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Experimental quantum estimation using NMR

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

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Experimental quantum estimation using NMR

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  1. Operational significance of discord in quantum metrology: Theory and Experiment* Experimental quantum estimationusing NMR Diogo de Oliveira Soares Pinto Instituto de Física de São Carlos Universidade de São Paulo (dosp@ifsc.usp.br) NMR – QIP in Rio November 2013 *Titleinspired in Nat.Phys. 8, 671 (2012)

  2. Outline: (Verybrief) Introductionto quantum metrology Results: Theory Result: Experiment Conclusions

  3. (Verybrief) Introductionto quantum metrology

  4. In the lab... Entangledstate? Quantum statetomography = experimental data Eigenvaluesof R orderedfromthehighesttothelowest Data analysis Entangledornot? Estimationproblem!

  5. 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 . 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*: * C.W. Helstrom Quantum DetectionandEstimationTheory (1976).

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

  7. 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)

  8. 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. Modiet al. Rev. Mod. Phys. 84, 1655 (2012).

  9. Results:Theory

  10. Let’s go back to the interferometric scheme. Suppose that the Hamiltonian HA 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:1309.1472.

  11. Invariant under local unitaries and nonincreasing under local operations on B; • Vanishes iffABis 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: idAB = 4x4 identity matrix): 1) Werner states 2) Bell diagonal states Details in ArXiv:1309.1472.

  12. Suppose two families of states*: classicallycorrelated. with quantum discord. *K. Modiet al. PRX 1, 021022 (2011).

  13. Results:Experiment

  14. Whatshallwemeasure? Whatshallwetestexperimentally? First:interferometricscheme Second:checkdiscord in theinitialstates Third:verifythemetrologicalquantities Compare andcheckifdiscordcanbeseen as a resourse for quantum metrology!

  15. NMR system: Target: Prepare @ CBPF Start preparing:

  16. Afterpreparingstate , weimplementthecircuitsbelowtoobtainthedesiredstates. It isimportantto note that Fidelityabove 99% for initialstates!

  17. How to implement unknown phase shift? Setting thephasetobeestimated as We can choose three directions to rotate

  18. 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. Buthow?

  19. The answer: Global rotationdependenton s and k!

  20. Example for s = C, Q and k = 1: Thiscanbedonealso for s = C, Q and k = 2, 3. ArXiv:1309.1472.

  21. Fromtheexperiment (ArXiv: 1309.1472):

  22. Conclusions

  23. Operationalinterpretationof quantum discord in termsof a resourse for quantum estimationproblemswhenisconsideredtheworst case scenario! In settings like NMR, wheredisorderis high, quantum correlationsevenwithoutentanglementcanbe a promisingresourse for quantum technology. Takingadvantageofthenameproposedfor theprotocol (blind quantum metrology), I canfinishciting: “Perhaps only in a world of the blind will things be what they truly are.” Saramago – Blindness. orbetter: “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.

  24. Theseguys are aroundhere! • People involved: • DavideGirolami – NUS (Singapore) • Vittorio Giovannetti – SNS (Italy) • TommasoTufarelli – 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)

  25. Thanks for the attention!

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