quantum metrology assisted by einstein-podolsky-rosen …sep 18, 2020 · quantum metrology...

Quantum metrology assisted by Einstein-Podolsky-Rosen steering

Benjamin Yadin,1, 2, ∗ Matteo Fadel,3, † and Manuel Gessner4, ‡1School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,

University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom2Wolfson College, University of Oxford, Linton Road, Oxford OX2 6UD, United Kingdom

3Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland4Laboratoire Kastler Brossel, ENS-Universit PSL, CNRS, Sorbonne Universit,

Collge de France, 24 Rue Lhomond, 75005, Paris, France(Dated: September 18, 2020)

The Einstein-Podolsky-Rosen (EPR) paradox plays afundamental role in our understanding of quantum me-chanics, and is associated with the possibility of predict-ing the results of non-commuting measurements with aprecision that seems to violate the uncertainty principle.This apparent contradiction to complementarity is madepossible by nonclassical correlations stronger than entan-glement, called steering. Quantum information recognisessteering as an essential resource for a number of tasks but,contrary to entanglement, its role for metrology has so farremained unclear. Here, we formulate the EPR paradox inthe framework of quantum metrology, showing that it en-ables the precise estimation of a local phase shift and of itsgenerating observable. Employing a stricter formulationof quantum complementarity, we derive a criterion basedon the quantum Fisher information that detects steering ina larger class of states than well-known uncertainty-basedcriteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to un-cover steering of non-Gaussian states in state-of-the-artexperiments.

In their seminal 1935 paper1, EPR presented a scenariowhere the position and momentum of one quantum system(B) can both be predicted with certainty from local measure-ments of another remote system (A). Based on this apparentviolation of the uncertainty principle, in 1989 Reid formulatedthe first practical criterion for an EPR paradox2, which has en-abled numerous experimental observations3: Steering from Ato B is revealed when measurement results of A allow to pre-dict the measurement results of B with errors that are smallerthan the limit imposed by the Heisenberg-Robertson uncer-tainty relation for B. More generally, an EPR paradox impliesthe failure of any attempt to describe the correlations betweenthe two systems in terms of classical probability distributionsand local quantum states for B, known as local hidden state(LHS) models, as was shown by Wiseman et al. in 2007 usingthe framework of quantum information theory4. Aside fromits fundamental interest, steering is recognised as an essentialresource for quantum information tasks5, such as one-sideddevice-independent quantum key distribution6,7 and quantumchannel discrimination8.

∗ [email protected]† [email protected]‡ [email protected]

Uncertainty relations describe the complementarity of non-commuting observables, but the complementarity principleapplies more generally to notions that are not necessarilyassociated with an operator. In this work we formulate asteering condition in terms of the complementarity of a phaseshift θ and its generating Hamiltonian H, using information-theoretic tools from quantum metrology9–12. We express oursteering condition in terms of the quantum Fisher information(QFI), the central tool for quantifying the precision of quan-tum parameter estimation. Besides its fundamental relevancefor quantum-enhanced precision measurements, the QFI is ofgreat interest for the characterisation of quantum many-bodysystems13,14 and gives rise to an efficient and experimentallyaccessible witness for multipartite entanglement11,12,15, butso far, its relation to steering has remained elusive. The moregeneral phase-generator complementarity principle repro-duces the Heisenberg-Robertson uncertainty relation in thespecial case where the phase is estimated from an observableM. Therefore, our metrological criterion is stronger than theuncertainty-based approach and allows us to uncover hiddenEPR paradoxes in experimentally relevant scenarios. It hasbeen a long-standing open question of whether quantumcorrelations stronger than entanglement, such as steeringor Bell correlations, play a role in metrology16. Our resultanswers this question positively for the case of steering, byidentifying it as a resource in quantum sensing applications.

Reid’s criterion for an EPR paradox. We first recallsome basic definitions by considering the following scenario(see Fig. 1a). Alice (A) performs on her subsystem a mea-surement and communicates her setting X and result a toBob (B). Based on this information, Bob uses an estimatorhest(a) to predict the result of his subsequent measurementof H =

∑h h|h〉〈h|. The average deviation between the pre-

diction and Bob’s actual result h is given by Var[Hest] :=∑a,h p(a, h|X,H) (hest(a) − h)2, often called the inference vari-

ance3, where p(a, h|X,H) is the joint probability distributionfor results a and h, conditioned on the measurement settings Xand H. The procedure is repeated with different measurementsettings Y and M, and Reid’s criterion2,3 for an EPR paradoxconsists of a violation of the local uncertainty limit

Var[Hest]Var[Mest] ≥|〈[H,M]〉ρB |2

4. (1)

From the perspective of quantum information theory, the con-dition (1) plays the role of a witness for steering, but it may

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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 Bob in the conditional quantum states ρBa|X(ρBb|Y ). Knowing Alice’s setting and result allows Bob to choose whatmeasurement to perform on his state, and to make a prediction for theresult. In an ideal scenario with strong quantum correlations, Alice’smeasurement of X (Y) steers Bob into an eigenstate of his observableH (M), allowing him to predict the result with certainty. When Hand M do not commute, this seems to contradict the complementarityprinciple. In practice, an EPR paradox is revealed whenever Bob’spredictions are precise enough to observe an apparent violation ofHeisenberg’s uncertainty relation, see Eq. (1). b) In our formulationof the EPR paradox as a metrological task, a local phase shift θ isgenerated by H on Bob’s state. Then, depending on Alice’s measure-ment setting and result, he decides whether to predict and measureH (as before), or to estimate θ from the measurement M. Here, Bobcan choose the observable M as a function of Alice’s measurementresult. The complementarity between θ and its generator H seems tobe contradicted if the lower bound on their estimation errors, Eq. (3),is violated. This gives a metrological criterion for observing the EPRparadox. Since the metrological complementarity is sharper than theuncertainty-based notion, this approach leads to a tighter criterion todetect steering. Both results coincide in the special case when Bobestimates θ only from the observable M.

not always succeed in revealing an EPR paradox.The most general way to formally model the joint statis-

tics p(a, h|X,H) is offered by the formalism of assemblages,i.e. functions A(a, X) = p(a|X)ρBa|X that map any possibleresult a of Alice’s measurement of X to a local probabilitydistribution p(a|X) and a (normalised) conditional quantumstate ρBa|X for Bob’s subsystem

17. This description avoids theneed to make assumptions about the nature of Alice’s system,which is key to one-sided device-independent quantum infor-mation processing5–7. We only impose a no-signalling condi-tion which requires that

∑aA(a, X) = ρB for all X, where

ρB is the reduced density matrix of Bob’s system. Basedon the assemblage A, the joint statistics are described asp(a, h|X,H) = p(a|X)〈h|ρBa|X |h〉.

The EPR paradox can now be formally defined as an ob-servation that rules out the possibility of modelling an as-semblage by a local hidden state (LHS) model. In such amodel, a classical random variable λ with probability distri-bution p(λ) determines both Alice’s statistics and Bob’s localstate: A(a, X) = ∑λ p(a|X, λ)p(λ)σBλ . Inequality (1) holdsfor arbitrary estimators and measurement settings whenever aLHS model exists. The sharpest formulation of Eq. (1) is thusobtained by optimising these choices to minimise the estima-tion error. The optimal estimator hest(a) = Tr{ρBa|XH} attainsthe lower bound3 Var[Hest] ≥ ∑a p(a|X)Var[ρBa|X ,H], whereVar[ρ,H] = 〈H2〉ρ − 〈H〉2ρ is the variance with 〈O〉ρ = Tr{ρO}.Optimising over Alice’s measurement setting X leads to thequantum conditional variance

VarB|AQ [A,H] := minX∑

a

p(a|X)Var[ρBa|X ,H], (2)

and the optimised version of Reid’s condition (1) readsVarB|AQ [A,H]VarB|AQ [A,M] ≥ |〈[H,M]〉ρB |2/4. Theuncertainty-based detection of the EPR paradox is based onthe fact that Alice’s choice of measurement can steer Bob’ssystem into conditional states that have small variances foreither one of the two non-commuting observables H and M.

