## Avoiding entanglement sudden death using single-qubit quantum measurement reversal |

Optics Express, Vol. 22, Issue 16, pp. 19055-19068 (2014)

http://dx.doi.org/10.1364/OE.22.019055

Acrobat PDF (2589 KB)

### Abstract

When two entangled qubits, each owned by Alice and Bob, undergo separate decoherence, the amount of entanglement is reduced, and often, weak decoherence causes complete loss of entanglement, known as entanglement sudden death. Here we show that it is possible to apply quantum measurement reversal on a single-qubit to avoid entanglement sudden death, rather than on both qubits. Our scheme has important applications in quantum information processing protocols based on distributed or stored entangled qubits as they are subject to decoherence.

© 2014 Optical Society of America

## 1. Introduction

2. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London) **404**, 247–255 (2000). [CrossRef]

3. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. **70**, 1895–1899 (1993). [CrossRef] [PubMed]

5. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

6. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. **93**, 140404 (2004). [CrossRef] [PubMed]

7. M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science **316**, 579–582 (2007). [CrossRef] [PubMed]

8. D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. **81**, 2594–2597 (1998). [CrossRef]

15. D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science **291**, 1013–1015 (2001). [CrossRef] [PubMed]

16. M. Koashi and M. Ueda, “Reversing measurement and probabilistic quantum Error correction,” Phys. Rev. Lett. **82**, 2598–2601 (1999). [CrossRef]

20. Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express **17**, 11978–11985 (2009). [CrossRef] [PubMed]

21. A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement reversal,” Phys. Rev. A **81**, 040103(R) (2010). [CrossRef]

24. Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. **8**, 117–120 (2012). [CrossRef]

21. A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement reversal,” Phys. Rev. A **81**, 040103(R) (2010). [CrossRef]

22. J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express **19**, 16309–16316 (2011). [CrossRef] [PubMed]

24. Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. **8**, 117–120 (2012). [CrossRef]

6. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. **93**, 140404 (2004). [CrossRef] [PubMed]

25. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A **53**, 2046–2052 (1996). [CrossRef] [PubMed]

28. J.-W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature (London) **410**, 1067–1070 (2001). [CrossRef]

## 2. Theory

*R*(

_{A}*p*) on her qubit as shown in Fig. 2(b). A reversing measurement is also a weak measurement but it partially collapses the state towards the opposite direction, i.e., |1〉

_{r}*. The reversing measurement operator is given as*

_{S}*p*is the strength of the reversing measurement. The reversing strength we used is

_{r}*p*=

_{r}*p*+

*D*where

_{A}p̄*p̄*≡ 1 −

*p*[21

21. A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement reversal,” Phys. Rev. A **81**, 040103(R) (2010). [CrossRef]

*W*(

_{B}*p*) and

*R*(

_{B}*p*). Note that the strength of the reversing measurement which Bob performs on his qubit is

_{r}*p*=

_{r}*p*+

*D*.

_{B}p̄*ρ*,

_{A}*C*, can be evaluated by where

_{A}*C*= Λ

_{A}*if Λ*

_{A}*> 0, otherwise*

_{A}*C*= 0. Note that

_{A}*C*>

_{A}*C*, meaning that

_{D}*ρ*is more entangled than

_{A}*ρ*. For the case with Bob’s single-qubit quantum measurement reversal,

_{D}*C*is the same as

_{B}*C*except the fact that

_{A}*D*and

_{A}*D*are interchanged, and

_{B}*C*and

_{A}*C*give the same value when

_{B}*D*=

_{A}*D*=

_{B}*D*.

## 3. Experiment

*β*-BaB

_{2}O

_{4}crystal is pumped by a 405 nm diode laser beam (100 mW) and a pair of down-converted photons centered at 810 nm are generated on the frequency-degenerate, noncollinear phase matching condition. Each down-converted photon is frequency-filtered using an interference filter with full width at half-maximum bandwidth of 5 nm. Then, the two-qubit entangled state |Φ〉 =

*α*|00〉 +

*β*|11〉 can be prepared using quantum interferometry [31

31. Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. **61**, 2921–2924 (1988). [CrossRef] [PubMed]

20. Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express **17**, 11978–11985 (2009). [CrossRef] [PubMed]

*H*〉) and vertically (|

*V*〉) polarized photons, respectively.

20. Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express **17**, 11978–11985 (2009). [CrossRef] [PubMed]

*p*and

*p*can be increased by adding more BPs. The amplitude damping decoherence for the photonic system can be realized using a displaced Sagnac interferometer [7

_{r}7. M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science **316**, 579–582 (2007). [CrossRef] [PubMed]

22. J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express **19**, 16309–16316 (2011). [CrossRef] [PubMed]

24. Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. **8**, 117–120 (2012). [CrossRef]

*H*〉 = |0〉

*, |*

_{S}*V*〉 = |1〉

*) and the environment qubit is encoded in the path of the single photon, |0〉*

_{S}*and |1〉*

_{E}*, see Fig. 3. Finally, the resulting states are analyzed by quantum state tomography.*

_{E}## 4. Results and analysis

*α*|00〉 +

*β*|11〉 is affected by the identical decoherence (

*D*=

_{A}*D*=

_{B}*D*). We evaluated the concurrence of the final state

*ρ*from the density matrices which are experimentally reconstructed by the quantum state tomography. For the two entangled input states with |

_{D}*α*| = |

*β*| and |

*α*| = 0.481 < |

*β*|, we measured

*C*(Λ

_{D}*) as increasing the strength of the decoherence*

_{D}*D*and the results are shown in Fig. 4(a). As expected,

*C*decreases as

_{D}*D*increases in both cases. Note that ESD is observed for the case of |

*α*| = 0.481.

## 5. Conclusion

## Appendix : Reversing measurement strength under decoherence

*ψ*

_{in}〉 =

*α*|0〉 +

*β*|1〉 does not suffer any decoherence (

*D*= 0), the reversing measurement for the weak measurement

*p*=

_{r}*p*[17

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

*ψ*

_{in}〉 =

*α*|0〉 +

*β*|1〉 is retrieved without errors only when

*p*=

_{r}*p*, and, thus,

*p*=

_{r}*p*is the optimal reversing measurement strength for

*D*= 0. Note that the optimal

*p*is independent on

_{r}*α*in this case.

*p*=

_{r}*p*is not the optimal measurement anymore and the “optimality” of the reversing measurement strength becomes highly non-trivial. In general,

*p*is dependent on several parameters such as

_{r}*D*,

*p*, and

*α*. Since the initial state cannot be retrieved perfectly, the optimal

*p*value is different for maximizing, e.g., the state fidelity, the success probability, the concurrence, etc. Moreover, the optimal

_{r}*p*value is also dependent on the condition of the decoherence suppression scenarios. In this section, we investigate several reversing measurement strategies and compare them with respect to the state fidelity and the success probability. Here, we consider three different situations i) a single qubit state suffers decoherence [21

_{r}**81**, 040103(R) (2010). [CrossRef]

22. J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express **19**, 16309–16316 (2011). [CrossRef] [PubMed]

**8**, 117–120 (2012). [CrossRef]

## A1. Single qubit case

*W*(

*p*) and reversing measurement

*R*(

*p*) measurements are performed before and after decoherence (

_{r}*D*), an initial qubit state |

*ψ*

_{in}〉 =

*α*|0〉 +

*β*|1〉, where |

*α*|

^{2}+ |

*β*|

^{2}= 1, is transformed to the final state

*ρ*

_{out}, given as where

*D̄*≡ 1 −

*D*,

*p̄*≡ 1 −

*p*,

*p̄*≡ 1 −

_{r}*p*. The success probability or channel transmittance is

_{r}*P*=

_{S}*p̄*(|

_{r}*α*|

^{2}+

*Dp̄*|

*β*|

^{2})+

*D̄p̄*|

*β*|

^{2}[21

**81**, 040103(R) (2010). [CrossRef]

**19**, 16309–16316 (2011). [CrossRef] [PubMed]

*ρ*

_{in}= |

*ψ*

_{in}〉 〈

*ψ*

_{in}| and the final state

*ρ*

_{out}, i.e.,

*F*= 〈

*ψ*

_{in}|

*ρ*

_{out}|

*ψ*

_{in}〉, which is calculated to be Then, the reversing measurement strength

*F*

_{max}is

*α*,

*D*, and

*p*.

*D*while we do not have any prior information about the initial state |

*ψ*

_{in}〉, i.e.,

*α*is an unknown parameter [22

**19**, 16309–16316 (2011). [CrossRef] [PubMed]

**8**, 117–120 (2012). [CrossRef]

*p*values. First, we obtain

_{r}*p*by substituting

_{r}*α*.

