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https://doi.org/10.1038/s41467-021-22353-3 OPEN Metrological complementarity reveals the Einstein- Podolsky-Rosen paradox ✉ Benjamin Yadin1,2,5, Matteo Fadel 3,5 & Manuel Gessner 4,5

The Einstein-Podolsky-Rosen (EPR) paradox plays a fundamental role in our understanding of quantum mechanics, and is associated with the possibility of predicting the results of non- commuting measurements with a precision that seems to violate the uncertainty principle.

1234567890():,; This apparent contradiction to complementarity is made possible by nonclassical correlations stronger than entanglement, called steering. Quantum information recognises steering as an essential resource for a number of tasks but, contrary to entanglement, its role for metrology has so far remained unclear. Here, we formulate the EPR paradox in the framework of quantum metrology, showing that it enables the precise estimation of a local phase shift and of its generating observable. Employing a stricter formulation of quantum complementarity, we derive a criterion based on the quantum Fisher information that detects steering in a larger class of states than well-known uncertainty-based criteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to uncover steering of non-Gaussian states in state-of-the-art experiments.

1 School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, UK. 2 Wolfson College, University of Oxford, Oxford, UK. 3 Department of Physics, University of Basel, Basel, Switzerland. 4 Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, Paris, France. 5These authors contributed equally: Benjamin Yadin, Matteo Fadel, ✉ Manuel Gessner. email: [email protected]

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n their seminal 1935 paper1, EPR presented a scenario where The most general way to formally model the joint statistics p(a, the position and momentum of one quantum system (B) can h∣X, H) is offered by the formalism of assemblages, i.e. functions I B ; ρ ’ both be predicted with certainty from local measurements of Aða XÞ¼pðajXÞ ajX that map any possible result a of Alice s another remote system (A). Based on this apparent violation of measurement of X to a local probability distribution p(a∣X) and a fi the uncertainty principle, in 1989 Reid formulated the rst (normalised) conditional quantum state ρB for Bob’s practical criterion for an EPR paradox2, which has enabled ajX subsystem18. This description avoids the need to make assump- numerous experimental observations3: Steering from A to B is tions about the nature of Alice’s system, which is key to one-sided revealed when measurement results of A allow one to predict the device-independent quantum information processing5–7. We only measurement results of B with errors that are smaller than the – impose a no-signalling condition which requires that limit imposed by the Heisenberg Robertson uncertainty relation ∑ ; ρB ρB aAða XÞ¼ for all X, where is the reduced density for B. More generally, an EPR paradox implies the failure of any ’ attempt to describe the correlations between the two systems in matrix of Bob s system. Based on the assemblage A, the joint ; ; ρB terms of classical probability distributions and local quantum statistics are described as pða hjX HÞ¼pðajXÞhjh ajXjih . states for B, known as local hidden state (LHS) models, as was The EPR paradox can now be formally defined as an shown by Wiseman et al.4 using the framework of quantum observation that rules out the possibility of modelling an information theory. Aside from its fundamental interest, steering assemblage by a LHS model. In such a model, a classical random is recognised as an essential resource for quantum information variable λ with probability distribution p(λ) determines both B tasks5, such as one-sided device-independent quantum key Alice’s statistics p(a∣X, λ) and Bob’s local state σλ , leading to the 6,7 8 B distribution and quantum channel discrimination . assemblage Aða; XÞ¼∑λpðajX; λÞpðλÞσλ . Inequality (1) holds Uncertainty relations describe the complementarity of non- for arbitrary estimators and measurement settings whenever a commuting observables, but the complementarity principle LHS model exists. The sharpest formulation of Eq. (1)isthus applies more generally to notions that are not necessarily asso- obtained by optimising these choices to minimise the estimation ciated with an operator. One generalisation9 involves the quan- ρB error. The optimal estimator hestðaÞ¼Tr½ ajXHŠ attains the lower tum Fisher information (QFI), the central tool for quantifying the bound3 Var½H Š ≥ ∑ pðajXÞVar½ρB ; HŠ, where Var½ρ; HŠ¼ precision of quantum parameter estimation10–13. Besides its est a ajX 2 2 fundamental relevance for quantum-enhanced precision mea- hH iρ ÀhHiρ is the variance with hOiρ ¼ Tr½ρOŠ. Optimising surements, the QFI is of great interest for the characterisation of over Alice’s measurement setting X leads to the quantum quantum many-body systems14,15 and gives rise to an efficient conditional variance and experimentally accessible witness for multipartite BjA ; : ∑ ρB ; ; entanglement12,13,16, but so far, its relation to steering has VarQ ½A HŠ ¼ min pðajXÞVar½ ajX HŠ ð2Þ X a remained elusive. It has been a long-standing open problem to determine if quantum