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Optimizing pointer states for dynamical evolution of quantum correlations under decoherence

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Title: Optimizing pointer states for dynamical evolution of quantum correlations under decoherence


1
Optimizing pointer states for dynamical evolution
of quantum correlations under decoherence
  • Bo You,Li-xiang Cen
  • Department of Physics, SiChuan University

2
Outline
1.Dissipative channels and pointer bases 2. Two
different optimizations in relation to quantum
discord 2.1.Bell-diagonal state
2.2.two-qubit state of two ranks 3. Optimizing
pointer bases to achieve maximal condition
entropy
3
Dissipative channels and pointer bases
  • Dynamical evolution under dissipative channel
  • In this case of dephasing channel
  • Pointer bases selected by the system-environment
    coupling

4
  • Quantum discord
  • Where
  • Geometry measure of Quantum Discord
  • Where is the set of zero-discord.
  • Entanglement of formation
  • where

H. Ollivier and W. H. Zurek, PhysRevLett.88.017901
B.Dakic,V.Vedral, and C. Brukner,Phys. Rev. Lett.
105, 190502
5
Schematic illustration dependence of decoherence
dynamics on pointer bases
6
Motivation and main concerns
  • We investigate the dynamics of decorrelation
    under different choices of the pointer states of
    system-reservoir couplings. In detail, we
    consider a two-qubit system,initially prepared in
    certain states with non-zero quantum
    correlations, subjected to local dissipative
    channels responsible for various pointer states.
    Dynamical evolution of entanglement, quantum
    discord,and the mutual information sharing
    between the two qubits, is depicted. We elucidate
    various optimizations of the pointer states,
    e.g., minimization and maximization of the
    conditional entropy, as well as the geometric
    optimization via minimizing discord,and analyze
    the properties of the corresponding behavior of
    decorrelation.

7
Case 1 Two different optimizations in relation
to quantum discord
It is proved that quantum discord and its
geometry measure can be rewritten
as Where is the projector state
achieving the minimize.
Therefore we could choose the two projector
states as our optimal bases,so that we could get
some physics about dissipate process with
different optimizations.
S.L. Luo and S.S. Fu,PhysRevA.82.034302
8
Bell-diagonal states
  • We can express the state as
  • For Bell-diagonal state, the measurements for
    minimizing the quantum discord and geometry
    measure of QD can be easily calculated, which are
    the same, that is

Thus, the dynamic with the two optimizations is
the same for Bell-diagonal states.
S.L.Luo,Phys. Rev. A 77, 042303 (2008)
9
Two qubits state of two ranks
  • The state express as
  • Where
  • The minimizing discord can be obtained by
    purifying the state to a three-qubit pure state
  • In the state, the minimal conditional entropy of
    AC-system is equal to the entanglement of
    BC-system

L.X.Cen, et.al., Phys. Rev. A 83, 054101 (2011)
10
  • Therefore, the minimizing conditional entropy of
    AC-system could be obtained by calculating the
    EoF of BC-system.
  • Thus through the method conducted by
    Wootters(phys.rev.let.80.2245),the decomposition
    achieving the EoF can be deduced.
  • So the measurement corresponding to minimizing
    conditional entropy can be determined by
  • Thus the measurement is
  • where is the normalized vector , and

11
  • The measurement for minimizing the geometry
    measure of can be obtained through the method
    conducted by B.Dakic(Phys.Rev.Lett.105.190502),wri
    te as
  • Where is the normalized eigenvector with the
    largest eigenvalue of
    ,and the state is expressed as
  • It is clear that ,generally,the measurement for
    minimizing condition entropy is not the one
    minimizing the geometry distance of discord
    ,unless it satisfying
  • but the state is not necessarily the
    Bell-diagonal state.

12
  • the figure shows the correlation evolution in
    dephasing process, which set

where dashed line is the evolution with minimal
geometry measure of discord,green line quantum
discord.
13
Case 2 Optimizing pointer bases to achieve
maximal condition entropy
Quantum discord
If we select the pointer bases without min,
otherwise, with max to conditional entropy
,i.e. ,how situation will be? Generally, the
evolution of mutual information is slower than
above choices, however the evolution of quantum
discord has not easy relation, that is , max
choice doesnt meaning the slower quantum discord
evolution.
14
The left figure show the evolution of quantum
correlation of two-qubit of two ranks above,the
red line is corresponding to max selection, we
can see QD line of max-selection is the top
one,while the mutual information line of max
selection is the bottle one.
The right figure show the evolution of the state
with we can see QD line of max-selection is
the bottle one,while the mutual information line
of max selection is also the bottle one.
15
  • We could consider the amplitude damping channel,
    which in Kraus operator is
  • Select the pointer bases above, generally, we can
    find the evolution of the quantum discord has
    increasing region in max selection in above
    setting, showing below

16
Thanks for your attention!
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