Entanglement of formation and concurrence for mixed states

Abstract

We review some results on analytical computations of the measures for quantum entanglement: entanglement of formation and concurrence. We introduce some estimations of the lower bounds for the entanglement of formation in bipartite mixed states, and of lower bounds for the concurrence in bipartite and tripartite systems. The results on lower bounds for the concurrence are also generalized to arbitrary multipartite systems.

References

1. Nielsen M A Chuang I L Quantum Computation and QuantumInformationCambridgeCambridge University Press 2000
2. Bennett C H DiVincenzo D P Smolin J A et al.Mixed state entanglement and quantum error correctionPhysical Review A 1996 5438243851. doi:10.1103/PhysRevA.54.3824
3. Horodecki M Horodecki P Horodecki R et al.Classical capacity of a noiseless quantum channelassisted by noisy entanglementQuantum Informationand Computation 2001 1(3)7078
4. Bru? D CharacterizingentanglementJournal of Mathematical Physics 2002 434237. doi: 10.1063/1.1494474
5. Plenio M B Virmani S An introduction to entanglementmeasuresQuantum Information and Computation 2007 7125
6. Uhlmann A Fidelityand concurrence of conjugated statesA PhysicalReview A 2000 62032307. doi: 10.1103/PhysRevA.62.032307
7. Albeverio S Fei S M A Note on Invariants and EntanglementsJ Opt B: Quantum Semiclass Opt 2001 3223227. doi:10.1088/1464‐4266/3/4/305
8. Rungta P Bu?ek V Caves C M et al.Universal state inversion and concurrence in arbitrarydimensionsPhysical Review A 2001 64042315. doi: 10.1103/PhysRevA.64.042315
9. Hill S Wootters W K Entanglement of a pair of quantumbitsPhysical Review Letters 1997 7850225025. doi:10.1103/PhysRevLett.78.5022
10. Wootters W K Entanglementof formation of an arbitrary state of two qubitsPhysical Review Letters 1998 8022452248. doi:10.1103/PhysRevLett.80.2245
11. Osterloh A Amico L Falci G et al.Scaling of entanglement close to a quantum phasetransitionsNature 2002 416608. doi: 10.1038/416608a
12. Wu L A Sarandy M S Lidar D A Quantum phase transitions and bipartite entanglementPhysical Review Letters 2004 93250404. doi: 10.1103/PhysRevLett.93.250404
13. Ghosh S Rosenbaum T F Aeppli G et al.Entangled quantum state of magnetic dipolesNature 2003 42548. doi: 10.1038/nature01888
14. Vedral V Quantumphysics. Entanglement hits the big timeNature 2003 42528. doi: 10.1038/425028a
15. Walborn S P Souto Ribeiro P H Davidovich L et al.Experimental determination of entanglement witha single measurementNature 2006 4401022. doi: 10.1038/nature04627
16. Terhal B M Vollbrecht K G H The entanglement of formationfor isotropic statesPhysical Review Letters 2000 8526252628. doi:10.1103/PhysRevLett.85.2625
17. Fei S M Jost J Li-Jost X Q et al.Entanglement of formation for a class of mixed statesPhysics Letters A 2003 310333338. doi:10.1016/S0375‐9601(03)00379‐7
18. Fei S M Li-Jost X Q A class of special matricesand quantum entanglementReports on MathematicalPhysics 2004 53195210. doi:10.1016/S0034‐4877(04)90012‐2
19. Fei S M Wang Z X Zhao H A note on entanglement of formation and generalized concurrencePhysics Letters A 2004 329414419. doi:10.1016/j.physleta.2004.07.030
20. Rungta P Caves C M I-concurrence and tangle forisotropic statesPhysical Review A 2003 67012307. doi: 10.1103/PhysRevA.67.012307
21. Chen P X Liang L M Li C Z et al.A lower bound on entanglement of formation of 2 ? n systemPhysics Letters A 2002 295175177. doi:10.1016/S0375‐9601(02)00175‐5
22. Gerjuoy E Lowerbound on entanglement of formation for the qubit-qudit systemPhysical Review A 2003 67052308. doi: 10.1103/PhysRevA.67.052308
23. ?oziński A Buchleitner A ?yczkowski K et al.Entanglement of 2 × K quantum systemsEurophysicsLetters 2003 62168
24. Audenaert K Verstraete F Moor B De Variational characterisations of separability and entanglementof formationPhysical Review A 2001 64052304. doi: 10.1103/PhysRevA.64.052304
25. Mintert F Ku? M Buchleitner A oncurrence of mixed bipartite quantum states in arbitrarydimensionsPhysical Review Letters 2004 92167902. doi: 10.1103/PhysRevLett.92.167902
26. Mintert F Ph.D. thesis Measures and dynamics of entangled statesMunichMunich University 2004
27. Chen K Albeverio S Fei S M Concurrence of arbitrary dimensional bipartite quantumstatesPhysical Review Letters 2005 95040504. doi: 10.1103/PhysRevLett.95.040504
28. Chen K Albeverio S Fei S M Entanglement of formation of bipartite quantum statesPhysical Review Letters 2005 95210501. doi: 10.1103/PhysRevLett.95.210501
29. Breuer H P Separabilitycriteria and bounds for entanglement measuresJournal of Physics A: Mathematical and General 2006 3911847. doi: 10.1088/0305‐4470/39/38/010
30. Breuer H P Optimalentanglement criterion for mixed quantum statesPhysical Review Letters 2006 97080501. doi: 10.1103/PhysRevLett.97.080501
31. Vicente J I de Lower bounds on concurrence and separability conditionsPhysical Review A 2007 75052320. doi: 10.1103/PhysRevA.75.052320
