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

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 : --conditional entropy: the residual entropy (unknown information) of S given the state of A The two definitions are equivalent for the classical world Definition I: Definition II: Classical correlations

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

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: Erasure of 1-bit information has an energy cost (Landauer’s principle) Information vs energy Fig. from Maruyama etc, RMP 2009

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 some particular cases owing to the difficulty to quantify Qd Werner states / Bell-diagonal states See: Maziero etc., PRA 2009 Werland etc., PRA 2009

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

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

Eof: arbitrary two-qubit states (Wootters, PRL 2008) Qd: Bell diagonal states (Luo etc., PRA 2008) Two-qubit X-states (Ali etc., PRA 2010) 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 Studies on quantification of Qd and Eof Previous results: Our results: Intrinsic relation between Qd and Eof

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

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

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

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

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:

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

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

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

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

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

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

Thanks!