EPR-assisted metrology. To express quantum mechanicalcomplementarity in the framework of quantum metrology9–12,we assume that the observable H imprints a local phase shiftθ on Bob’s system through the unitary evolution e−iHθ – seeFig. 1b. The phase shift θ is complementary to the generatingobservable H and we show that the violation of

Var[θest]Var[Hest] ≥ 14n (3)

implies an EPR paradox and reveals steering from A to B.Here, Var[θest] describes the error of an arbitrary estimator forthe phase θ, constructed from local measurements by Aliceand Bob on n copies of their state. Given any M, it is possibleto construct an estimator θest that achieves

Var[θest] =Var[Mest]

n|〈[H,M]〉ρB |2. (4)

For this specific estimation strategy, we thus recover theuncertainty-based formulation (1) of the EPR paradox fromthe more general expression (3).

In the following, we will derive our main result, which willallow us to prove the above statements. First note that thelocal phase shift acts on Bob’s conditional quantum statesbut has no impact on Alice’s measurement statistics due tono-signalling, and thus produces the assemblage Aθ(a, X) =p(a|X)ρBa|X,θ, where ρBa|X,θ = e−iHθρBa|XeiHθ. This implies thephase shift has no impact on the existence of LHS models,and Aθ(a, X) = ∑λ p(a|X, λ)p(λ)σBλ,θ. Without any assistancefrom Alice, Bob’s precision of the estimation of θ is deter-mined by his reduced density matrix ρBθ . In this case, the er-ror of an arbitrary unbiased estimator θBest for θ is bounded bythe quantum Cramér-Rao bound, Var[θBest] ≥ (nFQ[ρB,H])−1,the central theorem of quantum metrology10–12,18–20, where

3

FQ[ρB,H] is the QFI. The quantum Cramér-Rao bound canbe saturated by optimising both the estimator and the mea-surement observable18.

In the assisted phase-estimation protocol, Fig. 1b, Alicecommunicates to Bob her measurement setting and result, i.e.X and a. This additional knowledge allows Bob to adapt thechoice of his observable as a function of the conditional stateρBa|X and to achieve the maximal sensitivity FQ[ρ

Ba|X ,H] for an

estimation of θ. This way, he can attain an average sensitivityas large as the quantum conditional Fisher information

FB|AQ [A,H] := maxX∑

a

p(a|X)FQ[ρBa|X ,H]. (5)

As the main result of our paper, we show that in the absenceof steering the quantum conditional Fisher information (5) isalways bounded from above in terms of the quantum condi-tional variance (2): For any assemblage A that admits a LHSmodel, the following bound holds (see Methods):

FB|AQ [A,H] ≤ 4VarB|AQ [A,H]. (6)

Note that FQ[ρB,H] ≤ 4Var[ρB,H] holds for arbitrary ρB,and by means of the Cramér-Rao bound implies the phase-generator complementarity relation

Var[θBest]Var[ρB,H] ≥ 1

4n. (7)

This clearly shows how a violation of (3) implies an EPR para-dox. The result (6) has several important consequences thatwe discuss in the remainder of this article.

Useful steering for quantum metrology is identifiedby correlations that violate the condition (6). We notethat classical correlations between Alice and Bob maybe sufficient for having FQ[ρB,H] < F

B|AQ [ρ

AB,H] andVarB|AQ [ρ

AB,H] < Var[ρB,H]. This shows that assistanceis useful even in the absence of steering to improve theestimation precision for θ and H, but only with steering canthe limit defined by quantum mechanical complementarity (6)be overcome.

Comparison to Reid-type criteria. The metrologicalsteering condition (6) is stronger than standard criteria basedon Heisenberg-Robertson uncertainty relations. In fact, thelower bound

|〈[H,M]〉ρB |2VarB|AQ [A,M]

≤ FB|AQ [A,H] (8)

holds for arbitrary observables H,M and, besides no-signalling, does not require assumptions about the assemblageA (see Methods for the proof). Hence, the bound (6) impliesReid’s uncertainty-based condition (1) for all LHS models. Inexperimentally relevant situations where the observables Hand M are chosen as linear observables, such as quadraturemeasurements in quantum optics or collective spins in atomicsystems, the bound (8) can be interpreted as a Gaussianapproximation to the assisted sensitivity. In fact, violation

of criterion (1) (choosing the appropriate observables) isnecessary and sufficient for steering of Gaussian states byGaussian measurements4. The metrological approach thusprovides particular advantages for the highly challengingproblem of steering detection in non-Gaussian quantumstates. This is in analogy to the metrological detection ofentanglement that is known to be significantly more efficientin terms of the QFI instead of Gaussian quantifiers such asspin squeezing coefficients12,15,21,22.

Bounds for specific measurements. Experimental testsof the condition (6) are possible even without knowledge ofthe measurement settings that achieve the optimisations inEqs. (5) and (2). Any fixed choice of local measurement set-tings X and X′ for Alice and Bob, respectively, provides ajoint sensitivity quantified by the (classical) Fisher informa-tion FAB[Aθ, X, X′], and we obtain the hierarchy of inequali-ties (see Methods for a proof)

1nVar[θest]

≤ FAB[Aθ, X, X′] ≤ FA,BQ [Aθ] ≤ FB|AQ [A,H], (9)

where FA,BQ [Aθ] = maxX,X′ FAB[Aθ, X, X′] is the joint Fisherinformation, maximised over local measurement settings.Similarly, any fixed choice of X yields an upper bound on (2)and the inequalities

VarB|AQ [A,H] ≤∑

a

p(a|X)Var[ρBa|X ,H] ≤ Var[Hest] (10)

are saturated by an optimal measurement (2) and estimator,respectively3. These hierarchies reveal that any choice oflocal measurement settings leads to experimentally observ-able bounds for both sides of the inequality (6). They furthershow how the simpler condition (3) can be derived from (6).Note that a different choice of setting X must be used forestimating θ or H in order to observe any effect from steeringcorrelations. Both parties generally need to know which ofthe two settings is being used.

Bounds on FB|AQ and VarB|AQ . It is interesting to note that

both sides of the inequality (6) respect the same upper andlower bounds

FQ[ρB,H] ≤ FB|AQ [A,H](∗)≤ 4Var[ρB,H],

FQ[ρB,H](∗)≤ 4VarB|AQ [A,H] ≤ 4Var[ρB,H]. (11)

These inequalities hold for arbitrary assemblagesA.When we can assume Alice’s system to be quantum, we ob-

tain the assemblageA from the bipartite quantum state ρAB asA(a, X) = TrA[EAa|XρAB], where the EAa|X ≥ 0 form a positiveoperator-valued measure (POVM) for the measurement set-ting X, normalised by

∑a EAa|X = 1

A. The inequalities in (11)marked by (∗) are saturated when ρAB is a pure state, assumingAlice is able to perform any quantum measurement (see Meth-ods). This result is a consequence of the remarkable facts thatthe QFI is the convex roof of the variance23 while the varianceis its own concave roof24, in addition to Alice being able to

4

steer Bob’s system into any pure-state ensemble for the localstate ρB.25

We construct explicit measurement bases for Alice toachieve steering in the optimal ensembles that saturate theabove inequalities (Supplementary Section IV). We furtherobserve that the inequality (6), even with a fixed generatorH, is capable of witnessing steering correlations for almostany pure state ψAB. More precisely, (6) is violated for anyentangled ψAB whenever H is not constant on the support ofthe local state ρB.

