Abstract

We demonstrate the conditional reversal of a weak (partial-collapse) quantum measurement on a photonic qubit. The weak quantum measurement causes a nonunitary transformation of a qubit which is subsequently reversed to the original state after a successful reversing operation. Both the weak measurement and the reversal operation are implemented linear optically. The state recovery fidelity, determined by quantum process tomography, is shown to be over 94% for partial-collapse strength up to 0.9. We also experimentally study information gain due to the weak measurement and discuss the role of the reversing operation as an information erasure.

© 2009 Optical Society of America

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References

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  1. R. Shankar, Principles of Quantum Mechanics, 2nd ed., (Plenum Press, New York, 1994).
  2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
  3. M. Koashi and M. Ueda, "Reversing measurement and probabilistic quantum error correction," Phys. Rev. Lett. 82, 2598-2601 (1999).
    [CrossRef]
  4. A. N. Korotkov and A. N. Jordan, "Undoing a weak quantum measurement of a solid-state qubit," Phys. Rev. Lett. 97, 166805 (2006).
    [CrossRef] [PubMed]
  5. N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
    [CrossRef] [PubMed]
  6. The Physics of Quantum Information, edited by D. Bouwmeester, A. K. Ekert, and A. Zeilinger (Springer, Berlin, 2000).
  7. E. Knill, R. Laamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature (London) 409, 46-52 (2001).
    [CrossRef]
  8. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
    [CrossRef]
  9. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
    [CrossRef]
  10. C. K. Hong and L. Mandel, "Experimental realization of a localized one-photon state," Phys. Rev. Lett. 56, 58-60 (1986).
    [CrossRef] [PubMed]
  11. S.-Y. Baek, O. Kwon, and Y.-H. Kim, "Temporal shaping of a heralded single-photon wave packet," Phys. Rev. A 77, 013829 (2008).
    [CrossRef]
  12. M. Mitchell, J. Lundeen, and A. Steinberg, "Super-resolving phase measurements with a multi-photon entangled state," Nature 429, 161-164 (2004).
    [CrossRef] [PubMed]
  13. The calculated and measured p values for each BP set agreed well. For example, for two pieces of BP’s, they are 0.478 and 0.455, respectively, and for seven pieces of BP’s, they are 0.897 and 0.895.
  14. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits: Qubit state tomography," Phys. Rev. A 64, 052312 (2001).
    [CrossRef]
  15. J. Fiurasek and Z. Hradil, "Maximum-likelihood estimation of quantum processes," Phys. Rev. A 63, 020101(R) (2001).
    [CrossRef]
  16. K. Banaszek, "Fidelity Balance in Quantum Operations," Phys. Rev. Lett. 86, 1366-1369 (2001).
    [CrossRef] [PubMed]
  17. S.-Y. Baek, Y. W. Cheong, and Y.-H. Kim, "Minimum disturbance measurement without postselection," Phys. Rev. A 77, 060308(R) (2008).
    [CrossRef]

2008 (2)

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "Temporal shaping of a heralded single-photon wave packet," Phys. Rev. A 77, 013829 (2008).
[CrossRef]

2007 (1)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

2006 (1)

A. N. Korotkov and A. N. Jordan, "Undoing a weak quantum measurement of a solid-state qubit," Phys. Rev. Lett. 97, 166805 (2006).
[CrossRef] [PubMed]

2004 (1)

M. Mitchell, J. Lundeen, and A. Steinberg, "Super-resolving phase measurements with a multi-photon entangled state," Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

2001 (3)

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

K. Banaszek, "Fidelity Balance in Quantum Operations," Phys. Rev. Lett. 86, 1366-1369 (2001).
[CrossRef] [PubMed]

E. Knill, R. Laamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature (London) 409, 46-52 (2001).
[CrossRef]

1999 (1)

M. Koashi and M. Ueda, "Reversing measurement and probabilistic quantum error correction," Phys. Rev. Lett. 82, 2598-2601 (1999).
[CrossRef]

1986 (1)

C. K. Hong and L. Mandel, "Experimental realization of a localized one-photon state," Phys. Rev. Lett. 56, 58-60 (1986).
[CrossRef] [PubMed]

Ansmann, M.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Baek, S.-Y.

