Experimental quantum estimation using NMR Diogo de Oliveira Soares Pinto Instituto de Física de São Carlos Universidade de São Paulo NMR – QIP in Rio November 2013 Operational significance of discord in quantum metrology: Theory and Experiment * * 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

Entangled state? In the lab... Quantum state tomography = experimental data Eigenvalues of R ordered from the highest to the lowest Entangled or not? Estimation problem! Data analysis

Simplest version of a typical quantum estimation problem: → Recover the phase  introduced by the unitary operator H is a known Hamiltonian that generates the phase . 1) Prepare the N-probe system in a state  2) Apply the unitary transformation U  to the state  3) Measure the final state   = U   U   5) Check the estimation accuracy through the Root Mean Square Error * : Repeat these steps times to improve accuracy Stepwise process: 4) From the data find the estimator * C.W. Helstrom Quantum Detection and Estimation Theory (1976).

Two important limits for this “interferometric-measurement scheme” for phase estimation * (  1, g the largest Hamiltonian gap): * V. Giovannetti, S. Lloyd, L. Maccone, Nature Photonics 5, 222 (2011). N probes, repetitions. N-entangled probes, repetitions. Standard Quantum Limit (SQL) or “shot” noise limit Heisenberg limit

In usual estimation problems,  obey the Cramér-Rao bound: where F (  ) is the Fisher information. In quantum estimation problems, this bound (quantum Cramér-Rao bound) is given by: Symmetric Logarithm Derivative (optimal measurement)

Is entanglement the only resource for enhanced estimation that Quantum Mechanics can give us? Fortunately no! We also have... Nature 474, (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 H A that generate the phase  over the partition A is given by 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: 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:

Invariant under local unitaries and nonincreasing under local operations on B ; Vanishes iff  AB is classically correlated; Reduces to an entanglement monotone for pure states; It is analytically computable if A is a qubit. Characteristics of Examples for two qubits (obs: id AB = 4x4 identity matrix): 1) Werner states 2) Bell diagonal states Details in ArXiv:

Suppose two families of states * : with quantum discord. classically correlated. * K. Modi et al. PRX 1, (2011).

Results: Experiment

What shall we measure? What shall we test experimentally? First: interferometric scheme Second: check discord in the initial states Third: verify the metrological quantities Compare and check if discord can be seen as a resourse for quantum metrology!

NMR CBPF Target: Prepare Start preparing:

After preparing state, we implement the circuits below to obtain the desired states. It is important to note that Fidelity above 99% for initial states!

How to implement unknown phase shift? Setting the phase to be estimated as We can choose three directions to rotate

Ok. But what is the (optimal) measurement? We must measure in the eigenbasis of the symmetric logarithm derivative to obtain the maximum allowed precision. Since: We can map the eigenvectors onto the computational basis of two qubits. Doing so, the ensemble expectation values can be directly observed in the diagonal elements of the density matrix. 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:

From the experiment (ArXiv: ):

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.

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) These guys are around here!

Thanks for the attention!