Steering of GHZ states. Let us illustrate our criterion witha simple but relevant example. Consider a system composedof N + 1 qubits, partitioned into a single control qubit (Alice)and the remaining N qubits on Bob’s side, that are prepared ina Greenberger-Horne-Zeilinger (GHZ) state of the form

|GHZN+1φ 〉 =1√2

(|0〉 ⊗ |0〉⊗N + eiφ |1〉 ⊗ |1〉⊗N

), (12)

where |0〉 , |1〉 are eigenstates of the Pauli matrix σz. We takethe local Hamiltonian JBz =

12∑

i∈B σ(i)z , where the sum ex-

tends over the particles on Bob’s side. When Alice measuresher qubit in the σz basis, Bob attains the quantum conditionalvariance VarB|AQ [|GHZN+1φ 〉 , JBz ] = 0. GHZ states have theproperty26 |GHZN+1φ 〉 = 1√2

(|+〉 ⊗ |GHZNφ 〉 + |−〉 ⊗ |GHZNφ+π〉

),

where |+〉 , |−〉 are eigenstates of σx. This allows Alice to steerBob’s system into GHZ states by measuring in the σx basis,and we obtain

FB|AQ [|GHZN+1φ 〉 , Jz] (13)

=12

(FQ[|GHZNφ 〉 , JBz ] + FQ[|GHZNφ+π〉 , JBz ]

)= N2.

This measurement is optimal and achieves the maximumin (5) since FQ[ρ, JBz ] ≤ N2 holds for arbitrary quantumstates11,12. Steering is detected by the clear violation ofthe condition (6) for LHS models. So far the only knowncriteria able to detect steering in multipartite GHZ statesare based on nonlocal observables that require individ-ual addressing of the particles27,28, while our criterionis accessible by collective measurements. The crite-rion is moreover robust to white noise: For a mixtureρ = p |GHZN+1φ 〉〈GHZN+1φ | + (1 − p)1/2N+1, using the samemeasurements we obtain FB|AQ [ρ, Jz] ≥ p2N2/[p + 2(1 −p)/2N], 4VarB|AQ [ρ, Jz] ≤ (1− p)N/2N + p(1− p)N2. Wheneverp � 2−N , the criterion witnesses steering (SupplementarySection II).

Steering of atomic split twin Fock states. As an exampleof immediate practical relevance for state-of-the-art ultracold-atom experiments, consider N/2 spin excitations symmetri-cally distributed over N particles, i.e. a twin Fock state. Sepa-rating the particles into two addressable modes A and B with a50 : 50 beam splitter results in a split twin Fock state |STFN〉,which has been generated experimentally29. Similar experi-ments based on squeezed states were able to use Reid’s crite-rion to verify steering30,31, but the vanishing polarisation

a

b

c

y

A B

measured

measured

{steering

steering

z

x

kA = 70

kA = 40

kA = 90

kA = 0

JAz

JAx

kA = 90

4Var[ρB, JBz ]

FQ[ρBkA |JAz , JBz ]

FIG. 2. EPR-assisted metrology with twin Fock states. a) We con-sider a twin Fock state with N = 200 particles, that is split into twoparts with NA = NB = N/2, here represented by the Wigner functionon the Bloch sphere. b) The reduced state on either side is a mix-ture of Dicke states, resulting from tracing out the other half of thesystem. c) The two subsystems show perfect correlations for bothmeasurement settings Jx and Jz: When Alice measures JAz (J

Ax ) and

obtains the result kA, she steers Bob’s system into an eigenstate of JBz(JBx ) with eigenvalue N/2−kA. This can be used for assisted quantummetrology, and to reveal an EPR paradox. In the plot we show Bob’ssensitivity FQ[ρBkA |JAx

, JBz ] when Alice obtains the result kA from mea-

suring JAx (blue line). Alice’s results are all equally probable withp(kA|JAx ) = 2/(N + 2). Bob’s average sensitivity FB|AQ [|SDN,N/2〉 , JBz ]coincides with the variance for the reduced state 4Var[ρB, JBz ] (yel-low line), indicating that the measurement is optimal (SupplementarySection III).

5

〈JBx 〉ρB = 〈JBy 〉ρB = 0 makes this impossible for split twinFock states and so far only the entanglement between A andB could be detected29. We show that the criterion (6) success-fully reveals the EPR steering of split twin Fock states whenAlice measures local spin observables JAx , J

Az and a phase shift

θ is generated by JBz . We obtain VarB|AQ [|STFN〉 , JBz ] = 0 and

FB|AQ [|STFN〉 , JBz ] = N/4, leading to a violation of (6) thatscales linearly with N. This value is limited by the partitionnoise that is introduced by the beam splitter which generatesbinomial fluctuations of the particle number in each mode.

To overcome this limit, we propose the following al-ternative preparation of split Dicke states. Consider twoaddressable groups of N/2 atoms each. A collective measure-ment of the total number k of spin excitations projects thesystem into a split Dicke state |SDN,k〉 without partition noise.This can be realised, e.g., with arrays of cold atoms in acavity32. Using the same settings for Alice and Bob as before,these states still yield VarB|AQ [|SDN,k〉 , JBz ] = 0 while leading tosignificantly larger values of the quantum conditional Fisherinformation, and for the twin Fock case, k = N/2, we obtainthe quadratic scaling FB|AQ [|SDN,N/2〉 , JBz ] = N(N + 4)/12; seeFig. 2. For details on arbitrary split Dicke states with andwithout partition noise, see Supplementary Section III.

Multiparameter estimation. Our result reveals the role ofsteering for quantum-enhancements in the estimation of mul-tiple parameters θ = (θ1, . . . , θm) that are imprinted by a fam-ily of non-commuting generators, H = (H1, . . . ,Hm). Mul-tiparameter quantum metrology has recently drawn increasedattention due to the possibility of achieving a collective quan-tum gain from the simultaneous estimation of a set of param-eters33–38. So far most results are limited to the case of com-muting generators because it is generally not known how toidentify a single measurement that is optimal to extract mul-tiple parameters39,40. Here we avoid this problem by consid-ering a sequential scenario, where in each experimental trialonly a single parameter is estimated. The challenge consists ofpreparing a quantum state that is simultaneously sensitive tothe evolutions generated by a set H of non-commuting Hamil-tonians. To achieve this, Bob is assisted by steering from Al-ice who picks different measurement settings Xi as a functionof the acting Hamiltonian Hi. Alice includes in her communi-cation to Bob details of which parameter θi is to be estimated.

A suitable figure of merit for Bob’s average sensitivity is

FB|AQ [A,H] =m∑

i=1

FB|AQ [A,Hi]. (14)

Using the same techniques as for the main inequality (6), wefind that any assemblage admitting a LHS model satisfies (seeMethods)

FB|AQ [A,H] ≤ max|φ〉Bm∑

i=1

4Var[|φ〉〈φ|B,Hi]. (15)

An advantage over (6) is that the right-hand side is state-independent. For a system B of dimension d, we can take

the Hi to be a set of d2 − 1 Hilbert-Schmidt orthonormal gen-erators of SU(d), and this bound simplifies to

FB|AQ [A,H] ≤ 4(d − 1). (16)As a simple example, when Bob has a qubit (d = 2), we cantake the Pauli matrices as generators, Hi = σi/

√2, i = x, y, z.

Then (16) becomes FB|AQ [A,H] ≤ 4. For a shared maximallyentangled state, this inequality is violated since FB|AQ [A,H] =6. To interpret these numbers, note that any pure qubit state onBob’s side is optimal for sensing rotations about two orthogo-nal axes (each of which contributes a QFI of 2), but useless forthe remaining axis. With a maximally entangled state, Alicecan choose to steer Bob’s system into a state that is optimalfor whichever axis has been chosen, thus sensing about anygiven axis is optimal.

For continuous variables, a fully state-independent bound isnot possible, but we can instead assume a finite mean particlenumber N̄ = 〈N〉ρB . When B is a single mode, we may take apair of conjugate quadratures H = (x, p); then any LHS modelsatisfies

FB|AQ [A,H] ≤ 8N̄. (17)This can be violated by a two-mode squeezed state, forexample. Pick such a state in which the quadratures(xA − xB)/√2, (pA + pB)/√2 have variance 1/(2s2), and(xA + xB)/

√2, (pA− pB)/√2 have variance s2/2, where s ≥ 1.

