Abstract

Accurate characterization of two-qubit gates will be critical for any realization of quantum computation. We discuss a range of measurements for characterizing two-qubit gates. These measures are architecture-independent and span a range of complexity from simple measurement routines to full quantum-state and process tomography. Simple indicative measures, which flag but do not quantify gate operation in the quantum regime, include the fringe visibility, parity, Bell-state fidelity, and entanglement witnesses. Quantitative measures of gate output states include linear entropy and tangle; measures of, and error bounds to, whole-gate operation are provided by metrics such as process fidelity, process distance, and average gate fidelity. We discuss which measures are appropriate, given the stage of development of the gate, and highlight connections between them.

© 2007 Optical Society of America

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    [CrossRef]
  49. M. A. Nielsen, "A simple formula for the average gate fidelity of a quantum dynamical operation," Phys. Lett. A 303, 249-252 (2002).
    [CrossRef]
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2006 (1)

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, "Measurement of the entanglement of two superconducting qubits via state tomography," Science 313, 1423-1425 (2006).
[CrossRef] [PubMed]

2005 (3)

A. Gilchrist, N. K. Langford, and M. A. Nielsen, "Distance measures to compare real and ideal quantum processes," Phys. Rev. A 71, 062310 (2005).
[CrossRef]

N. K. Langford, T. J. Weinhold, R. Prevedel, A. Gilchrist, J. L. O'Brien, G. J. Pryde, and A. G. White, "Demonstration of a simple entangling optical gate and its use in Bell-state analysis," Phys. Rev. Lett. 95, 210504 (2005).
[CrossRef] [PubMed]

H. F. Hofmann, "Complementary classical fidelities as an efficient criterion for the evaluation of experimentally realized quantum operations," Phys. Rev. Lett. 94, 160504 (2005).
[CrossRef] [PubMed]

2004 (2)

J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, "Quantum process tomography of a controlled-NOT gate," Phys. Rev. Lett. 93, 080502 (2004).
[CrossRef] [PubMed]

N. A. Peters, J. B. Altepeter, D. A. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, "Maximally entangled mixed states: creation and concentration," Phys. Rev. Lett. 92, 133601 (2004).
[CrossRef] [PubMed]

2003 (6)

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, "Ancilla-assisted quantum process tomography," Phys. Rev. Lett. 93, 193601 (2003).
[CrossRef]

G. M. D'Ariano and P. L. Presti, "Imprinting a complete information about a quantum channel on its output state," Phys. Rev. Lett. 91, 047902 (2003).
[CrossRef] [PubMed]

F. De Martini, A. Mazzei, M. Ricci, and G. M. D'Ariano, "Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single qubit device," Phys. Rev. A 67, 062307 (2003).
[CrossRef]

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, "Realization of the Cirac-Zoller controlled-NOT quantum gate," Nature 422, 408-411 (2003).
[CrossRef] [PubMed]

J. L. O'Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum controlled-NOT gate," Nature 426, 264-267 (2003).
[CrossRef] [PubMed]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

2002 (4)

O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, "Detection of entanglement with few local measurements," Phys. Rev. A 66, 062305 (2002).
[CrossRef]

M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, "Practical scheme for quantum computation with any two-qubit entangling gate," Phys. Rev. Lett. 89, 247902 (2002).
[CrossRef] [PubMed]

J. L. Dodd and M. A. Nielsen, "A simple operational interpretation of the fidelity," Phys. Rev. A 66, 044301 (2002).
[CrossRef]

M. A. Nielsen, "A simple formula for the average gate fidelity of a quantum dynamical operation," Phys. Lett. A 303, 249-252 (2002).
[CrossRef]

2001 (6)

J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, "Entangling operations and their implementation using a small amount of entanglement," Phys. Rev. Lett. 86, 544-547 (2001).
[CrossRef] [PubMed]

A. M. Childs, I. L. Chuang, and D. W. Leung, "Realization of quantum process tomography in NMR," Phys. Rev. A 64, 012314 (2001).
[CrossRef]

W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239-1246 (2001).
[CrossRef]

W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat, "Maximizing the entanglement of two mixed qubits," Phys. Rev. A 64, 030302 (2001).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312 (2001).
[CrossRef]

