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Optimal measurements in phase estimation: simple examples

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Abstract

We identify optimal measurement strategies for phase estimation in different scenarios in which the interferometer acts on two-mode symmetric states. For pure states of a single qubit, we show that optimal measurements form a broad set parametrized with a continuous variable. When the state is mixed, this set reduces to merely two possible measurements. For two-qubit symmetric Werner state, we find the optimal measurement and show that estimation from the population imbalance is optimal only if the state is pure. We also determine the optimal measurements for a wide class of symmetric N-qubit Werner-like states. Finally, for a pure symmetric state of N qubits, we find under which conditions the estimation from the full N-body correlation and from the population imbalance is optimal.

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Notes

  1. Note that at \(\theta =0\), there is a simple way to reduce the optimal POVM to the simple measurement of population imbalance. Indeed, the following transformation

    $$\begin{aligned} \hat{V}=\exp \left( i\frac{\pi }{2}\frac{\hat{J}_x \hat{J}_y + \hat{J}_y \hat{J}_x}{2}\right) \exp \left( i \frac{\pi }{4} \hat{J}_y\right) . \end{aligned}$$

    applied to the states (31) gives \(\hat{V} |\Psi _1\rangle = |0,2\rangle \), \(\hat{V} |\Psi _2\rangle = |1,1\rangle \) and \(\hat{V} |\Psi _3\rangle = |2,0\rangle \). In this way, we obtain the eigenstates of the \(\hat{J}_z\) operator, and the optimal measurement is based on the determination of the population imbalance. Nevertheless, to accomplish this we needed an additional operation \(\hat{V}\) acting on the state. This transformation is non-local—it correlates the particles, since the product of two angular momentum operators cannot be written as a sum of operators acting on each qubit independently.

  2. The elements of the angular momentum matrix, \(d_{jk}(\theta ) \equiv \langle j \vert \hbox {e}^{-i \theta \hat{J}_y} \vert k \rangle \) are

    $$\begin{aligned} d_{jk}(\theta )= & {} \sqrt{\frac{j!(N-j)!}{k!(N-k)!}}\left[ \sin \frac{\theta }{2} \right] ^{j-k}\left[ \cos \frac{\theta }{2} \right] ^{j+k-N}\\&\times \, P_{N-j}^{j-k,j+k-N}(\cos \theta ), \end{aligned}$$

    where \(P_{n}^{\alpha ,\beta }(x)\) is the Jacobi polynomial.

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Acknowledgments

J. Ch. acknowledges the Foundation for Polish Science International TEAM Programme co-financed by the EU European Regional Development Fund and the support of the Polish NCBiR under the ERA-NET CHIST-ERA project QUASAR. T.W. acknowledges the Foundation for Polish Science International Ph.D. Projects Programme co-financed by the EU European Regional Development Fund and the National Science Center Grant No. DEC-2011/03/D/ST2/00200. L.P. acknowledges financial support by MIUR through FIRB Project No. RBFR08H058. This research was partially supported by the EU-STREP Project QIBEC.

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Authors and Affiliations

  1. Institute of Theoretical Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warsaw, Poland

    Tomasz Wasak & Jan Chwedeńczuk

  2. QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125, Florence, Italy

    Augusto Smerzi & Luca Pezzé

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  1. Tomasz Wasak
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Correspondence to Jan Chwedeńczuk.

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Wasak, T., Smerzi, A., Pezzé, L. et al. Optimal measurements in phase estimation: simple examples. Quantum Inf Process 15, 2231–2252 (2016). https://doi.org/10.1007/s11128-016-1248-5

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