The upper bound in (17) becomes 2(s2 + s−2), but we find

FB|AQ [A,H] = 4s2. Thus there is a violation for any nonzerosqueezing s > 1.

Steering quantification. Let us finally discuss how ourcriterion can be converted into a quantifier of steering. Oneway to achieve this is via the maximum possible violation of(6), given the ability to vary the generator H. Since a rescal-ing of H → rH scales the QFI and the variance by the samefactor r2, we fix the norm of H – a convenient choice is to takeTr[H2] = 1. Then the maximum violation of (6) is

Smax(A) := maxH,Tr[H2]=1

[FB|AQ [A,H] − 4VarB|AQ [A,H]

]+, (18)

where [x]+ = max{0, x}. For a bipartite pure quantum stateψAB, we have the easily computable formula (SupplementarySection V) Smax(ψAB) = 4λmax[diag(p) − ppT ], where p is thevector of eigenvalues of ρB (equivalently, the Schmidt coeffi-cients of ψAB) and λmax denotes the largest eigenvalue.

Alternatively, we can average over all H with Tr[H2] = 1.Formally, this (rescaled) average is defined by

Savg(A) = (d2−1)[∫

µ(dn)FB|AQ [A,n ·H] − 4VarB|AQ [A,n ·H]]+,

(19)where Hi is any basis of orthonormal SU(d) generators, and µis the uniform measure over the sphere of unit vectors |n| = 1.For pure states, we have Savg(ψAB) = 4 ∑i, j pi p j (1 + 2pi+p j ).

6

In the pure state case, steering correlations (as with all cor-relations) are equivalent to entanglement. We find that bothquantities Smax,Savg are indeed faithful indicators of entan-glement for pure states, meaning that they each vanish if andonly if ψAB is separable. However, only Savg is found to bea fully valid entanglement measure, being monotonically de-creasing under local operations and classical communication(LOCC). This brings up the question of whether Savg is a validmeasure in the resource theory of steering based on one-wayLOCC6.

METHODS

Fisher information. For a probability distribution p(x|θ)parameterised by θ ∈ R, the classical Fisher informationis F[p(x|θ)] :=

∫dx p(x|θ) [∂θ ln p(x|θ)]2. The quantum

version for a parameter-dependent state ρθ can be defined byFQ[ρθ] = Tr[ρθL2θ], where the symmetric logarithmic deriva-tive Lθ is defined implicitly through ∂θρθ = (Lθρθ + ρθLθ)/29.In the case of unitary parameter encoding ρθ = e−iθHρeiθH viaa fixed generator H, the QFI is independent of θ, so we denoteit by FQ[ρ,H]. Note that the QFI is equal to the maximalclassical FI that can be obtained from the statistics resultingfrom any possible POVM18.

Proof of the main result. Suppose A is described by aLHS model, then

FB|AQ [A,H] = maxX∑

a

p(a|X)FQ∑λ

p(a|X, λ)p(λ)p(a|X) σ

Bλ ,H

≤ max

X

∑λ

p(a|X, λ)p(λ)FQ[σBλ ,H]

=∑λ

p(λ)FQ[σBλ ,H], (20)

where we used the convexity of the QFI and∑

a p(a|X, λ) = 1,since λ and X are independent. Making use of the upperbound12,18 FQ[ρ,H] ≤ 4Var[ρ,H] that holds for arbitrarystates ρ, we obtain

FB|AQ [A,H] ≤ 4∑λ

p(λ)Var[σBλ ,H]. (21)

Moreover, following analogous steps, we obtain from the con-cavity of the variance3

VarB|AQ [A,H] ≥∑λ

p(λ)Var[σBλ ,H]. (22)

Inserting (22) into (21) proves the result (6).

Recovering Reid’s criterion. The QFI describes the sensi-tivity for a parameter θ generated by H that is achievable withan optimal measurement and estimation strategy. By using aspecific estimator, constructed from the expectation value ofsome observable M, one obtains the lower bound12,21

FQ[ρ,H] ≥|〈[H,M]〉ρ|2Var[ρ,M]

. (23)

Together with the Cauchy-Schwarz inequality, we obtain forallA

FB|AQ [A,H] ≥ maxX∑

a

p(a|X)|〈[H,M]〉ρBa|X |2Var[ρBa|X ,M]

≥ maxX

|∑a p(a|X) 〈[H,M]〉ρBa|X |2∑a p(a|X)Var[ρBa|X ,M]

= maxX

|〈[H,M]〉ρB |2∑a p(a|x)Var[ρBa|X ,M]

=|〈[H,M]〉ρB |2VarB|AQ [A,M]

. (24)

Inserting (24) into (6) yields Reid’s criterion. The formula-tion (1) follows by using that Var[Mest] ≥ VarB|AQ [A,M] for allM.

We can also directly recover Reid’s criterion from theweaker condition (3) by constructing an specific estimatorfrom the measurement data b,m of Alice and Bob, respec-tively. We assume that the dependence of the average value〈Mest−M〉θ = ∑b ∑m p(b,m|Y,M, θ)(mest(b)−m) on θ is knownfrom calibration, where p(b,m|Y,M, θ) = p(b|Y)〈m|ρBb|Y,θ|m〉.Given a sample of n measurement results, the value of θcan now be estimated as the one that yields 〈Mest − M〉θ =1n∑n

i=1(mest(bi) − mi). Without loss of generality we cali-brate the estimator around the fixed value θ = 0, such thatthe estimator for m is unbiased, i.e. 〈Mest〉 = 〈M〉θ=0 (anybiased estimator would lead to a larger error). The sampleaverage evaluated at θ = 0 has a variance of 1n Var[Mest].Note that only Bob’s results mi depend on θ, and therefore| ∂∂θ〈Mest − M〉θ| = | ∂∂θ 〈M〉θ|. In the central limit (n � 1), this

strategy therefore yields a sensitivity of

Var[θest] =Var[Mest]

n∣∣∣ ∂〈M〉θ

∂θ

∣∣∣2 , (25)giving the result Eq. (4).

Sensitivity for fixed local measurements. For fixed mea-surement settings X and X′, respectively, the joint statisticsof Alice and Bob are described by the probability distributionp(a, b|X, X′, θ) = p(a|X)Tr{Eb|X′ρBa|X,θ} where Eb|X′ is a pos-itive operator-valued measure (POVM) describing the mea-surement X′. The Cramér-Rao bound

nVar[θest] ≥ 1/FAB[Aθ, X, X′] (26)identifies the precision limit for any estimator that is con-structed from the local measurement results a and b and forany choice of X and X′ in terms of the Fisher information

FAB[Aθ, X, X′] =∑a,b

p(a, b|X, X′, θ)(∂

∂θlog p(a, b|X, X′, θ)

)2.

(27)

A straightforward calculation reveals that

FAB[Aθ, X, X′] =∑

a

p(a|X)FB[X′|ρBa|X,θ], (28)

7

i.e. for fixed settings, the joint sensitivity coincideswith Bob’s average conditional sensitivity FB[X′|ρBa|X,θ] =∑

b Tr{Eb|X′ρBa|X,θ}(∂∂θ

log Tr{Eb|X′ρBa|X,θ})2

since Alice’s data isindependent of θ. Maximising over the choice of measure-ment yields the hierarchy

FAB[Aθ, X, X′] ≤ maxX

maxX′

∑a

p(a|X)FB[X′|ρBa|X,θ]︸︷︷︸FA,BQ [Aθ]

≤ maxX

∑a

p(a|X) maxX′

FB[X′|ρBa|X,θ]︸︷︷︸FQ[ρBa|X ,H]

= FB|AQ [A,H]. (29)

This completes the proof for the set of inequalities (9).

Metrological steering for bipartite quantum states. Letus first note that if Alice’s system is quantum, the optimalmeasurements in (2) and (5) can always be implemented byrank-1 POVMs. This follows from the convexity of the QFIand the concavity of the variance (Supplementary Section I).