A. G. White, D. F. V. James, W. J. Munro, and P. G. Kwiat, "Exploring Hilbert space: accurate characterization of quantum information," Phys. Rev. A 65, 012301 (2001).
[CrossRef]

2000 (4)

B. M. Terhal, "Bell inequalities and the separability criterion," Phys. Lett. A 271, 319-326 (2000).
[CrossRef]

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, "Experimental entanglement of four particles," Nature 404, 256-259 (2000).
[CrossRef] [PubMed]

B. M. Terhal and P. Horodecki, "Schmidt number for density matrices," Phys. Rev. A 61, 040301(R) (2000).
[CrossRef]

V. Coffman, J. Kundu, and W. K. Wootters, "Distributed entanglement," Phys. Rev. A 61, 052306 (2000).
[CrossRef]

1999 (2)

M. Horodecki, P. Horodecki, and R. Horodecki, "General teleportation channel, singlet fraction, and quasidistillation," Phys. Rev. A 60, 1888-1898 (1999).
[CrossRef]

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally entangled states: production, characterization, and utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
[CrossRef]

1998 (2)

W. K. Wootters, "Entanglement of formation of an arbitrary state of two qubits," Phys. Rev. Lett. 80, 2245-2248 (1998).
[CrossRef]

M. A. Nielsen, E. Knill, and R. Laflamme, "Complete quantum teleportation using nuclear magnetic resonance," Nature 396, 52-55 (1998).
[CrossRef]

1997 (2)

I. L. Chuang and M. A. Nielsen, "Prescription for experimental determination of the dynamics of a quantum black box," J. Mod. Opt. 44, 732-744 (1997).

J. F. Poyatos, J. I. Cirac, and P. Zoller, "Complete characterization of a quantum process: the two-bit quantum gate," Phys. Rev. Lett. 78, 390-393 (1997).
[CrossRef]

1996 (1)

M. Horodecki, P. Horodecki, and R. Horodecki, "Separability of mixed states: necessary and sufficient conditions," Phys. Lett. A 223, 1-8 (1996).
[CrossRef]

1972 (1)

A. Jamiolkowski, "Linear transformations which preserve trace and positive semidefiniteness of operators," Rep. Math. Phys. 3, 275-278 (1972).
[CrossRef]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, "Proposed experiment to test local hidden-variable theories," Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

1964 (1)

J. S. Bell, "On the Einstein-Podolsky-Rosen paradox," Physics (Long Island City, N.Y.) 1, 195-200 (1964).
[CrossRef]

Alicki, R.

R. Alicki, "False qubits II. Entanglement of Josephson junctions," arXiv.org e-Print archive, quantum physics/0609122, 16 September 2006, http://www.arxiv.org/abs/quant-ph/0609122.

Altepeter, J. B.

N. A. Peters, J. B. Altepeter, D. A. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, "Maximally entangled mixed states: creation and concentration," Phys. Rev. Lett. 92, 133601 (2004).
[CrossRef] [PubMed]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, "Ancilla-assisted quantum process tomography," Phys. Rev. Lett. 93, 193601 (2003).
[CrossRef]

Ansmann, M.

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, "Measurement of the entanglement of two superconducting qubits via state tomography," Science 313, 1423-1425 (2006).
[CrossRef] [PubMed]

Barbieri, M.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

Becher, C.

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, "Realization of the Cirac-Zoller controlled-NOT quantum gate," Nature 422, 408-411 (2003).
[CrossRef] [PubMed]

Bell, J. S.

J. S. Bell, "On the Einstein-Podolsky-Rosen paradox," Physics (Long Island City, N.Y.) 1, 195-200 (1964).
[CrossRef]

Bialczak, R. C.

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, "Measurement of the entanglement of two superconducting qubits via state tomography," Science 313, 1423-1425 (2006).
[CrossRef] [PubMed]

Blatt, R.

F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, "Realization of the Cirac-Zoller controlled-NOT quantum gate," Nature 422, 408-411 (2003).
[CrossRef] [PubMed]

Blume-Kohout, R.

R. Blume-Kohout and P. Hayden, "Accurate quantum state estimation via 'Keeping the experimentalist honest'," arXiv.org e-Print archive, quantum physics/0603116, 14 March 2006, http://www.arxiv.org/abs/quant-ph/0603116.

Branning, D.

