Witnessing entanglement without entanglement witness operators Luca Pezzèa,b,c, Yan Lid, Weidong Lid,1, and Augusto Smerzia,b,c a Quantum Science and Technology in Arcetri, 50125 Florence, Italy; bNational Institute of Optics, National Research Council, 50125 Florence, Italy; cEuropean Laboratory for Non-Linear Spectroscopy, 50125 Florence, Italy; and dInstitute of Theoretical Physics and Department of Physics, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
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.
quantum entanglement entanglement detection Fisher information trapped ions
| quantum technology |
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 W such that Tr½ρsep W ≥ 0 for all separable states ρsep and Tr½ρW < 0 for, at least, one entangled state 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 W exp ≠ W such that Tr½ρsep W exp < 0 also for some separable states, therefore signaling entanglement in states that are only classically correlated (9, 10). The same problem arises, even more dramatically, when trying to detect entanglement via the tomographic reconstruction of the quantum state (10–15). 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, www.pnas.org/cgi/doi/10.1073/pnas.1603346113
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 Bellbased DIEW results reported in ref. 20, genuine multipartite entanglement up to six ions. 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. 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. Freely available online through the PNAS open access option. 1
To whom correspondence should be addressed. Email: [email protected]
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1603346113/-/DCSupplemental.
PNAS | October 11, 2016 | vol. 113 | no. 41 | 11459–11464
Edited by Anton Zeilinger, University of Vienna, Vienna, Austria, and approved July 26, 2016 (received for review March 7, 2016)
1 μ Pðμjθ0 Þ
, which coincides with the Fisher infor-
mation (33). The specific measurement observable and the parametric transformation entering in Eqs. 2 and 3 via the conditional probabilities are arbitrary but, in practice, chosen so to efficiency distinguish PðμjθÞ from Pðμjθ0 Þ. Extracting the statistical speed requires (at least) two settings, independent from the number of particles, the quantum state and the measurement observable. A statistical speed higher than the bound
Fig. 1. Statistical distinguishability and statistical speed. (A) N parties prepared in a quantum state ρ undergo a unitary nonlocal transformation with θ a tunable parameter. The completely positive trace-preserving map Λ includes arbitrary θ-independent decoherence effects. (B) The probability distribution PðμjθÞ is obtained by collecting the measurement results μ for different values of the parameter, here chosen to be θ0 (red line) and θ0 + δθ (green line). (C) To quantify the statistical distinguishability between the two distributions we pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ introduce unit vectors p0 = fPðμjθ0 Þ1=2 gμ (red) and pδθ = fPðμjθ + δθÞ1=2 gμ p0ﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃ (green) and measure the Euclidean distance among them: ℓ = 2 p − p 0
(dashed line), Eq. 2. The statistical speed, Eq. 3, is an entanglement witness.
Results We consider N qubits in a state ρ and apply the collective unitary transformation e−iHθ parameterized by a real number θ, where H=
N X αi
σ ðiÞ m +
N X Vij i, j=1
ðjÞ σ ðiÞ n σn .
