# Bound 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

Next, we concentrate on 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]

Strong evidence has been presented that 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]

## 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. arXiv:quant-ph/9801069. doi:10.1103/PhysRevLett.80.5239. S2CID 111379972.
2. ^ Bruß, Dagmar; Peres, Asher (4 February 2000). "Construction of quantum states with bound entanglement". Physical Review A. 61 (3): 030301. arXiv:quant-ph/9911056. doi:10.1103/PhysRevA.61.030301. S2CID 7019402.
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" (PDF). Physical Review Letters. 82 (26): 5385–5388. doi:10.1103/PhysRevLett.82.5385. S2CID 14688979.
4. ^ Breuer, Heinz-Peter (22 August 2006). "Optimal Entanglement Criterion for Mixed Quantum States". Physical Review Letters. 97 (8): 080501. arXiv:quant-ph/0605036. doi:10.1103/PhysRevLett.97.080501. PMID 17026285. S2CID 14406014.
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. arXiv:quant-ph/0607061. doi:10.1103/PhysRevA.75.012305. S2CID 55900164.
6. ^ Smolin, John A. (9 February 2001). "Four-party unlockable bound entangled state". Physical Review A. 63 (3): 032306. arXiv:quant-ph/0001001. doi:10.1103/PhysRevA.63.032306. S2CID 119474939.
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. arXiv:quant-ph/9910026. doi:10.1103/PhysRevA.61.062312. S2CID 37213011.
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. arXiv:quant-ph/9910122. doi:10.1016/S0304-3975(01)00376-0. S2CID 43737866.
9. ^ Horodecki, Michał; Horodecki, Paweł; Horodecki, Ryszard (1 September 1999). "General teleportation channel, singlet fraction, and quasidistillation". Physical Review A. 60 (3): 1888–1898. arXiv:quant-ph/9807091. doi:10.1103/PhysRevA.60.1888. S2CID 119532807.
10. ^ Sanpera, Anna; Bruß, Dagmar; Lewenstein, Maciej (9 April 2001). "Schmidt-number witnesses and bound entanglement". Physical Review A. 63 (5): 050301. arXiv:quant-ph/0009109. doi:10.1103/PhysRevA.63.050301. S2CID 44232418.
11. ^ Tóth, Géza; Gühne, Otfried (1 May 2009). "Entanglement and Permutational Symmetry". Physical Review Letters. 102 (17): 170503. arXiv:0812.4453. doi:10.1103/PhysRevLett.102.170503. PMID 19518768. S2CID 43527866.
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. arXiv:1203.3711. doi:10.1103/PhysRevA.85.060302. S2CID 118386611.
13. ^ Peres, Asher (1999). "All the Bell Inequalities". Foundations of Physics. 29 (4): 589–614. doi:10.1023/A:1018816310000. S2CID 9697993.
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. S2CID 5135148.
15. ^ Horodecki, Paweł; Horodecki, Michał; Horodecki, Ryszard (1 February 1999). "Bound Entanglement Can Be Activated". Physical Review Letters. 82 (5): 1056–1059. arXiv:quant-ph/9806058. doi:10.1103/PhysRevLett.82.1056. S2CID 119390324.
16. ^ Masanes, Lluís (17 April 2006). "All Bipartite Entangled States Are Useful for Information Processing". Physical Review Letters. 96 (15): 150501. arXiv:quant-ph/0508071. doi:10.1103/PhysRevLett.96.150501. PMID 16712136. S2CID 10914899.
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. S2CID 206306250.