Quantifying quantum discord and Entanglement of Formation via

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Quantifying quantum discord and Entanglement of Formation via Unified Purifications 岑理相 lixiangcen@scu. edu. cn

Quantifying quantum discord and Entanglement of Formation via Unified Purifications 岑理相 [email protected] edu. cn 四川大学 物理科学与技术学院

Outline Ø Background • A brief introduction to quantum discord • Recent studies on

Outline Ø Background • A brief introduction to quantum discord • Recent studies on quantum discord and related topics Ø 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 Ø Summary and acknowledgement

Different measures of quantum correlations Quantum correlations: the key resource to realize QIP Entanglement:

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

Introduction to quantum discord Conception of mutual information: Definition I: Definition II: Classical correlations

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

Introduction to quantum discord (cont. ) For a given quantum state of bipartite systems:

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

Quantum discord for particular examples 1. Classical correlated states: 2. Werner states: Separable when

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

Quantum discord and Maxwell’s demon Szilard’s engine (1929) Work produced in the isothermal expansion:

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) Fig. from Maruyama etc, RMP 2009 Information vs energy

Quantum discord and Maxwell’s demon (cont. ) Quantum demon (nonlocal): Classical demon (local): Difference

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

Dynamics of quantum correlations under decoherence separable Dynamics of quantum discord? Only investigated for

Dynamics of 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

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

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

Discord in deterministic quantum computation ar. Xiv: 10062460 Both quantum discord and entanglement are

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

Studies on quantification of Qd and Eof Previous results: Eof: arbitrary two-qubit states (Wootters,

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

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

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

Qd versus Eof in Unified Purifications (cont. ) C Conditional entropy: --Two different definitions:

Qd versus Eof in Unified Purifications (cont. ) C Conditional entropy: --Two different definitions: 1. von-Neumann projective measurement 2. Positive operator-valued measurement A B

Trilateral relationship of Qd and Eof in Unified purifications C A Eof of B

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

Quantifying quantum correlations: quantum discord versus entanglement of formation Quantifying Qd via Eof Quantifying

Quantifying quantum correlations: quantum discord versus entanglement of formation Quantifying Qd via Eof Quantifying Eof via Qd Eof of two-qubit systems: Wootters’ formula Discord of 1. Bell-diagonal States 2. Two-qubit X-states systems with rank two dim. C=2 dim. A=n 2 dim. B=2 4 Entanglement for corresponding mixed states (rank-2) of systems 2

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

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

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

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

Deriving quantum discord via entanglement of formation States of a systems with no more

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

Deriving quantum discord via entanglement of formation (cont. )

Deriving quantum discord via entanglement of formation (cont. )

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

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

Comparison: Amount of Qd and Eof C Trilateral relation A Result 1: Result 2:

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

Applications: dynamics under decoherence Initial state: Evolution under a phase-damping process Experimentally realizable via

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/ncomms 1005 (2010).

Entanglement of formaiton: Quantum discord: Eof is always larger than Qd in the specified

Entanglement of formaiton: Quantum discord: Eof is always larger than Qd in the specified dynamical process

Summary Ø The intrinsic relation is revealed between quantum discord and entanglement of formation

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

Acknowledgement Xin. Qi Li (Beijing Normal Univ. ) Jiu. Shu Shao (Beijing Normal Univ.

Acknowledgement Xin. Qi Li (Beijing Normal Univ. ) Jiu. Shu Shao (Beijing Normal Univ. ) Yi. Jing Yan (HKUST, Hong Kong) Dr. Jian. Wei Xu (Sichuan Univ. , Chengdu)

Thanks!

Thanks!