The coefficients αi (without loss of generality, 0 ≤ αi ≤ 1) account for (possibly) inhomogeneous linear couplings (29) or local/subgroups operations on the parties, and Vij = Vji provides a nonlocal ðiÞ interaction between different spins. Here, σ n ≡ σ ðiÞ · n is the Pauli matrix for the ith particle and n is a versor in the Bloch sphere. The unitary transformation is eventually followed by arbitrary noise and decoherence that, in full generality, we model as a completely positive trace-preserving map Λ that does not depend on θ. Finally, the output state is characterized by the statistical probability distribution PðμjθÞ of possible measurement results μ of a readout observable M. The statistical distinguishability between the two probability distributions Pðμjθ0 Þ and PðμjθÞ is quantified by the Hellinger distance (30) sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ X pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ℓðθ0 , θÞ ≡ 2 Pðμjθ0 Þ − PðμjθÞ ,
speed is defined as υQ ≡ maxM υ and its square can be calculated as υ2Q = Tr½ρL2 (32), where L is the symmetric logarithmic deLρ + ρL rivative defined via the relation dρ (32, 33). Recently, it dθ = 2 has been shown that the quantum statistical speed is linked to the dynamic susceptibility (34) and it is thus available in condensedmatter experiments. Notice that the quantum statistical speed can only remain constant or decrease under the action of a θ-independent map Λ following the unitary transformation (30). We remark that the Λ can be nonlocal, namely, it can generate entanglement. Because the bound 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 ψ sep i probed by the Hamiltonian in Eq. 1 is υ2Q = υ20 + υ21 + υ22 ,
11460 | www.pnas.org/cgi/doi/10.1073/pnas.1603346113
where υ20 =
D E2 , α2i 1 − σ ðiÞ m
υ21 = 2
h D ED EiD E Vij αi n · m − σ ðiÞ σ ðiÞ σ ðjÞ n m n ,
i, j=1 i≠j
that is, the rate at which ℓðθ0 , θÞ changes with θ around the reference point θ0. A Taylor expansion of Eq. 2 gives
witnesses entanglement in the state ρ. υsep is obtained by maximizing the statistical speed over all pure separable states ψ sep i = ψ ð1Þ i ⊗ . . . ⊗ ψ ðNÞ i, all possible maps Λ, and all (including noisy and biased) measurements 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 (i.e., for arbitrary classically correlated states P states ρsep = λ qλ ψ sep,λ ihψ sep,λ , with qλ ≥ 0). The lower bound in Eq. 4 is the maximum quantum statistical speed of separable states, υsep = maxjψ sep i υQ. For a generic state ρ, the quantum statistical
with the sum extending over all possible μ. ℓ is a statistical distance (31, 32): It ranges from p zero, ﬃﬃﬃ if and only if Pðμjθ0 Þ = PðμjθÞ ∀μ, to its maximum value ℓ = 2 2, if and only if Pðμjθ0 Þ × PðμjθÞ = 0 ∀μ, and it satisfies the triangular inequality.The Hellinger distance is pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ proportional to the Euclidean distance p0 − pθ between the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ unit vectors p0 = f Pðμjθ0 Þ gμ and pθ = f PðμjθÞ gμ. It is useful to introduce the notion of statistical speed: dℓ  υ ≡ , dθ θ0
max υ, jψ sep i, Λ, M
and υ22 =
N V2 D E2 D E2
X ij σ ðjÞ 1 − σ ðiÞ n n 2 i, j=1 i≠j
D E2 D ED E σ ðlÞ σ ðjÞ Vij Vil 1 − σ ðiÞ n n n .
i, j, l=1 i≠j≠l
For local Hamiltonians (Vij = 0), the maximization in Eq. P4 is readily ðiÞ 2 done: The optimal states have hσ m i = 0 ∀i, giving υ2sep = N i=1 αi . This 2 generalizes the bound υsep = N discussed in refs. 21 and 35 for homogeneous coupling αi = 1. For nonlocal Hamiltonians (Vij ≠ 0) the Pezzè et al.