correlations stronger than entanglement, and the optimised version of Reid’s condition (1) reads 17 BjA ; BjA ; ≥ ; 2= such as steering or Bell correlations, play a role in metrology . VarQ ½A HŠVarQ ½A MŠ jh½H MŠiρB j 4. The uncertainty- In this work we formulate a steering condition in terms of the based detection of the EPR paradox is based on the fact that θ complementarity of a phase shift and its generating Hamilto- Alice’s choice of measurement can steer Bob’s system into nian H, using information-theoretic tools from quantum conditional states that have small variances for either one of the metrology. We express our steering condition in terms of the QFI. two non-commuting observables H and M. The more general phase-generator complementarity principle – reproduces the Heisenberg Robertson uncertainty relation in the EPR-assisted metrology. To express quantum mechanical com- special case where the phase is estimated from an observable M. plementarity in the framework of quantum metrology10–13,we Therefore, our metrological criterion is stronger than the assume that the observable H imprints a local phase shift θ on uncertainty-based approach and allows us to uncover hidden EPR Bob’s system through the unitary evolution e−iHθ—see Fig. 1b. paradoxes in experimentally relevant scenarios. Our result The phase shift θ is complementary to the generating observable answers positively the question of whether steering can be a H and we show that the violation of resource in quantum sensing applications. 1 Var½θ ŠVar½H Š ≥ ð3Þ Results est est 4n Reid’s criterion for an EPR paradox.Wefirst recall some basic implies an EPR paradox and reveals steering from A to B. Here, fi de nitions by considering the following scenario (see Fig. 1a). Var[θest] describes the error of an arbitrary estimator for the Alice (A) performs a measurement on her subsystem and com- phase θ, constructed from local measurements by Alice and Bob municates her setting X and result a to Bob (B). Based on this on n copies of their state. Given any M, it is possible to construct θ ≫ information, Bob uses an estimator hest(a) to predict the result of an estimator est that achieves in the central limit (n 1) his subsequent measurement of H ¼ ∑ hhjihjh . The average h Var½M Š deviation between the prediction and Bob’s actual result h is given Var½θ Š¼ est : ð Þ ÀÁ est jh½ ; Ši j2 4 ½ Š :¼ ∑ ð ; j ; Þ ð ÞÀ 2 n H M ρB by Var Hest a;hp a h X H hest a h , often called the 3 ∣ inference variance , where p(a, h X, H) is the joint probability Essentially, we convert an n-sample average of Mest into an distribution for results a and h, conditioned on the measurement estimate of θ (see “Methods” fordetails).Forthisspecific estimation settings X and H. The procedure is repeated with different strategy, we thus recover the uncertainty-based formulation (1)of measurement settings Y and M, and Reid’s criterion2,3 for an EPR the EPR paradox from the more general expression (3). paradox consists of a violation of the local uncertainty limit In the following, we will derive our main result, which will allow us jh½ ; Ši j2 to prove the above statements. First note that the local phase shift acts ½ Š ½ Š ≥ H M ρB : ð1Þ Var Hest Var Mest 4 on Bob’s conditional quantum states but has no impact on Alice’s From the perspective of quantum information theory, the measurement statistics due to no-signalling, and thus produces the ; ρB ρB ÀiHθρB iHθ condition (1) plays the role of a witness for steering, but it may assemblage Aθða XÞ¼pðajXÞ ajX;θ,where ajX;θ ¼ e ajXe . not always succeed in revealing an EPR paradox. This implies the phase shift has no impact on the existence of LHS

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Fig. 1 Formulation of the EPR paradox as a metrological task. a In the standard EPR scenario, Alice’s measurement setting X (Y), and result a (b), leave B B Bob in the conditional quantum states ρajX (ρbjY ). Knowing Alice’s setting and result allows Bob to choose what measurement to perform on his state, and to make a prediction for the result. In an ideal scenario with strong quantum correlations, Alice’s measurement of X (Y) steers Bob into an eigenstate of his observable H (M), allowing him to predict the result with certainty. When H and M do not commute, this seems to contradict the complementarity principle. In practice, an EPR paradox is revealed whenever Bob’s predictions are precise enough to observe an apparent violation of Heisenberg’s uncertainty relation, see Eq. (1). b In our formulation of the EPR paradox as a metrological task, a local phase shift θ is generated by H on Bob’s state. Then, depending on Alice’s measurement setting and result, he decides whether to predict and measure H (as before), or to estimate θ from the measurement M. Here, Bob can choose the observable M as a function of Alice’s measurement result. The complementarity between θ and its generator H seems to be contradicted if the lower bound on their estimation errors, Eq. (3), is violated. This gives a metrological criterion for observing the EPR paradox. Since the metrological complementarity is sharper than the uncertainty-based notion, this approach leads to a tighter criterion to detect steering. Both results coincide in the special case when Bob estimates θ only from the observable M.

