Bound entanglement

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Short description: Weak form of quantum entanglement

Bound entanglement is a weak form of quantum entanglement, from which no singlets can be distilled with local operations and classical communication (LOCC). Bound entanglement was discovered by M. Horodecki, P. Horodecki, and R. Horodecki. Bipartite entangled states that have a non-negative partial transpose are all bound-entangled. Moreover, a particular quantum state for 2x4 systems has been presented.[1] Such states are not detected by the Peres-Horodecki criterion as entangled, thus other entanglement criteria are needed for their detection. There are a number of examples for such states.[2][3][4][5]

There are also multipartite entangled states that have a negative partial transpose with respect to some bipartitions, while they have a positive partial transpose to the other partitions, nevertheless, they are undistillable.[6]

The possible existence of bipartite bound entangled states with a negative partial transpose is still under intensive study.[7]

Properties of bound entangled states with a positive partial transpose

Bipartite bound entangled states do not exist in 2x2 or 2x3 systems, only in larger ones.

Rank-2 bound entangled states do not exist.[8]

Bipartite bound entangled states with a positive partial transpose are useless for teleportation, as they cannot lead to a larger fidelity than the classical limit.[9]

Bound entangled states with a positive partial transpose in 3x3 systems have a Schmidt number 2.[10]

It has been shown that bipartite bound entangled states with a positive partial transpose exist in symmetric systems. It has also been shown that in symmetric systems multipartite bound entangled states exists for which all partial transposes are non-negative.[11][12]

Asher Peres conjectured that bipartite bound entangled states with positive partial transpose cannot violate a Bell inequality.[13] After a long search for counterexamples, the conjecture turned out to be false.[14]

While no singlets can be distilled from bound entangled state, they can be still useful for some quantum information processing applications. Bound entanglement can be activated.[15] Any entangled state can enhance the teleportation power of some other state. This holds even if the state is bound entangled.[16] Bipartite entangled states with a non-negative partial transpose can be more useful for quantum metrology than separable states.[17] Families of bound entangled states known analytically even for high dimension that outperform separable states for metrology. For large dimensions they approach asymptotically the maximal precision achievable by bipartite quantum states.[18] There are bipartite bound entangled states that are not more useful than separable states, but if an ancilla is added to one of the subsystems then they outperform separable states in metrology.[19]

