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arXiv:2012.02441 · Physical Review A

Improving phase estimation using number-conserving operations

Huan Zhang, Wei Ye, Chaoping Wei, Cunjin Liu, Zeyang Liao, Liyun Hu

Published May 5, 2021• 25 citations
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Abstract

We propose a theoretical scheme to improve the resolution and precision of phase measurement with parity detection by using a nonclassical state generated by applying a number-conserving generalized superposition of products (GSP) operation, ${\left(sa{a}^{\ifmmode\dagger\else\textdagger\fi{}}+t{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\right)}^{m}$ with ${s}^{2}+{t}^{2}=1$, on a two-mode squeezed vacuum (TMSV) state as the input of Mach-Zehnder interferometer. Then, the statistical properties of the proposed GSP-TMSV are investigated via average photon number (APN), antibunching effect, and two-mode squeezing. Particularly, both higher order $m$ GSP operation and lower proportion parameter $s$ are beneficial for presenting larger total APN, which leads to the improvement of quantum Fisher information (QFI). In addition, we also compare the phase measurement precisions with and without photon losses between our scheme and the previous photon subtraction or photon addition schemes. It is found that our scheme, especially for the case of $s=0$, even in the presence of photon losses, has the best performance via the enhanced phase resolution, sensitivity, and QFI when compared to those previous schemes. Interestingly, without losses, the standard quantum-noise limit (SQL) is always broken through for our scheme and the phase uncertainty associated with the state of our scheme is closer to the corresponding Heisenberg limit (HL) than the TMSV in the larger total APN region, especially for the case of two-side GSP operation $\left(m,n\right)\ensuremath{\in}\left(1,1\right)$. However, in the presence of photon losses, the HL cannot be beaten, but the SQL can be broken through, particularly for the large total APN regimes.

