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Quantifying quantum discord and Entanglement of Formation via Unified Purifications 岑理相

Quantifying quantum discord and Entanglement of Formation via Unified Purifications 岑理相 lixiangcen@scu.edu.cn 四川大学 物理科学与技术学院. Outline. Background A brief introduction to quantum discord Recent studies on quantum discord and related topics

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Quantifying quantum discord and Entanglement of Formation via Unified Purifications 岑理相

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  1. Quantifying quantum discord and Entanglement of Formation via Unified Purifications 岑理相 lixiangcen@scu.edu.cn 四川大学 物理科学与技术学院

  2. Outline • Background • A brief introduction to quantum discord • Recent studies on quantum discord and related topics • Quantifying Qd and Eof via unified purifications • Trilateral relation between Qd and Eof in purifications • Qd and Eof in pure states of three qubits • Qd and Eof in rank-2 mixed states of 4 X 2 systems • Summary and acknowledgement

  3. Different measures of quantum correlations Quantum correlations: the key resource to realize QIP Entanglement: nonlocality Quantum discord: quantumness

  4. Introduction to quantum discord Conception of mutual information: Definition I: Definition II: Classical correlations --conditional entropy: the residual entropy (unknown information) of S given the state of A The two definitions are equivalent for the classical world

  5. Introduction to quantum discord (cont.) For a given quantum state of bipartite systems: Total correlations: Conditional entropy Classical correlations: Quantum discord: Generally not identical to entanglement! (except for pure states)

  6. Quantum discord for particular examples 1. Classical correlated states: 2. Werner states: Separable when Separable states could have nonzero discord! Ollivier & Zurek, PRL(2002)

  7. Quantum discord and Maxwell’s demon Szilard’s engine （1929） Work produced in the isothermal expansion: Erasure of 1-bit information has an energy cost (Landauer’s principle) Information vs energy Fig. from Maruyama etc, RMP 2009

  8. Quantum discord and Maxwell’s demon (cont.) Quantum demon (nonlocal): Difference between the efficiency defines the quantum discord Classical demon (local):

  9. Dynamicsof quantum correlations under decoherence separable Dynamics of quantum discord? Only investigated for some particular cases owing to the difficulty to quantify Qd Werner states / Bell-diagonal states See: Maziero etc., PRA 2009 Werland etc., PRA 2009

  10. Resources for quantum computation: quantum discord or entanglement? H(n) Uf Quantum discord or entanglement?

  11. Discord in deterministic quantum computation arXiv:10062460 Both quantum discord and entanglement are responsible for the QC speedup. DQC: See E. Knill and R. Laflamme, (1998)

  12. Studies on quantification of Qd and Eof Previous results: Eof: arbitrary two-qubit states (Wootters, PRL 2008) Qd: Bell diagonal states (Luo etc., PRA 2008) Two-qubit X-states (Ali etc., PRA 2010) Our results: Intrinsic relation between Qd and Eof Qd: Arbitrary two-qubit mixed states with rank two Arbitrary rank-2 mixed state of 4 X 2 systems Eof: A sort of rank-2 mixed states of 4 X 2 systems

  13. Qd versus Eof in Unified Purifications C B A Purification Set: Properties: Locally equivalent

  14. Qd versus Eof in Unified Purifications (cont.) C B A Conditional entropy: --Two different definitions: • von-Neumann projective • measurement • 2.Positive operator-valued • measurement

  15. Trilateral relationship of Qd and Eof in Unified purifications Eof of C B A Consequence: quantify quantum discord via Eof & vice versa

  16. Quantifying quantum correlations: quantum discord versus entanglement of formation 2 2 4 Quantifying Qd via Eof Quantifying Eof via Qd Eof of two-qubit systems: Wootters’ formula • Bell-diagonal States • Two-qubit X-states Discord of systems with rank two dimC=2 Entanglement for corresponding mixed states (rank-2) of systems dimB=2 dimA=n

  17. Qd and Eof in pure states of three qubits Entanglement of formation Concurrence:

  18. Qd and Eof in pure states of three qubits (cont.) 3-tangle:

  19. Deriving quantum discord via entanglement of formation States of a systems with no more than two nonzero eigenvalues: ( : Bell-diagonal state)

  20. Deriving quantum discord via entanglement of formation (cont.)

  21. Deriving entanglement of formation via quantum discord Analytical expression for Eof other than two-qubit systems!

  22. Comparison: Amount of Qd and Eof C Trilateral relation B A Result 1: Result 2: or Fanchini etc., arXiv:10062460

  23. Applications: dynamics under decoherence Initial state: Evolution under a phase-damping process Experimentally realizable via optical systems J.-S. Xu, etc., Nat. Commun. 1:7 doi: 10.1038/ncomms1005 (2010).

  24. Quantum discord: Entanglement of formaiton: Eof Eof is always larger than Qd in the speci-fied dynamical process

  25. Summary • The intrinsic relation is revealed between quantum discord • and entanglement of formation in unified purifications • Quantification of quantum discord for the systems • with rank two is obtained • Analytical expression of Eof for a sort of mixed states of • systems is achieved • Application to describe dynamical behavior of quantum • correlations of physical systems under decoherence

  26. Acknowledgement XinQi Li (Beijing Normal Univ.) JiuShu Shao (Beijing Normal Univ.) YiJing Yan (HKUST, Hong Kong) Dr. JianWei Xu (Sichuan Univ., Chengdu)

  27. Thanks!

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