B models, and Aθða; XÞ¼∑λpðajX; λÞpðλÞσλ;θ.Withoutanyassis- following bound holds: tance from Alice, Bob’s precision of the estimation of θ is determined ρB BjA ; ≤ BjA ; : by his reduced density matrix θ . In this case, the error of an FQ ½A HŠ 4VarQ ½A HŠ ð6Þ θB θ arbitrary unbiased estimator est for is bounded by the quantum – θB ≥ ρB; À1 The proof (see “Methods”) primarily follows from the fact that Cramér Rao bound, Var½ estŠ ðnFQ½ HŠÞ , the central theorem – – ρ ρ of quantum metrology11 13,19 21,whereF [ρB, H]istheQFI.F [ρB, the QFI FQ[ , H] is a convex function of the state , while the Q Q ρ 22 ρB ≤ H] can be thought of intuitively as measuring the sensitivity of the variance Var[ , H] is instead concave . Note that FQ[ , H] ρB ρB state ρB to evolution generated by H;see“Methods” for a formal 4Var[ , H] holds for arbitrary , and by means of the – definition and explicit expression in terms of the eigenvectors and Cramér Rao bound implies the phase-generator complementar- eigenvalues of ρB. The quantum Cramér-Rao bound can be saturated ity relation by optimising both the estimator and the measurement observable21. In the assisted phase-estimation protocol, Fig. 1b, Alice θB ρB; ≥ 1 : Var½ estŠVar½ HŠ ð7Þ communicates to Bob her measurement setting and result, i.e. 4n X and a. This additional knowledge allows Bob to adapt the ρB This clearly shows how a violation of (3) implies an EPR choice of his observable as a function of the conditional state ajX ρB ; paradox. The result (6) has several important consequences that and to achieve the maximal sensitivity FQ½ ajX HŠ for an we discuss in the remainder of this article. estimation of θ. This way, he can attain an average sensitivity Useful steering for quantum metrology is identified by as large as the quantum conditional Fisher information correlations that violate the condition (6). We note that classical fi BjA ; : ∑ ρB ; : correlations between Alice and Bob may be suf cient for having FQ ½A HŠ ¼ max pðajXÞFQ½ ajX HŠ ð5Þ X a ρB; BjA ρAB; BjA ρAB; ρB; FQ½ HŠ

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; A ρAB A ≥ Comparison to Reid-type criteria. The metrological steering Aða XÞ¼TrA½EajX Š, where the EajX 0 form a positive condition (6) is stronger than standard criteria based on operator-valued measure (POVM) for the measurement setting – Heisenberg Robertson uncertainty relations. In fact, the lower X, normalised by ∑ EA ¼ 1A. The inequalities in (11) marked bound a ajX by (*) are saturated when ρAB is a pure state, assuming Alice is 2 jh½H; MŠiρB j “ ” ≤ BjA½ ; Š able to perform any quantum measurement (see Methods ). This j F A H ð8Þ VarB A½A; MŠ Q result is a consequence of the remarkable facts that the QFI is the Q convex roof of the variance27 while the variance is its own holds for arbitrary observables H, M and, besides no-signalling, concave roof22, in addition to Alice being able to steer Bob’s does not require assumptions about the assemblage A (see system into any pure-state ensemble for the local state ρB28. “Methods” for the proof). Hence, the bound (6) implies Reid’s We construct explicit measurement bases for Alice to achieve uncertainty-based condition (1) for all LHS models. In experi- steering in the optimal ensembles that saturate the above mentally relevant situations where the observables H and M are inequalities (Supplementary Note 5). We further observe that chosen as linear observables, such as quadrature measurements in the inequality (6), even with a fixed generator H, is capable of quantum optics or collective spins in atomic systems, the bound witnessing steering correlations for almost any pure state ψAB. (8) can be interpreted as a Gaussian approximation to the assisted More precisely, (6) is violated for any entangled ψAB whenever H sensitivity. In fact, violation of criterion (1) (choosing the is not constant on the support of the local state ρB. appropriate observables) is