32. Zhang C J Zhang Y S Zhang S et al.Optimal entanglement witnesses based on local orthogonalobservablesPhysical Review A 2007 76012334. doi: 10.1103/PhysRevA.76.012334
33. Gao X H Fei S M Wu K Lower bounds of concurrence of tripartite quantum systemsPhysical Review A 2006 74050303(R)
34. Bennett C H Bernstein H J Popescu S et al.Concentrating partial entanglement by local operationsPhysical Review A 1996 5320462052. doi:10.1103/PhysRevA.53.2046
35. Werner R F Quantumstates with Einstein-Podolsky-Rosen correlations admitting a hidden-variablemodelPhysical Review A 1989 4042774281. doi:10.1103/PhysRevA.40.4277
36. Brun T A Measuringpolynomial functions of statesQuantum Informationand Computation 2004 4401
37. Mintert F Kus' M Buchleitner A Concurrence of mixed multi-partite quantum statesPhysical Review Letters 2005 95260502. doi: 10.1103/PhysRevLett.95.260502
38. Fei S M Li-Jost X R-function related to entanglementof formationPhysical Review A 2006 73024302. doi: 10.1103/PhysRevA.73.024302
39. Vollbrecht K G H Werner R F Entanglement measures undersymmetryPhysical Review A 2001 64062307. doi: 10.1103/PhysRevA.64.062307
40. Wootters W K Entanglementof formation and concurrenceQuantum Informationand Computation127 2001
41. Chen K Albeverio S Fei S M Concurrence-based entanglement measure for Werner statesReports on Mathematical Physics 2006 58325334. doi:10.1016/S0034‐4877(07)00003‐1
42. Albeverio S Fei S M Goswami D Separability of Rank Two Quantum StatesPhysics Letters A 2001 2869196. doi:10.1016/S0375‐9601(01)00413‐3
43. Fei S M Gao X H Wang X H et al.Separability of rank two quantum states on multiplequantum spacesPhysics Letters A 2002 300555562. doi:10.1016/S0375‐9601(02)00882‐4
44. Chen K Wu L A A matrix realignment methodfor recognizing entanglementQuantum Informationand Computation 2003 3193202
45. Horodecki M Horodecki P Horodecki R Separability of mixed quantum states: linear contractionsapproachOpen Systems and Information Dynamics 2006 13103. doi: 10.1007/s11080‐006‐7271‐8
46. Rudolph O Furtherresults on the cross norm criterion for separabilityQuantum Information Processing 2005 4219. doi: 10.1007/s11128‐005‐5664‐1
47. Ou Y C Fan H Fei S M Concurrence, distillability, and distributed entanglementfor arbitrary quantum statesarXiv:0711.2865v2 2007
48. Horodecki M Horodecki P Reduction criterion of separabilityand limits for a class of distillation protocolsPhysical Review A 1999 5942064216. doi:10.1103/PhysRevA.59.4206
49. Vollbrecht K G H Werner R F Entanglement measures undersymmetryPhysical Review A 2001 64062307. doi: 10.1103/PhysRevA.64.062307
50. Hofmann H F Takeuchi S Violation of local uncertaintyrelations as a signature of entanglementPhysical Review A 2003 68032103. doi: 10.1103/PhysRevA.68.032103
51. Hofmann H F Boundentangled states violate a nonsymmetric local uncertainty relationPhysical Review A 2003 68034307. doi: 10.1103/PhysRevA.68.034307
52. Gühne O Mechler M Toth G et al.Entanglement criteria based on local uncertaintyrelations are strictly stronger than the computable cross norm criterionPhysical Review A 2006 74010301(R)
53. Yu S X Liu N L Entanglement detection by localorthogonal observablesPhysical Review Letters 2005 95150504. doi: 10.1103/PhysRevLett.95.150504
54. Vicente J I de Separability criteria based on the Bloch representation of densitymatricesQuantum Information and Computation 2007 7624
55. Bennett C H DiVincenzo D P Mor T et al.Unextendible Product bases and bound entanglementPhysical Review Letters 1999 8253855388. doi:10.1103/PhysRevLett.82.5385
56. Fei S M Li-Jost X Sun B Z A Class of Bound Entangled StatesPhysics Letters A 2006 352321. doi: 10.1016/j.physleta.2005.12.038
57. Albeverio S Chen K Fei S M Generalized reduction criterion for separability of quantumstatesPhysical Review A 2003 68062313. doi: 10.1103/PhysRevA.68.062313
58. Horn R A Johnson C R Topics in Matrix AnalysisNew YorkCambridgeUniversity Press 1991
59. Acin A Andrianov A Costa L et al.Generalized Schmidt decomposition and classificationof three-quantum-bit statesPhysical ReviewLetters 2000 8515601563. doi:10.1103/PhysRevLett.85.1560
60. Rains E M Boundon distillable entanglement.Physical ReviewA 1999 60179. doi: 10.1103/PhysRevA.60.179
61. Khasin M Kosloff R Rise and fall of quantum andclassical correlations in opensystem dynamics.Physical Review A 2007 76012304. doi: 10.1103/PhysRevA.76.012304
62. Zhao M J Fei S M Wang Z X Entanglement of Multipartite Schmidt-correlated StatesPhysics Letters A 2008 37225522557
63. Pan F Lu G Y Draayer J P Classification and quantification of entangled bipartitequtrit pure statesInternational Journalof Modern Physics B 2006 2013331342. doi:10.1142/S0217979206033966
64. Lamata L León J Salgado D et al.Inductive classification of multipartite entanglementunder SLOCCPhysical Review A 2006 74052336. doi: 10.1103/PhysRevA.74.052336