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Generating and Harnessing Photonic Entanglement

This talk explores the generation and utilization of photonic entanglement for quantum information processing. Topics include qubits, CNOT gates, quantum process tomography, generalized quantum measurements, qutrits, and qudits.

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Generating and Harnessing Photonic Entanglement

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  1. Generating and harnessing photonic entanglement Geoff Pryde Quantum Technology Lab Rohan Dalton Michael Harvey Nathan Langford Till Weinhold Jeremy O’Brien Geoff Pryde Andrew White www.quantinfo.org Theory Colleagues Stephen Bartlett Aggie Branczyk Michael Bremner Jen Dodd Andrew Doherty Alexei Gilchrist Gerard Milburn Michael Nielsen Tim Ralph Funding:

  2. Alexei Gilchrist Agatha Brancyzk Stephen Bartlett Rohan Dalton Andrew White Jeremy O’Brien Michael Harvey Geoff Pryde Nathan Langford www.quantinfo.org Till Weinhold Gerard Milburn Tim Ralph

  3. www.quantinfo.org Generating and harnessing photonic entanglement Talk outline • Qubits • CNOT gate • Quantum process tomography  • Generalized quantum measurements with photons • Qutrits and Qudits • Gaussian spatial modes • Constructing and measuringqutrits • Use in quantum bit commitment

  4. Polarization qubits 3Single qubit gates Hadamard gate |0ñ |1ñ |Hñ /2 (|Hñ|Vñ)/√2 V H Arbitrary rotation gate |Hñ / /2 / |Hñ |Vñ Poincaré Sphere 3Single qubits • ?Two-qubit gates

  5. Basic photonic CNOT HT HT CNOT = HT + CSIGN + HT CSIGN gate C C 0 0 C C 1 1 p phase shift T T 0 0 T T 1 1

  6. 2-photon CNOT operation CSIGN gate -1/3 1/3 1/3 Ralph, Langford, Bell & White, PRA65, 062324 (2002)Hofmann & Takeuchi, PRA66, 024308 (2002)

  7. 2-photon CNOT operation both reflected both transmitted CSIGN gate -1/3 1/3 1/3 Ralph, Langford, Bell & White, PRA65, 062324 (2002)Hofmann & Takeuchi, PRA66, 024308 (2002)

  8. 2-photon CNOT operation CSIGN gate -1/3 1/3 1/3 Ralph, Langford, Bell & White, PRA65, 062324 (2002)Hofmann & Takeuchi, PRA66, 024308 (2002)

  9. Polarization 2-photon CNOT CNOT gate -1/3 Control out Control in 1/3 Target out Target in 1/3 Ralph, Langford, Bell & White, PRA65, 062324 (2002)Hofmann & Takeuchi, PRA66, 024308 (2002)

  10. Concatenating CNOTs T. C. Ralph, quant-ph/0306190 J. L. Dodd et al., quant-ph/0306081

  11. 2-photon CNOT in the context of scalable QC LOQC = “Linear Optics Quantum Computing” Knill, Laflamme and Milburn, Nature409, 46 (2001)

  12. 2-photon CNOT circuit Non-classical interference Very stable: insensitive to x-y-z translation J. L. O’Brien, G. J. Pryde, et al., Nature 426, 264 (2003)

  13. Truth table Ideal Measured Average logical fidelity = Tr [MidealMmeas]/4 = 94 ± 2 % O’Brien, Pryde, et al., Nature 426, 264 (2003) O’Brien, Pryde, et al., PRL 93, 080502 (2004)

  14. State tomography Populations Coherences Fidelity = 92 % Ideal Measured: Real Measured: Imaginary |Cñ|Tñin = |0 -1ñ|1ñ = |H -Vñ|Vñ

  15. Characterizing the gate itself Decompose into “basis” operations, e.g. rotations ?

  16. Quantum process tomography Any physical process can be written as a completely positive map: For a CNOT: +0.34 Ideal Measured (Re)

  17. Quantum process tomography Measured (Re) Measured (Im) • Physical interpretation? Change basis CNOT • (II, IX, IY, IZ, XI, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ) = 87%

  18. Gate measures 95 ± 1% • Average gate fidelity where d is the Hilbert space dimension • Average error rate ≤ C2 = 1-Fp where C is process “closeness” 7 ± 1% • Direct measurement of process fidelity 71 measurements 93 ± 1% • Chief source of non-ideal gate operation Gilchrist, Langford, and Nielsen, quant-ph/0408063 O’Brien, Pryde, et al., PRL 93, 080502 (2004)

  19. QND and Generalized Measurement Measurement outcome is correlated with the signal input The measurement does not alter the value of the measured obsevable Repeated measurement yields the same result - quantum state preparation (QSP)  Grangier et al. Nature 396, 537

  20. CNOT gate as a QND device |Controlin |Controlout CNOT |Targetout |Targetin = |0 Measure

  21. Experimental scheme for QND 1/3 Non-deterministic: when 1 photon is detected in the meter output the measurement is known to have succeeded

  22. Experimental realization 0.46 0.51 0.90 0.47

  23. Fidelity measures • Each compares two probability distributions p and q using the classical fidelity: F > 85% for all input states