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, "Ancilla-assisted quantum process tomography," Phys. Rev. Lett. 93, 193601 (2003).
[CrossRef]

J. L. O'Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum controlled-NOT gate," Nature 426, 264-267 (2003).
[CrossRef] [PubMed]

Branning, D. A.

N. A. Peters, J. B. Altepeter, D. A. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, "Maximally entangled mixed states: creation and concentration," Phys. Rev. Lett. 92, 133601 (2004).
[CrossRef] [PubMed]

Bremner, M. J.

M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, "Practical scheme for quantum computation with any two-qubit entangling gate," Phys. Rev. Lett. 89, 247902 (2002).
[CrossRef] [PubMed]

Bruß, D.

O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, "Detection of entanglement with few local measurements," Phys. Rev. A 66, 062305 (2002).
[CrossRef]

Brylinski, J. L.

J. L. Brylinski and R. K. Brylinski, "Computational mathematics," in Mathematics of Quantum Computation, R.K.Brylinski and G.Chen, eds. (Chapman & Hall/CRC Press, 2002), Chap. 2.

Brylinski, R. K.

J. L. Brylinski and R. K. Brylinski, "Computational mathematics," in Mathematics of Quantum Computation, R.K.Brylinski and G.Chen, eds. (Chapman & Hall/CRC Press, 2002), Chap. 2.

Childs, A. M.

A. M. Childs, I. L. Chuang, and D. W. Leung, "Realization of quantum process tomography in NMR," Phys. Rev. A 64, 012314 (2001).
[CrossRef]

Chuang, I. L.

A. M. Childs, I. L. Chuang, and D. W. Leung, "Realization of quantum process tomography in NMR," Phys. Rev. A 64, 012314 (2001).
[CrossRef]

I. L. Chuang and M. A. Nielsen, "Prescription for experimental determination of the dynamics of a quantum black box," J. Mod. Opt. 44, 732-744 (1997).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

Cirac, J. I.

J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, "Entangling operations and their implementation using a small amount of entanglement," Phys. Rev. Lett. 86, 544-547 (2001).
[CrossRef] [PubMed]

J. F. Poyatos, J. I. Cirac, and P. Zoller, "Complete characterization of a quantum process: the two-bit quantum gate," Phys. Rev. Lett. 78, 390-393 (1997).
[CrossRef]

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, "Proposed experiment to test local hidden-variable theories," Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

Cleland, A. N.

M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, "Measurement of the entanglement of two superconducting qubits via state tomography," Science 313, 1423-1425 (2006).
[CrossRef] [PubMed]

Coffman, V.

V. Coffman, J. Kundu, and W. K. Wootters, "Distributed entanglement," Phys. Rev. A 61, 052306 (2000).
[CrossRef]

D'Ariano, G. M.

F. De Martini, A. Mazzei, M. Ricci, and G. M. D'Ariano, "Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single qubit device," Phys. Rev. A 67, 062307 (2003).
[CrossRef]

G. M. D'Ariano and P. L. Presti, "Imprinting a complete information about a quantum channel on its output state," Phys. Rev. Lett. 91, 047902 (2003).
[CrossRef] [PubMed]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
[CrossRef] [PubMed]

Dawson, C. M.

