Entanglement-Assisted Differential Phase Measurement

• Entanglement-assisted differential phase measurement is a quantum metrology method utilizing entangled states to estimate phase differences beyond the standard quantum limit in various sensing contexts.
• It exploits distinct entanglement topologies—such as jointly entangled atomic clouds versus independent entangled pixels—to achieve sensitivity enhancements up to 9.7 times better than classical benchmarks.
• The approach leverages noise immunity, decoherence-free subspaces, and advanced state-engineering techniques like one-axis twisting to enable precise measurements in atomic, photonic, and nonlocal interferometric systems.

arXiv search: entanglement-assisted differential phase measurement (Yu et al., 2022, Landini et al., 2014, Kaubruegger et al., 11 Jun 2025, Liu et al., 2021, Colangelo et al., 2017, He et al., 2011)

Entanglement-assisted differential phase measurement denotes a class of quantum-metrological protocols in which the parameter of interest is a phase difference, frequency difference, or spatial shift between two locations, pixels, modes, or sensor nodes, and quantum entanglement is used to surpass the standard quantum limit imposed by quantum projection noise and photon shot noise. In the atomic-clock and imaging-spectroscopy setting emphasized by "Entanglement-enhanced Synchronous differential comparison" (Yu et al., 2022), the central task is a synchronous comparison between two spatially separated regions, with the aim of detecting small effective spatial frequency shifts relevant to applications such as gravitational redshift detection by means of \emph{in situ} imaging spectroscopy. Closely related formulations appear in differential interferometry under strong common-mode phase noise (Landini et al., 2014), distributed quantum sensing with photonic networks (Liu et al., 2021), and two-node decoherence-free sensor architectures (Kaubruegger et al., 11 Jun 2025).

1. Measurement model and operational definition

In synchronous differential comparison, two spatial regions are interrogated simultaneously and the differential observable is extracted from the population difference

$P_d = P_2 - P_1 .$

For the effective spatial frequency shift, the measurement relation reported in (Yu et al., 2022) is

$\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$

where $N$ is the number of atoms per pixel, $T$ is the Ramsey time, and $\phi$ is the interrogation phase. The associated sensitivity is

$$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$

with

$$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$

where $G(P_1,P_2)$ is the inter-pixel correlation (Yu et al., 2022).

This formulation makes the role of entanglement highly explicit. The metrological gain can come either from reducing the single-pixel projection noise $\Delta P_k$, from exploiting correlations $G(P_1,P_2)$, or from both. In the coherent-spin-state benchmark, the projection-noise scale per pixel is

$\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$0

so sub-SQL differential performance requires a genuine reduction in the normalized differential noise (Yu et al., 2022).

A closely related operational statement appears in two-mode interferometry with fluctuating particle number, where the smallest detectable phase shift in a single measurement is written as

$\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$1

using normalized phase observables immune to number fluctuations (He et al., 2011). This places differential phase estimation within the broader framework of operational phase sensitivity rather than solely within a state-classification language.

2. Entanglement topology in synchronous differential comparison

The principal result of (Yu et al., 2022) is that the sensitivity gain depends strongly on how the entanglement is distributed across the two regions being compared. Two configurations are studied.

First, both pixels can belong to a single jointly entangled atomic cloud. In that case, the difference measurement benefits from inter-pixel quantum correlations, but the improvement saturates. The reported best-case ratio is

$\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$2

corresponding to a sensitivity enhancement factor of $\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$3 over the standard quantum limit, and increasing the atom number hardly further improves the sensitivity (Yu et al., 2022).

Second, the two pixels can be independent, with each pixel individually prepared as an entangled ensemble. In that case there is no inter-pixel correlation, $\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$4, but each pixel is internally squeezed, so that

$\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$5

For $\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$6 entangled atoms in each pixel, the reported minimum ratio is approximately $\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$7, corresponding to a sensitivity enhancement factor of $\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$8 and to a reduction of the averaging time by a factor of about $\Delta_{\textrm{eff}} = \frac{P_d}{N T \sin\phi},$9 (Yu et al., 2022).

