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week ending PRL 112, 210401 (2014) PHYSICAL REVIEW LETTERS 30 MAY 2014

Quantum Discord Determines the Interferometric Power of Quantum States

Davide Girolami,1,2,8 Alexandre M. Souza,3 Vittorio Giovannetti,4 Tommaso Tufarelli,5 Jefferson G. Filgueiras,6 Roberto S. Sarthour,3 Diogo O. Soares-Pinto,7 Ivan S. Oliveira,3 and Gerardo Adesso1,* 1School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 2Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, 117583 Singapore, Singapore 3Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, 22290-180 Rio de Janeiro, Brazil 4NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, Piazza dei Cavalieri 7, I-56126 Pisa, Italy 5QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom 6Fakultät Physik, Technische Universität Dortmund, 44221 Dortmund, Germany 7Instituto de Física de São Carlos, Universidade de São Paulo, P.O. Box 369, São Carlos, 13560-970 São Paulo, Brazil 8Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom (Received 11 February 2014; published 27 May 2014) Quantum metrology exploits quantum mechanical laws to improve the precision in estimating technologically relevant parameters such as phase, frequency, or magnetic fields. Probe states are usually tailored to the particular dynamics whose parameters are being estimated. Here we consider a novel framework where quantum estimation is performed in an interferometric configuration, using bipartite probe states prepared when only the spectrum of the generating Hamiltonian is known. We introduce a figure of merit for the scheme, given by the worst-case precision over all suitable Hamiltonians, and prove that it amounts exactly to a computable measure of discord-type quantum correlations for the input probe. We complement our theoretical results with a metrology experiment, realized in a highly controllable room-temperature nuclear magnetic resonance setup, which provides a proof-of-concept demonstration for the usefulness of discord in sensing applications. Discordant probes are shown to guarantee a nonzero phase sensitivity for all the chosen generating Hamiltonians, while classically correlated probes are unable to accomplish the estimation in a worst-case setting. This work establishes a rigorous and direct operational interpretation for general quantum correlations, shedding light on their potential for quantum technology.

DOI: 10.1103/PhysRevLett.112.210401 PACS numbers: 03.65.Ud, 03.65.Wj, 03.67.Mn, 06.20.-f

ρφ φ~ All quantitative sciences benefit from the spectacular square error Var AB ð Þ that measures the statistical distance −1 developments in high-accuracy devices, such as atomic between φ~ and φ,Varρφ ðφ~ Þ ≥ ½νFðρAB; HAÞ , where F is clocks, gravitational wave detectors, and navigation sen- AB the quantum Fisher information (QFI) [7], which quantifies sors. Quantum metrology studies how to harness quantum φ how much information about φ is encoded in ρAB. The mechanics to gain precision in estimating quantities not ν → ∞ amenable to direct observation [1–5]. The phase estimation inequality is asymptotically tight as , provided the paradigm with measurement schemes based on an inter- most informative quantum measurement is carried out at ferometric setup [6] encompasses a broad and relevant class the output stage. Using this quantity as a figure of merit for of metrology problems, which can be conveniently cast in independent and identically distributed trials, and under H terms of an input-output scheme [1]. An input probe state the assumption of complete prior knowledge of A, then “ ” H ρAB enters a two-arm channel, in which the reference [8] in the eigenbasis of A is the essential subsystem B is unaffected while subsystem A undergoes a resource for the estimation [2]; as maximal coherence in a local unitary evolution, so that the output density matrix known basis can be reached by a superposition state of φ † A can be written as ρAB ¼ðUA ⊗ IBÞρABðUA ⊗ IBÞ , with subsystem only, there is no need for a correlated (e.g., −iφH B UA ¼ e A , where φ is the parameter we wish to estimate entangled) subsystem at all in this conventional case. and HA is the local Hamiltonian generating the unitary We show that the introduction of correlations is instead dynamics. Information on φ is then recovered through unavoidable when the assumption of full prior knowledge an estimator function φ~ constructed upon possibly joint of HA is dropped. More precisely, we identify, in correlations measurements of suitable dependent per- commonly referred to as “quantum discord” between A and φ B formed on the output ρAB. For any input state ρAB and [9,10], the necessary and sufficient resources rendering generator HA, the maximum achievable precision is deter- physical states able to store phase information in a unitary mined theoretically by the quantum Cramér-Rao bound [3]. dynamics, independently of the specific Hamiltonian that Given repetitive interrogations via ν identical copies of ρAB, generates it. Quantum discord is an indicator of quantum- this fundamental relation sets a lower limit to the mean ness of correlations in a composite system, usually revealed