necessary and sufficient for steering of 4 Gaussian states by Gaussian measurements . The criterion (1)is Steering of GHZ states 23,24 . Let us illustrate our criterion with a also able to detect the steering of some non-Gaussian states , simple but relevant example. Consider a system composed of N + 1 but its ability to capture complex distributions is ultimately fi qubits, partitioned into a single control qubit (Alice) and the limited by only considering rst and second moments. The remaining N qubits on Bob’s side, that are prepared in a metrological approach thus provides particular advantages for the Greenberger–Horne–Zeilinger (GHZ) state of the form highly challenging problem of steering detection in non-Gaussian E ÀÁ quantum states. This is in analogy to the metrological detection of Nþ1 1 N iϕ N GHZϕ ¼ pffiffiffi ji0 ji0 þ e ji1 ji1 ; ð12Þ entanglement that is known to be significantly more efficient in 2 terms of the QFI instead of Gaussian quantifiers such as spin ji; ji σ squeezing coefficients13,16,25,26. where 0 1 are eigenstates of the Pauli matrix z.Wetake B ¼ 1 ∑ σðiÞ the local Hamiltonian Jz 2 i2B z ,wherethesumextends Bounds for specific measurements. Experimental tests of the over the particles on Bob’s side. When Alice measures her σ condition (6) are possible even without knowledge of the mea- qubit in the z basis, Bob attains the quantum conditional BjA½j Nþ1i; BŠ¼ surement settings that achieve the optimisations in Eqs. (5) and variance VarQ GHZϕ Jz 0. GHZ states have the fi 0 þ (2). Any xed choice of local measurement settings X and X for property29 jGHZN 1i¼p1ffiffi ðjþi jGHZN iþjÀi jGHZN iÞ, fi ϕ 2 ϕ ϕþπ Alice and Bob, respectively, provides a joint sensitivity quanti ed jiþ ; jiÀ σ AB½ ; ; 0Š where are eigenstates of x. This allows Alice to steer by the (classical) Fisher information F Aθ X X , and we ’ σ “ ” Bob s system into GHZ states by measuring in the x basis, and obtain the hierarchy of inequalities (see Methods for a proof) we obtain 1 ≤ AB½ ; ; 0Š ≤ A;B½ Š ≤ BjA½ ; Š; ð Þ hi E hi E hi E F Aθ X X FQ Aθ FQ A H 9 BjA Nþ1 1 N B N B 2 nVar½θ Š F GHZϕ ; J ¼ F GHZϕ ; J þ F GHZϕþπ ; J ¼ N : est Q z 2 Q z Q z A;B½ Š¼ AB½ ; ; 0Š ð Þ where FQ Aθ maxX;X0 F Aθ X X is the joint Fisher 13 information, maximised over local measurement settings. Simi- fi This measurement is optimal and achieves the maximum in (5) larly, any xed choice of X yields an upper bound on (2) and the ρ; B ≤ 2 12,13 inequalities since FQ½ Jz Š N holds for arbitrary quantum states . Steering is detected by the clear violation of the condition (6)forLHS BjA½ ; Š ≤ ∑ ð j Þ ½ρB ; Š ≤ ½ Š VarQ A H p a X Var ajX H Var Hest ð10Þ models. So far the only known criteria able to detect steering in a multipartite GHZ states are based on nonlocal observables that are saturated by an optimal measurement (2) and estimator, require individual addressing of the particles (see e.g. refs. 30,31), 3 respectively . These hierarchies reveal that any choice of local while our criterion is accessible by collective measurements. The measurement settings leads to experimentally observable bounds criterion is moreover robust to white noise: For a mixture for both sides of the inequality (6). They further show how the Nþ1 Nþ1 Nþ1 ρ ¼ pjGHZϕ ihGHZϕ jþð1 À pÞ1=2 ,usingthesame simpler condition (3) can be derived from (6). Note that a dif- BjA ρ; ≥ 2 2= = N ; ferent choice of setting X must be used for estimating θ or H in measurements we obtain FQ ½ JzŠ p N ½p þ 2ð1 À pÞ 2 Š BjA ρ; ≤ = N 2 ≫ −N order to observe any effect from steering correlations. Both par- 4VarQ ½ JzŠ ð1 À pÞN 2 þ pð1 À pÞN . Whenever p 2 , ties generally need to know which of the two settings is the criterion witnesses steering. See Supplementary Note 2 for being used. details and Supplementary Note 3 for an additional example involving a Schrödinger cat state. FBjA BjA Bounds on Q and VarQ . It is interesting to note that both sides of the