Now suppose that ρAB is pure. Since the optimal POVM forFB|AQ is rank-1, the corresponding conditional states ρ

Ba|X are

pure. An important fact about bipartite pure states is that anypure-state ensemble on Bob’s side (consistent with the averagestate ρB) may be realised by an appropriate rank-1 POVM onAlice’s side25. Thus the optimisation can be reduced to

FB|AQ [ρAB,H] = max

{p(a), |φa〉}a :∑a p(a)|φa〉〈φa |=ρB

∑a

p(a)FQ[|φa〉〈φa| ,H]

= max{p(a), |φa〉}a :∑

a p(a)|φa〉〈φa |=ρB4∑

a

p(a)Var[|φa〉〈φa| ,H]

= 4Var[ρB,H]. (30)

In the last line we used that the variance is its own concaveroof24. For VarB|AQ the minimisation is the same as taking theconvex roof, resulting in23 FQ[ρB,H]. Hence, for a pure stateρAB, we obtain the equalities FB|AQ [ρ

AB,H] = 4Var[ρB,H] and4VarB|AQ [ρ

AB,H] = FQ[ρB,H]. For arbitrary assemblages,we obtain the upper bounds FB|AQ [A,H] ≤ 4Var[ρB,H]and 4VarB|AQ [A,H] ≥ FQ[ρB,H] as a consequence ofconvexity of the QFI, concavity of the variance, andFQ[ρ,H] ≤ 4Var[ρ,H]. For the same reason, we obtain thatFB|AQ [A,H] ≥ FQ[ρB,H] and VarB|AQ [A,H] ≤ Var[ρB,H] forarbitrary assemblages A, including those obtained from ρAB.This concludes the proof of (11).

Multiple parameters. One can ask whether there is a (po-tentially weaker) steering witness involving only the QFI. Itis clear that the right-hand side of (6) cannot be made state-independent: the best one can do is to replace VarB|AQ [A,H]by maxσ Var[σ,H], leading to an inequality that holds for allcases, even non-steerable.

Instead, we turn to the quantity (14). Without any assistancefrom Alice, the best achievable precision would be

FQ[ρB,H] :=m∑

i=1

FQ[ρB,Hi]. (31)

Following the same technique as for a single parameter, anyLHS model satisfies

FB|AQ [A,H] ≤ F∗Q[H]:= max

σBFQ[σB,H]

= max|φ〉B

FQ[|φ〉〈φ|B,H]

= max|φ〉B

∑i

4Var[|φ〉〈φ|B,Hi]. (32)

The fact that pure states achieve the maximum on the right-hand side follows from convexity of the QFI. This bound is ofcourse only possible when the Hi are bounded.

Using the same techniques as for Savg (Supplementary Sec-tion V), we can take Hi to be a set of d2 − 1 traceless gen-erators of SU(d) satisfying Tr[HiH j] = δi, j, and compute

F∗Q[H] = FQ[|φ〉〈φ|B,H] = 4(d − 1) (which actually holdsfor any |φ〉). Thus, for this set of H in d dimensions, the LHSbound is

FQ[A,H] ≤ 4(d − 1). (33)For a pure state ψAB,

FB|AQ [ψAB,H] =

∑i

4Var[ρB,Hi]

= 4(d − 1) + 4∑i, j

pi p j, (34)

so that (33) is violated if and only if ψAB is entangled.For continuous variables, using a single mode on Bob’s side

with the quadratures H = q = (x̂, p̂), an LHS satisfies

FB|AQ [A,q] ≤ 4∑λ

p(λ) (Var[σλ, x̂] + Var[σλ, p̂])

≤ 4∑λ

p(λ)(〈x̂2〉σλ + 〈 p̂2〉σλ

)= 8

∑λ

p(λ) 〈N〉σλ= 8N̄. (35)

8

AcknowledgmentsWe thank G. Adesso, L. Pezzè, A. Smerzi, and P. Treutleinfor discussions. BY acknowledges financial support from theEuropean Research Council (ERC) under the Starting GrantGQCOP (Grant No. 637352) and grant number (FQXi FFFGrant number FQXi-RFP-1812) from the Foundational Ques-tions Institute and Fetzer Franklin Fund, a donor advised fundof Silicon Valley Community Foundation. MF was partially

supported by the Swiss National Science Foundation, and bythe Research Fund of the University of Basel for ExcellentJunior Researchers. MG acknowledges funding by the LabExENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02PSL*.

Author contributionsAll authors contributed equally to this work.

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11

When p � 1/d = 2−N , we can neglect the terms involving d. Then FB|AQ [ρ, Jz] ' pN2, VarB|AQ [ρ, Jz] / p(1 − p)N2 < pN2 forp < 1.

III. ATOMIC SPLIT DICKE STATES

In this Section, we apply our criterion to detect steering between two addressable atomic ensembles with a fixed numberof total excitations. We first consider in III A the deterministic distribution of N atoms in two modes with a fixed number ofexcitations, as proposed in the main text of our manuscript. Then, in III B we analyse a Dicke state that is sent onto a spatialbeam splitter to separate each of its mode in two, as was done experimentally with an ensemble of N = 5 000 atoms in Ref.29.

A. Dicke states with fixed splitting NA : NB

Consider N atoms split into two addressable modes A and B, with respectively NA and NB = N − NA particles. Assumethat we know that the internal spin degree of freedom of a total number of 0 ≤ k ≤ N atoms is excited (e.g., from a collectivemeasurement), but we do not know the distribution of the excited atoms into the two modes. The system is described by the splitDicke state

|SDk,NA:NB〉 = N∑kA,kB

kA+kB=k

|kA〉 ⊗ |kB〉 , (42)

where we introduced the eigenstates with kX excitations of the NX-particle spin observable JXz , for X = A, B,

JXz |kX〉 = (kX − NX/2)|kX〉 . (43)The range of kA and kB in the sum depends on the values of k, NA and NB. We can formulate constaints, e.g., in terms of kA:Since (i) if there are k > NB excitations in total, the number of excitations in A must be at least kA = k − NB and (ii) NA atomscan show at most NA excitations. These constraints can be taken into account explicitly as

|SDk,NA:NB〉 =1√

kmax − kmin + 1kmax∑

kA=kmin

|kA〉 ⊗ |k − kA〉 , (44)

where

kmin = max{0, k − NB}kmax = min{k,NA} . (45)

1. Alice measures JAz , Bob measures JAz

To determine the conditional variance, we consider the projection of Alice’s system (A) onto eigenstates |kA〉 of JAz . Aliceobtains any of the results kA = kmin, . . . , kmax with probability p(kA|JAz ) = 1/(kmax − kmin + 1), while other results have probabilityzero. Bob’s conditional state |ΨkA |JAz 〉 = |k − kA〉 has zero variance for JBz , and we obtain

VarB|AQ [|SDk,NA:NB〉, JBz ] =∑kA

p(kA|JAz )Var[|k − kA〉, JBz ] = 0 . (46)

This measurement is therefore optimal in the sense that it achieves the minimum in the definition of VarB|AQ [|SDk,NA:NB〉, JBz ] [seeEq. (2) in the main text]. This result can be understood intuitively: knowing the total number k of excitations along with the factthat kA of them are found in Alice’s subsystem, allows us to predict with certainty that Bob will measure kB = k− kA excitations.

