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2003 | 1 | 4 | 695-707
Article title

On entanglement distillation and quantum error correction for unknown states and channels

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EN
Abstracts
EN
We consider the problem of invariance of distillable entanglement D and quantum capacities Q under erasure of information about single copy of quantum state or channel respectively. We argue that any 2 ⊗N two-way distillable state is still two-way distillable after erasure of single copy information. For some known distillation protocols the obtained two-way distillation rate is the same as if Alice and Bob knew the state from the very beginning. The isomorphism between quantum states and quantum channels is also investigated. In particular it is pointed out that any transmission rate down the channel is equal to distillation rate with formal LOCC-like superoperator that uses in general nonphysical Alice actions. This allows to we prove that if given channel Λ has nonzero capacity (Q
→ or Q
⟺) then the corresponding quantum state ϱ(Λ) has nonzero distillable entanglement (D
→ or D
⟺). Follwoing the latter arguments are provided that any channel mapping single qubit into N level system allows for reliable two-way transmission after erasure of information about single copy. Some open problems are discussed.
Publisher

Journal
Year
Volume
1
Issue
4
Pages
695-707
Physical description
Dates
published
1 - 12 - 2003
online
1 - 12 - 2003
Contributors
  • Facully of Applied Physics and Mathematics, Gdańsk University of Technology, 80-952, Gdańsk, Poland
References
  • [1] C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin and W.K. Wootters: “Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels”, Phys. Rev. Lett., Vol. 76, (1996), pp. 722–725. http://dx.doi.org/10.1103/PhysRevLett.76.722[Crossref]
  • [2] A. Ekert: “Quantum cryptography based on Bell's theorem”, Phys. Rev. Lett., Vol. 67, (1991), pp. 661–663. http://dx.doi.org/10.1103/PhysRevLett.67.661[Crossref]
  • [3] C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters: “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels”, Phys. Rev. Lett., Vol. 70, (1993), pp. 1895–1899. http://dx.doi.org/10.1103/PhysRevLett.70.1895[Crossref]
  • [4] C.H. Bennett, D.P. Di Vincenzo, J. Smolin and W.K. Wootters: “Mixed-state entanglement and quantum error correction”, Phys. Rev. A, Vol. 54, (1997), pp. 3824–3851. http://dx.doi.org/10.1103/PhysRevA.54.3824[Crossref]
  • [5] H. Barnum, E. Knill and M. Nielsen: “On quantum fidelities and channel capacities”, IEEE T. Inform. Theory, Vol. 46, (2000), pp. 1317–1329. http://dx.doi.org/10.1109/18.850671[Crossref]
  • [6] A. Albert et al.: Quantum information: basic concepts and experiments, Springer, Berlin, 2001.
  • [7] R.F. Werner: “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model”, Phys. Rev. A, Vol. 40, (1989), pp. 4277–4281. http://dx.doi.org/10.1103/PhysRevA.40.4277[Crossref]
  • [8] E. Rains: “Rigorous treatment of distillable entanglement”, Phys. Rev. A, Vol. 60, (1999), pp. 173–178. http://dx.doi.org/10.1103/PhysRevA.60.173[Crossref]
  • [9] M. Horodecki, P. Horodecki and R. Horodecki: “Inseparable Two Spin- 1/2 Density Matrices Can Be Distilled to a Singlet Form”, Phys. Rev. Lett., Vol. 78, (1997), pp. 574–577. http://dx.doi.org/10.1103/PhysRevLett.78.574[Crossref]
  • [10] M. Horodecki, P. Horodecki and R. Horodecki: “Mixed-State. Entanglement and Distillation: Is there a “Bound” Entanglement in Nature?”, Phys. Rev. Lett., Vol. 80, (1998), pp. 5239–5242. http://dx.doi.org/10.1103/PhysRevLett.80.5239[Crossref]