  24. Scheme for generalized measurement Half wave plate 1/3 V 

  25. Complementarity Most advanced general measurement of a qubit: non-destructive; arbitrary strength; any basis; BUT non-deterministic Pryde, O’Brien, White, Bartlett & Ralph PRL 92, 190402 (2004)

  26. Z-measurement error correction Syndrome measured but not corrected Teleported gates fail by making a Z-measurement Knill, LaFlamme and Milburn, Nature 409, 46 (2001) Pittman, Jacobs and Franson, PRA, 64, 062311 (2001) Average Fidelity 96 ± 3 % Nielsen, PRL, 93, 040503 (04) Browne & Rudolph,quant-ph/0405157 LOQC cluster states O’Brien, Pryde White and Ralph, quant-ph/0408064 a (|HH> + |VV>) + b |VH> + |HV>) “V” a |V> + b |H>

  27. QUTRITS

  28. Encoding information in single photons Polarisation qubits: What if we want to create photonic qudits? • Gaussian spatial modes: • Infinite dimensional • Discrete • Orthogonal • Can describe any paraxial beam

  29. Gaussian spatial modes What about other spot shapes? … for paraxial beams Gaussian optical mode:

  30. Gaussian spatial modes Siegman, Lasers (1986) Bandres et al., Optics Letters29, 144 (2004) Gouy phase shift Non-vortex Mode Families … Hermite-Gauss: (HG) rectangular Ince-Gauss: (IG) elliptical Laguerre-Gauss: (LGN) cylindrical Vortex Mode Families (carrying orbital angular momentum) Laguerre-Gauss: (LGV)

  31. Encoding q. information in spatial modes HG LGV Degenerate Qubit Non-degenerate Qubit H V G R • same Gouy phase shift • stationary superposition patterns • different Gouy phase shifts • evolving superposition patterns D R Non-degenerate Qutrit |0=L |1=G |2=R Langford et al., PRL 93, 053601 (2004) Mair et al., Nature412, 313 (2001); Vaziri et al., PRL 89, 240401 (2002)

  32. Hologram production

  33. The holograms - how do they work? Spatial Mode Conversion Medium Spatial Frequency High Spatial Frequency Diffraction Orders Distribution Envelope Low Spatial Frequency

  34. Spatial mode quantum state tomography Type-I Spontaneous Parametric Down-conversion (SPDC) Conceptual Experimental Diagram • Post-select coincident photon pairs using counting electronics. • Two-photon QST: all possible pairs of single-photon measurements. • Analyse the spatial mode using holograms and single-mode fibres. • What is the spatial mode quantum state of the photon pairs?

  35. Spatial mode analyzer (SMA) • Analyser extinction efficiency: ~1:103 • Uses holograms and single-mode fibres (SMFs):

  36. Spatial mode quantum state tomography

  37. Spatial mode quantum state tomography populations

  38. Spatial mode quantum state tomography coherences

  39. Spatial mode quantum state tomography coherences

  40. Two-qutrit quantum state tomography Non-degenerate Qutrit Re(r) Im(r) • EOF < 0.704 • SL = 0.18 • Fy = 0.88 |0=L |1=G |2=R Langford et al., quant-ph/0312072

  41. Quantum bit commitment ? 0 29-39-5 • Communication between mistrustful parties • Basis of other protocols, e.g. quantum coin flipping 0 29-39-5 • Alice should commit to a message and not be able to change it. • Bob should not be able to decode the message until Alice reveals it. • Quantum bit-commitment with arbitrarily good security is impossible • Qutrits offer the best-known BC security levels, whereas qubits do not!

  42. Quantum bit commitment A Simulated Purification Protocol • Uses an entangled, two-part system: • (a) the proof and (b) the token subsystems. • Assumption: initial state is only source of imperfection. Re(r) Im(r) Spekkens and Rudolph, PRA 65, 012310 (2001) Langford et al., quant-ph/0312072 Step 1: Alice starts with our experimentally measured two-qutrit state.

  43. Quantum bit commitment Step 2: Alice prepares her chosen logical bit. Step 3: Alice sends the token to Bob (the commitment). orthogonal two-qutrit states non-orthogonal token states

  44. Quantum bit commitment Step 4: Alice sends the proof subsystem to Bob to complete the BC protocol. He decodes the message with a two-qutrit projective measurement: • Distinguishability of token states limits Alice’s control (trace-distance): • Non-orthogonal token states limit Bob’s possible knowledge gain (fidelity):

  45. Quantum bit commitment classical r = p/3I + (1-p)rideal p = 0.29 p = 0.19 p = 0.09 • Fidelity: impossible achievable with qubits qutrits inaccessible to best known BC protocols KEY POINTS • Orthogonal two-qutrit states result in non-orthogonal reduced token states. • Ideal case provides optimal security, but simulated case still does not! Langford et al., quant-ph/0312072

  46. Conclusions • Quantum process tomography of CNOT– fully characterize the process in the 2-qubit space– high fidelity operation  useful for q. info and q. physics tests • Generalized measurement and QND – non-destructive; arbitrary strength; any basis • Qutrit entanglement– measured, characterized for use in communications protocols

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