $\mid S_x=s\hslash {⟩}_q$ as a function of the outcomes $m$ and $m^{\prime }$ ($j=50$, $s=10$, $g=0.01$, $\theta =\pi ∕12$, $\varphi =\pi ∕4$).

Figure 11

Fidelity $F_{m{m}^{\prime }}$ after the two measurements on the state $\mid S_x=s\hslash {⟩}_q$ as a function of the outcomes $m$ and $m^{\prime }$ ($j=50$, $s=10$, $g=0.01$, $\theta =\pi ∕12$, $\varphi =\pi ∕4$).