1 / 41

Daniel F. V. JAMES Department of Physics & Center for Quantum Information and Quantum Control

Measuring and Characterizing Quantum States and Processes. Daniel F. V. JAMES Department of Physics & Center for Quantum Information and Quantum Control University of Toronto QELS ’10, San Jose CA QFF-Quantum State Reconstruction/QFF-1 21 May 2010. Outline.

Download Presentation

Daniel F. V. JAMES Department of Physics & Center for Quantum Information and Quantum Control

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Measuring and Characterizing Quantum States and Processes Daniel F. V. JAMES Department of Physics & Center for Quantum Information and Quantum Control University of Toronto QELS ’10, San Jose CA QFF-Quantum State Reconstruction/QFF-1 21 May 2010

  2. Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits). 3. Positivity: The first big problem, and its fixes. 4. Characterizing quantum states. 5. Quantum Processes. 6. Scalability: the second big problem. 7. Conclusions and resources.

  3. Measure Single Copy by projecting on to : get answer “click” or “no click” • One bit of information about a and b: you know one of them is non-zero §1: State of a Single Qubit • Photon polarization based qubits • Measure multiple (assumed identical) copies: frequency of “clicks” gives estimate of |b|2

  4. Find relative phase of a and b by performing a unitary operation before beam splitter: e.g.: • Frequency of “clicks” now gives an estimate of • Systematic way of getting all the data needed….

  5. (i) 50% intensity (ii) H-polarizer (iii) 45o polarizer (iv) RCP Stokes Parameters Measure intensity with four different filters: G. G. Stokes, Trans Cambridge Philos Soc9 399 (1852)

  6. Stokes Parameters • These 4 parameters completely specify polarization of beam • Beam is an ensemble of photons… Pauli matrices

  7. Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits).

  8. §2: Two Qubit Quantum States • Pure states • Ideal case • Mixed states • Quantum state is random: need averages and correlations of coefficients

  9. Arbitrary state: change basis… State Creation by OPDC

  10. source measurement Two-Qubit Quantum State Tomography Coincidence Rate measurements for two photons

  11. Linear combination of na,b yields the two-photon Stokes parameters: From the two-photon Stokes parameters, we can get an estimate of the density matrix: • Doesn’t give the right answer….

  12. Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits). 3. Positivity: The first big problem, and its fixes.

  13.   2/3 - must try harder! What about: ? “sea” of negative matrices “happy beach” of positive states experimental data (with error bars) §3: The First Problem… • Properties of Density matrices: - Hermitian: - unit trace: - non-negative definite  all eigenvalues are non-negative: Q: how to find the “best” positive from noisy data?

  14. Goodness? • Data is random, with (say) a Gaussian distribution, with mean given by the expectation values determined by the density matrix: • find the “best”  by ensuring this probability is a maximum

  15. • We need a  such that • “Cholesky decomposition” - André-Louis Cholesky (French Army Officer, 1875-1918) - non-negative matrices can be written as follows: Enforcing Positivity • Incorporate constraints in numerical search so that the eigenvalues are positive -tedious mucking about finding the eigenvalues each step -a bunch of Lagrange multipliers to find

  16. where: r = TT†/Tr{TT†}  and Maximum Likelihood Tomography* • Numerically Minimize the function: • Maximum Likelihood fit to "physical" density matrix • Density matrix must be Hermitian, normalized, non-negative • * D. F. V. James, et al., Phys Rev A64, 052312 (2001).

  17. Example: Measured Density Matrix

  18. Quantum State Tomography I • Sublevels of Hydrogen (partial) (Ashburn et al, 1990) • Optical mode (Raymer et al., 1993) • Molecular vibrations (Walmsley et al, 1995) • Motion of trapped ion (Wineland et al., 1996) • Motion of trapped atom (Mlynek et al., 1997) • Liquid state NMR (Chaung et al, 1998) • Entangled Photons (Kwiat et al, 1999) • Entangled ions (Blatt et al., 2002; 8 ions: 2005) • Superconducting qubits (Martinis et al., 2006)

  19. Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits). 3. Positivity: The first big problem, and its fixes. 4. Characterizing quantum states.

  20. Fidelity: how close are two states? Pure states: Mixed states: doesn’t work: §4: Characterizing the State Purity

  21. Entropy of reduced density matrix of one photon • Concurrence: • Concurrence is equivalent to Entanglement : • C=0 implies separable state • C=1 implies maximally entangled state (e.g. Bell states) Measures of Entanglement • Pure states • How much entanglement is in this state?

  22. “Average” Concurrence: dependent on decomposition • “Minimized Average Concurrence”: • Independent of decomposition • C=0 implies separable state • C=1 implies maximally entangled state (e.g. Bell states) • Analytic expression (Wootters, ‘98) makes things very convenient! Entanglement in Mixed States • Mixed states can be de-composed into incoherent sums of pure (non-orthogonal) states:

  23. Two Qubit Mixed State Concurrence “spin flip matrix” Transpose (in computational basis) Eigenvalues of R (in decreasing order) W.K. Wootters, Phys. Rev. Lett.80, 2245 (1998)

  24. MEMS states *W. J. Munro et al., Phys Rev A 64, 030302-1 (2001) “Map” of Hilbert Space* * D.F.V. James and P.G .Kwiat, Los Alamos Science, 2002

  25. Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits). 3. Positivity: The first big problem, and its fixes. 4. Characterizing quantum states. 5. Quantum Processes.

