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

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.

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

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  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

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