0031-9007=14=112(21)=210401(5) 210401-1 © 2014 American Physical Society week ending PRL 112, 210401 (2014) PHYSICAL REVIEW LETTERS 30 MAY 2014 via the state disturbance induced by local measurements correlated, i.e., Alice and Bob prepare a density matrix [9–12]; recent results suggest that discord might enable ρAB diagonal with respect to a local basis on A [11,13,17], quantum advantages in specific computation or communi- then no precision in the estimation is guaranteed; indeed, in cation settings [13–19]. In this Letter a general quantitative this case there is always a particularly adverse choice for HA, equivalence between discord-type correlations and the such that ½ρAB;HA ⊗ IB¼0 and no information about φ guaranteed precision in quantum estimation is established can be imprinted on the state, resulting in a vanishing IP. theoretically, and is observed experimentally in a liquid- Conversely, the degree of discord-type correlations of the ρ state nuclear magnetic resonance (NMR) proof-of-concept state AB not only guarantees but also directly quantifies, via implementation [20,21]. Eq. (1), its usefulness as a resource for estimation of a φ H Theory.—An experimenter Alice, assisted by her partner parameter , regardless of the generator A of a given ρ Bob, has to determine, as precisely as possible, an unknown spectral class. For generic mixed probes AB, this is true even parameter φ introduced by a “black box” device. The black in the absence of entanglement. −iφH box implements the transformation UA ¼ e A and is Remarkably, we can obtain a closed formula for the IP controlled by a referee Charlie, see Fig. 1. Initially, only the of an arbitrary of a bipartite system when A spectrum of the generator HA is publicly known, and it is subsystem is a . Deferring the proof to [27], this reads PA ρ ς M assumed to be nondegenerate. For instance, the experi- ð ABÞ¼ min½ (2) menters might be asked to monitor a remote (uncooper- where ς ½M is the smallest eigenvalue of the 3 × 3 matrix ative) target whose interaction with the probing signals is min M of elements partially incognito [22]. Alice and Bob prepare ν copies of ρ 1 X q − q 2 a bipartite (generally mixed) probe state AB of their choice. M ð i lÞ Charlie then distributes ν identical copies of the black m;n ¼ 2 q q i;l∶q q ≠0 i þ l box, and Alice sends each of her subsystems through one iþ l iteration of the box. After the transformations, Charlie × hψ ijσmA ⊗ IBjψ lihψ ljσnA ⊗ IBjψ ii reveals the Hamiltonian HA used in the box, prompting q ; ψ Alice and Bob to perform the best possible joint measure- with f i j iig being, respectively,P the eigenvalues and φ ρ ρ q ψ ψ ment on the transformed state ðρ Þ⊗ν in order to estimate eigenvectors of AB, AB ¼ i ij iih ij. This renders AB PA ρ φ [23]. Eventually, the experimenters infer a probability ð ABÞ an operational and computable indicator of general distribution associated to an optimal estimator φ~ for φ nonclassical correlations for practical purposes. — saturating the Cramér-Rao bound (for ν ≫ 1), so that the Experiment. We report an experimental implementation corresponding QFI determines exactly the estimation pre- of black box estimation in a room temperature liquid-state cision. For a given input probe state, a relevant figure of NMR setting [20,21]. Here, quantum states are encoded in the spin configurations of magnetic nuclei of a 13C-labeled merit for this protocol is then given by the worst-case QFI d 1 over all possible black box settings, chloroform (CHCl3) sample diluted in 6 acetone. The H and 13C nuclear spins realize A and B, respectively; ρ A 1 the states AB are engineerable as pseudopure states [26,28] P ðρABÞ¼ minFðρAB; HAÞ; (1) 4 HA by controlling the deviation matrix from a fully thermal ensemble [21]. A highly reliable implementation of unitary where the minimum is intended over all Hamiltonians with phase shifts can be obtained by means of radio frequency (rf) given spectrum (see also [27]), and we inserted a normali- pulses. Referring to [27] for further details of the sample A zation factor 1=4 for convenience. We shall refer to P ðρABÞ preparation and implementation, we now discuss the plan as the “interferometricpower” (IP) oftheinputstateρAB,since (Fig. 2) and the results (Fig. 3) of the experiment. it naturally quantifies the guaranteed sensitivity that such a We compare two scenarios, where Alice and Bob prepare state allows in an interferometric configuration (Fig. 1). input probe states ρAB either with or without discord. The We prove that the quantity in Eq. (1) is a rigorous measure chosen families of states are, respectively [25,29,30], of discord-type quantum correlations of an arbitrary bipar- 0 1 1 p2 002p tite state ρAB (see Supplemental Material [27]). If (and only þ B 2 C if) the probe state is uncorrelated or only classically Q 1 B 01− p 00C ρ ¼ B C; AB 4 @ 001− p2 0 A Alice 2p 001þ p2 0 1 1 p2 pp B C 1 B p2 1 ppC Charlie ρC ¼ B C: (3) AB 4 @ pp1 p2 A 2 Bob ppp 1 Both classes of states have the same purity, given by Q;C 2 2 2 FIG. 1 (color online). Black box quantum estimation. Tr½ðρAB Þ ¼ð1=4Þð1 þ p Þ , where 0 ≤ p ≤ 1. This 210401-2 week ending PRL 112, 210401 (2014) PHYSICAL REVIEW LETTERS 30 MAY 2014

