Informationally Complete Distributed Metrology Without a Shared Reference Frame

TL;DR

This paper tackles distributed quantum metrology without a shared reference frame by showing that RF misalignment induces a G-twirling decoherence that erases locally encoded information under 1-local operations. It introduces the 2-LUI-RE protocol, which applies reversed encoding on two copies of a local-unitary-invariant network state and uses local twirling, breaking copy-space SWAP symmetry to recover the full quantum Fisher information and preserve Heisenberg-limited scaling. The authors prove that local Bell-state measurements saturate the QFI, outperforming naive direct measurements, and demonstrate HL scaling for distributed phase estimation with GHZ states. The work provides a practical, experimentally feasible path to high-precision, RF-misaligned distributed sensing, with broad implications for space-based quantum networks and clock synchronization.

Abstract

In quantum information processing, implementing arbitrary preparations and measurements on qubits necessitates precise information to identify a specific reference frame (RF). In space quantum communication and sensing, where a shared RF is absent, the interplay between locality and symmetry imposes fundamental restrictions on physical systems. A restriction on realizable unitary operations results in a no-go theorem prohibiting the extraction of locally encoded information in RF-independent distributed metrology. Here, we propose a reversed-encoding method applied to two copies of local-unitary-invariant network states. This approach circumvents the no-go theorem while simultaneously mitigating decoherence-like noise caused by RF misalignment, thereby enabling the complete recovery of the quantum Fisher information (QFI). Furthermore, we confirm local Bell-state measurements as an optimal strategy to saturate the QFI. Our findings pave the way for the field application of distributed quantum sensing, which is inherently subject to unknown RF misalignment and was previously precluded by the no-go theorem.

Figures (5)

• Figure 1: The Role of RFs and the Impact of Their Misalignment in Quantum Information Processing.(a) Demonstration of RF Misalignment for Classical and Quantum Objects. The left side shows objects defined in a local RF at Site-$i$, while the right side shows the same objects as viewed from a common RF. For a classical object (upper), e.g., a cat, a misalignment between frames is described by a 3D rotation $g_i \in \mathrm{SO(3)}$. An orientation vector $\vec{\mathbf{r}}_i$ in the local frame is perceived as $g_i\vec{\mathbf{r}}_i$ in the common frame. For a quantum object (down), such as a photon in a horizontally-polarized state $|H_i\rangle$, the same physical rotation corresponds to a transformation $\hat{V}_i(g_i) \in \mathrm{SU(2)}$ on its quantum state. This results in a different state, such as an elliptical polarization, in the common frame. (b) Demonstration of the Effects of Drifting RFs on the State Shared in Two Sites. From the view of the common reference point, the coordination of site A and B is changing with $g_\text{A}(t),g_\text{B}(t)\in \mathrm{SO(3)}$. The corresponding shared state $\rho_{\text{AB}}$ is experiencing the unitary rotation given $\hat{V}_\text{A}(g_\text{A})\otimes \hat{V}_\text{B}(g_\text{B})$. The average effect over the measuring time is the $G$-twirling. (c) Demonstration of $\boldsymbol{k}$-copy Quantum States Shared in $\boldsymbol{N}$ sites. The connected dots represent an $N$-qudit state shared in $N$ sites. And the shared $k$ copies mean that $k$$N$-qudit states are included. At each site, there are $k$ qudits ($k$ balls), one for each copy.
• Figure 2: Comparison of Encoding Strategies for 2-copy LUI States.(a) 2-LUI-IE Protocol: Both copies undergo the same encoding operation, $\Theta_{\theta}$, and in this case, only nonlocal interactions can give rise to a non-vanishing QFI. (b) 2-LUI-RE Protocol: The two copies are reversely encoded with $\Theta_\theta$ and $\Theta_{-\theta}$. This structure explicitly breaks the SWAP symmetry between the copies. Crucially, while $\Theta_{\theta}$ is depicted as a global unitary for generality, this strategy is effective even when the encoding is generated by purely local interactions at a single site.
• Figure 3: 2-LUI-RE Protocol with Independent Encodings. Two copies of the GHZ state ($\rho_0$) are shared by separated sites in a network, and then each site applies reversed-encoding on two on-hand photons with opposing encoding $\Theta_i$ and $\Theta_i^\dagger$, forming two reversely encoded network states $\rho_{\vec{\boldsymbol{\theta}}}$ and $\rho_{-\vec{\boldsymbol{\theta}}}$. Afterward, each site performs local randomized rotations $\hat{U}=\bigotimes_{i=1}^N\hat{U}_i$ to the two photons to generate the LUI state $\tilde{\rho}_{\vec{\boldsymbol{\theta}}}$. The averaged parameter $\bar{\theta}=\frac{1}{N}\sum_{i=1}^N\theta_i$ can be estimated through a specific measurement strategy geared toward $\tilde{\rho}_{\vec{\boldsymbol{\theta}}}$.
• Figure 4: Local Measurement Strategies for the 2-LUI-RE Protocol.(a) Direct Computational Basis Measurement (DM). Each qudit is measured directly in the computational basis. This strategy suffers an exponential loss of information. (b) Local SWAP Test (LST). An ancillary qubit is introduced at each site to perform a controlled-SWAP operation between the two copies. (c) Local Bell-state Measurement (LBM). For qubit systems ($d=2$), the local SWAP test is operationally equivalent to a Bell-state measurement. This is implemented efficiently using a CNOT gate and a Hadamard gate at each site, followed by Z-basis measurements. LST and LBM constitute optimal measurement strategies.
• Figure 5: Fisher Information for Different Measurement Strategies in the 2-LUI-RE Protocol. The QFI of the protocol (blue dashed-dotted line) represents the ultimate precision bound. The CFI from the optimal LBM (orange solid line) perfectly saturates the Cramér-Rao bound, while the CFI from a naive DM (purple solid line) performs dramatically worse, falling far below the SQL.