inequality (6) respect the same upper and lower Steering of atomic split twin Fock states. As an example of bounds immediate practical relevance for state-of-the-art ultracold-atom ðÃÞ experiments, consider N/2 spin excitations symmetrically dis- ½ρB; Š ≤ BjA½ ; Š ≤ ½ρB; Š; FQ H FQ A H 4Var H tributed over N particles, i.e., a twin Fock state. Separating the ð11Þ ðÃÞ particles into two addressable modes A and B with a 50:50 beam ½ρB; Š ≤ BjA½ ; Š ≤ ½ρB; Š: FQ H 4VarQ A H 4Var H splitter results in a split twin Fock state STFN , which has been These inequalities hold for arbitrary assemblages A. generated experimentally32. Similar experiments based on When we can assume Alice’s system to be quantum, we obtain squeezed states were able to use Reid’s criterion to verify ρAB 33,34 h Bi ¼h Bi ¼ the assemblage A from the bipartite quantum state as steering , but the vanishing polarisation Jx ρB Jy ρB 0

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Phase estimation with multiple generators. Our result reveals the role of steering for generating probe states that are highly sensitive to the evolutions generated by a family of non- = … commuting generators, H (H1, , Hm). We focus on a sequential scenario, where in each experimental trial a single parameter is generated by one of the elements of H. Bob’s esti- mation of the phase is assisted by steering from Alice who picks different measurement settings Xi as a function of the acting Hamiltonian Hi and includes details of Hi in her communication to Bob. Achieving high sensitivity for multiple generators is relevant for multiparameter quantum metrology36–41, but the identification of a single measurement observable that is suitable for all parameters42,43 provides an additional complication that is not considered in our scenario. A suitable figure of merit for Bob’s average sensitivity is m BjA ; ∑ BjA ; : FQ ½A HŠ¼ FQ ½A HiŠ ð14Þ i¼1 Using the same techniques as for the main inequality (6), we find that any assemblage admitting a LHS model satisfies (see “Methods”) m BjA ; ≤ ∑ ϕ B; : FQ ½A HŠ max 4Var½ji HiŠ ð15Þ jiϕ B i¼1 An advantage over (6) is that the right-hand side is state- Fig. 2 EPR-assisted metrology with twin Fock states. a We consider a twin Fock state with N = 200 particles, that is split into two parts with independent. For a system B of dimension d, we can take the Hi N = N = N to be a set of d2 − 1 Hilbert–Schmidt orthonormal generators of A B /2, here represented by the Wigner function on the Bloch SU(d), and this bound simplifies to sphere. b The reduced state on either side is a mixture of Dicke states, resulting from tracing out the other half of the system. c The two BjA ; ≤ : ð16Þ subsystems show perfect correlations for both measurement settings Jx FQ ½A HŠ 4ðd À 1Þ A A and Jz: When Alice measures Jz (Jx ) and obtains the result kA,shesteers As a simple example, when Bob has a qubit (d = 2), we can take B B pffiffiffi Bob’ssystemintoaneigenstateofJz (Jx ) with eigenvalue N/2 − kA.This σ = ; ; ; the Pauli matrices as generators, Hi ¼ i 2 i ¼ x y z.Then can be used for assisted quantum metrology, and to reveal an EPR BjA ; ≤ ’ F ½ρB ; JBŠ (16) becomes F ½A HŠ 4. For a shared maximally entangled paradox. In the plot we show Bob ssensitivity Q k jJA z when Alice Q A x BjA A ; obtains the result kA from measuring Jx (blue line). Alice’sresultsareall state, this inequality is violated since FQ ½A HŠ¼6. To interpret pðk jJAÞ¼ =ðN þ Þ ’ these numbers, note that any pure qubit state on Bob’ssideis equally probable with A x 2 2 .Bobs average sensitivity FBjA½j i; JBŠ optimal for sensing rotations about two orthogonal axes (each of Q SDN;N=2 z coincides with the variance for the reduced state B B which contributes a QFI of 2), but useless for the remaining axis. 4Var½ρ ; Jz Š (yellow line), indicating that the measurement is optimal With a maximally entangled state, Alice can choose to steer Bob’s (Supplementary Note 4).