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "Temporal shaping of a heralded single-photon wave packet," Phys. Rev. A 77, 013829 (2008).
[CrossRef]

Banaszek, K.

K. Banaszek, "Fidelity Balance in Quantum Operations," Phys. Rev. Lett. 86, 1366-1369 (2001).
[CrossRef] [PubMed]

Bialczak, R. C.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Cleland, A. N.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Hofheinz, M.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Hong, C. K.

C. K. Hong and L. Mandel, "Experimental realization of a localized one-photon state," Phys. Rev. Lett. 56, 58-60 (1986).
[CrossRef] [PubMed]

James, D. F. V.

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

Jordan, A. N.

A. N. Korotkov and A. N. Jordan, "Undoing a weak quantum measurement of a solid-state qubit," Phys. Rev. Lett. 97, 166805 (2006).
[CrossRef] [PubMed]

Katz, N.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Kim, Y.-H.

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "Temporal shaping of a heralded single-photon wave packet," Phys. Rev. A 77, 013829 (2008).
[CrossRef]

Knill, E.

E. Knill, R. Laamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature (London) 409, 46-52 (2001).
[CrossRef]

Koashi, M.

M. Koashi and M. Ueda, "Reversing measurement and probabilistic quantum error correction," Phys. Rev. Lett. 82, 2598-2601 (1999).
[CrossRef]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

Korotkov, A. N.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

A. N. Korotkov and A. N. Jordan, "Undoing a weak quantum measurement of a solid-state qubit," Phys. Rev. Lett. 97, 166805 (2006).
[CrossRef] [PubMed]

Kwiat, P. G.

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

Kwon, O.

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "Temporal shaping of a heralded single-photon wave packet," Phys. Rev. A 77, 013829 (2008).
[CrossRef]

Laamme, R.

E. Knill, R. Laamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature (London) 409, 46-52 (2001).
[CrossRef]

Lucero, E.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Lundeen, J.

M. Mitchell, J. Lundeen, and A. Steinberg, "Super-resolving phase measurements with a multi-photon entangled state," Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

Mandel, L.

C. K. Hong and L. Mandel, "Experimental realization of a localized one-photon state," Phys. Rev. Lett. 56, 58-60 (1986).
[CrossRef] [PubMed]

Martinis, J. M.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

E. Knill, R. Laamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature (London) 409, 46-52 (2001).
[CrossRef]

Mitchell, M.

M. Mitchell, J. Lundeen, and A. Steinberg, "Super-resolving phase measurements with a multi-photon entangled state," Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

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

Neeley, M.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

Oconnell, A.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Steinberg, A.

M. Mitchell, J. Lundeen, and A. Steinberg, "Super-resolving phase measurements with a multi-photon entangled state," Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Ueda, M.

M. Koashi and M. Ueda, "Reversing measurement and probabilistic quantum error correction," Phys. Rev. Lett. 82, 2598-2601 (1999).
[CrossRef]

Wang, H.

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

White, A. G.

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

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Nature (1)

M. Mitchell, J. Lundeen, and A. Steinberg, "Super-resolving phase measurements with a multi-photon entangled state," Nature 429, 161-164 (2004).
[CrossRef] [PubMed]

Nature (London) (1)

E. Knill, R. Laamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature (London) 409, 46-52 (2001).
[CrossRef]

Phys. Rev. A (2)

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "Temporal shaping of a heralded single-photon wave packet," Phys. Rev. A 77, 013829 (2008).
[CrossRef]

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

Phys. Rev. Lett. (5)

