Bound entanglement explained

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]

Notes and References

  1. Horodecki . Michał . Horodecki . Paweł . Horodecki . Ryszard . Mixed-State Entanglement and Distillation: Is there a "Bound" Entanglement in Nature? . Physical Review Letters . 15 June 1998 . 80 . 24 . 5239–5242 . 10.1103/PhysRevLett.80.5239. quant-ph/9801069 . 1998PhRvL..80.5239H . 111379972 .
  2. Bruß . Dagmar . Peres . Asher . Construction of quantum states with bound entanglement . Physical Review A . 4 February 2000 . 61 . 3 . 030301 . 10.1103/PhysRevA.61.030301. quant-ph/9911056 . 2000PhRvA..61c0301B . 7019402 .
  3. Bennett . Charles H. . DiVincenzo . David P. . Mor . Tal . Shor . Peter W. . Smolin . John A. . Terhal . Barbara M. . Unextendible Product Bases and Bound Entanglement . Physical Review Letters . 28 June 1999 . 82 . 26 . 5385–5388 . quant-ph/9808030. 10.1103/PhysRevLett.82.5385. 1999PhRvL..82.5385B . 14688979 .
  4. Breuer . Heinz-Peter . Optimal Entanglement Criterion for Mixed Quantum States . Physical Review Letters . 22 August 2006 . 97 . 8 . 080501 . 10.1103/PhysRevLett.97.080501. 17026285 . quant-ph/0605036 . 2006PhRvL..97h0501B . 14406014 .
  5. Piani . Marco . Mora . Caterina E. . Class of positive-partial-transpose bound entangled states associated with almost any set of pure entangled states . Physical Review A . 4 January 2007 . 75 . 1 . 012305 . 10.1103/PhysRevA.75.012305. quant-ph/0607061 . 2007PhRvA..75a2305P . 55900164 .
  6. Smolin . John A. . Four-party unlockable bound entangled state . Physical Review A . 9 February 2001 . 63 . 3 . 032306 . 10.1103/PhysRevA.63.032306. quant-ph/0001001 . 2001PhRvA..63c2306S . 119474939 .
  7. DiVincenzo . David P. . Shor . Peter W. . Smolin . John A. . Terhal . Barbara M. . Thapliyal . Ashish V. . Evidence for bound entangled states with negative partial transpose . Physical Review A . 17 May 2000 . 61 . 6 . 062312 . 10.1103/PhysRevA.61.062312. quant-ph/9910026 . 2000PhRvA..61f2312D . 37213011 .
  8. Horodecki . Pawel . Smolin . John A . Terhal . Barbara M . Thapliyal . Ashish V . Rank two bipartite bound entangled states do not exist . Theoretical Computer Science . January 2003 . 292 . 3 . 589–596 . 10.1016/S0304-3975(01)00376-0. quant-ph/9910122 . 43737866 .
  9. Horodecki . Michał . Horodecki . Paweł . Horodecki . Ryszard . General teleportation channel, singlet fraction, and quasidistillation . Physical Review A . 1 September 1999 . 60 . 3 . 1888–1898 . 10.1103/PhysRevA.60.1888. quant-ph/9807091 . 1999PhRvA..60.1888H . 119532807 .
  10. Chen . Lin . Tang . Wai-Shing . Schmidt number of bipartite and multipartite states under local projections . Quantum Information Processing . 2 February 2017 . 16 . 75 . 75 . 10.1007/s11128-016-1501-y. 1609.05100. 2017QuIP...16...75C . 34893860 .
  11. Tóth . Géza . Gühne . Otfried . Entanglement and Permutational Symmetry . Physical Review Letters . 1 May 2009 . 102 . 17 . 170503 . 10.1103/PhysRevLett.102.170503. 19518768 . 0812.4453 . 2009PhRvL.102q0503T . 43527866 .
  12. Tura . J. . Augusiak . R. . Hyllus . P. . Kuś . M. . Samsonowicz . J. . Lewenstein . M. . Four-qubit entangled symmetric states with positive partial transpositions . Physical Review A . 22 June 2012 . 85 . 6 . 060302 . 10.1103/PhysRevA.85.060302. 1203.3711 . 2012PhRvA..85f0302T . 118386611 .
  13. Peres . Asher . All the Bell Inequalities . Foundations of Physics . 1999 . 29 . 4 . 589–614 . 10.1023/A:1018816310000. 9697993 .
  14. Vértesi . Tamás . Brunner . Nicolas . Disproving the Peres conjecture by showing Bell nonlocality from bound entanglement . Nature Communications . December 2014 . 5 . 1 . 5297 . 10.1038/ncomms6297. 25370352 . 1405.4502 . 2014NatCo...5.5297V . 5135148 . free .
  15. Horodecki . Paweł . Horodecki . Michał . Horodecki . Ryszard . Bound Entanglement Can Be Activated . Physical Review Letters . 1 February 1999 . 82 . 5 . 1056–1059 . 10.1103/PhysRevLett.82.1056. quant-ph/9806058 . 1999PhRvL..82.1056H . 119390324 .
  16. Masanes . Lluís . All Bipartite Entangled States Are Useful for Information Processing . Physical Review Letters . 17 April 2006 . 96 . 15 . 150501 . 10.1103/PhysRevLett.96.150501. 16712136 . quant-ph/0508071 . 2006PhRvL..96o0501M . 10914899 .
  17. Tóth . Géza . Vértesi . Tamás . Quantum States with a Positive Partial Transpose are Useful for Metrology . Physical Review Letters . 12 January 2018 . 120 . 2 . 020506 . 10.1103/PhysRevLett.120.020506. 29376687 . 1709.03995 . 2018PhRvL.120b0506T . 206306250 . free .
  18. Pál . Károly F. . Tóth . Géza . Bene . Erika . Vértesi . Tamás . Bound entangled singlet-like states for quantum metrology . Physical Review Research . 10 May 2021 . 3 . 2 . 023101 . 10.1103/PhysRevResearch.3.023101. 2002.12409. 2021PhRvR...3b3101P . free .
  19. Tóth . Géza . Vértesi . Tamás . Horodecki . Paweł . Horodecki . Ryszard . Activating Hidden Metrological Usefulness . Physical Review Letters . 7 July 2020 . 125 . 2 . 020402 . 10.1103/PhysRevLett.125.020402. 32701319 . 1911.02592 . 2020PhRvL.125b0402T . free .

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