Title: Optimizing pointer states for dynamical evolution of quantum correlations under decoherence
1Optimizing pointer states for dynamical evolution
of quantum correlations under decoherence
- Bo You,Li-xiang Cen
- Department of Physics, SiChuan University
2Outline
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
3Dissipative 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
5Schematic illustration dependence of decoherence
dynamics on pointer bases
6Motivation 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.
7Case 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
8Bell-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)
9Two 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.
13Case 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.
14The 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
16Thanks for your attention!