K. Banaszek, "Fidelity Balance in Quantum Operations," Phys. Rev. Lett. 86, 1366-1369 (2001).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, "Experimental realization of a localized one-photon state," Phys. Rev. Lett. 56, 58-60 (1986).
[CrossRef] [PubMed]

M. Koashi and M. Ueda, "Reversing measurement and probabilistic quantum error correction," Phys. Rev. Lett. 82, 2598-2601 (1999).
[CrossRef]

A. N. Korotkov and A. N. Jordan, "Undoing a weak quantum measurement of a solid-state qubit," Phys. Rev. Lett. 97, 166805 (2006).
[CrossRef] [PubMed]

N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. Oconnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, "Reversal of the weak measurement of a quantum state in a superconducting phase qubit," Phys. Rev. Lett. 101, 200401 (2008).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-173 (2007).
[CrossRef]

Other (6)

R. Shankar, Principles of Quantum Mechanics, 2nd ed., (Plenum Press, New York, 1994).

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

The Physics of Quantum Information, edited by D. Bouwmeester, A. K. Ekert, and A. Zeilinger (Springer, Berlin, 2000).

The calculated and measured p values for each BP set agreed well. For example, for two pieces of BP’s, they are 0.478 and 0.455, respectively, and for seven pieces of BP’s, they are 0.897 and 0.895.

S.-Y. Baek, Y. W. Cheong, and Y.-H. Kim, "Minimum disturbance measurement without postselection," Phys. Rev. A 77, 060308(R) (2008).
[CrossRef]

J. Fiurasek and Z. Hradil, "Maximum-likelihood estimation of quantum processes," Phys. Rev. A 63, 020101(R) (2001).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of the experimental setup. The heralded single-photon state is used for encoding the polarization qubit |ψo 〉 with a set of quarter and half-wave plates (WP). The partial-collapse measurement is implemented with a set of Brewster-angle glass plates (BP) and the partial-collapse strength p is varied by increasing the number of BP’s. The reversing operation requires two half-wave plates (HWP), in addition to the BP. Qubit state tomography is performed with WP and a polarizer (P).

Fig. 2.
Fig. 2.

The initial states (first row), the states after the partial-collapse measurement (second row), and the recovered states (third row) are represented on the Bloch sphere, as measured by quantum state tomography. For the second and the third rows, the points on the Bloch sphere correspond to varied partial-collapse strength p. The fourth row shows the fidelities between the initial states and the recovered states as functions of the partial-collapse strength. The input states are (a) |H〉, (b) |L〉, (c) |V〉, and (d) |A〉.

Fig. 3.
Fig. 3.

(a) Quantum process tomography matrix χ for both the partial-collapse measurement and the reversing operation together at partial-collapse strength p=0.895. It is clear that the quantum process due to the partial-collapse measurement and the reversing operation together is mainly of the identity operation acting on the qubit. (b) The fidelity of the quantum process is over 94% for all the partial-collapse strength p tested in the experiment.

Fig. 4.
Fig. 4.

Information gain via the weak measurement (quantified as G avg) for the two guessing strategies discussed in the text as functions of the partial-collapse strength p. The random guess and the projection measurement correspond to p=0 and p=1, respectively. Solid lines are theory plots.

Equations (15)

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

ψo =α 0 +β 1 ,
0 =00 =(1000)
1 =11 =(0001)
M1 =P 11 =(000P) .
M2 =00+1p 11 =(1001p) .
ψm =M2 ψo ψoM2M2ψo =α 0 +β 1 ,
M21 =11p (1p001) ,
M21 =11p (0110) (1001p) (0110) 11p M2rev ,
F =T r [χexpχideal] ,
Gavg = ψoρGψodψo.
ρGI =P1 1 1 +P2 {p00+(1p)11} ,
GavgI =(3+p2) 6 .
ρGII =P1 1 1 +P2 0 0
GavgII =(3+p) 6 .
M2rev M2 =1 1p .

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