  26. set of basis matrices, e.g.: Trace orthogonality: §5: Process Tomography • Trace Preserving Completely Positive Maps: Every thing that could possibly happen to a quantum state “operator-sum formalism” “Kraus operators”

  27. then- where- c is a Hermitian, positive 16x16 matrix(“error correlation matrix”), with the constraints- Decompose the Kraus operators:  is almost like “Choi-Jamiolkowski isomorphism”

  28. Process Tomography * 16 Input states 16 Projection states • Estimate probability from counts 16x16 = 256 data: • Recover cmn by linear inversion - problematic in constraining positivity - close analogy with state tomography… *I. L. Chuang and M. A. Nielsen, J. Mod Op.44, 2455 (1997)

  29. Maximum Likelihood Process Tomography • Numerically optimize where: (256 free parameters) • Constraints on cmn : - positive - Hermitian - additional constraint for physically allowed process:

  30. Process Tomography of UQ Optical CNOT* Most Likely cmn matrix Actual CNOT *J. L. O’Brien et al., “Quantum process tomography of a controlled-NOT gate,” Phys Rev Lett,93, 080502 (2004); quant-ph/0402166.

  31. Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits). 3. Positivity: The first big problem, and its fixes. 4. Characterizing quantum states. 5. Quantum Processes. 6. Scalability: the second big problem.

  32. §6: Scalability? • record: 8 qubit W-state (Blatt et al., 2006) • Why not more? N qubit state tomography requires 4N-1 measurements (& numerical optimization in a 4N-1 dimensional space)

  33. Fixes? • Measurements: you can get a good guess at the density matrix with fewer measurements (it still requires exponential searching) (Aaronson, 2006) • Direct Characterization: In some cases you can get the information you need more directly, without the tedious mucking around with the density matrix (e.g. entanglement witnesses; noise characterization) • Push the envelope: How far can we go using smart computer science before we hit the wall? - convex optimization - improved data handling and processing -other approaches to ‘optimziation’

  34. Is “the best” the enemy of “good enough” Are we being too pedantic in looking for the optimal density matrix to fit a given data set, when a simpler numerical technique produces a good estimation (i.e. within the error bars)? Alternatives: - ‘quick and dirty’: zero out the negative eigenvalues rather than perform an exhaustive optimization. - ‘forced purity’: we are trying to make specifc states, so why not use that fact? Both give positive matrices quickly: but are they the actual states in question? • M. Kaznady and D. F. V. James, Phys Rev A79, 022109 (2009); arXiv:0809.2376.

  35. Numerical experiments • choose a state • simulate measurement data with a Poisson RNG • estimate state using code • compare estimated and actual state Q&D - Fidelities for 2-qubit states: MLE FP

  36. Bayesian Approach* *Robin Blume-Kohout, “Optimal,reliable estimation of quantum states,” New Journal of Physics12 043034 (2010)

  37. Polynomial Time Tomography?* Choose a few-parameter set of states suitable for what you are trying to do. Find a protocol to find the best set of parameters to fit your data. Check that the state thus recovered is a good approximation to the actual data *S. T. Flammia et al., “Heralded Polynomial-Time Quantum State Tomography” arXiv:1002:3839; see also M. Cramer and M. Plenio, arXiv:1002.3780

  38. Outline 1. Single qubit tomography = Polarimetry 2. Two (and more) photons (and other qubits). 3. Positivity: The first big problem, and its fixes. 4. Characterizing quantum states. 5. Quantum Processes. 6. Scalability: the second big problem. 7. Conclusions and resources.

  39. Conclusions • Maybe these techniques can give reasonably good characterization of a dozen or so qubits…. • Beyond that, how an we know quantum computers is doing what it’s meant to? - well characterized components. - error correction: you can’t know if it’s bust or not, so you’d best fix it anyway. - answers are easy to check.

  40. Resources • A gentle introduction: D.F.V. James and P.G .Kwiat, “Quantum State Entanglement: Creation, characterization, and application,” Los Alamos Science, 2002 • More details (books, compilation volumes etc.): U. Leonhardt,Measuring the Quantum State of Light (Cambridge, 1997) [tomography of harmonic oscillator modes via inverse Radon transforms] Quantum State Estimation, Lecture Notes in Physics, Vol. 649, M. G. A. Paris and J.Řeháček, eds. (Springer, Heidelberg, 2004) Asymptotic Theory of Quantum Statistical Inference: SelectedPapers, edited by M. Hayashi (World Scientific, Singapore,2005)

  41. Thanks to... • My Group: Dr. René Stock Asma Al-Qasimi Omal Gamel Max Kaznady Ardavan Darabi Faiyaz Hasan Timur Rvachov Bassam Helou • Funding Agencies: • Collaborators/helpers: Paul Kwait Andrew White Bill Munro HartmutHäffner Robin Blume-Kohout Steve Flammia Christian Roos

More Related