2. Alice measures JAx , Bob estimates θ

For the estimation of a phase shift θ generated by JBz on Bob’s subsystem, let us now consider the measurement of JAx by

Alice, described by projection onto the eigenstates |kA〉x = e−i π2 JAy |kA〉. The assemblage is given byA(kA, JAx ) = TrA{(|kA〉x〈kA|x ⊗ 1)|SDk,NA:NB〉〈SDk,NA:NB |}

= p(kA|JAx )|ΨkA |JAx 〉〈ΨkA |JAx |, (47)

13

FIG. 3. Split Dicke state without partition noise. Dicke state with N = 100 particles and k excitations, deterministically split into NA =NB = N/2. Plot of Eq. (46) (blue), Eq. (55) (yellow), Eq. (59) (green dashed) and Eq. (60) (red dashed).

and for the second moments, we obtain

〈(JBz )2〉ρB =1


kA=kmin

(k − kA − NB2

)2=

112

(3(NB − 2k)2 + 4k2min + 2kmax(3NB − 6k + 2kmax + 1) + 2kmin(3NB − 6k + 2kmax − 1)

)(58)

We obtain

Var[ρB, JBz ] =1

12(kmax − kmin + 2)(kmax − kmin). (59)

Since the state ρB is invariant under transformations generated by JBz , i.e. [ρB, JBz ] = 0, we obtain that the quantum Fisher

information of Bob’s reduced state vanishes, FQ[ρB, JBz ] = 0. This can be confirmed explicitly using the expression

FQ[ρB, JBz ] = 2∑i, j

(pi − p j)2pi + p j

|〈ψi|JBz |ψ j〉|2, (60)

where ρB =∑

i pi|ψi〉〈ψi| is the spectral decomposition of ρB with eigenvalues pi = 1/(kmax − kmin + 1) and eigenvectors|ψi〉 = |k − i〉.

4. Results for a twin Fock state divided into NA = NB = N/2

When the initial state is a twin Fock state, i.e. k = N/2, that is split in two equal parts with NA = NB = N/2, the aboveexpressions simplifies further. First of all, we obtain kmin = 0, kmax = N/2, and

∑kmaxk′A=kmin

|〈kA|ei π2 JAy |k′A〉|2 = 1 (as the sum runsover the full basis of the NA = N/2 particle state), giving for the probabilities p(kA|JAx ) = 2/(N + 2). To simplify the conditionalstates (48), note from (50) that 〈kA|e−iJAy φ|k′A〉 = 〈NA − kA|e−iJ

Ay φ|NA − k′A〉 and that 〈kA|e−iJ

Ay φ|k′A〉 = 〈k′A|eiJ

Ay φ|kA〉. Moreover, the

14

matrix elements of JAy and JBy coincide since both operators are of the same length. We use this in Eq. (48) to write

|ΨkA |JAx 〉 =N/2∑k′A=0

〈kA|ei π2 JAy |k′A〉|N/2 − k′A〉

=

N/2∑k′A=0

〈k′A|e−iπ2 J

Ay |kA〉|N/2 − k′A〉

=

N/2∑k′A=0

〈N/2 − k′A|e−iπ2 J

Ay |N/2 − kA〉|N/2 − k′A〉

=

N/2∑k′A=0

〈N/2 − k′A|e−iπ2 J

By |N/2 − kA〉|N/2 − k′A〉

=

N/2∑k′A=0

|N/2 − k′A〉〈N/2 − k′A|︸︷︷︸

1B

e−iπ2 J

By |N/2 − kA〉

= |N/2 − kA〉x. (61)Hence, just like in the early examples by EPR and Bohm1,3, the state shows perfect correlations in two non-commuting mea-surement bases, and we may express (44) as

|SD N2 ,

N2 :

N2〉 =

√2

N + 2

N/2∑kA=0

|kA〉 ⊗ |N/2 − kA〉

=

√2

N + 2

N/2∑kA=0

|kA〉x ⊗ |N/2 − kA〉x. (62)

Using (61), we determine the first and second moments of JBz for the conditional states to be

〈JBz 〉kA |JAx = 0, (63)and

〈(JBz )2〉kA |JAx =18

(2kA(N − 2kA) + N). (64)

This leads to a quantum conditional Fisher information of

FB|AQ [|SD N2 , N2 : N2 〉, JBz ] = 4

N/2∑kA=0

p(kA|JAx )(〈(JBz )2〉kA |JAx =N/2∑kA=0

1N + 2

(2kA(N − 2kA) + N) = 112 N(4 + N). (65)

By comparison with Eq. (59), which yields Var[ρB, JBz ] =1

48 N(4 + N), we notice that the upper bound FB|AQ [|SDk,NA:NB〉, JBz ] =

4Var[ρB, JBz ] [see Eq. (11) in the main text] is indeed saturated by this choice of measurement. This shows that no othermeasurement by Alice could yield a higher average sensitivity on Bob’s side. The measurement of JAx is optimal for assistedmetrology with split twin Fock states as it achieves the maximum in the definition of FB|AQ [|SD N2 , N2 : N2 〉, JBz ] [see Eq. (5) in themain text].

B. Splitting a Dicke state into two modes

We now focus on a preparation of split Dicke states by a beam splitter operation. Consider a Dicke state with k excitations inthe modes a and b, described as

|Dk,N〉 = (a†)k(b†)N−k√k!(N − k)! |0〉. (66)

15

By sending this state onto a beam splitter with ratio p : 1 − p, both modes are split by into two modes as

a† =√

pa†A +√

1 − pa†B,b† =

√pb†A +

√1 − pb†B. (67)

As a consequence of the partition noise, the total number of particles in each mode fluctuates. By expanding the binomials thatappear upon inserting (67) into (66), this state can be written as

|SDk,N,p〉 =k∑

kA=0

N−k+kA∑NA=kA

√(kkA

)(N − k

NA − kA

)√pNA

√1 − pN−NA |kA〉NA ⊗ |k − kA〉N−NA

=

N∑NA=0

kmax∑kA=kmin

√(kkA

)(N − k

NA − kA

)√pNA

√1 − pN−NA |kA〉NA ⊗ |k − kA〉N−NA , (68)

where |kA〉NA is an eigenstate of the spin-NA/2 observable JAz , and similarly for subsystem B.Alice’s measurements of JAz or J

Ax , provide simultaneous information about the spin quantum number and the number of

particles NA, whose observable commutes with all spin components. Typically, after a suitable rotation of the state, one measureshow many spins point up/down, such that the information about the total number of particles is provided simultaneously. Alicecould ignore the information provided by NA, but this coarse-graining would lead to sub-optimal results for the conditionalvariance and quantum Fisher information; see Sec. I.

1. Alice measures JAz , Bob measures JBz

A measurement of JAz with the result kA for the magnetic quantum number kA and NA for the number of particles occurs withprobability

p(kA,NA|JAz ) =(

kkA

)(N − k

NA − kA

)pNA (1 − p)N−NA (69)

for all kmin ≤ kA ≤ kmax and with zero probability otherwise. This event produces the conditional state|SDk,N,p〉kA,NA |JAz = |k − kA〉N−NA (70)

on Bob’s side. Since these are eigenstates of JBz we obtain that

VarB|AQ [|SDk,N,p〉, JBz ] = 0 . (71)

This reflects the fact that a measurement of JAz and NA allows to predict with certainty the measurement results for JBz and NB.

2. Alice measures JAx , Bob estimates θ

For the estimation of a phase shift generated by JAz , we consider measurements of JAx , together with NA, with results (kA,NA).

