Operational significance of discord in Experimental quantum estimation quantum metrology: using NMR Theory and Experiment* Diogo de Oliveira Soares Pinto Instituto de Física de São Carlos Universidade de São Paulo ([email protected]) NMR – QIP in Rio November 2013 *Title inspired in Nat.Phys. 8, 671 (2012) Outline: 1) (Very brief) Introduction to quantum metrology 2) Results: Theory 3) Result: Experiment 4) Conclusions (Very brief) Introduction to quantum metrology In the lab... Entangled state? Quantum state tomography = experimental data R y Data analysis 1 y 2 * y 3 4 C m ax 0, Entangled or not? Estimation problem! y Eigenvalues of R ordered from the highest to the lowest Simplest version of a typical quantum estimation problem: → Recover the phase introduced by the unitary operator U e i H H is a known Hamiltonian that generates the phase . Stepwise process: 1) Prepare the N-probe system in a state Repeat these steps times to improve accuracy 2) Apply the unitary transformation U to the state 3) Measure the final state =U U 4) From the data find the estimator 5) Check the estimation accuracy through the Root Mean Square Error*: 2 1 2 H ; H 2 H * C.W. Helstrom Quantum Detection and Estimation Theory (1976). 2 H 2 Two important limits for this “interferometric-measurement scheme” for phase estimation* ( 1, g the largest Hamiltonian gap): a b in 2 a e i 2 p in | N probes, repetitions. in a 2 N a 1 cos Standard Quantum Limit (SQL) or “shot” noise limit N b N 2 N e iN b N 2 p in | N-entangled probes, repetitions. 2 g b g N * V. Giovannetti, S. Lloyd, L. Maccone, Nature Photonics 5, 222 (2011). 2 1 cos N 2 Heisenberg limit In usual estimation problems, obey the Cramér-Rao bound: 1 F ( ) F ( ) , p | 1 p | 2 where F( ) is the Fisher information. In quantum estimation problems, this bound (quantum Cramér-Rao bound) is given by: 1 F ( ; H ) Symmetric Logarithm Derivative (optimal measurement) , F ( ; H ) tr L , L 2 l j 1 2 L L j j j , Is entanglement the only resource for enhanced estimation that Quantum Mechanics can give us? Fortunately no! We also have... Nature 474, 24-26 (2011). For a review see: K. Modi et al. Rev. Mod. Phys. 84, 1655 (2012). Results: Theory Let’s go back to the interferometric scheme. Suppose that the Hamiltonian HA that generate the phase over the partition A is given by H A n A and we don’t know a priori the direction ‘n’. Consequently the Hamiltonian itself is unknown for us (blind quantum metrology). From the worst case scenario we can define a figure of merit for this interferometric scheme: P A AB 1 4 inf F A B ; H A HA Interferometric Power of the input state AB Guarantees the usefulness of the input state for quantum estimation and is a measure of discord! Discord as a resourse for quantum metrology! Details in ArXiv:1309.1472. • Invariant under local unitaries and nonincreasing under local operations on Characteristics of B; • Vanishes iff is classically AB 1 A correlated; P A B inf F A B ; H A • Reduces to an entanglement monotone 4 H for pure states; • It is analytically computable if A is a qubit. Examples for two qubits (obs: idAB = 4x4 identity matrix): 1) Werner states A W AB f B e ll AB 1 f 4 id A B ; 0 f 1 . P A AB 2f 2 1 f 2) Bell diagonal states BD AB 1 id A B 4 3 C ij i , j 1 iA jB P A C Details in ArXiv:1309.1472. C AB 2 2 2 2 C 2 C 2 1 C c1 c 2 c 3 , 2 2 2 m ax c1 , c 2 , c 3 . 2 2 2 2 det C 2 ; Suppose two families of states*: Q AB 1 p 2 1 0 4 0 2p 0 1 p 0 2 0 1 p 0 0 0 2 2p 0 0 2 1 p D A A p , log 1 p p log 1 p Q AB A Q AB 2 4 1 Q AB p p 2 2 p 1 p p p 1 p p p 2 p p 2 p 1 2 4 p log 4 1 p , U 1 classically correlated. with quantum discord. P C AB 1 4 tr A B Q ,C 2 1 (1 p 2 ) 2 ; 4 0 p 1. 1 p . 2 *K. Modi et al. PRX 1, 021022 (2011). Results: Experiment What shall we measure? What shall we test experimentally? First: interferometric scheme A B tw o fa m ilie s o f sta te s UA e i n A , † A B U A in U A Second: check discord in the initial states P A AB 1 4 inf F A B ; H A HA Third: verify the metrological quantities j j , F A B ; H A tr A B L , L l j 2 j 1 2 L L Compare and check if discord can be seen as a resourse for quantum metrology! NMR system: Q AB Target: Prepare @ CBPF 1 p 2 1 0 4 0 2p C AB 1 4 1 p 2 0 0 1 p 2 0 1 p 0 0 p 2 0 p 1 p p p 1 p p p Start preparing: 2 p p 2 p 1 2 2p 0 0 2 1 p After preparing state A B 0 A 0 B 0 A 0 B , we implement the circuits below to obtain the desired states. It is important to note that p cos . AB C p 0.5 AB Q Fidelity above 99% for initial states! How to implement unknown phase shift? Setting the estimated as ,C AB phase to be 4 We can choose three directions to rotate ,Q AB Ok. But what is the (optimal) measurement? We must measure A B, s in the eigenbasis of the symmetric logarithm derivative to obtain the maximum allowed precision. Since: k ,s L k ,s lj k ,s k ,s j j j F s AB ;H k A L k ,s 2 k ,s lj 2 k ,s dj j k ,s dj ,s k ,s tr A B j k ,s j s C , Q ; k 1, 2, 3. We can map the eigenvectors onto the computational basis of two qubits. Doing so, the ensemble expectation values d can be directly observed in the diagonal elements of the density matrix. k ,s j But how? The answer: Global rotation dependent on s and k! Example for s = C, Q and k = 1: This can be done also for s = C, Q and k = 2, 3. ArXiv:1309.1472. From the experiment (ArXiv: 1309.1472): H A zA HA xA yA 2 H A xA Conclusions Operational interpretation of quantum discord in terms of a resourse for quantum estimation problems when is considered the worst case scenario! In settings like NMR, where disorder is high, quantum correlations even without entanglement can be a promising resourse for quantum technology. Taking advantage of the name proposed for the protocol (blind quantum metrology), I can finish citing: “Perhaps only in a world of the blind will things be what they truly are.” Saramago – Blindness. or better: “Perhaps only in a [quantum mixed] world of the blind will things be what they truly are.” Fisher Information as a Measure of Quantum Discord. These guys are around here! People involved: • Davide Girolami – NUS (Singapore) • Vittorio Giovannetti – SNS (Italy) • Tommaso Tufarelli – Imperial College (UK) • Jefferson G. Filgueiras – TUD (Germany) • Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira – CBPF (Brazil) • Me – IFSC/USP (Brazil) • Gerardo Adesso – UoN (UK) Thanks for the attention!