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Theory of Compression Channels for Postselected Quantum Metrology

Jing Yang

TL;DR

This work develops a unified theory of lossless postselected quantum metrology by introducing a prior structure for compression channels that preserve quantum Fisher information (QFI) while discarding information-bearing outcomes. It shows that lossless compression channels (LCCs) must decompose the postselection POVMs as $E_{\omega}=q_{\omega}\rho_x^{\perp}+\Lambda_{\omega}$ and saturate generalized optimal measurement conditions, enabling substantial reduction in measurement costs without sacrificing precision. The authors provide explicit constructions for regular and null POVMs, prove a general LCC structure, analyze the sensitivity of postselected states, and connect these results to weak-value amplification. They further demonstrate two classes of near-lossless, subsystem-only compression for bipartite entangled states under non-interacting Hamiltonians, illustrating practical pathways to distribute quantum measurements in scalable sensing tasks. These insights offer a comprehensive framework to design compression channels that dramatically lower final-detection costs in quantum metrology, with implications for optical phase estimation, interferometry, and distributed quantum sensing.

Abstract

Postselected quantum metrological scheme is especially advantageous when the final measurements are either very noisy or expensive in practical experiments. In this work, we put forward a general theory on the compression channels in postselected quantum metrology. We define the basic notions characterizing the compression quality and illuminate the underlying structure of lossless compression channels. Previous experiments on Postselected optical phase estimation and weak-value amplification are shown to be particular cases of this general theory. Furthermore, for two categories of bipartite systems, we show that the compression loss can be made arbitrarily small even when the compression channel acts only on one subsystem. These findings can be employed to distribute quantum measurements so that the measurement noise and cost are dramatically reduced.

Theory of Compression Channels for Postselected Quantum Metrology

TL;DR

This work develops a unified theory of lossless postselected quantum metrology by introducing a prior structure for compression channels that preserve quantum Fisher information (QFI) while discarding information-bearing outcomes. It shows that lossless compression channels (LCCs) must decompose the postselection POVMs as and saturate generalized optimal measurement conditions, enabling substantial reduction in measurement costs without sacrificing precision. The authors provide explicit constructions for regular and null POVMs, prove a general LCC structure, analyze the sensitivity of postselected states, and connect these results to weak-value amplification. They further demonstrate two classes of near-lossless, subsystem-only compression for bipartite entangled states under non-interacting Hamiltonians, illustrating practical pathways to distribute quantum measurements in scalable sensing tasks. These insights offer a comprehensive framework to design compression channels that dramatically lower final-detection costs in quantum metrology, with implications for optical phase estimation, interferometry, and distributed quantum sensing.

Abstract

Postselected quantum metrological scheme is especially advantageous when the final measurements are either very noisy or expensive in practical experiments. In this work, we put forward a general theory on the compression channels in postselected quantum metrology. We define the basic notions characterizing the compression quality and illuminate the underlying structure of lossless compression channels. Previous experiments on Postselected optical phase estimation and weak-value amplification are shown to be particular cases of this general theory. Furthermore, for two categories of bipartite systems, we show that the compression loss can be made arbitrarily small even when the compression channel acts only on one subsystem. These findings can be employed to distribute quantum measurements so that the measurement noise and cost are dramatically reduced.
Paper Structure (9 sections, 2 theorems, 110 equations, 2 figures, 1 table)

This paper contains 9 sections, 2 theorems, 110 equations, 2 figures, 1 table.

Table of Contents

  1. Analytic expressions of the QFIs
  2. Low and upper bounds of $I_{\omega}$ and $I_{\omega}^{\text{cl}}\left(p(\omega|x)\right)$
  3. Regular POVM operator with $\sqrt{E_{\omega}}\ket{\psi_{x}}\neq0$ and $\sqrt{E_{\omega}}\ket{\partial_{x}^{\perp}\psi_{x}}\neq0$
  4. Null POVM operator with $\sqrt{E_{\omega}}\ket{\psi_{x}}=0$ and $\sqrt{E_{\omega}}\ket{\partial_{x}^{\perp}\psi_{x}}\neq0$
  5. Proof of Theorem \ref{['thm: LCC-structure']}
  6. Sensitivity of the Postselected state
  7. Calculation details on weak-value amplification
  8. Entangled initial state with the non-interacting Hamiltonian
  9. Expression for $\ket{\partial_{x}^{\perp}\psi_{x}}\bra{\partial_{x}^{\perp}\psi_{x}}$ and $\ket{\partial_{x}^{\perp}\psi_{x}}\bra{\psi_{x}}$ in terms of $\rho_{x}$

Key Result

Theorem 1

For a pure state $\ket{\psi_{x}}$, the POVM operators in an efficient LCC must satisfy with $p(\omega|x)>0$ for $\omega\in\checkmark$ and $\sum_{\omega\in\checkmark}p(\omega|x)<1$, where $\ket{\psi_{x}^{\perp}}\equiv\ket{\partial_{x}^{\perp}\psi_{x}}/\sqrt{g(\rho_{x})}$ is the normalized vector along $\ket{\partial_{x}^{\perp}\psi_{x}}$ direction.

Figures (2)

  • Figure 1: The protocol of Postselected quantum metrology. "A" denotes the ancilla. The unitary operation combined with the following projective measurement on the ancilla implement the post-selection channel $K_{\omega}$. After post-selection, measurements can be further performed (not shown), as in standard metrology, on these Postselected states to extract information about the estimation parameter.
  • Figure 2: In both figures, $K_{\checkmark}\!=\!\sqrt{E_{\checkmark}}$. (a) The solid lines correspond to $p(\checkmark|x)I(\sigma_{x|\checkmark})/I(\rho_{x})$ with $I(\rho_{x})\!=\!1/\sigma^{2}$, where the deviation from $1$ represents the loss $\gamma$, while the dashed lines correspond to the gain $\eta=I(\sigma_{x|\checkmark})/I(\rho_{x})$. Each color represents one post-selection scheme: Red-LCC on the two-level system with $\theta_{*}\!=\!-\theta\!=\!-\pi/3$; Blue-WVA with $\theta_{*}\!=\!-2\pi/3\!+\!10^{-2}$; Purple-LCC on the meter discussed in the main text with $\varepsilon\!=\!10^{-4}$.(b) Post-selection on the three-qubit entangled state. The LCC is given by Eq. (\ref{['eq:E-entangled']}) with $x=10^{-5}$, $\theta\!=\!\pi/3$, $p_{1}=1\!-\!p_{2}\!=\!2/3$, and $\varepsilon\!=\!10^{-4}$. The red solid line corresponds to the plot of $1-\gamma$ (left frame ticks)while the blue round dots correspond to the gain $\eta$ (right frame ticks).

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2