A straightforward calculation shows that the event (kA,NA) occurs with probability

p(kA,NA|JAx ) = pNA (1 − p)N−NAkmax∑

k′A=kmin

(kk′A

)(N − k

NA − k′A

)|〈kA|ei π2 JAy |k′A〉|2 , (72)

for 0 ≤ NA ≤ N and 0 ≤ kA ≤ NA. Bob’s conditional state in this case reads

|SDk,N,p〉kA,NA |JAx =1√

p(kA,NA|JAx )kmax∑

k′A=kmin

√(kk′A

)(N − k

NA − k′A

)√pNA

√1 − pN−NA〈kA|ei π2 JAy |k′A〉|k − k′A〉N−NA

=1√∑kmax

k′A=kmin

(k

k′A

)(N−k

NA−k′A)|〈kA|ei π2 JAy |k′A〉|2

kmax∑k′A=kmin

√(kk′A

)(N − k

NA − k′A

)〈kA|ei π2 JAy |k′A〉|k − k′A〉N−NA . (73)

16

These states have the expectation value

〈JBz 〉kA,NA |JAx =1∑kmax

k′A=kmin

(k

k′A

)(N−k


kmax∑k′A=kmin

(kk′A

)(N − k

NA − k′A

)|〈kA|ei π2 JAy |k′A〉|2

(k − k′A −

NB2

), (74)

and second moment

〈(JBz )2〉kA,NA |JAx =1∑kmax

k′A=kmin

(k

k′A

)(N−k


kmax∑k′A=kmin

(kk′A

)(N − k

NA − k′A

)|〈kA|ei π2 JAy |k′A〉|2

(k − k′A −

NB2

)2, (75)

yielding the quantum Fisher information

FQ[|SDk,N,p〉kA,NA |JAx , JBz ] = 4Var[|SDk,N,p〉kA,NA |JAx , JBz ] = 4(〈(JBz )2〉kA,NA |JAx − 〈JBz 〉2kA,NA |JAx ) . (76)

This choice of measurements leads to the conditional Fisher information [see Eq. (54)]:

FB|A[|SDk,N,p〉, JAx , JBz ] =k∑

kA=0

N−k+kA∑NA=kA

p(kA,NA|JAx )FQ[|SDk,N,p〉kA,NA |JAx , JBz ] . (77)

3. Reduced quantum Fisher information and variance

Bob’s reduced state is given by

ρB = TrA{|SDk,N,p〉〈SDk,N,p|} (78)

=

k∑kA=0

N−k+kA∑NA=kA

(kkA

)(N − k

NA − kA

)pNA (1 − p)N−NA |k − kA〉N−NA〈k − kA|N−NA . (79)

It yields the average value

〈JBz 〉ρB =k∑

kA=0

N−k+kA∑NA=kA

(kkA

)(N − k

NA − kA

)pNA (1 − p)N−NA〈k − kA|JBz |k − kA〉N−NA

=

k∑kA=0

N−k+kA∑NA=kA

(kkA

)(N − k

NA − kA

)pNA (1 − p)N−NA

(k − kA − N − NA2

)=

(N2− k

)(1 − p). (80)

For the second moment, we obtain

〈(JBz )2〉ρB =k∑

kA=0

N−k+kA∑NA=kA

(kkA

)(N − k

NA − kA

)pNA (1 − p)N−NA

(k − kA − N − NA2

)2=

(N2− k

)2(1 − p)2 + N

4p(1 − p), (81)

and the variance reads

Var[ρB, JBz ] =N4

p(1 − p). (82)

Since the state is again diagonal in the eigenbasis of JBz , we obtain

FQ[ρB, JBz ] = 0. (83)

The data shown in Fig. 4 shows that the measurement of S Ax is again optimal for a split twin Fock state, as the conditionalFisher information (77) reaches its upper bound (82).

17

FIG. 4. Dicke state split with partition noise. Dicke state with N = 100 particles and k excitations, split into two modes with 50 : 50 ratio.Plot of Eq. (71) (blue), Eq. (77) (yellow), Eq. (82) (green dashed) and Eq. (83) (red dashed).

IV. BIPARTITE PURE STATES

A. Witnessing with a fixed H

For a shared pure state we can use the saturation of the inequalities labeled by (*) in (11), to express the condition (6) forLHS models as 4Var[ρB,H] ≤ FQ[ρB,H], whereas in general, this inequality holds in reverse. Hence, steering in this scenario isrevealed whenever 4Var[ρB,H] and FQ[ρB,H] do not coincide. Even for a fixed H (i.e. without optimisation), this condition isclose to being a faithful witness of steering: it is satisfied precisely when H is constant on the support of ρB. This follows fromLemma 1 below.

Lemma 1. FQ[ρ,H] = 4Var[ρ,H] if and only if ΠρHΠρ ∝ Πρ, where Πρ is the projector on the support of ρ.

Proof. Using the spectral decomposition ρ =∑

i pi |i〉〈i| and with Hi j := 〈i|H| j〉, we can express41

Var[ρ,H] − 14

FQ[ρ,H] = 2∑i, j

pi p jpi + p j

|Hi j|2 +∑

i

piH2ii −∑

i

piHii

2 , (84)

where the second bracketed term, and the terms in the first sum, are all non-negative. When this quantity vanishes, we thereforesee that the off-diagonals Hi j = 0 whenever pi, p j , 0. In addition the bracketed term must vanish, and this is simply the varianceof the diagonals Hii in the distribution pi. This variance vanishes if and only if the Hii are constant over the range of i such thatpi , 0. Since

Πρ =∑

i: pi,0

|i〉〈i| ,

ΠρHΠρ =∑i, j:

pi,p j,0

Hi j |i〉〈 j| , (85)

these conditions can be equivalently expressed neatly as ΠρHΠρ ∝ Πρ. �

As will be shown in Section V, varying over H can make this witness faithful. When Bob’s system is a qubit (d = 2), withoutloss of generality we can take the observable to be a Pauli matrix: H = n.σ, and then

FB|AQ [ψAB,n.σ] − 4VarB|AQ [ψAB,n.σ] = 8

(1 − Tr[(ρB)2]

), (86)

which is a function of the purity of ρB and notably independent of the direction n.

18

B. Optimal measurements

In the case of an overall pure state, it is also possible to determine the optimal measurements needed on Alice’s side:

Theorem 1. i) For a shared pure state ψAB, an optimal measurement for Alice to achieve FB|AQ [A,H] is

|x̃∗k〉 :=1√d

d−1∑l=0

e2πikl/d |x∗l 〉 , (87)

where |xl〉 are the eigenstates ofX :=

√ρBH

√ρB − 〈H〉ρB ρB, (88)

and ∗ denotes complex conjugation in the Schmidt basis of ψAB.

ii) Similarly, an optimal measurement to achieve VarB|AQ [A,H] is |y∗k〉, where |yk〉 are the eigenstates of the operator

Y :=∑

i

2√pi p jpi + p j

Hi j |i〉〈 j| . (89)

Proof. We follow the proof of Ref.23, which found a construction for the optimal pure state ensemble in the concave roof of thevariance, but stopped short of giving explicit expressions. As shown there, it is sufficient to find a basis in which the diagonalsof X vanish. We show that the basis |x̃k〉 (Fourier transformed with respect to the eigenbasis of X) is such a basis. First note thatTr X = 0, then writing the spectral decomposition X =

∑l xl |xl〉〈xl|,

〈x̃k |X|x̃k〉 = 1d∑l,m

e2πik(m−l)/d 〈l|X|m〉

=1d

∑l

xl = 0. (90)

The remainder proceeds as in Ref.23, which we include for completeness. The optimal ensemble is constructed by√

qk |ψk〉 :=√ρ |x̃k〉 (where we write ρB = ρ for brevity). It follows that qk = 〈x̃k |ρ|x̃k〉, and that this is indeed a valid ensemble decomposition

for ρ: ∑k

qk |ψk〉〈ψk | =∑

k

√ρ |x̃k〉〈x̃k | √ρ = ρ. (91)

Now we have

0 = 〈x̃k |X|x̃k〉 = 〈x̃k |(√ρH√ρ − 〈H〉ρ ρ

)= qk 〈ψk |H|ψk〉 − 〈H〉ρ 〈x̃k |ρ|x̃k〉 , (92)

so 〈ψk |H|ψk〉 = 〈H〉ρ whenever qk , 0. Thus∑k

qkVar[ψk,H] =∑

k

qk 〈H2〉ψk −∑

k

qk 〈H〉2ψk

= 〈H2〉ρ − 〈H〉2ρ = Var[ρ,H], (93)thus providing the concave roof of the variance.