References

  1. ↑ Horodecki, Michał; Horodecki, Paweł; Horodecki, Ryszard (15 June 1998). "Mixed-State Entanglement and Distillation: Is there a "Bound" Entanglement in Nature?". Physical Review Letters 80 (24): 5239–5242. doi:10.1103/PhysRevLett.80.5239. Bibcode: 1998PhRvL..80.5239H. 
  2. ↑ Bruß, Dagmar; Peres, Asher (4 February 2000). "Construction of quantum states with bound entanglement". Physical Review A 61 (3): 030301. doi:10.1103/PhysRevA.61.030301. Bibcode: 2000PhRvA..61c0301B. 
  3. ↑ Bennett, Charles H.; DiVincenzo, David P.; Mor, Tal; Shor, Peter W.; Smolin, John A.; Terhal, Barbara M. (28 June 1999). "Unextendible Product Bases and Bound Entanglement". Physical Review Letters 82 (26): 5385–5388. doi:10.1103/PhysRevLett.82.5385. Bibcode: 1999PhRvL..82.5385B. https://pure.uva.nl/ws/files/3112710/7608_77862y.pdf. 
  4. ↑ Breuer, Heinz-Peter (22 August 2006). "Optimal Entanglement Criterion for Mixed Quantum States". Physical Review Letters 97 (8): 080501. doi:10.1103/PhysRevLett.97.080501. PMID 17026285. Bibcode: 2006PhRvL..97h0501B. 
  5. ↑ Piani, Marco; Mora, Caterina E. (4 January 2007). "Class of positive-partial-transpose bound entangled states associated with almost any set of pure entangled states". Physical Review A 75 (1): 012305. doi:10.1103/PhysRevA.75.012305. Bibcode: 2007PhRvA..75a2305P. 
  6. ↑ Smolin, John A. (9 February 2001). "Four-party unlockable bound entangled state". Physical Review A 63 (3): 032306. doi:10.1103/PhysRevA.63.032306. Bibcode: 2001PhRvA..63c2306S. 
  7. ↑ DiVincenzo, David P.; Shor, Peter W.; Smolin, John A.; Terhal, Barbara M.; Thapliyal, Ashish V. (17 May 2000). "Evidence for bound entangled states with negative partial transpose". Physical Review A 61 (6): 062312. doi:10.1103/PhysRevA.61.062312. Bibcode: 2000PhRvA..61f2312D. 
  8. ↑ Horodecki, Pawel; Smolin, John A; Terhal, Barbara M; Thapliyal, Ashish V (January 2003). "Rank two bipartite bound entangled states do not exist". Theoretical Computer Science 292 (3): 589–596. doi:10.1016/S0304-3975(01)00376-0. 
  9. ↑ Horodecki, Michał; Horodecki, Paweł; Horodecki, Ryszard (1 September 1999). "General teleportation channel, singlet fraction, and quasidistillation". Physical Review A 60 (3): 1888–1898. doi:10.1103/PhysRevA.60.1888. Bibcode: 1999PhRvA..60.1888H. 
  10. ↑ Chen, Lin; Tang, Wai-Shing (2 February 2017). "Schmidt number of bipartite and multipartite states under local projections". Quantum Information Processing 16 (75): 75. doi:10.1007/s11128-016-1501-y. Bibcode: 2017QuIP...16...75C. 
  11. ↑ Tóth, Géza; Gühne, Otfried (1 May 2009). "Entanglement and Permutational Symmetry". Physical Review Letters 102 (17): 170503. doi:10.1103/PhysRevLett.102.170503. PMID 19518768. Bibcode: 2009PhRvL.102q0503T. 
  12. ↑ Tura, J.; Augusiak, R.; Hyllus, P.; Kuś, M.; Samsonowicz, J.; Lewenstein, M. (22 June 2012). "Four-qubit entangled symmetric states with positive partial transpositions". Physical Review A 85 (6): 060302. doi:10.1103/PhysRevA.85.060302. Bibcode: 2012PhRvA..85f0302T. 
  13. ↑ Peres, Asher (1999). "All the Bell Inequalities". Foundations of Physics 29 (4): 589–614. doi:10.1023/A:1018816310000. 
  14. ↑ Vértesi, Tamás; Brunner, Nicolas (December 2014). "Disproving the Peres conjecture by showing Bell nonlocality from bound entanglement". Nature Communications 5 (1): 5297. doi:10.1038/ncomms6297. PMID 25370352. Bibcode: 2014NatCo...5.5297V. 
  15. ↑ Horodecki, Paweł; Horodecki, Michał; Horodecki, Ryszard (1 February 1999). "Bound Entanglement Can Be Activated". Physical Review Letters 82 (5): 1056–1059. doi:10.1103/PhysRevLett.82.1056. Bibcode: 1999PhRvL..82.1056H. 
  16. ↑ Masanes, Lluís (17 April 2006). "All Bipartite Entangled States Are Useful for Information Processing". Physical Review Letters 96 (15): 150501. doi:10.1103/PhysRevLett.96.150501. PMID 16712136. Bibcode: 2006PhRvL..96o0501M. 
  17. ↑ Tóth, Géza; Vértesi, Tamás (12 January 2018). "Quantum States with a Positive Partial Transpose are Useful for Metrology". Physical Review Letters 120 (2): 020506. doi:10.1103/PhysRevLett.120.020506. PMID 29376687. Bibcode: 2018PhRvL.120b0506T. 
  18. ↑ Pál, Károly F.; Tóth, Géza; Bene, Erika; Vértesi, Tamás (10 May 2021). "Bound entangled singlet-like states for quantum metrology". Physical Review Research 3 (2): 023101. doi:10.1103/PhysRevResearch.3.023101. Bibcode: 2021PhRvR...3b3101P. 
  19. ↑ Tóth, Géza; Vértesi, Tamás; Horodecki, Paweł; Horodecki, Ryszard (7 July 2020). "Activating Hidden Metrological Usefulness". Physical Review Letters 125 (2): 020402. doi:10.1103/PhysRevLett.125.020402. PMID 32701319. Bibcode: 2020PhRvL.125b0402T. 



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