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Quantum Discord Determines the Interferometric Power of Quantum States

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Quantum Discord Determines the Interferometric Power of Quantum States 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 coherence [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 observables 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 quantum state of a bipartite system when A spectrum of the generator HA is publicly known, and it is subsystem is a qubit. 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 qubits 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.
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