NATURE COMMUNICATIONS | (2021) 12:2410 | https://doi.org/10.1038/s41467-021-22353-3 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-22353-3 system into a state that is optimal for whichever axis has been such correlations can be useful for quantum-enhanced mea- chosen, thus sensing about any given axis is optimal. surement protocols, thus having the potential to enable new sensing applications in quantum technologies. Steering quantification. We may expect that the degree of vio- lation of (6), in a suitable sense, measures the amount of steering Methods ∣θ θ 2 R correlations. The proposed resource theory of steering7 gives a set Fisher information. For a probability distributionR p(x ) parameterisedÂà by , ½ ð jθފ :¼ ð jθÞ ∂ ð jθÞ 2 of criteria to be satisfied by a valid measure of steering. A general the classical Fisher information is F p x dxpx θln p x . The ρ ρ fi steering monotone S assigns a non-negative real number to each quantum version FQ[ θ] for a parameter-dependent state θ may be de ned as the maximum classical Fisher information associated with statistics obtained from any assemblage. First, we require (i) SðAÞ¼0 for every assemblage A ð jθÞ¼ ½ρ Š21 possible POVM {Ex} via p x Tr θE . In the case of unitary parameter − θ θ x with a LHS model. Next, (ii) S must be non-increasing (on encoding ρθ = e i Hρei H with a fixed generator H, the QFI is independent of θ,so ρ ψ average) when A is operated upon by local operations and one- we denote it by FQ[ , H]. This can be computed from the eigenvectors i and λ ρ way classical communication from Bob to Alice (1W-LOCC). eigenvalues i of : ð þ½ À 2 Finally, we may optionally require (iii) convexity: S pA1 1 ðλ À λ Þ 2 ½ρ; Š¼ ∑ i j hψ j jψ i : ð Þ pŠA Þ ≤ pSðA Þþð1 À pÞSðA Þ for any pair of assemblages FQ H 2 i H j 19 2 1 2 i;j: λ þλ ≠0 λ þ λ classically mixed with probability p. Here, we propose two i j i j potential quantifiers and address whether they satisfy these Proof of the main result. Suppose A is described by a LHS model, then criteria.  ð j ; λÞ ðλÞ One quantity is the maximum possible violation of (6), given BjA½ ; Š¼ ∑ ð j Þ ∑ p a X p σB; FQ A H max p a X FQ ð j Þ λ H the ability to vary the generator H. Since a rescaling of H → rH X a λ p a X 2 fi ≤ ∑ ∑ ð j ; λÞ ðλÞ ½σB; Š ð Þ scales the QFI and the variance by the same factor r ,we x the max p a X p FQ λ H 20 – 2 X a λ norm of H a convenient choice is to take Tr½H Š¼1. Then the B ¼ ∑ pðλÞF ½σλ ; HŠ; maximum violation of (6)is λ Q  ∑ ∣ λ = λ þ where we used the convexity of the QFI and ap(a X, ) 1, since and X are 1 BjA BjA 13,21 ρ ≤ ρ : ; ; ; independent. Making use of the upper bound FQ[ , H] 4Var[ , H] that holds SmaxðAÞ ¼ max FQ ½A HŠÀVarQ ½A HŠ ð17Þ ρ H;Tr½H2Š¼1 4 for arbitrary states , we obtain BjA½ ; Š ≤ ∑ ðλÞ ½σB; Š: þ FQ A H 4 p Var λ H ð21Þ where ½Šx ¼ maxf0; xg. For a bipartite pure quantum state ψAB, λ we have the easily computable formula (Supplementary Note 6) Moreover, following analogous steps, we obtain from the concavity of the ψAB λ T variance3 Smaxð Þ¼ max½diagðpÞÀpp Š, where p is the vector of ρB fi ψAB BjA ; ≥ ∑ λ σB; : eigenvalues of (equivalently, the Schmidt coef cients of ) Var ½A HŠ pð ÞVar½ λ HŠ ð22Þ λ Q λ and max denotes the largest eigenvalue. Alternatively, we can average over all H with Tr½H2Š¼1. Inserting (22) into (21) proves the result (6). fi Formally, this (rescaled) average is de