# Witnessing entanglement without entanglement witness operators

^{a}Quantum Science and Technology in Arcetri, 50125 Florence, Italy;^{b}National Institute of Optics, National Research Council, 50125 Florence, Italy;^{c}European Laboratory for Non-Linear Spectroscopy, 50125 Florence, Italy;^{d}Institute of Theoretical Physics and Department of Physics, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

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Edited by Anton Zeilinger, University of Vienna, Vienna, Austria, and approved July 26, 2016 (received for review March 7, 2016)

## Significance

The experimental detection and theoretical characterization of entanglement in many-body systems is a mostly unsolved problem. Here we relate entanglement to the statistical speed measuring how quickly nearby states become distinguishable under the action of an arbitrary many-body Hamiltonian. The method is remarkably simple: We witness multipartite entanglement, just elaborating on published experimental data with ions and photons. The unveiled connection between the statistical speed and entanglement provides a unifying framework that can shed new light on quantum information science and quantum critical phenomena.

## Abstract

Quantum mechanics predicts the existence of correlations between composite systems that, although puzzling to our physical intuition, enable technologies not accessible in a classical world. Notwithstanding, there is still no efficient general method to theoretically quantify and experimentally detect entanglement of many qubits. Here we propose to detect entanglement by measuring the statistical response of a quantum system to an arbitrary nonlocal parametric evolution. We witness entanglement without relying on the tomographic reconstruction of the quantum state, or the realization of witness operators. The protocol requires two collective settings for any number of parties and is robust against noise and decoherence occurring after the implementation of the parametric transformation. To illustrate its user friendliness we demonstrate multipartite entanglement in different experiments with ions and photons by analyzing published data on fidelity visibilities and variances of collective observables.

A central problem in quantum technologies is to detect and characterize entanglement among correlated parties (1⇓–3). A most popular approach is based on the implementation of entanglement witness operators (EWs). EW is a Hermitian operator *ρ* (4⇓⇓⇓–8). The power of this method relies on the algebraic fact that for each multipartite entangled state there exists (at least) one EW that recognizes it (4). EWs are device-dependent: Experimental mischaracterization may lead to false positives, namely, to the unwitting realization of operators

Device-independent entanglement witness operators (DIEWs) can be constructed by exploiting Bell-like inequalities testing the correlations between measurement data. These are obtained for different optimized configurations and demand that the parties be addressed locally and do not interact during operations and measurements. DIEWs recognize an important class of entangled states without relying on any hypothesis about the measurement actually performed (16⇓⇓–19). The specific configurations required to witness entanglement are only known for particular cases; the extension to an arbitrary state can vary from computationally hard to prohibitive. A recent experiment (20) has exploited Bell-based DIEWs to demonstrate genuine multipartite entanglement up to six trapped ions. Cross-talk among the particles affected the DIEW of larger systems.

An alternative approach to detect entanglement that does not require the construction of a witness operator has been proposed (21⇓–23). This exploits the Fisher information (or “statistical speed”) quantifying how quickly two slightly different quantum states become statistically distinguishable under a parametric transformation that, as in the case of DIEW, is local. However, local probing preserves the amount of entanglement in the system and belongs to a restricted subset of all possible unitary transformations. As a consequence, this method recognizes an important but relatively small class of entangled states. Furthermore, the strict condition of local probing might not be satisfied experimentally, thus preventing the use of this method.

In this paper we demonstrate that a quite broader class of entangled states can be detected by the statistical distinguishability of two quantum states connected by a transformation that is nonlocal, namely, that couples different qubits (Fig. 1). Our results show that the statistical speed of an initially classically correlated system is strictly upper-bounded even if the transformation generates entanglement. A higher speed witnesses entanglement on the initial state. This can open the way to study multipartite entanglement near quantum phase transition critical points by quenching the parameters of Ising-like Hamiltonians. Furthermore, because statistical distinguishability is directly related to the sensitivity of parameter estimation, our study allows us to characterize entanglement-enhanced precision measurements in many-body systems. The method also allows us to explicitly take into account residual cross-talking between different qubits. However, as will be explained below, our approach is not device-independent because it requires the experimental control of the nonlocal transformation. However, any noise, decoherence, or finite efficiency readout occurring after the parametric transformation (Fig. 1) does not lead to false positives. A characteristic trait of witnessing entanglement via statistical distinguishability is its simplicity. We indeed demonstrate, by just elaborating on published experimental data (24⇓⇓⇓–28), entanglement up to 14 ions and 10 photons, and, in agreement with Bell-based DIEW results reported in ref. 20, genuine multipartite entanglement up to six ions.