Configuration	Minimum ratio	Consequence
Jointly entangled single cloud	$N$0	factor $N$1, saturates with $N$2
Two independent entangled pixels	$N$3 for $N$4; $N$5 for $N$6	factor $N$7 to $N$8, improves with $N$9

A recurrent misconception is that any inter-pixel entanglement must automatically provide the best differential sensitivity. The comparison in (Yu et al., 2022) shows otherwise: entanglement topology is decisive. Joint entanglement across the two compared regions can create useful correlations in the difference channel, but individually squeezed pixels provide the scalable route in the reported atomic setting. This suggests that the relevant design question is not merely whether a state is entangled, but whether the entanglement geometry matches the estimator.

3. State engineering and metrologically useful squeezing

The state-preparation schemes in (Yu et al., 2022) are based on one-axis twisting, often realized via cavity-mediated interactions. For the jointly entangled-cloud configuration, the squeezed-spin state can be written as

$T$0

where $T$1 is the coherent spin state after the initial $T$2 pulse and $T$3 quantifies one-axis twisting. For two independent entangled pixels,

$T$4

(Yu et al., 2022).

Beyond this atomic implementation, number-fluctuation-robust criteria clarify what kind of entanglement is operationally relevant for phase measurement. In "Entanglement, number fluctuations and optimized interferometric phase measurement" (He et al., 2011), normalized spin operators

$T$5

are introduced via the Moore-Penrose generalized inverse of the number operator. The phase-entanglement criterion

$T$6

and the phase-squeezing criterion

$T$7

are both immune to number fluctuations and are proportional to enhanced phase-measurement sensitivity (He et al., 2011).

A distinct but closely related resource is planar quantum squeezing, which reduces noise in two orthogonal spin directions simultaneously and therefore allows noise reduction over all phase-angles simultaneously (He et al., 2011). In a cold $T$8Rb ensemble, experimentally generated planar quantum squeezed states were reported to give a metrological advantage of at least $T$9 dB relative to classical states for an arbitrary phase, and to beat traditional squeezed states generated with the same QND resources except for a narrow range of phase values (Colangelo et al., 2017). For differential phase tasks with limited or absent prior information, this is a particularly relevant distinction.

4. Noise, common-mode rejection, and decoherence-free structure

Differential phase sensing is especially attractive when the dominant disturbance is common-mode phase noise. "Phase-noise protection in quantum-enhanced differential interferometry" (Landini et al., 2014) generalizes differential interferometry to entangled probe states and shows that, for perfectly correlated interferometers and arbitrary large phase noise, sub-shot-noise sensitivities up to the Heisenberg limit are still possible in the ideal lossless scenario.

The key object is the effective input state

$\phi$0

which converts the noisy problem into an equivalent noiseless one. When the relative noise vanishes, $\phi$1, a non-trivial decoherence-free subspace exists, spanned by states with $\phi$2, and entanglement can be passively protected (Landini et al., 2014). In that regime, separable inputs satisfy $\phi$3, whereas general entangled inputs can satisfy $\phi$4.

The optimal DFS state identified in (Landini et al., 2014) is

$\phi$5

which achieves $\phi$6. Product states of two NOON states give $\phi$7, while spin-squeezed and twin-Fock states can also offer sub-SN sensitivity when relative noise is suppressed (Landini et al., 2014).

A more recent two-node formulation appears in "Lieb-Mattis states for robust entangled differential phase sensing" (Kaubruegger et al., 11 Jun 2025). There the phase is encoded by

$\phi$8

while immunity to common-mode noise follows from operating within a DFS of $\phi$9. Two preparation strategies are analyzed: a unitary bosonic-two-mode-squeezing analogue yielding $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$0 and estimator variance $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$1, and a dissipative preparation via collective emission into a shared cavity mode that offers a $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$2 improvement over the SQL, with numerically observed variance scaling $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$3 (Kaubruegger et al., 11 Jun 2025). This suggests a shift from fragile maximally entangled probes toward DFS-compatible many-body states that retain utility under local noise.