DOI
10.1103/physreva.103.052602

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We propose a theoretical scheme to improve the resolution and precision of phase measurement with parity detection by using a nonclassical state generated by applying a number-conserving generalized superposition of products (GSP) operation, (saa\ifmmode†\else\textdagger\fi+ta\ifmmode†\else\textdagger\fia)m{\left(sa{a}^{\ifmmode\dagger\else\textdagger\fi{}}+t{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\right)}^{m}(saa\ifmmode†\else\textdagger\fi+ta\ifmmode†\else\textdagger\fia)m with s2+t2=1{s}^{2}+{t}^{2}=1s2+t2=1, on a two-mode squeezed vacuum (TMSV) state as the input of Mach-Zehnder interferometer. Then, the statistical properties of the proposed GSP-TMSV are investigated via average photon number (APN), antibunching effect, and two-mode squeezing. Particularly, both higher order mmm GSP operation and lower proportion parameter sss are beneficial for presenting larger total APN, which leads to the improvement of quantum Fisher information (QFI). In addition, we also compare the phase measurement precisions with and without photon losses between our scheme and the previous photon subtraction or photon addition schemes. It is found that our scheme, especially for the case of s=0s=0s=0, even in the presence of photon losses, has the best performance via the enhanced phase resolution, sensitivity, and QFI when compared to those previous schemes. Interestingly, without losses, the standard quantum-noise limit (SQL) is always broken through for our scheme and the phase uncertainty associated with the state of our scheme is closer to the corresponding Heisenberg limit (HL) than the TMSV in the larger total APN region, especially for the case of two-side GSP operation (m,n)\ensuremath∈(1,1)\left(m,n\right)\ensuremath{\in}\left(1,1\right)(m,n)\ensuremath∈(1,1). However, in the presence of photon losses, the HL cannot be beaten, but the SQL can be broken through, particularly for the large total APN regimes.
We propose a theoretical scheme to improve the phase sensitivity by introducing the number-conserving generalized superposition of product into squeezed thermal state of the traditional SU(1,1) interferometer with intensity detection. We investigate the effect of different parameter and photon losses on phase sensitivity under the condition of the ideal and the realistic phase estimation process, respectively. Moreover, compared with the previous methods, our scheme can not only enhance the phase sensitivity, but also significantly suppress the photon losses with only first-order GSP operation. It means that our scheme needs less resources and less operations to obtain higher phase sensitivity. These above results indicate that our scheme is viable in the phase sensitivity improvement, which may have practical applications in quantum information processing.
Quantum metrology is to estimate accurately the value of an unknown parameter with the assistance of the quantum effects, in order to break through the standard quantum limit, even reach the Heisenberg limit. In this work, we study the performance of a general photon-added two-mode squeezed vacuum state that is taken as a detection state of a Mach-Zehnder interferometer. Based on quantum Fisher information, within the constraint on the total mean photon number, symmetric and asymmetric photon addition cannot improve the ultimate phase sensitivity. However, for a given initial squeezing parameter, on this occasion, the symmetric and asymmetric photon addition can improve the ultimate phase sensitivity. Compared with the asymmetric photon-added two-mode squeezed vacuum state, the symmetric one can well improve the ultimate phase sensitivity. This may be because it is always better to implement the symmetric photon addition rather than the asymmetric one in order to increase the mean photon number of the resulting state. On the other hand, via parity detection, the symmetric and asymmetric photon-added two-mode squeezed vacuum state can indeed improve the phase sensitivity of a Mach-Zehnder interferometer for a given initial squeezing parameter. Based on the parity detection, within a constraint on the mean photon number, although the two-mode squeezed vacuum state can give the better phase sensitivity at the optimal phase shift (φ = 0), the phase sensitivity offered by the symmetric and asymmetric photon-added two-mode squeezed vacuum states are both more stable around φ = 0 than by the two-mode squeezed vacuum state. In addition, we show that for the symmetric photon-added two-mode squeezed vacuum state, parity detection is an optimal detection only when the optimal phase shift approaches to zero. When the phase shift slightly deviates from zero, the parity detection is not an optimal detection scheme. Finally, for all values of the phase shift, our results also clearly show that the parity detection is not an optimal detection scheme for the asymmetric photon-added two-mode squeezed vacuum state serving as an interferometer state.
We investigate the sensitivity and resolution of phase estimation in a Mach–Zehnder interferometer with photon-added two-mode squeezed vacuum states (PA-TMSVS) with a photon-number parity measurement scheme. Our results indicate that, when the phase to be estimated is small, the PA-TMSVS offers an improved phase resolution over both the two-mode squeezed vacuum and the photon-subtracted two-mode squeezed vacuum for a given initial squeezing level. In addition, we study the behavior of the quantum Fisher information of the PA-TMSVS, and show that the optimum phase uncertainty predicted by the corresponding Cramer–Rao bound can be achieved with a parity measurement.
Abstract The nonclassicality is the prerequisite for quantum states to be applied into quantum information, especially for quantum metrology. Here we theoretically investigate the non-classical properties of the non-Gaussian state generated by repeatedly operating a number-conserving generalized superposition of products (GSP), i.e., ( s 1 aa † + t 1 a † a ) m with s 1 2 + t 1 2 = 1 , on the squeezed thermal state (STS), in terms of second-order correlation function, Mandel’s Q parameter, quadrature squeezing and Wigner function (WF). It is shown that, compared to the cases of the STS, the GSP-STS with the high-order GSP operations ( m > 1) at the small-squeezing levels can be beneficial to the existence of the photon-antibunching effect, the sub-Poissonian distribution and the partial negativity of the WF, apart from the quadrature squeezing. In addition, for the case of m = 1, we also compare with the non-classical properties of several different non-Gaussian resources, involving the photon-subtracted-then-added (PSTA) STS, the GSP-STS and the photon-added-then-subtracted (PATS) STS. It is found that the PSTA-STS with respect to the sub-Poissonian distribution and the partial negativity of the WF has a better performance than others. Significantly, the generated GSP-STS has an obvious advantage of showing the photon-antibunching effects, compared to the PSTA-STS and the PATS-STS, which means that our scheme may have an excellent guidance for the practical implementations in quantum information.
Abstract The precision of quantum measurements can be effectively improved by using both photon-added non-Gaussian operations and Kerr nonlinear phase shift. Here, we employ a coherent state mixed with a photon-added squeezed vacuum state as the input to a Mach–Zehnder interferometer with parity detection, thereby achieving a significant enhancement in phase measurement precision. Our research focuses on the phase sensitivity of linear phase shift under both ideal conditions and photon loss, as well as quantum Fisher information (QFI). The results demonstrate that employing the photon addition operations can markedly enhance phase sensitivity and QFI, and under optimal conditions, the measurement precision can approach the Heisenberg limit for the linear phase shift case. In addition, we delve deeper into the scenario of replacing the linear phase shift with a Kerr nonlinear one and systematically analyze the QFI under both ideal and photon loss conditions. By comparison, it is evident that employing both the photon addition operations and the Kerr nonlinear phase shift can further significantly enhance phase measurement precision while effectively improving the system’s robustness against photon loss. These findings are instrumental in facilitating the development and practical application of quantum metrology.
Abstract The non-Gaussian operation, which can be easily implemented by current techniques, is an effective way for the improvement of the continuous-variable entanglement. Here, we theoretically propose a scheme for generating a number-conserving two-mode generalized superposition of products (TM-GSP) state by performing the -order GSP operations i.e. on an input two-mode squeezed vacuum (TMSV) state. Then, the entanglement properties of the TM-GSP state are analyzed detailedly by means of logarithmic negative, Einstein–Podolsky–Rosen (EPR) correlation and two-mode squeezing property. It is shown that, in contrast to the TMSV case, for the optimal choices of s 1 and s 2 , the entanglement properties of the TM-GSP state can be improved by using symmetric GSP operations ( ) and single-side GSP operations . Furthermore, the fidelity effect is also considered when the TM-GSP state is used as the entangled resource under the Braunstein and Kimble scheme. In terms of the optimal teleportation fidelity, it is found that the effect of the single GSP operation is equivalent to that of the case . This phenomenon also exists in logarithmic negative, EPR correlation and two-mode squeezing property. These results show that the prepared TM-GSP state may provide a well application in the fields of quantum information processing.