Quantum estimation through a bottleneck

We study the estimation of a single parameter characterizing families of unitary transformations acting on two systems. We consider the situation with the presence of bottleneck, i.e., only one of the systems can be measured to gather information. The estimation capabilities are related to unitaries’ generators. In particular, we establish continuity of quantum Fisher information with respect to generators. Furthermore, we find conditions on the generators to achieve the same maximum quantum Fisher information we would have in the absence of bottleneck. We also discuss the usefulness of initial entanglement across the two systems as well as across multiple estimation instances.

1. In passing, we notice that the error defined in Eq. (1) of Ref. [5], to be consistent with the results reported there, should have been written with a square root, i.e., \(\delta \varphi =\left\langle {\left( \varphi _{est}/| \frac{\partial \langle \varphi _{est}\rangle _{av}}{\partial \varphi }|-\varphi \right) ^{2}}\right\rangle ^{\frac{1}{2}}\).

2. Clearly, sampling a discontinuous function on a discrete set of points cannot be representative of the behavior of the function, while it can for a continuous function.

3. We can always factor out e.g., \(\Vert \hat{\varvec{m}}\Vert \) from \(G_1\), which will cause a rescaling of the parameter \(t_1\), and incorporate it into the parameter \(\alpha \).

4. It is worth mentioning that the set of optimal input states when \(t_1=t_2=0\) extends to \(\left( \cos \theta |0\rangle \pm i \sin \theta |1\rangle \right) _A |\varphi \rangle _E\), \(\forall \theta \in [0,2\pi ]\) and \(\forall |\varphi \rangle \in {\mathbb {C}}^2\).

5. Notice that in (91) we cannot take the limit \(t_1\rightarrow 0\) (or \(t_2\rightarrow 0\)), because otherwise we should consider the case ii) (or i), respectively).

The work of M.R. is supported by China Scholarship Council.

Authors and Affiliations

1. Department of Mathematics and Physics, Xinjiang Agricultural University, Ürümqi, 830053, China

   Milajiguli Rexiti

2. School of Advanced Studies, University of Camerino, 62032, Camerino, Italy

   Milajiguli Rexiti

3. School of Science and Technology, University of Camerino, 62032, Camerino, Italy

   Stefano Mancini

4. INFN-Sezione di Perugia, 06123, Perugia, Italy

   Stefano Mancini

Corresponding author

Correspondence to Milajiguli Rexiti.

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Appendix A

Following up Hurwitz parametrization [26], we can write N-qubit states as

where \([n]_2\) stands for the binary representation of n. We also have

with

Now searching the maximum of a function over the set of states (101) can be done by randomly sampling such states according to the Haar measure of \(U(2^N)\) [27]. However, in such a way, we cannot account for separable states, as this subset of states has a vanishing probability measure [28]. Therefore, we opted for sampling on a grid of 50 points for \(\vartheta _n\) in \([0,\pi /2]\) and 200 points for \(\varphi _n\) in \([0,2\pi ]\).

Appendix B

Consider a unitary with generator

where \(t_{22}, t_{33}\in {\mathbb {R}}\). The eigenvalues of G result \(\{-1-t_{22}-t_{33}, 1+t_{22}-t_{33}, 1-t_{22}+t_{33}, -1+t_{22}+t_{33}\}\), hence, the maximum Fisher information we can get when accessing both systems B and F is:

We can obtain:

• \({\overline{J}}_{B}={\overline{J}}_{BF}=4\left( 1+t_{33}\right) ^2\), with input \(\frac{1}{2}\left( |00\rangle +|01\rangle -|10\rangle +|11\rangle \right) \);

• \({\overline{J}}_{B}={\overline{J}}_{BF}=4\left( 1-t_{33}\right) ^2\), with input \(\frac{1}{2}\left( |00\rangle +|01\rangle +|10\rangle -|11\rangle \right) \);

• \({\overline{J}}_{B}={\overline{J}}_{BF}=4\left( t_{22}+t_{33}\right) ^2\), with input \(\frac{1}{2}\left( |00\rangle +|01\rangle -|10\rangle -|11\rangle \right) \);

• \({\overline{J}}_{B}={\overline{J}}_{BF}=4\left( t_{22}-t_{33}\right) ^2\), with input \(\frac{1}{2}\left( |00\rangle +|01\rangle +|10\rangle +|11\rangle \right) \);

• \({\overline{J}}_{B}={\overline{J}}_{BF}=4\left( 1+t_{22}\right) ^2\), with input \(|01\rangle \);

• \({\overline{J}}_{B}={\overline{J}}_{BF}=4\left( 1-t_{22}\right) ^2\), with input \(|00\rangle \).

Appendix C

Corollary 6.1

Given \(\rho _i(\alpha )=e^{\alpha {{\mathcal {L}}}_i}\rho \), with \({{\mathcal {L}}}_i\) Liuovillian superoperators, we have (dropping the dependence from \(\alpha \) for a lighter notation):

where \(C_1,C_2\) are as in Corollary 3.2, and \(\Vert \cdot \Vert _{1\rightarrow 1}\) is the induced 1-norm on the superoperators, i.e., \(\left\| {{\mathcal {L}}}\right\| _{1\rightarrow 1}:=\sup _{\rho : \Vert \rho \Vert _1=1} \Vert {{\mathcal {L}}}\rho \Vert _1\).

Proof

Moving on from Theorem 3.1, for the first term at the right hand side of Eq. (14), we have

where from (109) to (110) we have used the property (27) together with the fact that \(e^{\alpha {{\mathcal {L}}}_i}\) is trace preserving.

For the second term at the right hand side of Eq. (14), instead, we have

where from (115) to (116) we have used the property (27) together with the fact that \(e^{\alpha {{\mathcal {L}}}_i}\) is trace preserving. Notice that we could have reversed the role of \( {{\mathcal {L}}}_1\) and \( {{\mathcal {L}}}_2\). Thus, by inserting Eqs. (110), (116) into (14) and taking into account that \(\alpha \in [0,2\pi ]\) we get the desired result. \(\square \)

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