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N. A. Peters, J. B. Altepeter, D. A. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, "Maximally entangled mixed states: creation and concentration," Phys. Rev. Lett. 92, 133601 (2004).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, "Experimental entanglement of four particles," Nature 404, 256-259 (2000).
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N. A. Peters, J. B. Altepeter, D. A. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, "Maximally entangled mixed states: creation and concentration," Phys. Rev. Lett. 92, 133601 (2004).
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J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, "Ancilla-assisted quantum process tomography," Phys. Rev. Lett. 93, 193601 (2003).
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A. G. White, D. F. V. James, W. J. Munro, and P. G. Kwiat, "Exploring Hilbert space: accurate characterization of quantum information," Phys. Rev. A 65, 012301 (2001).
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A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally entangled states: production, characterization, and utilization," Phys. Rev. Lett. 83, 3103-3107 (1999).
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F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, "Realization of the Cirac-Zoller controlled-NOT quantum gate," Nature 422, 408-411 (2003).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, "Experimental entanglement of four particles," Nature 404, 256-259 (2000).
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A. Gilchrist, N. K. Langford, and M. A. Nielsen, "Distance measures to compare real and ideal quantum processes," Phys. Rev. A 71, 062310 (2005).
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N. K. Langford, T. J. Weinhold, R. Prevedel, A. Gilchrist, J. L. O'Brien, G. J. Pryde, and A. G. White, "Demonstration of a simple entangling optical gate and its use in Bell-state analysis," Phys. Rev. Lett. 95, 210504 (2005).
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J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, "Quantum process tomography of a controlled-NOT gate," Phys. Rev. Lett. 93, 080502 (2004).
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J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein, "Entangling operations and their implementation using a small amount of entanglement," Phys. Rev. Lett. 86, 544-547 (2001).
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M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
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O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, "Detection of entanglement with few local measurements," Phys. Rev. A 66, 062305 (2002).
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M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, "Measurement of the entanglement of two superconducting qubits via state tomography," Science 313, 1423-1425 (2006).
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M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D'Ariano, and C. Macchiavello, "Detection of entanglement with polarized photons: experimental realization of an entanglement witness," Phys. Rev. Lett. 91, 227901 (2003).
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F. De Martini, A. Mazzei, M. Ricci, and G. M. D'Ariano, "Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single qubit device," Phys. Rev. A 67, 062307 (2003).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, "Experimental entanglement of four particles," Nature 404, 256-259 (2000).
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M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, "Practical scheme for quantum computation with any two-qubit entangling gate," Phys. Rev. Lett. 89, 247902 (2002).
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A. G. White, D. F. V. James, W. J. Munro, and P. G. Kwiat, "Exploring Hilbert space: accurate characterization of quantum information," Phys. Rev. A 65, 012301 (2001).
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A. Gilchrist, N. K. Langford, and M. A. Nielsen, "Distance measures to compare real and ideal quantum processes," Phys. Rev. A 71, 062310 (2005).
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I. L. Chuang and M. A. Nielsen, "Prescription for experimental determination of the dynamics of a quantum black box," J. Mod. Opt. 44, 732-744 (1997).

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J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, "Ancilla-assisted quantum process tomography," Phys. Rev. Lett. 93, 193601 (2003).
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N. K. Langford, T. J. Weinhold, R. Prevedel, A. Gilchrist, J. L. O'Brien, G. J. Pryde, and A. G. White, "Demonstration of a simple entangling optical gate and its use in Bell-state analysis," Phys. Rev. Lett. 95, 210504 (2005).
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J. L. O'Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum controlled-NOT gate," Nature 426, 264-267 (2003).
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J. L. O'Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, "Quantum process tomography of a controlled-NOT gate," Phys. Rev. Lett. 93, 080502 (2004).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, "Experimental entanglement of four particles," Nature 404, 256-259 (2000).
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J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, "Ancilla-assisted quantum process tomography," Phys. Rev. Lett. 93, 193601 (2003).
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N. A. Peters, J. B. Altepeter, D. A. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, "Maximally entangled mixed states: creation and concentration," Phys. Rev. Lett. 92, 133601 (2004).
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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, "Experimental entanglement of four particles," Nature 404, 256-259 (2000).
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Phys. Rev. A (11)

O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, "Detection of entanglement with few local measurements," Phys. Rev. A 66, 062305 (2002).
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V. Coffman, J. Kundu, and W. K. Wootters, "Distributed entanglement," Phys. Rev. A 61, 052306 (2000).
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W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat, "Maximizing the entanglement of two mixed qubits," Phys. Rev. A 64, 030302 (2001).
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D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312 (2001).
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A. G. White, D. F. V. James, W. J. Munro, and P. G. Kwiat, "Exploring Hilbert space: accurate characterization of quantum information," Phys. Rev. A 65, 012301 (2001).
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F. De Martini, A. Mazzei, M. Ricci, and G. M. D'Ariano, "Exploiting quantum parallelism of entanglement for a complete experimental quantum characterization of a single qubit device," Phys. Rev. A 67, 062307 (2003).
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[CrossRef] [PubMed]

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W. K. Wootters, "Entanglement of formation of an arbitrary state of two qubits," Phys. Rev. Lett. 80, 2245-2248 (1998).
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N. A. Peters, J. B. Altepeter, D. A. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, "Maximally entangled mixed states: creation and concentration," Phys. Rev. Lett. 92, 133601 (2004).
[CrossRef] [PubMed]

