1. Quantum entanglement measures and detection
Zheng-Wei Zhou (周正威)
中国科学院中国科学技术大学量子信息重点实验室

2. 合作者
Univ. of Sci. & Tech. of China
蔡建明，孙方稳，张顺，郭光灿
Penn State
周幸祥

3. 1 Background: Quantum Entanglement 2 Entanglement Measures
Outlines
1 Background: Quantum Entanglement
2 Entanglement Measures
(a) Reviews for entanglement measures
(b) Information-theoretic measure of entanglement
based on information complementary relation
(c) Compatibility conditions from multipartite entanglement
3 Detection for Entanglement Measures

4. 1 Background: Quantum Entanglement 2 Entanglement Measures
(a) Reviews for entanglement measures
(b) Information-theoretic measure of entanglement
based on information complementary relation
(c) Compatibility conditions from multipartite entanglement
3 Detection for Entanglement Measures

5. EPR’s Paradox and Bell’s Inequality
Bell Inequality (1960)
CHSH-Bell Inequality (1969)

6. What is Entanglement ?
Werner (1989): A state is called entangled if it cannot be prepared by local operations (and classical communication) out of a product state.
Separable states

7. Entanglement in Quantum Information and Physics
Quantum Teleportation
Quantum Dense Coding
Quantum Cryptography
Quantum Communication Protocol
Theory on Quantum Channel
Quantum Error Correction Coding
Quantum Computation via Quantum entanglement
Entanglement and Quantum Phase Transition

8. Studies on Entanglement
How to distinguish entanglement and separability?
How to quantify quantum entanglement?
How to classify many body entangled states?

9. 1 Background: Quantum Entanglement 2 Entanglement Measures
(a) Reviews for entanglement measures
(b) Information-theoretic measure of entanglement
based on information complementary relation
(c) Compatibility conditions from multipartite entanglement
3 Detection for Entanglement Measures

10. Postulates for Entanglement Measures
does not change under local unitary operations
Convexity

11. Entanglement Measures
Entanglement Cost
Distillable Entanglement
Relative Entropy of Entanglement
Negativity and Logarithmic Negativity
NOT a convex function

12. Entanglement Measures
Squashed Entanglement
Robustness Entanglement
Entanglement of Formation

13. Entanglement of Two Qubits
Entanglement of Formation
Concurrence
--- Wootters PRL 78, 5022 (1997) PRL 80, 2245 (1998)

14. Concurrence of Bipartite States in Arbitrary Dimensions
Pure States
linear entropy
I-concurrence
I-tangle
Mixed States
--- F. Mintert etc. PRL 92, (2004)
--- Kai Chen etc. PRL 95, (2005) PRL 95, (2005)

15. Entanglement of Three and multi-qubit
Three qubits can be entangled in two inequivalent ways
GHZ class, W class True three-qubit entanglement
W. Dur etc. PRA 62, (2000)
Essential three-qubit entanglement
Wootters etc. PRA 61, (2000)
SLOCC invariant
General monogamy Inequality for multi-qubit pure states
Osborne PRL (2006)

16. Information Complementary Relation
Operationally Invariant Information measure (PRL 83, 3354(1999))
If a quantum measurement (P) on some quantum system results in n kinds of outputs, with probabilities(p1,p2,…,pn):
The uncertainty for nj is:
The average uncertainty in an individual measurement is:
Information revealed by measurement (P) is :
A normalized measure of information in an individual measurement is defined:

17. Information Complementary Relation
Total information is defined as the sum of information based on mutually complementary observables measurements:
1-qubit:
2-qubit:
n-qubit pure states :

18. Information Complementary Relation
Pure States of Two Qubits
Pure States of Three Qubits
A hypothesis for n-qubit pure States
(J.-M. Cai，Z.-W. Zhou, G.-C. Guo, Phys. Lett. A 363, 392, 2007.)

19. Information-theoretic Measure of Entanglement
Pure States of Two Qubits and Three Qubits
A “linear” hypothesis :

20. Information-theoretic Measure of Entanglement
Pure States of Even Number Qubits
For Example: N=4
Polynomial SLOCC invariant
Entanglement Monotone
for product states
J. -M. Cai, Z. –W. Zhou, X. Zhou, G. –C. Guo, PRA 74, (2006)

21. Mintert’s measure for multipartite entanglement
symmetric and anti-symmetric projectors
N-partite pure state
L. Aolita, F. Mintert, PRL 97, (2006)

22. Mintert’s measure for multipartite entanglement
L. Aolita, F. Mintert, PRL 97, (2006)

23. An Entanglement Monotone for pure states with even qubits
--- F. Mintert ’s symmetric projection
J. -M. Cai, Z. –W. Zhou, S. Zhang, G. –C. Guo, PRA 75, (2007) .

24. Compatibility Conditions from multipartite entanglement
What’s compatibility conditions for density matrix of pure states?

25. Compatibility Conditions from multipartite entanglement
For Instance: (A. Higuchi, et al, PRL 90, (2003))
A necessary and sufficient condition for as reduced density
matrices of :
where are eigenvalues of reduced density matrix , and

26. Compatibility Conditions from multipartite entanglement

27. Compatibility Conditions from multipartite entanglement
For example:

28. 1 Background: Quantum Entanglement 2 Entanglement Measures
(a) Reviews for entanglement measures
(b) Information-theoretic measure of entanglement
based on information complementary relation
(c) Compatibility conditions from multipartite entanglement
3 Detection for Entanglement Measures

29. Detection for entanglement measures
Two-particle interferometer
Werner states
Concurrence
single-particle fringe visibility for particle 1
Two particles coincidence
J. -M. Cai, Z. –W. Zhou, G. –C. Guo, PRA 73, (2006)

30. Detection for quantum entanglement
--- F. Mintert etc. PRL 97, (2005) Nature 440, 1022 (2006)

31. Experimental measurement for high dimensional entanglement
Two-photon state produced from optical parametric down conversion:
Symmetric and asymmetric projection for two-fold copy of

32. Experimental measurement for high dimensional entanglement
However, the coincidence counts do not indicate exactly the symmetric and asymmetric measurement. Because a full form for the states produced in optical PDC is:
is not:
But, they have the following relation:

33. Experimental measurement for high dimensional entanglement=

34. Experimental measurement for high dimensional entanglement

35. Experimental measurement for high dimensional entanglement

36. Summary
Quantifying entanglement measure for multi-partite pure states from the viewpoint of information
Compatibility Conditions from multipartite entanglement
Experimental measurement for high dimensional entanglement

37. References （1）quantum entanglement measure
Jian-Ming Cai，Zheng-Wei Zhou, Guang-Can Guo, Phys. Lett. A 363, 392, 2007.
Jian-Ming Cai，Zheng-Wei Zhou, Xingxiang Zhou , Guang-Can Guo, Phys. Rev. A 74, , 2006.
Jian-Ming Cai， Zheng-Wei Zhou, Shun Zhang，Guang-Can Guo, Phys. Rev. A 75, , 2007.
（2）detection for quantum entanglement
Jian-Ming Cai，Zheng-Wei Zhou, Guang-Can Guo, Phys. Rev. A 73, , 2006.
Fang-Wen Sun，Jian-Ming Cai，Jin-Shi Xu, Geng Chen, Bi-Heng Liu, Chuan-Feng Li, Zheng-Wei Zhou, Guang-Can Guo, submitted.

38. Thank You!
Questions and comments are welcome