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Lossless Postselected Quantum Metrology with Quasi-pure Mixed States

Jing Yang

Abstract

Postselection can compress the metrological information and improve sensitivity in the presence of certain types of technical noise. Postselected quantum metrology with pure states has been significantly advanced recently. However, extending this framework to mixed states leads to formidable challenges, such as the difficulty in searching for lossless postselection measurements or even the loss of metrological information. In this work, we leverage the intuition for the lossless postselection of pure states and generalize the theory to the lossless postselection of a class of mixed states, dubbed quasi-pure states. We illustrate our findings in postselected quantum imaging, unitary estimation problems, and show that the quasi-pure structure can be universally engineered through only classical correlation with an ancilla. Our findings extend the utility of postselection techniques to scenarios with decoherence and also offer new perspectives to foundational questions in quantum information geometry.

Lossless Postselected Quantum Metrology with Quasi-pure Mixed States

Abstract

Postselection can compress the metrological information and improve sensitivity in the presence of certain types of technical noise. Postselected quantum metrology with pure states has been significantly advanced recently. However, extending this framework to mixed states leads to formidable challenges, such as the difficulty in searching for lossless postselection measurements or even the loss of metrological information. In this work, we leverage the intuition for the lossless postselection of pure states and generalize the theory to the lossless postselection of a class of mixed states, dubbed quasi-pure states. We illustrate our findings in postselected quantum imaging, unitary estimation problems, and show that the quasi-pure structure can be universally engineered through only classical correlation with an ancilla. Our findings extend the utility of postselection techniques to scenarios with decoherence and also offer new perspectives to foundational questions in quantum information geometry.
Paper Structure (9 sections, 1 theorem, 74 equations, 4 figures)

This paper contains 9 sections, 1 theorem, 74 equations, 4 figures.

Table of Contents

  1. Supplementary Note 1: Challenges in the lossless postselection theory for mixed states
  2. Supplementary Note 2: Proof of Theorem thm:QP-PS
  3. Supplementary Note 3: Details about the criteria of a quasi-pure structure
  4. Supplementary Note 4: Proofs of the properties of quasi-pure states

Key Result

Theorem 1

Quasi-pure states can be lossless postselected through the postselection measurement given by Eq. (eq:QP-E-kernel) in the limit $x_{*}\to x$.

Figures (4)

  • Figure 1: The postselection measurement can be realized by entangling the probe system with an ancilla and a following projective measurements on the ancilla.
  • Figure 2: The errors of the post-selection measurements Eq. (\ref{['eq:QP-E-kernel']}) and Eq. (\ref{['eq:E-2qu']}) for (a) superresolution and (b) two-qubit example in the limit $x\to0$, respectively. The norms used in the numerical calculation of Eq. (\ref{['eq:vareps-def']}) are the $L^{2}$-norm and matrix-$2$ norm for (a) and (b), respectively. Values of parameters: (a) $\lambda=10^{-2}$, $\sigma=1$, $q=0.3$. (b)$\lambda=10^{-4}$, $q=0.3$.
  • Figure 3: The universal protocol to create the quasi-pure structure by introducing ancilla. The dimension of the Hilbert space of the ancilla is at least $d_{r}$. The initial state of the ancilla is prepared in $\ket{1}$. After the control sum gate, the state become Eq. (\ref{['eq:sig-init']}) in the main text.
  • Figure S1: The postselection of a probe system in mixed states can be artificially described as the postselection on the joint system consisting of the probee system and the optimal environment. An ancilla is introduced to implement the postselection measurement.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • proof
  • proof
  • proof
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  • proof