FIG. 2 (color online). Experimental scheme for black box parameter estimation with NMR. The protocol is divided in three steps: probe state preparation (leftmost frame, yellow), black box transformation (middle frame, red), and optimal measurement (rightmost frame, green). Starting from a thermal equilibrium distribution, we initialize the two-qubit system in a pseudopure −5 state of the form ρ ¼½ð1 − ϵÞ=4I þ ϵρAB, with ϵ ∼ 10 and FIG. 3 (color online). Experimental results. Each column HðkÞ k 1 ρAB ¼j00ih00jAB. This is done by applying the pulse sequence corresponds to a different black box setting A , ¼ ,2,3, φ A ðπ=2Þx → UJð1=4JÞ → ðπ=2Þy → UJð1=4JÞ → ðπ=2Þ−x → Gz → generating a rotation on qubit around a Bloch sphere direction !n ðkÞ ðπ=4Þy → UJð1=2JÞ → ðπ=6Þx → Gz, where ðθÞα is a rotation of ; the set directions are depicted in the insets of row (b). Empty ρC each qubit by an angle θ in the direction α, UJðτÞ is a free evolution red squares refer to data from classical probes AB, filled blue ρQ under the scalar interaction between the spins for a time τ,andGz is circles refer to data from discordant probes AB; error bars, due to a pulsed field gradient (which dephases all the spins along the z small pulse imperfections in state preparation and tomography axis). We then proceed to prepare two types of probe states: the [27], are smaller than the size of the points. The lines refer to C Q classically correlated ones ρAB (a), and the discordant ones ρAB (b), theoretical predictions. Both families of states depend on a purity defined in Eq. (3). We first apply rf pulses, with a flip angle θ, parameter p, experimentally tuned by a flip angle (see Fig. 2). The followed by a pulsed field gradient Gz; this allows us to tune the top row (a) shows the precision achieved by each probe in purity parameter p ¼ cos θ, by varying θ between 0° and 90° in estimating φ for the different settings: the respective QFIs (divided steps of 2.5°. The subsequent circuits differ for each type of state: by 4) as obtained from the output measured data are plotted and C PA ρQ for ρAB (a), a CNOT gate followed by Hadamard gates H on both compared with the IP ð ABÞ of the discordant states Q qubits A and B are implemented, while for ρAB (b), the CNOT is (black crosses) measured from initial state tomography. The H A Q ð1Þ C ð1Þ 2 followed by a Hadamard on qubit only and by a second CNOT. theoretical predictions are: FthðρAB;HA Þ¼FthðρAB;HA Þ¼8p = 1 p2 F ρQ ; Hð2Þ 4p2 F ρC ;Hð2Þ 4p2= 1 p2 ð þ Þ, thð AB A Þ¼ , thð AB A Þ¼ ð þ Þ, Q ð3Þ 2 C ð3Þ A Q 2 allows us to focus on the role of initial correlations for the FthðρAB; HA Þ¼4p , FthðρAB; HA Þ¼0, and P ðρABÞ¼p . subsequent estimation, at tunable common degree of The middle row (b) depicts the measured variances of the optimal φ~ ν ≈ 1015 mixedness mimicking realistic environmental conditions. estimators exp over the ensemble of molecules, together C While the states ρAB are classically correlated for all values with the theoretical predictions corresponding to the saturation Q of the quantum Cramér-Rao bound. The upper bound limiting of p, the states ρAB have discord increasing monotonically the estimation uncertainty for the discordant states is shown as with p>0. The probes are prepared by applying a chain of Q well, given by 4 νPA ρ −1 as calculated from the input states control operations to the initial Gibbs state (Fig. 2). We ½ ð ABÞ (crosses). The bottom row (c) depicts the inferred mean value of perform a full tomographical reconstruction of each input ~ the optimal estimator hφexpi for the various settings. Both classical state to validate the quality of our state preparation, and quantum probes allow us to obtain a consistently unbiased 99 7 0 2 % obtaining a mean fidelity of ð . . Þ with the guess for φ (in the experiment, the true value of φ was set at theoretical density matrices of Eq. (3) [27]. In the case φ π=4 ρC k 3 Q 0 ¼ ), apart from the unreliable results of AB for ¼ , of ρAB, we measure the degree of discord-type correlations which demonstrates that classical probes cannot return any in the probes by evaluating the closed formula (2) for the IP estimation in the worst-case scenario. on the tomographically reconstructed input density matri- ces; this is displayed as black crosses in the top panel of Fig. 3 and is found in excellent agreement with the experiments without any loss of generality), we implement A Q 2 theoretical expectation P ðρABÞ¼p . three different choices of Charlie’s black box transforma- ðkÞ −iφ HðkÞ ð1Þ φ U e 0 A ⊗ I H σ Then, for each fixed input probe, and denoting by 0 tion A ¼ pffiffiffi B. These are given by A ¼ zA, φ ð2Þ ð3Þ the true value of the unknown parameter to be estimated HA ¼ðσxA þ σyAÞ= 2, and HA ¼ σxA, and are engi- by Alice and Bob (which we set to φ0 ¼ π=4 in the neered by applying, respectively, the pulse sequences 210401-3 week ending PRL 112, 210401 (2014) PHYSICAL REVIEW LETTERS 30 MAY 2014