  • [11] If the state of bipartite system is defined on Hilbert space \(\mathcal{H}_A \otimes \mathcal{H}_B \) and \(d_A = dim\mathcal{H}_A \) , \(d_B = dim\mathcal{H}_B \) then we deal withd A⊗d B system.
  • [12] A. Peres: “Separability Criterion for Density Matrices”, Phys. Rev. Lett., Vol. 77, (1996), pp. 1413–1415. http://dx.doi.org/10.1103/PhysRevLett.77.1413[Crossref]
  • [13] B. Kraus: J.I. Cirac, S. Karnas, and M. Lewenstein: “Separability in 2N composite quantum systems”, Phys. Rev. A, Vol. 61, (2000), 062302. http://dx.doi.org/10.1103/PhysRevA.61.062302[Crossref]
  • [14] D.P. DiVincenzo, P.W. Shor, J.A. Smolin, B. Terhal and A.W. Thapliyal: “Evidence for bound entangled states with negative partial transpose”, Phys. Rev. A, Vol. 61, (2000), 062312; D. Dür, J.I. Cirac, M. Lewenstein and D. Bruss: “Distillability and partial transposition in bipartite systems”, Phys. Rev. A, Vol. 61, (2000), 062312. http://dx.doi.org/10.1103/PhysRevA.61.062312[Crossref]
  • [15] P.W. Shor, J.A. Smolin and B.M. Terhal: “Nonadditivity of Bipartite Distillable Entanglement Follows from a Conjecture on Bound Entangled Werner States”, Phys. Rev. Lett., Vol. 86, (2001), pp. 2681–2684. http://dx.doi.org/10.1103/PhysRevLett.86.2681[Crossref]
  • [16] A. Jamiołkowski: “Linear transformations which preserve trace and positive semidefiniteness of operators”, Rep. Math. Phys., Vol. 3, (1972), pp. 275–278. http://dx.doi.org/10.1016/0034-4877(72)90011-0[Crossref]
  • [17] R. Jozsa: “Fidelity for Mixed Quantum States”, J. Mod. Opt., Vol. 41, (1994), pp. 2315–2323. [Crossref]
  • [18] P. Horodecki, M. Horodecki and R. Horodecki: “Binding entanglement channels”, J. Mod. Opt., Vol. 47, (2000), pp. 347–354. D. DiVincenzo, T. Mor, P. Shor, J. Smolin, and B.M. Terhal: “Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement”, Commun. Math. Phys., Vol. 238, pp. 379–410. http://dx.doi.org/10.1080/095003400148259[Crossref]
  • [19] P. Horodecki: “On Mixed States Entanglement and Quantum Communication: Aspects of Quantum Channels Theory”, Acta Phys. Polon., Vol. 101, (2002), pp. 339.
  • [20] P. Horodecki, M. Horodecki and R. Horodecki: “Bound Entanglement Can Be Activated”, Phys. Rev. Lett., Vol. 82, (1999), pp. 1056–1059. http://dx.doi.org/10.1103/PhysRevLett.82.1056[Crossref]
  • [21] T. A. Brun, C. M. Caves, R. Schack: “Entanglement purification of unknown quantum states”, Phys. Rev. A, Vol. 63, (2001), (PRA 63 0402309).
  • [22] Here, in general \(\vec p \in R^{15} \) since two qubit density matrix is described by 15 parameters. In analysis of hashing of Bell diagnonal states [21] effectively one can choose \(\vec p \in R^{15} \) since Bell diagnonal states depend effectively on 3 parameters.
  • [23] G.M. D'Ariano, M.G.A. Paris, M.F. Sacchi: “Quantum Tomography”, (quantph/0302028), http://arxiv.org/abs/quant-ph/0302028.
  • [24] A.S. Holevo and R.F. Werner: “Evaluating capacities of bosonic Gaussian channels”, Phys. Rev. A, Vol. 63, (2001), (PRA 63 032312). [Crossref]
  • [25] C.A. Fuchs, J. van de Graaf: “Cryptographic Distinguishability Measures for Quantum Mechanical States”, (quant-ph/9712042), http://arxiv.org/abs/quantph/9712042.
  • [26] P. Horodecki, PhD thesis, Technical University of Gdansk, Gdansk 1999.
  • [27] N. Gisin: “Hidden quantum nonlocality revealed by local filters”, Phys. Lett. A, Vol. 210, (1996), pp. 151–156. http://dx.doi.org/10.1016/S0375-9601(96)80001-6[Crossref]
Document Type
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.-psjd-doi-10_2478_BF02475911
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