where j↑i and j↓i are eigenstates of σ n, φi are arbitrary phases, ðiÞ and hσ n i as a function of γ is reported in Fig. 2A. Fig. 2B shows υ2sep as a function of γ. The numerical analysis in the homogeneous case αi = 1 reveals that, for γ smaller than a ðiÞ critical value γ c, the speed υ2Q is maximized when hσ n i are all ðiÞ 2 equal. In particular, for γ 1, υQ is the highest when hσ n i = γ 2 5 2 4 ∀i, giving υsep =N = 1 + 4γ + Oðγ Þ (solid blue line in Fig. 2B). The value γ c = 0.7302 is found analytically, as discussed in Materials and Methods. For γ > γ c, υ2Q ðγÞ is maximized by alterðiÞ ði+1Þ nating hσ n i = 1 and hσ n i = 0. In this limit, we find υ2sep =N = 1 1 2 + γ + γ (solid blue line in Fig. 2B). An upper bound to υ2sep, 2 2 valid in the inhomogeneous case (αi ≠ 0) and for every γ, can be obtained by maximizing each term in Eq. 5 separately. This gives ! N X X X N 2 2 υsep ≤  αi + γ max αi , αi + γ 2 , 2 even i i=1 odd i which is shown as the dashed red line in Fig. 2B. It is important to emphasize that different Hamiltonians H detect different subsets of entangled states. Local Hamiltonians,
P ðiÞ H0 = 12 N i=1 σ m , are well-suited to detect entangled states that are symmetric under particle exchange. For instance, the square quantum statistical speed of the Dicke state j↑i⊗N=2−ν j↓i⊗N=2+ν probed by a local Hamiltonian H0 is υ2Q = N 2 =2 − 2ν2 + N (here we consider n perpendicular to m). That is υ2Q > υ2sep = N for ν ≠ ± N=2: All Dicke states are detected as entangled except the (separable) spin polarized. Entangled nonsymmetric states are better detected by nonlocal Hamiltonians. Let us consider, forpinﬃﬃﬃ stance, the state of N spins jχi = ðj↑↓i⊗N=2 + j↑i⊗N=2 j↓i⊗N=2 Þ= 2. It is possible to demonstrate that, when applying e−iH0 θ, the quantum speed of jχi is smaller than the bound υ2sep = N for N > 6, even when optimizing the direction m in the Hamiltonian H0. Therefore, jχi cannot be detected as entangled when probed by only local Hamiltonians. Conversely, when by a nonlocal nearestP probed ðiÞ ði+1Þ neighbor Hamiltonian H1 = 14 N , the state jχi has a i=1 σ n σ n square quantum statistical speed equal to N 2 =4 − N + 1 that surpasses the bound υ2sep = N=2 if N ≥ 6. jχi can thus be detected as entangled when probed by the Ising Hamiltonian. The opposite is also true. For instance the Greenberger–Horne–Zeilinger (GHZ) pﬃﬃﬃ state jφi = ðj↑i⊗N + j↓i⊗N Þ= 2 has a null statistical speed when probed with H1. Nevertheless jφi can reach a statistical speed N 2 > N and it can thus be detected as entangled when probed by the local Hamiltonian H0. 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 υ2 > sk2 + r 2 ,
signals ðk + 1Þ-partite entanglement (i.e., among N parties, at least k are entangled), where s is the largest integer smaller than or equal to N=k and r = N − sk. In particular υ2 > ðN − 1Þ2 + 1, obtained from Eq. 11 with k = N − 1, is a witness of genuine N-partite entanglement. Statistical Speed from Dichotomic Measurements. We consider here the simplest (but experimentally relevant) case where the measurement results can only takeptwo values, = ±1. In this case, ﬃ Eqs. ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pμﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 and 3 simplify to ℓ2 = 8½1 − P0 Pδθ − ð1 − P0 Þð1 − Pδθ Þ and Fig. 2. Witness of entanglement with the Ising Hamiltonian. (A) Mean spin ðiÞ values hσ n i for the separable state maximizing υ2Q, as a function of γ. (B) Maximum statistical speed of separable states, υ2sep (dots), probed by the Ising Hamiltonian. Entanglement is witnessed by a statistical speed υ2 > υ2sep, in the gray region. The solid blue lines are analytical limits discussed in the main text. The dashed line is the upper bound Eq. 10.
Pezzè et al.