5. Distributed, photonic, and nonlocal realizations

In photonic networks, differential or averaged phase variables can also be estimated with entanglement distributed across modes and particles. "Distributed quantum phase estimation with entangled photons" (Liu et al., 2021) distinguishes four resource classes—MePe, MePs, MsPe, and MsPs—and reports Heisenberg-limit phase measurements for distributed sensing with discrete variables. For an averaged phase $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$4, the fully mode-and-particle-entangled strategy achieves $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$5, while fully separable sensing gives $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$6 (Liu et al., 2021). Experimentally, the reported error reductions are up to $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$7 dB below SNL for individual phase shifts, $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$8 dB below SNL for the averaged phase shift with MePe, and $$\sigma(\Delta_{\textrm{eff}})=\frac{\Delta P_d}{N T},$$9 dB below SNL in a combined strategy using six entangled photons with a total number of photon passes $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$0 (Liu et al., 2021).

A nonlocal weak-light version was demonstrated in "Entanglement Assisted Non-local Optical Interferometry in a Quantum Network" (Stas et al., 11 Sep 2025). The protocol combines event-ready remote quantum entanglement, photon mode erasure, and non-local, non-destructive photon heralding to perform a proof-of-concept entanglement-assisted differential phase measurement of weak incident light between two spatially separate stations. Successful operation was reported with a fiber-link baseline up to $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$1 km, and the Fisher information scales as $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$2 with ideal heralding, in contrast to $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$3 without heralding (Stas et al., 11 Sep 2025).

At the level of direct phase readout of entangled photonic states, "Phase-resolved measurement of entangled states via common-path interferometry" (Voitiv et al., 2024) uses an entangled mode as a collinear reference to measure the phase of biphoton states, including direct measurement of geometric phase accumulation. Because the method is applicable to any pure quantum system containing an exploitable reference state in its entanglement spectrum, it provides a route to phase-resolved characterization without an external reference arm (Voitiv et al., 2024).

6. Conceptual boundaries, misconceptions, and adjacent problems

Several limits recur across the literature. First, entanglement assistance is not synonymous with universal Heisenberg scaling. In the atomic synchronous-comparison setting, joint inter-pixel entanglement gives only a factor $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$4 enhancement and saturates with atom number, whereas independent entangled pixels can reach a factor $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$5 for $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$6 atoms per pixel (Yu et al., 2022). Second, robustness is highly task-dependent: under strong common-mode phase noise, the decisive structure is occupation of a decoherence-free subspace rather than the mere presence of large entanglement entropy (Landini et al., 2014).

Third, not every entangling measurement improves every multiparameter estimation problem. In "Entangling measurements for multiparameter estimation with two qubits" (Roccia et al., 2017), collective measurements on two copies improve the joint estimation of phase and phase diffusion, with experimental $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$7, but no enhancement is observed for multiple phases. This is an important boundary condition for differential phase metrology in settings where nuisance parameters are co-estimated (Roccia et al., 2017).

Fourth, unknown-phase operation requires different resources from local-phase tracking. Conventional spin-squeezing is only applicable when the measured phase is already known to a good approximation, or can be measured iteratively, whereas planar quantum squeezing allows noise reduction over all phase-angles simultaneously (He et al., 2011). An adaptive entangled optical protocol for a completely unknown phase reported variance improvement over $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$8 dB below the SQL at $$(\Delta P_d)^2 = (\Delta P_1)^2 + (\Delta P_2)^2 - 2G(P_1,P_2),$$9 photons, using all detection outcomes in a Bayesian update (Xiang et al., 2010). A plausible implication is that differential phase platforms intended for fast, non-iterative operation may favor PQS-like resources or adaptive entangled strategies over narrowly aligned one-axis-squeezed probes.

Taken together, these results define entanglement-assisted differential phase measurement as a family of estimation strategies in which the metrological benefit is controlled by entanglement topology, noise symmetry, and readout architecture. In the most favorable regimes, the reported gains range from modest constant-factor improvements in jointly entangled atomic clouds to Heisenberg-scaling or near-Heisenberg networked protocols, with direct relevance to gravitational redshift detection, \emph{in situ} imaging spectroscopy, distributed sensing, and nonlocal interferometry (Yu et al., 2022).