ned by ’ θ Z Recovering Reid s criterion. The QFI describes the sensitivity for a parameter 1 þ generated by H that is achievable with an optimal measurement and estimation 2 μ BjA ; BjA ; ; fi SavgðAÞ ¼ ðd À 1Þ ðdnÞ FQ ½A n Á HŠÀVarQ ½A n Á HŠ strategy. By using a speci c estimator, constructed from the expectation value of 4 some observable M, one obtains the lower bound13,25 2 ð18Þ jh½H; MŠiρj F ½ρ; HŠ ≥ : ð23Þ μ Q Var½ρ; MŠ where Hi is any basis of orthonormal SU(d) generators, and is the uniform measure over the sphere of unit vectors ∣n∣ = 1. For Together with the Cauchy-Schwarz inequality, we obtain for all A ψAB ∑ 2 ; 2 pure states, we have Savgð Þ¼ i≠jpipjð1 þ p þp Þ. jh½H MŠiρB j i j BjA ; ≥ ∑ ajX FQ ½A HŠ max pðajXÞ B It follows immediately from (6) that both Smax and Savg satisfy X a Var½ρ ; MŠ fi ajX criterion (i). Moreover, we nd that both are faithful indicators ∑ ; 2 j apðajXÞh½H MŠiρB j ψAB ajX for pure states, meaning that they each vanish if and only if is ≥ max X ∑ pðajXÞVar½ρB ; MŠ separable. Convexity is also straightforward to prove (see a ajX ð24Þ ; 2 Supplementary Note 6 for all details). Criterion (ii) can be ruled jh½H MŠiρB j ¼ max X ∑ ρB ; out for Smax by again considering pure states: in this case, steering apðajXÞVar½ ajX MŠ correlations (as with all correlations) are equivalent to 2 jh½H; MŠi B j 4,44 ρ : entanglement . Smax is found not to be an entanglement ¼ BjA½A; Š monotone; nevertheless, it remains an important quantity to VarQ M consider if one is interested in observing the maximum possible Inserting (24) into (6) yields Reid’s criterion. The formulation (1) follows by using that Var½M Š ≥ VarBjA½A; MŠ for all M. violation. On the other hand, we prove that Savg is a pure state est Q In the case of a Gaussian quantum bipartite state with Gaussian measurements entanglement monotone. Thus it remains an open question by Alice, Eq. (24) can be saturated. First, note that the quadrature variances (in fact, whether S is in general a steering monotone. ρB 45 avg the whole covariance matrix) are identical for each conditional state ajX .A suitable pair of conjugate quadratures H, M can be chosen such that Eq. (23)is Discussion saturated26, thus the first inequality in (24) is saturated. For the second inequality, We formulated the EPR paradox in the framework of quantum note that all variances in the denominator are identical. We can also directly recover Reid’s criterion from the weaker condition (3)by metrology, showing that it can be interpreted as an apparent constructing a specific estimator from the measurement data b, m of Alice and Bob, violation of the complementarity relation between a local phase respectively. We assume that the dependence of the average value hMest À Miθ ¼ ∑ ∑ ; ; ; θ θ shift and its generator. This idea allowed us to derive a criterion b mpðb mjY M ÞðmestðbÞÀmÞ on is known from calibration, where ; ; ; θ ρB to detect EPR correlations which is based on the QFI, and thus pðb mjY M Þ¼pðbjYÞhjm bjY;θjim . Given a sample of n measurement results, stronger than known criteria based on the Heisenberg uncertainty the value of θ can now be estimated as the one that yields hM À Mi ¼ 1 ∑n ðm ðb ÞÀm Þ. Without the loss of generality we calibrate relation. We illustrated this with concrete examples of non- est θ n i¼1 est i i the estimator around the fixed value θ = 0, such that the estimator for m is Gaussian states in optical and atomic systems that are of 〈 〉 = 〈 〉 unbiased, i.e., Mest M θ=0 (any biased estimator would lead to a larger error). immediate interest for experimental studies. By expressing the θ = 1 The sample average evaluated at 0 has a variance of n Var½MestŠ. Note that only ’ θ EPR paradox as a metrological task, our results demonstrate that the distribution of Bob s results mi depends on , and therefore