*p*to be

_{r}**81**, 040103(R) (2010). [CrossRef]

**19**, 16309–16316 (2011). [CrossRef] [PubMed]

*α*and ii) the state fidelity

*F*asymptotically approaches to 1 as the weak measurement strength

*p*→ 1 regardless of

*α*.

*F*between the initial and final states for two different reversing measurement strength

*F*

^{max}can be obtained by substituting

*F*

^{max}is always higher than both of

*F*

^{fix}and

*F*

^{exp}regardless of |

*α*|,

*D*, and

*p*. However, to achieve

*F*

^{max}, we need to know prior information about the initial state, meaning that the decoherence suppression scheme becomes state dependent. Thus, here, we focus on comparing

*F*

^{exp}with

*F*

^{fix}.

*D*= 0.617 are shown in Fig. 5(a). Note that

*F*

^{exp}>

*F*

^{fix}when |

*α*| is larger than |

*β*|. Since our decoherence suppression scheme exploits weak measurement, the success probability of the scheme is not unity. Hence, the success probability is also an important parameter for quantifying the performance of the scheme. The success probabilities for

*P*of Eq. (3). Note that, in most cases,

_{S}*F*.

*α*| and

*p*for different reversing measurement strengths when

*D*= 0.617. As shown in Fig. 5(b),

*α*| and

*p*. In the single qubit case, the reversing measurement strength

## A2. Two-qubit case: Both subsystems suffer from decoherence and two pairs of weak and reversing measurements are performed

*α*|00〉 +

*β*|11〉, where |

*α*|

^{2}+ |

*β*|

^{2}= 1, suffers from separate decoherence and two pairs of weak and reversing measurements are applied to suppress decoherence [24

**8**, 117–120 (2012). [CrossRef]

*D*=

_{A}*D*=

_{B}*D*) and the strengths of weak measurements

*W*(

_{A}*p*) and

*W*(

_{B}*p*), and reversing measurements

*R*(

_{A}*p*) and

_{r}*R*(

_{B}*p*) performed on each subsystem are the same. Then, the output state

_{r}*ρ*

_{out}is given as

*P*=

_{S}*p*|

_{r}D̄*α*|

^{2}(

*Dp*+

_{r}*p*− 2) + (

_{r}*Dp*− 1)

_{r}^{2}(1 −

*p*

^{2}|

*β*|

^{2}+ 2

*p*|

*β*|

^{2}is the success probability. The state fidelity between the initial state and the output state is obtained to be

*F*is if i)

*Dp̄*|

*β*|

^{2}< |

*α*|

^{2}<

*p̄*|

*β*|

^{2}. Otherwise,

*p*= 1 maximizes

_{r}*F*, meaning that the reversing measurement

*p*should be projection measurement regardless of the weak measurement strength

_{r}*p*.

*F*

^{fix}for

*F*

^{exp}for

*F*for

*F*

^{fix}and

*F*

^{exp}are similar to the results of the single qubit case, see Fig. 5(a).

*F*

^{exp}is generally larger than

*F*

^{fix}when |

*α*| is larger than |

*β*|.

*P*for various reversing measurement strength

_{S}*p*. As shown in Fig. 6(b),

_{r}*α*| and

*p*. The general tendency of the success probabilities are similar to the single qubit case in that

## A3. Two-qubit case: Both subsystems suffer from decoherence and one pair of weak and reversing measurements are performed

*α*|00〉 +

*β*|11〉, where |

*α*|

^{2}+ |

*β*|

^{2}= 1, undergoes separate decoherence and a set of weak measurement

*W*(

_{A}*p*) and reversing measurement

*R*(

_{A}*p*) is performed only on the subsystem undergoing decoherence (A). If we assume each subsystem undergoes the identical decoherence (

_{r}*D*=

_{A}*D*=

_{B}*D*), the output state

*ρ*

_{out}is given as where

*P*= |

_{S}*α*|

^{2}[

*p*−

*p*(

_{r}*D̄*+

*Dp*)] +

*p̄*(1 −

*Dp*) and the corresponding state fidelity between the initial state and the final state is calculated to be

_{r}*F*is rather complicated and cannot be simplified, we provide

*F*

^{fix}and

*F*

^{exp}and

*F*

^{fix}and

*F*

^{fix}and

*F*

^{exp}and

*F*cannot achieve unity even in the case that

*p*→ 1. Note that

*F*→ 1 as

*p*→ 1 in both of Fig. 5(a) and Fig. 6(b). Another thing to note is that

*F*

^{fix}is larger than

*F*

^{exp}when |

*α*| is larger than |

*β*| in Fig. 7(a) while

*F*

^{exp}is larger than

*F*

^{fix}when |

*α*| is larger than |

*β*| in both of Fig. 5(a) and Fig. 6(a). Interestingly, the difference between

*F*

^{fix}, and

*F*

^{exp}is very small in any combination of |

*α*| and

*p*in Fig. 7(a).

