# Detecting metrologically useful asymmetry and entanglement by a few local measurements

Abstract: Important properties of a quantum system are not directly measurable, but they can be disclosed by how fast the system changes under controlled perturbations. In particular, asymmetry and entanglement can be verified by reconstructing the state of a quantum system. Yet, this usually requires experimental and computational resources which increase exponentially with the system size. Here we show how to detect metrologically useful asymmetry and entanglement by a limited number of measurements. This is achieved by studying how they affect the speed of evolution of a system under a unitary transformation. We show that the speed of multiqubit systems can be evaluated by measuring a set of local observables, providing exponential advantage with respect to state tomography. Indeed, the presented method requires neither the knowledge of the state and the parameter-encoding Hamiltonian nor global measurements performed on all the constituent subsystems. We implement the detection scheme in an all-optical experiment.

## Summary (2 min read)

### Introduction

- Yet, this usually requires experimental and computational resources which increase exponentially with the system size.
- Indeed, the presented method requires neither the knowledge of the state and the parameter-encoding Hamiltonian nor global measurements performed on all the constituent subsystems.
- Also, verifying their presence is necessary, but not sufficient to guarantee a computational advantage.

### II. RELATING ASYMMETRY TO OBSERVABLES

- Note that tomography demands to prepare O(22n) system copies and perform a measurement on each of them [2].
- It is also possible to extract the swap value by single qubit interferometry [23–25].
- Experimentally measuring coherence, and in particular asymmetry, is hard [27,28].
- In fact, the quantum Fisher information quantifies the instantaneous response to a perturbation [11,30].

### III. WITNESSING METROLOGICALLY USEFUL ENTANGLEMENT

- The authors extend the analysis to multipartite systems, proving that nonlinear speed scaling witnesses useful entanglement.
- Given Eq. (4), entanglementenhanced precision in estimating a phase shift τ is verified if Sτ (ρ,Hn) > n/4. (5) The overlap detection network for n-qubit systems and additive Hamiltonians is depicted in Fig.
- Evaluating the SLDF is an appealing strategy to verify an advantage given by entanglement, rather than just detecting quantum correlations [22,33–38].
- The authors proposal has two peculiar advantages.
- First, it is applicable to any probe state ρ without a priori information and assumptions, e.g., invariance under permutation of the subsystems.

### IV. EXPERIMENTAL ASYMMETRY AND ENTANGLEMENT DETECTION

- While employing state tomography would require fifteen measurements, the authors verify that the proposed protocol needs six.
- Two photon pairs (photons 3–6) are generated via single BBO crystals (beamlike type-II phase matching).
- A 90◦ rotated QWP swaps the Bell states, |φ±〉 → |φ∓〉, generating a π phase shift between H,V polarizations.
- That is, three projections are required for evaluating purity and overlap, respectively.

### 1. Error sources

- The authors discuss the efficiency of the experimental setup.
- This poses the problem to rule out the case of BSMs measuring two photon pairs emitted by a single SPDC source [46].
- Single source double down conversion can also occur because of high-order emission noise, which has been minimized by setting a low pump power.
- Here the main error source is the imperfection of the three Hang-Ou-Mandel interferometers (one for the PBS and each BSM), which have a visibility of 0.91.
- This is due to the temporal distinguishability between the interfering photons, determined by the pulse duration.

### A. Quantum Fisher information as measures of state sensitivity

- Quantum information geometry studies quantum states and channels as geometric objects.
- This means that they have the appealing feature to be the unique class of contractive Riemannian metrics under completely positive trace preserving (CPTP) maps : d( (ρ), (σ )) d(ρ,σ ),∀ρ,σ, [51,52].
- The dynamics of the quantum Fisher information for closed and open quantum systems has been studied in Ref. [7].
- All such metrics reduce to the classical Fisher-Rao metric∑ i (dtλi(t)) 2/(λi(t)) for stochastic dynamics of probability distributions {λi(t)}, represented at any time by a diagonal density matrix.

### VI. CONCLUSION

- The authors showed how to extract quantitative bounds to metrologically useful asymmetry and entanglement in multipartite systems from a limited number of measurements, demonstrating the method in an all-optical experiment.
- The scalability of the scheme may make possible to certify quantum speedup in large scale registers [1,11,38], and to study critical properties of many-body systems [14,15,39], by limited laboratory resources.
- On this hand, the authors remark that they here compared their method with state tomography, as the two approaches share the common assumption that no a priori knowledge about the input state and the Hamiltonian is given.
- An interesting followup work would test the efficiency of their entanglement witness against two-time measurements of the classical Fisher information, when local measurements on the subsystems are only available.
- A further development would be to investigate macroscopic quantum effects via speed detection, as they have been linked to quadratic precision scaling in phase estimation [IF (ρ,Hn) = O(n2)] [20,31,40].

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### Cites background from "Detecting metrologically useful asy..."

...Upper and lower bounds on the geometric coherence have been investigated in (Zhang et al., 2017a)....

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### "Detecting metrologically useful asy..." refers background in this paper

...This means that distant laboratories can verify quantum speed-up due to entanglement in a shared system S by local operations and classical communication [1], providing each laboratory with two copies of a subsystem S i....

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...The sensitivity of a quantum system to a quantum operation described by a parametrized channel Φt [1], where t is the time, is determined by how fast its state ρt := Φt(ρ0) evolves....

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...Consider a phase estimation protocol, a building block of quantum computation and metrology schemes [1, 3, 11]....

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...Quantum coherence and entanglement can generate nonclassical speed-up in information processing [1]....

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...The scalability of the scheme may make possible to certify quantum speed-up in large scale registers [1, 11, 39], and to study critical properties of many-body systems [14, 15, 38], by limited laboratory resources....

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3,781 citations

### "Detecting metrologically useful asy..." refers methods in this paper

...To quantify the sensitivity of a probe state ρ = ∑ i λi|i〉〈i| to the unitary transformation Ut, we employ the symmetric logarithmic derivative quantum Fisher information (SLDF), a widely employed quantity in quantum metrology and quantum information [29]:...

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2,293 citations

### "Detecting metrologically useful asy..." refers background in this paper

...Consider a phase estimation protocol, a building block of quantum computation and metrology schemes [1, 3, 11]....

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###### Q2. What have the authors stated for future works in "Detecting metrologically useful asymmetry and entanglement by a few local measurements" ?

The scalability of the scheme may make possible to certify quantum speedup in large scale registers [ 1,11,38 ], and to study critical properties of many-body systems [ 14,15,39 ], by limited laboratory resources. A further development would be to investigate macroscopic quantum effects via speed detection, as they have been linked to quadratic precision scaling in phase estimation [ IF ( ρ, Hn ) = O ( n2 ) ] [ 20,31,40 ].