2 1 ∂Pθ , υ = P0 ð1 − P0 Þ ∂θ θ0 2
respectively, where P0 ≡ Pð+1jθ0 Þ and Pδθ ≡ Pð+1jθ0 + δθÞ. For instance, if PNAS | October 11, 2016 | vol. 113 | no. 41 | 11461
bound depends on the explicit form of Vij and αi. It can be calculated either numerically or, as shown below, analytically in many cases of interest. In the following we consider, as an example, the Ising model δ +δ having nearest-neighbor interaction Vij = γ j,i+1 2 j,i−1 and n · m = 1. In Supporting Information we also report the results for the Lipkin–Meshkov–Glick model where Vij = γ. The states that maximize Eq. 5 are vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D Eﬃ D Eﬃ u u u1 + σ ðiÞ u1 − σ ðiÞ n n t t N  j↑ii + e−iφi j↓ii , jψi = ⊗ 2 2 i=1
1 ± V cos Nθ , 2
where 0 ≤ V ≤ 1 is the visibility of the oscillating probabilities, we can straightforwardly calculate Eq. 12, obtaining V 2 N 2 sin2 ðNθÞ . 1 − V 2 cos2 ðNθÞ
. Notice It is thus possible to detected entanglement when V > p1ﬃﬃﬃ N that, with an increasing number of qubits N the required minimum visibility to detect entanglement decreases. Genuineﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 N-partite entanglement is detected when V > 1 − N1 + N12, which requires a visibility increasing with N. measurement results are Pnot always available, but only some averaged moments hμiθ = μ μPðμjθÞ. We can extend the notion of Hellinger distance and statistical speed to the probability distribuP μﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ and μ1 , p . . .ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ μm are measurement tion PðμjθÞ, where μ = m1 m i i=1 p ﬃ P results. We find ℓ2mom = 4 μ ð Pðμjθ0 Þ − Pðμjθ0 + δθÞ Þ2 and X μ
1 dPðμjθÞ 2 , Pðμjθ0 Þ dθ θ0
where the sum extends over all possible values of μ. Using a Cauchy–Schwarz inequality it is possible to demonstrate (Suppﬃﬃﬃﬃ porting Information) that υmom ≤ υ m. For m 1, the central limit theorem provides PðμjθÞ =
mðμ−hμiθ Þ 2ðΔμÞ2 θ
20 10 0
Statistical Speed from Average Moments. The probability of different
P where ðΔμÞ2θ = μ ðμ − hμiθ Þ2 PðμjθÞ. To the leading order in m, replacing Eq. 16 into Eq. 15, we obtain m dhμiθ 2 2 υmom = .  dθ θ0 ðΔμÞ2θ The entanglement criteria thus becomes υ2mom =m > υ2sep. When probing with local Hamiltonians, the inequality υ2mom =m > sk2 + r 2 witness ðk + 1Þ-partite entanglement from the experimental measurements of average moments. These bounds generalize to arbitrary observables the bounds to detect entanglement (36) and multipartite entanglement (37) from the estimation of the mean collective spin (1, 38). Witnessing Multipartite Entanglement in Trapped-Ion Experiments.
Several recent efforts pﬃﬃﬃ have been devoted to create GHZ states ðj↑i⊗N + j↓i⊗N Þ= 2 with trapped ions (24–26). In ref. 24 the creation of the state has been followed by a collective rotation ðjÞ ðjÞ ðjÞ ðjÞ iπ2σ θ ⊗N , with σ θ = σ x cos θ + σ y sin θ. The output state has j=1 e been characterized by dichotomic measurements of the parity, Π = ð−1ÞN↑ , with N↑ being the number of qubits measured in one of the two modes. The reported results are the oscillations of the average parity (and, therefore, of the probability to obtain the ±1 result) as a function of 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 N = f2 . . . 6,8,10,12,14g, open blue circles from the data of ref. 25 for N = 3, and of ref. 26 for N = 4,5,6. We first notice that all data satisfy υ2 > N: We thus detect entanglement in all of the states created in refs. 24–26. The different colored regions correspond to different k-partite entanglement detection 11462 | www.pnas.org/cgi/doi/10.1073/pnas.1603346113
9 10 11 12 13 14 15
N Fig. 3. Witness of multipartite entanglement. Squared statistical speed as a function of the number of qubits obtained analyzing published ions (circles) and photon (squares) experimental data: ref. 24, filled circles; ref 25 for N = 3 and ref. 26 for N = 4,5,6, open circles; ref. 27, open square; ref. 28, filled squares. When not visible, error bars are smaller than the symbol size. The upper thick line is the upper bound υ2 = N2, and the lower thick line is the separability bound υ2 = N. The different lines are bound for k-partite entanglement, Eq. 11. In particular, the darker red region stands for genuine N-partite entanglement.