## A4. Two-qubit case: One subsystem suffers from decoherence and one pair of weak and reversing measurements are performed

*D*=

_{A}*D*and

*D*= 0. Here only the subsystem A of the initial entangled state |Φ〉 =

_{B}*α*|00〉 +

*β*|11〉, where |

*α*|

^{2}+ |

*β*|

^{2}= 1, suffers from decoherence and a set of weak measurement

*W*(

_{A}*p*) and reversing measurement

*R*(

_{A}*p*) is performed on the subsystem A. Then, the output state

_{r}*ρ*

_{out}becomes where

*P*= |

_{S}*β*|

^{2}

*D̄p̄*+ |

*β*|

^{2}

*Dp̄p̄*+ |

_{r}*α*|

^{2}

*p̄*and the state fidelity between the input state and the output state is given as and the reversing measurement strength

_{r}*p*

^{max}that maximizes

*F*is

*F*

^{exp}>

*F*

^{fix}when |

*α*| is larger than |

*β*| and

*α*| and

*p*.

## A5. Conclusion

*F*and the success probabilities

*P*for various decoherence suppression schemes. Our results suggest that there is no reversing measurement which maximizes the state fidelity and the success probability simultaneously. In many cases, a trade-off relation between

_{S}*F*and

*P*holds, in other words, the higher the state fidelity, the lower the success probability, vice versa. Therefore, one should choose a proper

_{S}*p*value for a specific application, after carefully considering the trade-off relation.

_{r}## Acknowledgments

## References and links

1. | M. A. Nielsen and I. L. Chuang, |

2. | C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London) |

3. | C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. |

4. | D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature (London) |

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6. | T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. |

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8. | D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. |

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16. | M. Koashi and M. Ueda, “Reversing measurement and probabilistic quantum Error correction,” Phys. Rev. Lett. |

17. | A. N. Korotkov and A. N. Jordan, “Undoing weak quantum measurement of a solid-state qubit,” Phys. Rev. Lett. |

18. | Q. Sun, M. Al-Amri, and M. S. Zubairy, “Reversing the weak measurement of an arbitrary field with finite photon number,” Phys. Rev. A |

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20. | Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, “Reversing the weak quantum measurement for a photonic qubit,” Opt. Express |

21. | A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement reversal,” Phys. Rev. A |

22. | J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, “Experimental demonstration of decoherence suppression via quantum measurement reversal,” Opt. Express |

23. | Q. Sun, M. Al-Amri, L. Davidovich, and M. S. Zubairy, “Reversing entanglement change by a weak measurement,” Phys. Rev. A |

24. | Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, “Protecting entanglement from decoherence using weak measurement and quantum measurement reversal,” Nature Phys. |

25. | C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A |

26. | C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. |

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28. | J.-W. Pan, C. Simon, C. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature (London) |

29. | J. P. Groen, D. Ristè, L. Tornberg, J. Cramer, P. C. de Groot, T. Picot, G. Johansson, and L. DiCarlo, “Partial-measurement backaction and nonclassical weak values in a superconducting circuit,” Phys. Rev. Lett. |

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31. | Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. |

**OCIS Codes**

(270.5565) Quantum optics : Quantum communications

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: June 27, 2014

Manuscript Accepted: July 11, 2014

Published: July 29, 2014

**Citation**

Hyang-Tag Lim, Jong-Chan Lee, Kang-Hee Hong, and Yoon-Ho Kim, "Avoiding entanglement sudden death using single-qubit quantum measurement reversal," Opt. Express **22**, 19055-19068 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19055

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### References

- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, Cambridge, 2000).
- C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London)404, 247–255 (2000). [CrossRef]
- C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.70, 1895–1899 (1993). [CrossRef] [PubMed]
- D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature (London)390, 575–579 (1997). [CrossRef]
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys.74, 145–195 (2002). [CrossRef]
- T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett.93, 140404 (2004). [CrossRef] [PubMed]
- M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science316, 579–582 (2007). [CrossRef] [PubMed]
- D. A. Lidar, I. L. Chuang, and K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett.81, 2594–2597 (1998). [CrossRef]
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