The measurement basis for Alice to steer Bob into this ensemble follows straightforwardly. Using the Schmidt decomposition|ψ〉AB =

∑i√

pi |i〉 |i〉, the measurement basis for Alice is |x̃∗k〉, where ∗ denotes complex conjugation in the Schmidt basis. Thisis seen from

〈x̃∗k |A|ψ〉AB =∑

i

√pi |i〉〈x̃k |i〉∗

=∑

i

√pi |i〉〈i|x̃k〉

=√ρ |x̃k〉

=√

qk |ψk〉 . (94)The corresponding statement for VarB|AQ is similarly given by the optimal convex roof ensemble found in Ref.

23, namely√qk |ψk〉 = √ρ |yk〉. Exactly as above, the measurement required to steer into this ensemble is given by complex conjugation in

the Schmidt basis. �

19

V. MAXIMAL AND AVERAGE VIOLATION

Here, we define two quantities involving variation over the generator, namely the maximal and the average violation of themain inequality:

Smax(A) = maxH :

Tr[H2]=1

[14

FB|AQ [A,H] − VarB|AQ [A,H]]+, (95)

Savg(A) = (d2 − 1)[∫

µ(dn)14

FB|AQ [A,n ·H] − VarB|AQ [A,n ·H]]+, (96)

where [·]+ = max{0, ·} denotes the positive part, and the Hi provide a basis of SU(d) generators satisfying Tr[Hi] = 0, Tr[HiH j] =δi, j, and µ is the uniform measure over the sphere of unit vectors |n| = 1.

Note that both quantities are invariant under unitaries on Bob’s side. This is immediately evident for Smax. To see it for Savg,we express the action of some U on the generators as U†HiU =

∑j Ri jH j. This action preserves the Hilbert-Schmidt inner

product between generators, from which it is found that R must be an orthogonal matrix. Thus U†(n ·H)U = (RT n) ·H. Using

FB|AQ [UBAUB†,n ·H] = FB|AQ [A,U†(n ·H)U]

= FB|AQ [A, (RT n) ·H], (97)

and similarly for VarB|AQ , it follows that the integral over n is invariant.We now compute both quantities for a joint pure state ψAB.

Theorem 2. For a joint pure state ψAB, we have

Smax(ψAB) = λmax[diag(p) − ppT ], (98)

Savg(ψAB) =∑i, j

pi p j

(1 +

2pi + p j

), (99)

where pi are the eigenvalues of ρB and λmax[M] is the largest eigenvalue of a given matrix M.

Proof. From the Methods section, we have,

14

FB|AQ [ψAB,H] − VarB|AQ [ψAB,H] = Var[ρB,H] −

14

FQ[ρB,H]. (100)

This quantity has been studied by Tóth41, who shows that

Var[ρ,H] − 14

FQ[ρ,H] = 2∑i, j

pi p jpi + p j

|Hi j|2 +∑

i

piH2ii −∑

i

piHii

2 , (101)

and computes the average needed for Savg.For Smax, we turn (101) into a matrix expression. We encode the components Hi j into a vector whose first d components are

the diagonals and remaining d(d − 1)/2 components are the off-diagonals:

v = (H11,H22, . . . ,√

2H12,√

2H13, . . . ) (102)

such that |v|2 = ∑i j|Hi j|2 = Tr[H2] = 1. Similarly define a matrix M = MD ⊕ MO split into diagonal and off-diagonal parts:[MD]i, j = δi, j pi − pi p j, (103)

[MO](i j),(kl) = δi,kδ j,l

(2pi p jpi + p j

). (104)

We then see that

Var[ρ,H] − 14

FQ[ρ,H] = v†Mv

⇒ Smax(ψAB) = λmax[M]. (105)

20

Finally, we will show that λmax[M] = λmax[MD] ≥ λmax[MO]. Note that the diagonals of MO are the harmonic means of eachpair of pi, p j, (i , j), and so

λmax[MO] =2p1 p2p1 + p2

=: m∗, (106)

where without loss of generality we have ordered p1 ≥ p2 ≥ · · · ≥ pd. Note that m∗ ≥ p2.From the Weyl inequalities on eigenvalues42 (Theorem III.2.1), we can bound

λmax[MD] ≤ λmax[diag(p)] + λmax[−ppT ]= p1, (107)

λmax[MD] ≥ λ2[diag(p)] + λd−1[−ppT ]= p2, (108)

where λk denotes the kth largest eigenvalue. So we have p2 ≤ λmax[MD] ≤ p1 – we will now obtain a stronger lower bound. Letus inspect the eigenvector condition MDw = mw:

piwi − pi∑

j

p jw j = mwi ∀i

⇒ wi = piw̄pi − m , w̄ =∑

j

p jw j. (109)

This determines the eigenvector corresponding to the eigenvalue m. Multiplying by pi and summing over i,

w̄ =∑

i

p2i w̄pi − m

⇒ w̄ = 0 or∑

i

p2ipi − m = 1. (110)

Now w̄ , 0 or else we would have w = 0 by (109). Therefore m satisfies g(m) :=∑

i p2i /(pi − m) = 1. The function g is strictlyincreasing in regions where it is continuous:

g′(m) =∑

i

p2i(pi − m)2 > 0. (111)

Evaluating g(m∗) from (106),

g(m∗) =p21

p1 − 2p1 p2/(p1 + p2) +p22

p2 − 2p1 p2/(p1 + p2)

+∑i>2

p2ipi − m∗

= (p1 + p2)(

p1p1 − p2 +

p2p2 − p1

)+

∑i>2

p2ipi − m∗

= p1 + p2 +∑i>2

p2ipi − m∗

≤ p1 + p2≤ 1, (112)

since m∗ ≥ p2 ≥ p3 ≥ . . . . We are working in the region p2 ≤ m ≤ p1 as shown above, in which g is continuous and thus strictlyincreasing. So in order to have g(m) = 1, it must be that m ≥ m∗. In other words, λmax[MD] ≥ λmax[MO]. �

For pure states, steering is equivalent to entanglement, therefore any steering measure must reduce to an entanglement measurewhen evaluated on pure states.

21

FIG. 5. Steering quantifiers for pure bipartite states. A plot of the quantities Savg (blue) and Smax (orange) for a pure state with Schmidtcoefficients x, y, 1 − x − y. Apart from the extreme points, they coincide at x = y = 1/3, corresponding to the maximally entangled state withd = 3.

Corollary 1. Savg is a full entanglement measure for pure states. Smax is a faithful witness of entanglement for pure states, butnot a monotone under LOCC.

Proof. It is easy to see that Savg vanishes if and only if pi p j = 0 ∀i , j, i.e. p1 = 1, p2 = p3 = · · · = 0, meaning that ρB is pure.A function f (ρB) satisfies strong monotonicity under LOCC if and only if (i) f is a symmetric function of the eigenvalues

of ρB, (ii) f is expansible (meaning that zero eigenvalues can be appended without changing the value) and (iii) f is con-cave43,44. Properties (i) and (ii) are evident from the expression (99), while (iii) follows from concavity of V and convexity of FQ.

Smax vanishes if and only if ψAB is separable, and it is a symmetric, expansible function of p. However, it is not an entangle-ment monotone, since its maximal value is attained for p = (1/2, 1/2, 0, . . . ) (observed numerically; see Fig. 5), which in d > 2does not correspond to the maximally entangled state. �

For general mixed states, the values in Theorem 2 are upper bounds to Savg, Smax.

Quantum metrology assisted by Einstein-Podolsky-Rosen steering Methods References Supplementary InformationI Optimal POVMs for assisted metrologyII GHZ states with white noiseIII Atomic split Dicke statesA Dicke states with fixed splitting NA:NB1 Alice measures JzA, Bob measures JzA2 Alice measures JxA, Bob estimates 3 Reduced quantum Fisher information and variance4 Results for a twin Fock state divided into NA=NB=N/2

B Splitting a Dicke state into two modes1 Alice measures JzA, Bob measures JzB2 Alice measures JxA, Bob estimates 3 Reduced quantum Fisher information and variance

IV Bipartite pure statesA Witnessing with a fixed HB Optimal measurements

V Maximal and average violation

quantum metrology assisted by einstein-podolsky-rosen …sep 18, 2020 · quantum metrology...

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