(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 = 6 ions. The maximum value of υ2 is obtained for N = 8 particles, corresponding to seven-partite entanglement. The number of entangled particles in the system slowly decreases for increasing N: We have four-partite entanglement for the states of N = 10 ions and three-partite entanglement for the state of N = 12 and N = 14 ions. It is interesting to notice that a recent experiment (20) has investigated a Bell-based DIEW reporting genuine multipartite entanglement up to N = 6 ions, which is in agreement with our finding. 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 σ ⊗N x , whose mean value shows high-frequency oscillations, hσ ⊗N x i = V cos Nθ (28). Noticing that ðΔσ x⊗ N Þ2 = 1 − hσ x⊗ N i2, we can calculate the correυ2
V N sin ðNθÞ sponding statistical speed from Eq. 17 to obtain mom m = 1 − V 2 cos2 ðNθÞ. Also in this case, the witness of multipartite entanglement is solely based on the visibility of the interference signal. Results are shown in Fig. 3 (filled squares). We witness four-partite entanglement for the state of N = 8 photons, giving the highest value of the statistical speed reached with photons. Because υ2mom is a lower bound of Eq. 3, the filled squares in Fig. 3 are a lower bound for multipartite entanglement. 2
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 Pezzè et al.
Pðμjθ0 Þ . Pðμjθ0 + δθÞ
The KL entropy grows quadratically for small δθ with a coefficient proportional to the squared statistical speed (3), DKL = υ2 δθ2 =2. In Fig. 4A we illustrate the method using experimental data of ref. 24. We focus on the case N = 8 and calculate DKL around θ0 ≈ π=ð2NÞ according to Eq. 18. A quadratic fit provides υ2 = 44.6 ± 7.7, in agreement with the result (υ2 = 39.6 ± 0.8) obtained using Eq. 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 2π interval). 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 δθ and the sample size, taking, as a testing ground the parity measurements with probability (13). In Fig. 4B we plot Eq. 18 in the case N = 8 and V = 0.787, around θ0 ≈ π=ð2NÞ. The figure shows that the quadratic behavior is obtained at sufficiently large δθ (Supporting Information). For comparison, we also show the Hellinger distance as a function of δθ, which can also be exploited to extract experimentally the Fisher information (43). Notice that, for these values of the visibility, due to higher-order terms, the quadratic approximation of DKL holds at larger values of δθ than the ℓ2 expansion. A source of noise in the extraction of υ2 is the limited statistics of the measurement data (other sources of noise, such as detection noise and decoherence, result in a reduced visibility). Let us indicate as m the sample size (we consider m measurements performed at
Fig. 4. KL entropy and statistical speed. (A) KL entropy as a function of δθ obtained from an analysis of the experimental data of ref. 24 for N = 8 ions. Blue dots are calculated using Eq. 18. The blue line is a parabolic fit, DKL = δθ2 υ2 =2. The colored region corresponds to multipartite entanglement level, with color scale as in Fig. 3. (B) Squared statistical distance, ℓ2 (red line), KL entropy, 2DKL (green line), and their common low-order approximation, δθ2 υ2 (blue line), as a function of δθ. (C and D) Numerical simulation of the KL entropy (dots) as a function of the sample size m, for 2Nδθ=π = 0.4. Solid lines are analytical predictions, valid for m 1 (see text), for the statistical bias (C) and the statistical fluctuation of DKL (D). In B–D we used Eq. 13 for the probability, with N = 8 and V = 0.787, consistently with the experimental data of A.
Pezzè et al.