ð1Þ ð2Þ φ U ¼ðπ=2Þx → ðπ=2Þ−y → ðπ=2Þ−x, U ¼ðπ=2Þx y, and and we calculate the value of that minimizes the least- A A þ P k;s k;s 2 3 ðk;sÞ 4 thð Þ expð Þ ð Þ squares function Υ ðφÞ¼ j 1 ½dj ðφÞ − dj UA ¼ðπ=2Þx. A theoretical analysis asserts that the ¼ chosen settings encompass the best (setting k ¼ 1) and (equivalent to maximizing the log-likelihood assuming that d worst (setting k ¼ 3) case scenarios for both types of each j is Gaussian distributed over the ensemble). For each k s φ probes, while the setting k ¼ 2 is an intermediate case [27]. setting ( , ), the value of that solves the least-squares φ~ ðk;sÞ For each input state and black box setting, we carry problem is chosen as the expected value h exp i of our out the corresponding optimal measurement strategy for the estimator. These values are plotted in row (c) of Fig. 3.One φ π=4 estimation of φ. This is given by projections on the eigenbasis can appreciate the agreement with the true value 0 ¼ λ j 1; …; 4 of the unknown phase shift for all settings but the patho- fj jig ( ¼ P ) of the symmetric logarithmic derivative φ logical one (C, 3). In the latter case, the estimation is (SLD) Lφ ¼ jljjλjihλjj, an operator satisfying ∂φρAB ¼ 1=2 ρφ L L ρφ completely unreliable because the classical probes commute ð Þð AB φ þ φ ABÞ [7].TheQFIisthengivenby[31] with the corresponding Hamiltonian generator, thus failing the estimation task. φ 2 FðρAB; HAÞ¼Tr½ρABLφ Next, by expanding the SLD in its eigenbasis (see Table SI X q − q 2 in [27]), we obtain the experimental QFIs measured from ð i lÞ 2 k k;s P k;s k;s ¼ 4 jhψ ijðHA ⊗ IBÞjψ lij ; F ρs ;Hð Þ Lð Þ 2 lð Þ 2dexpð Þ q q our data, expð AB A Þ¼hð φ0 Þ i¼ jð j Þ j . i

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