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∂ ∂ ≫ j ∂θ hMest À Miθj¼j∂θ hMiθj. In the central limit (n 1), this strategy therefore Instead, we turn to the quantity (14). Without any assistance from Alice, the yields a sensitivity of best achievable precision would be m ρB; : ∑ ρB; : Var½MestŠ FQ½ HŠ ¼ FQ½ HiŠ ð31Þ Var½θ Š¼ ; i¼1 est ∂ 2 ð25Þ hMiθ fi n ∂θ Following the same technique as for a single parameter, any LHS model satis es BjA ; ≤ à which can be shown from a maximal likelihood analysis of the sample average FQ ½A HŠ FQ½HŠ distribution or from Gaussian error propagation46. We obtain the result Eq. (4). :¼ max F ½σB; HŠ σB Q ϕ B; ð32Þ Sensitivity for fixed local measurements fi ¼ max FQ½ji HŠ . For xed measurement settings X and ϕ B X0, respectively, the joint statistics of Alice and Bob are described by the probability ji 0 B B ð ; j ; ; θÞ¼ ð j Þ ½ 0 ρ Š 0 ¼ max ∑ 4Var½jiϕ ; H Š: distribution p a b X X p a X Tr EbjX ajX;θ where EbjX is a POVM B i jiϕ i describing the measurement X0. The Cramér–Rao bound The fact that pure states achieve the maximum on the right-hand side follows AB 0 from convexity of the QFI. This bound is of course only possible when the H are nVar½θ Š ≥ 1=F ½Aθ; X; X Š ð26Þ i est bounded. identifies the precision limit for any estimator that is constructed from the local Using the same techniques as for Savg (Supplementary Note 6), we can take Hi 0 2 − δ measurement results a and b and for any choice of X and X in terms of the Fisher to be a set of d 1 traceless generators of SU(d) satisfying Tr½HiHjŠ¼ i;j, and information à ϕ ϕ B; ϕ compute FQ½HŠ¼FQ½jihj HŠ¼4ðd À 1Þ (which actually holds for any ji). Thus, for this set of H in d dimensions, the LHS bound is ∂ 2 AB 0 0 0 ð Þ F ½Aθ; X; X Š¼∑ pða; bjX; X ; θÞ ln pða; bjX; X ; θÞ : 27 ; ≤ : a;b ∂θ FQ½A HŠ 4ðd À 1Þ ð33Þ For a pure state ψAB, A straightforward calculation reveals that BjA ψAB; ∑ ρB; FQ ½ HŠ¼ 4Var½ HiŠ AB ; ; 0 ∑ B 0 ρB ; i F ½Aθ X X Š¼ pðajXÞF ½X j ajX;θŠ ð28Þ ð34Þ a ¼ 4ðd À 1Þþ4 ∑ p p ; ≠ i j fi ’ i j i.e., for xed settings, the joint sensitivity coincides with Bob s average conditional 2 so that (33) is violated if and only if ψAB is entangled. B 0 ρB ∑ ρB ∂ ρB ’ sensitivity F ½X j ajX;θŠ¼ bTr½EbjX0 ajX;θŠ ∂θ ln Tr½EbjX0 ajX;θŠ since Alice s data is independent of θ. Maximising over the choice of measurement yields the hierarchy Data availability All relevant data are available from the authors. AB ; ; 0 ≤ ∑ B 0 ρB F ½Aθ X X Š max max apðajXÞF ½X j ajX;θŠ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}X X0 A;B FQ ½Aθ Š Code availability ≤ ∑ B 0 ρB Source codes of the plots are available from the authors upon request. max pðajXÞ max F ½X j ajX;θŠ ð29Þ X a |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}X0 F ½ρB ;HŠ Q ajX Received: 12 October 2020; Accepted: 11 March 2021; BjA ; : ¼ FQ ½A HŠ This completes the proof for the set of inequalities (9).

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