phase θ0 and m measurements performed at phase θ = θ0 + δθ). The experiment gives access to frequencies rather than probabilities, and ~ KL = f0 ln f0 + ð1 − f0 Þln 1 − f0 , where f0 = neven,0 =m Eq. 18 extends as D fδθ 1 − fδθ is the frequency of even parity results obtained at phase θ0 (and analogous definition for fδθ). Numerically, we can calculate f0 and fδθ by a Monte Carlo sampling of the probabilities P0 and Pδθ, respectively. In the large-m limit, we can calculate statistical fluctuations of the Hellinger distance by taking f0 = P0 + δf0 and ~ KL in Taylor series for small fδθ = Pδθ + δfδθ, and then expanding D δfδθ. We calculate the bias of the squared Hellinger distance, ~ KL i − DKL, where brackets indicate statistical averaging. In b ≡ hD Supporting Information we show that the bias is positive but decreases as b ∼ 1=2m with the sample size m. Fig. 4C shows a comparison between a numerical Monte Carlo analysis and the analytical prediction. Similarly, we can evaluate statistical fluctu~ KL = hD ~ 2 i − hD ~ KL i2. We obtain ations of the KL entropy, Δ2 D KL ~ KL ∼ 1=m, showing, also in this case, a scaling inversely proΔ2 D portional to the sample size. Details of our analytical calculations are reported in Supporting Information, and a comparison with analytical calculations is shown in Fig. 4D. Overall, the simulations show that few hundred measurements are sufficient to extract υ2 with a small statistical bias and large signal-to-noise. 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 υsep, Eq. 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 e−iθH, the quantum statistical speed is given by υ2Q = 4ðΔHÞ2. Taking H = H0 + H1, υ2Q is given by Eq. 5 with υ20 = 4ðΔH0 Þ2, υ21 = 4ðÆfH0 , H1 gæ − 2ÆH0 æÆH1 æÞ and υ22 = 4ðΔH1 Þ2. We detail here the calculation of υ22 for product pure states, where P P Vij ðiÞ ðjÞ Vij Vkl ðiÞ ðjÞ ðkÞ ðlÞ ðiÞ ðjÞ ðkÞ ðlÞ 2 H1 = N i,j,k,l 4 ½Æσ n σ n σ n σ n æ − Æσ n σ n æÆσ n σ n æ. i,j=1 4 σ n σ n . We have υ2 = Notice that the terms i = j and/or k = l do not contribute to υ22. When i ≠ j and ðiÞ ðjÞ ðkÞ ðlÞ ðiÞ ðjÞ ðkÞ ðlÞ ðiÞ ðjÞ ðkÞ ðlÞ k ≠ l, we have Æσ n σ n æÆσ n σ n æ = Æσ n æÆσ n æÆσ n æÆσ n æ, whereas Æσ n σ n σ n σ n æ = ðiÞ ðjÞ ðkÞ ðlÞ Æσ n æÆσ n æÆσ n æÆσ n æ only if the indexes i, j, k, l are all different. Therefore, only terms where at least two indexes are equal contribute to υ22. The terms k = i, l = j P ðiÞ ðjÞ and k = j, l = i both contribute with i,j Vij2 ð1 − Æσ n æ2 Æσ n æ2 Þ=4. After straightforward algebra, taking into account all contributing terms, one arrives at Eq. 8.
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without fitting the probability distribution. This can be done experimentally by estimating the Kullback–Leibler (KL) entropy (42)
Repeating the same procedure for υ20 and υ21, where H0 = Eq. 5.
αi ðiÞ i=1 2 σ m ,
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 αi = 1 and n = m. For γ ≤ γ c we find numerically that υ2Q is maximized by ðiÞ taking equal Æσ n æ for all i = 1, . . . , N (Fig. 2A). The optimization is thus done ðiÞ by replacing Æσ n æ = a in Eqs. 5 and 8. This provides the equation γ2 υ2Q = 1 − a2 + 2γ a − a3 + 1 + 2a2 − 3a4 , N 4
giving υ2sep =N = 1 + 54γ 2 + Oðγ 4 Þ. For γ > γ c we obtain numerically that υ2Q ðγÞ is ðiÞ ði+1Þ maximized when Æσ n æ = 1 and Æσ n æ = 0, giving υ2sep N
1 γ2 = +γ+ . 2 2
Indeed, in the limit γ 1, Eq. 20 goes as ∼ γ 2 =2 and thus overcomes Eq. 19, which goes as ∼ γ 2 =3 at best, as obtained by maximizing ð1 + 2a2 − 3a4 Þ=4 over a. The value of γ for which Eq. 20 is equal to the maximum over a of Eq. 19 gives the critical γ c .
which can be maximized over a at fixed value of γ. The exact analytical expression is long and not reported here. For γ 1 we find a = γ + Oðγ 3 Þ,
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).
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