Solid state realisation of Werner quantum states via Kondo spins

1
Solid state realisation of Werner quantum states
via Kondo spins

• Ross McKenzie
• Sam Young Cho

Reference S.Y. Cho and R.H.M,
Phys. Rev. A 73, 012109 (2006)
2
Thanks to

• Discussions with
• Briggs (RKKY in nanotubes)
• Doherty and Y.-C. Liang (Werner states)
• Dawson, Hines, and Milburn (decoherence and
  entanglement sharing)

3
Big goals for quantum nano-science

• Create and manipulate entangled quantum states in
  solid state devices
• Understand the quantum-classical boundary, e.g.,
  test quantum mechanics versus macro-realism
  (Leggett)
• Understand the competition between entanglement
  and decoherence

4
Entanglement vs. decoherence

• Interaction of a qubit with its environment leads
  to decoherence and entanglement of qubit with
  environment.
• Interactions between qubits entangles them with
  one another.
• We will also see that the environment can
  entangle the qubits with one another.

5
Outline

• Classical correlations vs. entanglement vs.
  violation of Bell inequalities (Werner states)
• Experimental realisations of two impurity Kondo
  model
• Competition between Kondo effect and RKKY
  interaction
• Entanglement between the two Kondo spins
• How to create Werner states in the solid state.

6
Quantum correlations in different regions of
Hilbert space
Entangled states
No correlations
Violate Bell inequalities
Correlations but no entanglement
7
Werner states
Mixed states of two qubits
In the Bell basis
Reduced density matrix
is probability of a singlet
No entanglement Bell-CSSH inequalities satisfied
8
Model system two Kondo spins interact with
metallic environment via Heisenberg exchange
interaction
9
Experimental realisation I

• N. J. Craig et al., Science 304, 565 (2004)
• 2DEG between spins
• in quantum dots
• induces an
• RKKY interaction
• between spins.
• Gates vary J

10
Experimental realisation II

• Endohedral fullerenes inside nanotubes

A. Khlobystov et al. Angewandte Chemie
International Edition 43, 1386-1389 (2004)
11
Single impurity Kondo model
Hamiltonian
Conduction electrons
J is the spin exchange coupling
Conduction-electron spin density at impurity site
R 0
Low temperature properties determined by single
energy scale . Kondo temperature
Band width D and the single particle density of
state at the Fermi surface
12
Tuneable quantum many-body states Kondo effect
in quantum dots
For a review, L. Kouwenhoven and L. Glazman,
Physics World 14, 33 (2001)
Conduction electron spin
Impurity spin
Kondo temperature can be varied over many orders
of magnitude
13
Two impurity Kondo model

Hamiltonian


To second order J, the indirect RKKY (Ruderman
Kittel-Kasuya-Yosida) interaction is
Ground state determined by competition between
Kondo of single spins and RKKY
14
Entanglement in single impurity Kondo model
T. A. Costi and R. H. McKenzie, Phys. Rev. A 68,
034301 (2003)
J
S1/2
Subsystem A
Subsystem B
Total system AB
Ground state
Reduced density matrix for the impurity
von Neumann entropy
The impurity spin is maximally entangled with the
conduction electrons
c.f., Yosidas variational wavefunction
K. Yosida, Phys. Rev. 147, 233 (1966)
15
Entanglement between the two Kondo spins

• Given by concurrence of the reduced density
  matrix for the two localised spins (Wootters)
• Ground state is a total spin singlet (S0) and
  thus invariant under global spin rotations
• Entanglement is determined by

.
\langle \vecS_A . \vecS_B \rangle
16
Reduced density matrix for the impurities
In the Bell basis
17
Low temperature behaviour of two impurity Kondo
model
B. A. Jones, C. M. Varma, and J. W. Wilkins,
Phys. Rev. Lett. 61, 125 (1988)
Numerical renormalization group calculation shows
that
Left
the staggered susceptibility and the specific
heat coefficients diverge.
The spin-spin correlation is continuously varying
and approaches at the critical value of
around the divergence of
susceptibility.
Right
18
Entanglement Quantum Phase transition
19
Unstable fixed point

• At the fixed point
• Gan, Ludwig, Affleck, and Jones
• Thus, for the critical coupling there is no
  entanglement between two qubits.

20
Questions for future

• Can the competition between Kondo and RKKY be
  better understood in terms of entanglement
  sharing?
• Why does the entanglement between Kondo spins
  vanish at the quantum critical point?
• What effect does temperature have?

21
Conclusions

• Two spin Kondo model provides a model system to
  study competition between entanglement of two
  qubits with each other and entanglement of each
  qubit with environment
• Entanglement between the two Kondo spins vanishes
  at the unstable fixed point.
• Varying system parameters will produce all the
  Werner states
• S.Y. Cho and RHM, Phys. Rev. A 73, 012109
  (2006)

22
Low temperature behaviours of two impurity Kondo
model
B. A. Jones, C. M. Varma, and J. W. Wilkins,
Phys. Rev. Lett. 61, 125 (1988)
Numerical renormalization group calculation shows
that
Left
the staggered susceptibility and the specific
heat coefficients diverge.
The spin-spin correlation is continuously varying
and approaches at the critical value of
around the divergence of
susceptibility.
Right
23
Unstable fixed point
B. A. Jones and C. M. Varma, Phys. Rev. B 40,
324 (1989)
Renormalization group flows
24
Three types of entanglements
(i)
and
(ii)
and
(iii)
and
Subsystem A
Subsystem B
25
Probabilities for spin singlet/triplet states
for singlet state
for triplet state
For P(S)P(T)1/2, the state for the two spins
can be regarded as an equal admixture of the
total spin of impurities Simp0 and Simp1.
at ps1/2
spin-spin correlation
26
Entanglement (ii) between the impurities
(ii)
and
Total system ABC
Although the total system is in a pure state,
the two impurity spins are in a mixed state.
Need to calculate the concurrence as a measure
of entanglement
W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)
27
Concurrence Critical Correlation
In terms of the Werner state
Hence, at ps1/2, there exists a critical value
of the spin-spin correlation separating
entangled state from disentangled state.
28
Comparison of criteria
singlet fidelity
42 R. Horodecki, P. Horodecki, and M.
Horodecki, Phys. Lett. A 200, 340 (1995)
48 S. Popescu, Phys. Rev. Lett. 72, 797 (1994)
29
Entanglement (iii)
S1/2
Subsystem A and B
Subsystems C
Total system ABC
von Neumann entropy