Optimal probe states for the estimation of Gaussian unitary channels

Dominik Šafránek and Ivette Fuentes

Phys. Rev. A 94, 062313 – Published 12 December 2016

Abstract

We construct a practical method for finding optimal Gaussian probe states for the estimation of parameters encoded by Gaussian unitary channels. This method can be used for finding all optimal probe states, rather than focusing on the performance of specific states as shown in previous studies. As an example, we apply this method to find optimal probes for the channel that combines the phase-change and squeezing channels, and for generalized two-mode squeezing and mode-mixing channels. The method enables a comprehensive study of temperature effects in Gaussian parameter estimation. It has been shown that the precision in parameter estimation using single-mode states can be enhanced by increasing the temperature of the probe. We show that not only higher temperature, but also larger temperature differences between modes of a Gaussian probe state can enhance the estimation precision.

Received 2 October 2016

DOI:https://doi.org/10.1103/PhysRevA.94.062313

Physics Subject Headings (PhySH)

Optical interferometry Quantum information processing with continuous variables Quantum information with atoms & light Quantum measurements Quantum state engineering

Atomic, Molecular & OpticalQuantum Information, Science & TechnologyParticles & FieldsAccelerators & Beams

Authors & Affiliations

Dominik Šafránek^* and Ivette Fuentes^†

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
and Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

^*dominik.safranek@univie.ac.at
^†Previously known as Fuentes-Guridi and Fuentes-Schuller.

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Vol. 94, Iss. 6 — December 2016

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Images

Figure 1
Scheme of the usual metrology setting illustrated on a one-mode Gaussian probe state. First, we prepare the state by using various Gaussian operations, then we feed the state into the channel we want to estimate, perform an appropriate measurement, and an estimator $\hat{ε}$ gives us an estimate of the true value of the parameter. In this paper, we are interested in optimizing over the preparation stage for a given encoding Gaussian unitary channel $\hat{U} (ε)$ .
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Figure 2
Estimation of the phase-changing channel $\hat{R} (ε)$ around point $ε = 0$ using various one-mode Gaussian probe states parametrized by Eq. (20). (1) squeezed vacuum $[r = - 0.88, | d | = 0, λ_{1} = 1, H (ε) \approx 16]$ , (2) squeezed thermal state $[r = - 0.88, | d | = 0, λ_{1} = 2, H (ε) \approx 25]$ , (3) displaced vacuum $[r = 0, | d | = 1, λ_{1} = 1, H (ε) = 4]$ , and (4) displaced thermal state $[r = 0, | d | = 1, λ_{1} = 2, H (ε) = 2]$ . The squeezing parameter $r = - 0.88$ was chosen in such a way that the squeezed vacuum and the displaced vacuum have the same mean energy $n = 1$ . We plot covariance matrices in the real form phase-space, before (blue with full line) and after (orange with dashed line) the phase change $\hat{R} (0.2)$ has been applied. The quantum Fisher information has been calculated from Eq. (26). The relative overlap of covariance matrices of squeezed states is the same in both cases (1) and (2), however, the covariance matrix being larger in (2) allows for a better precision in estimation given by the factor $\frac{λ_{1}^{2}}{1 + λ_{1}^{2}}$ . Thermal fluctuations in a squeezed state help the estimation. In contrast, the relative overlap of covariance matrices of displaced states is considerably larger in (4) as compared to (3). Higher thermal fluctuations in a displaced state is detrimental for the estimation. This decrease in precision is given by the factor $\frac{1}{λ_{1}}$ .
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Figure 3
Estimation of the beam splitter $\hat{B} (ε)$ around point $ε = 0$ using one of the optimal states, $\hat{ρ} = \hat{S} (r) | 0 〉 〈 0 | {\hat{S}}^{†} (r) \otimes \hat{S} (- r) | 0 〉 〈 0 | {\hat{S}}^{†} (- r)$ representing two one-mode squeezed states that are squeezed in orthogonal directions, and the one-mode squeezed state $\hat{ρ} = \hat{S} (r_{1}) | 0 〉 〈 0 | {\hat{S}}^{†} (r_{1}) \otimes | 0 〉 〈 0 |$ , both with the same mean energy $n = 2$ . We plot the real form marginal covariance matrices showing correlations between positions in the first and the second mode $x_{1}$ and $x_{2}$ , and momenta $p_{1}$ and $p_{2}$ in the real form phase space, before (blue with full line) and after (orange with dashed line) beam splitter $\hat{B} (0.1)$ has been applied. There are no correlations between position and momentum. Clearly, the optimal state is more sensitive to the channel allowing for a better estimation of the parameter $ε$ .
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