Quantum Entanglement and Distillation in Information

Processing

Shao-Ming Fei

2011.04.18

Quantum Entanglement

Pure state: vector on

Separable!

bipartite tripartite multipartite

a) PPT criterion:

Separability: no general criteria

b) Reduction:

Mixed state: density matrix

Separable!

Realignment, Covariance matrix criterion, …

Measure: Bipartite

Entanglement of Formation (EoF)

Pure state

Measurable bounds for EoF M. Li, S.M. Fei, Phys. Rev. A 82 (2010) 044303

Lower Bound for EoFK. Chen, S. Albeverio, S.M. Fei, Phys. Rev. Lett. 95(2005)210501

S.M. Fei, X. Li-Jost, Phys. Rev. A 73(2006)024302

Mixed state

Relative Entropy, Negativity, ……..

Lower Bound for ConcurrenceK. Chen, S. Albeverio, S.M. Fei, Phys. Rev. Lett. 95(2005)040504

X.H. Gao, S.M. Fei and K. Wu, Phys. Rev. A 74 (Rapid Comm.)(2006)050303M. Li, S.M. Fei and Z.X. Wang, J. Phys. A 42 (2009) 145303

X.S. Li, X.H. Gao and S.M. Fei, Phys. Rev. A 83 (2011) 034303

Concurrence

Two-qubit:

A. Buchleitner et.al. Nature 440(2006)1022

Experimental measurementof entanglement

Three qubits

Two qubits

Theorem (multi-qubit):

S.M. Fei, M.J. Zhao, K. Chen and Z.X. Wang, Phys. Rev. A 80 (2009) 032320

N-partite M-dimensional case:

SU(M) generators

Entanglement as resource in quantum formation processing

Quantum computation Quantum teleportation Dense coding Quantum cryptography Quantum error correction Quantum repeater

Phase transition

1 qubit 2 bit 1 qubit

Alice Bob

Entanglement enhanced

Alice Bob

01102

1

Entanglement enhanced

1 qubit1 qubit

Alice Bob

2 bit

Entanglement enhanced

1 qubit 2 bit 1 qubit

Alice Bob

Entanglement enhanced

Optimal Teleportation

M. Li, S.M. Fei and Z.X. Wang, Phys. Rev. A 8 (2008) 032332M.J. Zhao, Z.G. Li, S.M. Fei and Z.X. Wang, J. Phys. A 43, (2010) 275203

Fully entangled fraction:

M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A 60, 1888 (1999)S. Albeverio, S.M. Fei, Phys. Lett. A 276(2000)8S. Albeverio, S.M. Fei and W.L. Yang, Phys. Rev. A 66 (2002) 012301

$

~ ?

Bipartite system with one subsystem undergoes a noisy channel

Evolution of quantum entanglement:

Z.G. Li, S.M. Fei, Z.D. Wang and W.M. Liu, Phys. Rev. A,79 (2009) 024303

Two-qubit: Buchleitner et. al., Nature Physics 4, 99 (2008)

Kraus operator

E.g. generalized amplitude damping channel

Atom-Cavity System

zero detuning

Evolution of quantum entanglement:(Hamiltonian ~ master equation)

J.H. Eberly and T. Yu, Science 316 (2007) 555

L.L. Lan, X.B. Wang and S.M. Fei, Int. J. Quant. Inform. 8 (2010)1239

Entanglement of Assistance (EoA)

A

B CZ.G. Li, S.M. Fei, S. Albeverio, W.M. Liu, Phys. Rev. A 80 (2009) 034301

Evolution of EoAZ.G. Li, M.J. Zhao, S.M. Fei and W. M. Liu, Phys. Rev. A 81 (2010) 042312

Distillation:

U U

Entangled, but not distillable: bound entangled

Measurement

Separable PPT (positive partial transpose)

All PPT entangled states are bound entangled!

Non PPT distillable ? no answer yet

Violation of reduction criterion distillable

Distillability from Bell inequalities

M. Li, S.M. Fei, Phys. Rev. Lett. 104 (2010) 240502

Improving capacity of noisy channels

1 2

max. entangled

$

21

bound entangled

1 2

43

21

43

$

bound entangled!

on 1 and 3 by Alice

Alice measures 1 Bob measures 2 => |0>

Maximally Entangled !

Quantum Discord (quantum correlation besides entanglement)

Quantum mutual information

project measurement performed on subsystem B

Quantum conditional entropy

Associated quantum mutual information

Classical correlation

Quantum discord

S. Luo, Phys. Rev. A 77 (2008) 042303

Undergoes the phase flip channel

Concurrence (dashed line) and quantum discord (solid line) under phase flip channel for

B. Li, Z.X. Wang, S.M. Fei, Phys. Rev. A 83 (2011) 022321

H. Dong, D. Z. Xu, C. P. Sun, Coherent excitation transferring via dark state in light-harvesting process, arXiv:1102.0960

Entanglement in Biological structure