## Results

We consider *N* qubits in a state *ρ* and apply the collective unitary transformation *θ*, where*i*th particle and *θ*. Finally, the output state is characterized by the statistical probability distribution *μ* of a readout observable *M*. The statistical distinguishability between the two probability distributions *μ*. *θ* around the reference point **2** gives **2** and **3** via the conditional probabilities are arbitrary but, in practice, chosen so to efficiency distinguish *ρ*. *M*. Because of the convexity of the Fisher information (21), the bound **4** holds not only for product pure states but also for any statistical mixture of product states (i.e., for arbitrary classically correlated states **4** is the maximum quantum statistical speed of separable states, *ρ*, the quantum statistical speed is defined as *L* is the symmetric logarithmic derivative defined via the relation *θ*-independent map **4** is obtained by maximizing over all generalized readout, no false positives are possible in the presence of arbitrary noise and decoherence affecting the state after the parametric transformation.

As derived in *Materials and Methods*, the quantum statistical speed of a pure product state **1** is**4** is readily done: The optimal states have *Supporting Information* we also report the results for the Lipkin–Meshkov–Glick model where **5** are*A*. Fig. 2*B* shows *γ*. The numerical analysis in the homogeneous case *γ* smaller than a critical value *B*). The value *Materials and Methods*. For *B*). An upper bound to *γ*, can be obtained by maximizing each term in Eq. **5** separately. This gives*B*.

It is important to emphasize that different Hamiltonians *H* detect different subsets of entangled states. Local Hamiltonians, *N* spins

Beside the possibility to witness a larger class of entangled states, the measurement of the statistical speed generated by nonlocal Hamiltonians allows to take into account the residual coupling among neighboring spins. This, in contrast, can limit the experimental implementation of Bell-based DIEWs, in particular when dealing with a large number of ions (20).

## Applications

Our method to witness entanglement requires one to experimentally extract the statistical speed. We show below that this can be obtained from the visibility of fringe oscillating as a function of *θ*, from moments of the probability distribution or, more generally, by exploiting a basic relation between the statistical speed and the Kullback–Leibler entropy. In the absence of experimental data obtained with a nonlocal probing Hamiltonian, we apply our protocol to extract the statistical speed from published data in ions and photons experiments where a local transformation was used. In this case, the above method can be extended as a witness of multiparticle entanglement (22, 23): The inequality*N* parties, at least *k* are entangled), where *s* is the largest integer smaller than or equal to **11** with *N*-partite entanglement.

### Statistical Speed from Dichotomic Measurements.

We consider here the simplest (but experimentally relevant) case where the measurement results can only take two values, **2** and **3** simplify to **12**, obtaining*N* the required minimum visibility to detect entanglement decreases. Genuine *N*-partite entanglement is detected when *N*.

### Statistical Speed from Average Moments.

The probability of different measurement results are not always available, but only some averaged moments *Supporting Information*) that *m*, replacing Eq. **16** into Eq. **15**, we obtain

### Witnessing Multipartite Entanglement in Trapped-Ion Experiments.

Several recent efforts have been devoted to create GHZ states *θ* (cf. Eq. **13**), and we can directly analyze the experimental data with our multipartite entanglement witness. In Fig. 3, filled blue circles are obtained from data reported in ref. 24 for *k*-partite entanglement detection (delimited by solid thin lines given by Eq. **11**). In particular, genuine *N*-partite entanglement is marked by the darker red region that, from the data of ref. 24, is reached up to *N*: We have four-partite entanglement for the states of

### Witnessing Multipartite Entanglement in Photon Experiments.

Several experiments have demonstrated the creation of multipartite entanglement in photonic systems (27, 28, 39⇓–41). In particular, ref. 28 reports on the creation of a GHZ state up to 10 photons. After the creation of the state by parametric down-conversion, a phase shift is applied to each qubit, according to the scheme of Fig. 1. The state is finally characterized by measuring the operator **17** to obtain **3**, the filled squares in Fig. 3 are a lower bound for multipartite entanglement.