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J. F. Poyatos, J. I. Cirac, and P. Zoller, "Complete characterization of a quantum process: the two-bit quantum gate," Phys. Rev. Lett. 78, 390-393 (1997).
[CrossRef]

J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. OBrien, M. A. Nielsen, and A. G. White, "Ancilla-assisted quantum process tomography," Phys. Rev. Lett. 93, 193601 (2003).
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M. Steffen, M. Ansmann, R. C. Bialczak, N. Katz, E. Lucero, R. McDermott, M. Neeley, E. M. Weig, A. N. Cleland, and J. M. Martinis, "Measurement of the entanglement of two superconducting qubits via state tomography," Science 313, 1423-1425 (2006).
[CrossRef] [PubMed]

Other (15)

R. Alicki, "False qubits II. Entanglement of Josephson junctions," arXiv.org e-Print archive, quantum physics/0609122, 16 September 2006, http://www.arxiv.org/abs/quant-ph/0609122.

The partial transpose of ^p , with respect to an orthonormal basis {∣b›} of qubit two, is given by ^p T2=Sumbb'∣b′›‹b∣^p∣b′›‹b∣.

R.J.Hughes, ed., "A quantum information science and technology roadmap," Rep. LA-UR-04-1778, ARDA, April 2, 2004, http://qist.lanl.gov/.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

J. L. Brylinski and R. K. Brylinski, "Computational mathematics," in Mathematics of Quantum Computation, R.K.Brylinski and G.Chen, eds. (Chapman & Hall/CRC Press, 2002), Chap. 2.

On the qubit characteristic sphere the equal-weight superpositions form a great circle midway between the logical states. For example, on the Bloch sphere they lie on the equator, which can be accessed with π/2 pulses; on the Poincaré sphere they lie on the great circle containing the diagonally and circularly polarized states.

To be conclusive in this regard, Bell's inequality requires high-efficiency spacelike separated measurements, a feat not yet achieved in any experimental architecture.

Note that an operator sum representation is not unique and that there is unitary freedom in its choice.

Assuming that count statistics were the dominant source of uncertainty, 1000 Poissonian-distributed data sets (and hence reconstructed density matrices) were generated from each observed data set. The tangle, entropy, and fidelity were calculated for all 1000, yielding the standard deviation of each measure about the mean.

A. Doherty and A. Gilchrist, Department of Physics, University of Queensland, Brisbane QLD 4072, Australia are preparing a manuscript to be called "Quantum tomography using convex optimisation."

R. Blume-Kohout and P. Hayden, "Accurate quantum state estimation via 'Keeping the experimentalist honest'," arXiv.org e-Print archive, quantum physics/0603116, 14 March 2006, http://www.arxiv.org/abs/quant-ph/0603116.

D. W. Leung, "Towards robust quantum computation," arXiv.org e-Print archive, computer science/10012017, 20 December 2002, http://arxiv.org/abs/cs.CC/0012017.

M. Mohseni and D. A. Lidar, "Direct characterization of quantum dynamics: I. General theory," arXiv.org e-Print archive, quantum physics/0601033, 5 January 2006, http://www.arxiv.org/abs/quant-ph/0601033.

M. Ziman, "Notes on optimality of direct characterisation of quantum dynamics," arXiv.org e-Print archive, quantum physics/0603151, 17 March 2006, http://www.arxiv.org/abs/quant-ph/0603151.

H. F. Hofmann, R. Okamoto, and S. Takeuchi, "Analysis of an experimental quantum logic gate by complementary classical operations," arXiv.org e-Print archive, quantum physics/0608005, 1 August 2006, http://www.arxiv.org/abs/quant-ph/0608005.

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Figures (2)

Fig. 1
Fig. 1

Tangle-entropy plane. Measurements were made of 11 different two-qubit states (in this case, polarization-entangled photons). For each state, density matrices were obtained using the tomographic methods of Ref. [20] (gray points, error bars calculated analytically) and Ref. [21] [black points (red online), error bars calculated via a Monte Carlo technique[29]]. The upper curved line indicates maximally entangled mixed states[27] (values above this bound are unphysical); the lower curve indicates Werner states [Eq. (8)].

Fig. 2
Fig. 2

Bell-state fidelities for Fig. 1, where F > 1 2 indicates entanglement. For each state, the left and center bars were calculated using density matrices obtained via the tomographic methods of Refs. [20, 21], respectively; right bars were calculated from the entanglement witness obtained by six measurements [Eq. (7)]. Uncertainties for each fidelity are shown by thin vertical lines. Note that the Bell-state fidelity is an entanglement indicator, not a measure; e.g., states 1 and 9 have similar fidelities but very different tangles, 0.01 and 0.54 , respectively (data in Table 4 in Appendix B).

Tables (4)

Tables Icon

Table 1 Input–Output Values of a Reversible XOR Gate a

Tables Icon

Table 2 Truth Table of a Reversible XOR Gate a

Tables Icon

Table 3 Table of Two-Qubit Gate Measures a

Tables Icon

Table 4 Values for Figs. 1 and 2 a

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

V L = P even P odd P even + P odd = P 00 P 01 P 00 + P 01 = ( 1 + ε 2 ) 1 ( 1 + ε 2 ) 1 = 1 ,
P θ c θ t = θ c θ t ψ out 2 = cos θ c cos θ t + ε * e i ( ϕ c + ϕ t ) sin θ c sin θ t 2 ( 1 + ε 2 ) .
V E ( ϕ c , ϕ t ) = P ϕ c , ϕ t π 4 P ϕ c , ϕ t + π π 4 P ϕ c , ϕ t π 4 + P ϕ c , ϕ t + π π 4 = 2 ε 1 + ε 2 cos ( ϕ c + ϕ t + ξ ) ,
V E ( ϕ ) = 2 ε 1 + ε 2 cos ( 2 ϕ + ξ ) .
Π ( ϕ ) = P ϕ , ϕ π 4 + P ϕ + π , ϕ + π π 4 P ϕ , ϕ + π π 4 P ϕ + π , ϕ π 4 .
F ϕ ± = ϕ ± ρ ̂ ϕ ± = Tr ( ρ ̂ ϕ ± ϕ ± ) ,
F ψ ± = ψ ± ρ ̂ ψ ± = Tr ( ρ ̂ ψ ± ψ ± ) .
F ϕ ± = ( P H H + P V V ± P D D ± P A A P R R P L L ) 2 ,
F ψ ± = ( 1 1 P H H P V V ± P D D ± P A A ± P R R ± P L L ) 2 ,
( 1 p ) 4 I I + p ϕ ϕ ,
W ϕ + = 1 2 [ 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 ] .
W ϕ ± = ( I I X X ± Y Y Z Z ) 4 ,
W ψ ± = ( I I X X Y Y + Z Z ) 4 ,
W ϕ ± = ( P H V + P V H P D D P A A ± P R R ± P L L ) 2 ,
W ψ ± = ( P H H + P V V P D D P A A ± P R L ± P L R ) 2 .
W ϕ ± , ψ ± = 1 2 F ϕ ± , ψ ± .
ρ ̂ = 1 2 i = 0 3 S i S 0 σ ̂ i ,
ρ ̂ = 1 2 2 i 1 , i 2 = 0 3 S i 1 , i 2 S 0 , 0 σ ̂ i 1 σ ̂ i 2 ,
S i 1 , i 2 = j 1 , j 2 3 ( Υ 1 ) i 1 , j 1 ( Υ 1 ) i 2 , i 2 P i 1 , i 2 ,
Υ 1 = [ 1 0 0 0 1 2 0 0 1 0 2 0 1 0 0 2 ] , μ ̂ i μ ̂ j ( i , j = 0 , 1 , 2 , 3 )
T = V E ( n π 2 ) 2 = Π ( n π 2 ) 2 ,
T = ( max { λ 1 λ 2 λ 3 λ 4 , 0 } ) 2 .
E ( ρ ) = k E k ρ E k ,
E ( ρ ̃ i ) = j λ i j ρ ̃ j .
E ( ρ ) = m n χ m n E ̃ m ρ E ̃ n ,
E ( ρ ̃ i ) = j m n χ m n β i j m n ρ ̃ j .