### Statistical Speed from the Kullback–Leibler Entropy.

So far we have extracted the statistical speed by fitting the experimental probabilities of the different detection events. This simple approach can be implemented when the probabilities can be accurately fitted with a single parameter function, as in the ions and photons experiments discussed above. In general, it might be necessary to extract the statistical speed directly from the bare data without fitting the probability distribution. This can be done experimentally by estimating the Kullback–Leibler (KL) entropy (42)*A* we illustrate the method using experimental data of ref. 24. We focus on the case **18**. A quadratic fit provides **13** (Fig. 3). The large error bars are due to the finite sample statistics of the published data and can be reduced by increasing the sample size and concentrating the measurements around a few phase values (rather than for the whole

To supply to the lack of available experimental data and for illustration purposes, here we implement a numerical Monte Carlo analysis of Eq. **18** to evaluate the role of *B* we plot Eq. **18** in the case *Supporting Information*). For comparison, we also show the Hellinger distance as a function of

A source of noise in the extraction of *m* the sample size (we consider *m* measurements performed at phase *m* measurements performed at phase **18** extends as *m* limit, we can calculate statistical fluctuations of the Hellinger distance by taking *Supporting Information* we show that the bias is positive but decreases as *m*. Fig. 4*C* shows a comparison between a numerical Monte Carlo analysis and the analytical prediction. Similarly, we can evaluate statistical fluctuations of the KL entropy, *Supporting Information*, and a comparison with analytical calculations is shown in Fig. 4*D*. Overall, the simulations show that few hundred measurements are sufficient to extract

## Discussion

The statistical speed reveals and quantifies entanglement among *N* parties. This requires one to probe a quantum state with a generic but known parametric transformation. After the state has been transformed, any coupling with a decoherence source or a nonoptimal choice (or noisy implementation) of the readout measurement does not lead to a false detection of entanglement. A false positive can be obtained as the consequence of a mischaracterization of the Hamiltonian *H*, or of the value of the parameter *θ*. Notice that the upper bound on the statistical speed **4**, can be further maximized over all possible direction of the Pauli matrices acting on each qubit. This bound can be used in the presence of systematic errors in the direction of Pauli matrices of the Hamiltonian implemented experimentally. The protocol also includes generic nonlocal interactions due to experimental tuning or accidental cross-talk effects. The number of operations required to witness entanglement does not increase with the number of parties: The statistical speed is extracted from the knowledge of, at least, two probability distributions obtained at nearby values of *θ*. Several experiments have shown the feasibility of controlled collective parametric transformations with cold (44, 45) and ultracold atoms (43, 46, 47), ions (24⇓–26, 48, 49), photons (28, 41), and superconducting circuits (50). It should be emphasized that not all entangled states probed by Eq. **1** have a statistical speed larger than all possible separable states, even in a noiseless scenario and with optimized output measurements. However, the entangled states violating Eq. **4** with a nonlocal Hamiltonian are those, and only those, overcoming the maximum interferometric phase sensitivity limit achievable with separable states and a phase-encoding transformation generated by the given Hamiltonian.

## Materials and Methods

### Derivation of Eqs. 5–8.

For pure states and unitary transformation **5** with **8**. Repeating the same procedure for **5**.

### Statistical Speed for the Ising Model.

We provide here details on the analysis of the Ising model discussed in the main text. We consider the homogeneous case *A*). The optimization is thus done by replacing **5** and **8**. This provides the equation

which can be maximized over *a* at fixed value of *γ*. The exact analytical expression is long and not reported here. For

Indeed, in the limit **20** goes as **19**, which goes as *a*. The value of *γ* for which Eq. **20** is equal to the maximum over *a* of Eq. **19** gives the critical

## Statistical Speed for the Lipkin–Meshkov–Glick Model

We derive here **1** with *γ*, *A*). Replacing **5**–**8** we find*a* analytically for each *N* (the explicit expression is long and not reported here). For **S1** separately, we can find a upper bound to *B* shows the comparison between the numerical evaluation of **S2** and **S3**, and the bound Eq. **S4**.

## Statistical Speed for Multiple Measurements

We can extend the notion of Hellinger distance (given in Eq. **2** for a single measurement) to the case of *m* measurements:*i*th measurement) when the phase shift is equal to *θ*, and the sum runs over all possible sequences. For independent measurements we have

## Statistical Bias and Fluctuations of the KL Entropy

We calculate the statistical bias of the KL entropy *m* we have

## Acknowledgments

This work was supported by the National Natural Science Foundation of China Grant 11374197, Program for Changjiang Scholars and Innovative Research Team Grant IRT13076, and The Hundred Talent Program of Shanxi Province (2012).

## Footnotes

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^{1}To whom correspondence should be addressed. Email: wdli{at}sxu.edu.cn.

Author contributions: L.P., Y.L., W.L., and A.S. performed research and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1603346113/-/DCSupplemental.

Freely available online through the PNAS open access option.

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