E k = D k k m U m k E ̃ m .
χ = K T [ E ( ρ ( 11 ) ) E ( ρ ( 12 ) ) E ( ρ ( 21 ) ) E ( ρ ( 22 ) ) ] K ,
P ( { p a b } E x ) a b exp [ ( p a b Tr { M b E x ( ρ a ) } ) 2 2 Tr { M b E x ( ρ a ) } ] .
min x a b ( p a b Tr { M b E x ( ρ a ) } ) 2 2 Tr { M b E x ( ρ a ) } .
P ¯ e D p ( E , U )
1 F p ( E , U ) .
P e D s ( E , U )
1 F s ( E , U ) ,
F ¯ ( E , F ) d ψ F ( E ( ψ ) , F ( ψ ) ) .
F ¯ ( E , U ) = d F p ( E , U ) + 1 d + 1 ,
E ( ρ ̂ ) = ( K 0 I ) ρ ̂ ( K 0 I ) + ( K 1 X ) ρ ̂ ( K 1 X ) ,
E ( ρ ̂ ) = ( I H K 0 H ) ρ ̂ ( I H K 0 H ) + ( Z H K 1 H ) ρ ̂ ( Z H K 1 H ) .
W ϕ ± = ( ψ ± ψ ± ) T 2 = 1 2 I I ϕ ± ϕ ± ,
W ψ ± = ( ϕ ± ϕ ± ) T 2 = 1 2 I I ψ ± ψ ± .
Tr ( W ψ , ϕ ρ ̂ ) = 1 2 Tr ( ρ ̂ ψ , ϕ ψ , ϕ )
= 1 2 F ψ ± , ϕ ± .
( ϕ + ϕ + ) T 2 ψ = w ψ .
U I ( ϕ + ϕ + ) T 2 ψ = w U I ψ .
( U I ϕ + ϕ + U I ) T 2 U I ψ = w U I ψ ;
[ ρ ( 11 ) ρ ( 12 ) ρ ( 13 ) ρ ( 14 ) ρ ( 21 ) ρ ( 22 ) ρ ( 23 ) ρ ( 24 ) ρ ( 31 ) ρ ( 32 ) ρ ( 33 ) ρ ( 34 ) ρ ( 41 ) ρ ( 42 ) ρ ( 43 ) ρ ( 44 ) ] = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a a 1 i 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 a 0 0 0 1 0 0 0 i 0 0 0 i 2 i 2 a a * i 2 i 2 a a * a a 1 i a * a * i 1 a * a * 1 i 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 a a * 1 2 1 2 a a * a * a * 1 i a a i 1 0 a 0 0 0 a 0 0 0 1 0 0 0 i 0 0 a * 0 0 0 a * 0 0 0 1 0 0 0 i 0 0 0 1 2 1 2 a * a 1 2 1 2 a * a a a 1 i a * a * i 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a a 1 i 0 0 0 0 0 0 0 0 i 2 i 2 a * a i 2 i 2 a * a a * a * 1 i a a i 1 0 a * 0 0 0 a * 0 0 0 1 0 0 0 i 0 0 0 0 0 0 a * a * 1 i 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ] [ ρ ( H H ) ρ ( H V ) ρ ( H D ) ρ ( H R ) ρ ( V H ) ρ ( V V ) ρ ( V D ) ρ ( V R ) ρ ( D H ) ρ ( D V ) ρ ( D D ) ρ ( D R ) ρ ( R H ) ρ ( R V ) ρ ( R D ) ρ ( R R ) ] .
ρ ̂ ( H ) = [ 1 0 0 0 ] , ρ ̂ ( V ) = [ 0 0 0 1 ] ,
ρ ̂ ( D ) = 1 2 [ 1 1 1 1 ] , ρ ̂ ( R ) = 1 2 [ 1 i i 1 ] ,
ρ ̂ ( D R ) = 1 4 [ 1 i 1 i i 1 i 1 1 i 1 i i 1 i 1 ] .
E ( ρ ̂ ( 13 ) ) = a E ( ρ ̂ ( H H ) ) a E ( ρ ̂ ( V H ) ) + E ( ρ ̂ ( D H ) ) + i E ( ρ ( R H ) ) .
E ( ρ ̂ ) = p CNOT ρ ̂ CNOT + ( 1 p ) ρ ̂ ,
χ = 1 4 [ 4 3 p p 0 0 p p 0 0 p p 0 0 p p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p p 0 0 p p 0 0 p p 0 0 p p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] .

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